Subdimensional Topological Quantum

Phases of Matter

Trithep Devakul

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Physics

Advisor: David Huse

January 2021

c© Copyright by Trithep Devakul, 2021.

All rights reserved.

Abstract

This Dissertation concerns phases of matter with various “subdimensional” prop-

erties. We start with topologically ordered phases with subdimensional properties.

This includes a discussion of fracton topologically ordered phases, resonating va-

lence plaquette phases, and floating topological phases. We then move on to dis-

cuss systems with symmetries that act on subdimensional subsystems, in particular

those with line-like subsystems in 2D or planar subsystems in 3D. A classification of

symmetry-protected topological (SPT) phases protected by such symmetries is pre-

sented. Finally, we discuss symmetries which act on fractal dimensional subsystems

and a classification of such phases.

iii

Acknowledgements

I would like to thank my academic advisor David Huse for all of his insight and

support over the years, even as I was often off on my own investigations. I am equally

thankful to Shivaji Sondhi for inspiring the line of research leading to this dissertation

and much more, and for always being a source of sage support and guidance, without

whom my career would not have been possible.

I am thankful to my collaborators on projects which contributed to this disserta-

tion: Yizhi You, Dominic Williamson, Wilbur Shirley, Juven Wang, Fiona Burnell,

Sid Parameswaran, Steve Kivelson, and Erez Berg. I am also thankful to my collab-

orators on other projects throughout my PhD: Yves Kwan, Sanjay Moudgalya, Curt

Von Keyserlingk, Dan Arovas, Titus Neupert, Debayan Mitra, Peter Brown, Elmer

Guardardo-Sanchez, Stanimir Kondov, Peter Schauss, Waseem Bakr, Phuc Nguyen,

Matthew Halbasch, Michael Zaletel, Brian Swingle, Satya Majumdar, Vedika Khe-

mani, Frank Pollmann, and Liangsheng Zhang. I am indebt to my undergraduate

advisors Don Heiman, Adrian Feiguin, and Rajiv Singh, who got me started on the

right path through academia. I am very grateful to Kate Brosowsky who is always

there to help whenever I needed it. My years as a graduate student at Princeton have

been the most impactful of my life, both academically and personally, due in great

part to all my wonderful friends and colleagues in Jadwin Hall.

I could not have made it to this point without the support of my lovely girlfriend

Yuwen, my cute cat Bow, and, of course, my ever supportive parents Tri and Tam.

iv

To my parents.

v

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I Subdimensional Topological Orders 8

1 Preliminaries 9

1.1 Topological order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Fracton topological order . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Correlation function diagnostics 26

2.1 Ising gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Euclidean Path Integral and Wilson Loops . . . . . . . . . . . . . . . 30

2.3 Diagnostic behaviors . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Phase Diagram and Quantum Monte Carlo . . . . . . . . . . . . . . . 38

3 Resonating Plaquette Phases 47

3.1 FCC Plaquette model . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 The Hard-Core constraint . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3 ZN Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4 Generalized Models on other lattices . . . . . . . . . . . . . . . . . . 65

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vi

4 Floating topological phases 77

4.1 Gapped topological floating phases . . . . . . . . . . . . . . . . . . . 78

4.2 Floating phases via the Fredenhagen-Marcu order parameter . . . . . 82

4.3 Gapless floating topological phases . . . . . . . . . . . . . . . . . . . 87

II Regular Subsystem Symmetric Phases 92

5 Preliminaries 93

5.1 Symmetry-protected topological phases . . . . . . . . . . . . . . . . . 93

5.2 Linear subsystem symmetries . . . . . . . . . . . . . . . . . . . . . . 99

6 Classifying 2D linear subsystem SPTs 102

6.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Standard SPT phase equivalence . . . . . . . . . . . . . . . . . . . . 107

6.3 Strong equivalence of SSPT phases . . . . . . . . . . . . . . . . . . . 113

6.4 Example: 2D cluster model . . . . . . . . . . . . . . . . . . . . . . . 128

6.5 Other Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Classifying 3D planar subsystem SPTs 146

7.1 Review of 2D SPTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 3D Planar SSPTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.3 Strong models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4 Fracton duals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

III Fractal Subsystem Symmetric Phases 176

8 Fractal symmetric phases 177

vii

8.1 Cellular Automata Generate Fractals . . . . . . . . . . . . . . . . . . 178

8.2 Fractal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.3 Spontaneous fractal symmetry breaking . . . . . . . . . . . . . . . . . 191

8.4 Fractal symmetry protected topological phases . . . . . . . . . . . . . 193

8.5 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

9 Classification of 2D Fractal SPTs 220

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

9.3 Fractal Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.4 Local phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

9.5 Constructing commuting models for arbitrary phases . . . . . . . . . 247

9.6 Irreversibility and Pseudosymmetries . . . . . . . . . . . . . . . . . . 253

9.7 Identifying the phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

9.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.9 Proof of main result . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

10 Conclusion 276

Bibliography 277

viii

0.1 Introduction

The idea that many-body systems may be succinctly described and categorized by

their phase of matter is one of the core concepts in condensed matter physics. Two

systems belonging to the same phase of matter are qualitatively similar: there exists

a path in some parameter space along which one can be smoothly deformed into

the other. It was recognized by Landau [1] that symmetry played a pivotal role in

the distinction between phases of matter. Even though a system may nominally be

described by a theory with a particular symmetry, it is possible for the symmetry to

be spontaneously broken in the state of the system. One example is ferromagnetism.

A simple model for a ferromagnet (say a bar of iron) is as a collection of localized

magnetic moments (spins) which prefer to be aligned with their neighbors. At low

enough temperatures the free energy will be minimized by a state in which all spins

align along one direction, resulting in ferromagnetism: a non-zero macroscopic net

magnetic moment and spontaneous symmetry breaking of spin rotation symmetry.

At high temperatures, entropy favors a paramagnetic state with zero net magnetic

moment. The ferromagnetic and paramagnetic phases differ fundamentally in their

pattern of broken symmetry (one breaks spin rotation symmetry, the other does not),

and therefore are necessarily separated by a phase transition.

The story remains largely unchanged even at zero temperature where quantum

effects are strong. At T = 0, spontaneous symmetry breaking becomes a statement

about the ground states of the system’s Hamiltonian H under a symmetry operation

S. Take the quantum Ising chain in a transverse field:

H = −J∑

i

σzi σzi+1 − h

∑

i

σxi (1)

where σαi are Pauli matrices acting on the spin-1/2 degree of freedom on site i. The

Ising model has a Z2 symmetry S =∏

i σxi which commutes with the Hamiltonian,

1

[S,H] = 0. When h J , H has a unique paramagnetic ground state |PM〉 which

is symmetric under S, S |PM〉 = |PM〉. On the other hand, when J h, H has

two degenerate ferromagnetic ground states which may distinguished by the overall

sign of σz magnetization, |FM ↑〉 and |FM ↓〉. The Z2 symmetry is spontaneously

broken as the ground states transform non-trivially under S, S |FM ↑〉 = |FM ↓〉.

Deep in either phase, the ground states are separated from the remaining spectrum

by a finite energy gap. As the ground state degeneracy cannot change continuously,

any attempt to tune smoothly between the two phases must be met by some point at

which the gap goes to zero: a quantum phase transition. An exception arises when

the symmetry is broken explicitly (such as by an external applied magnetic field).

Without symmetry, the distinction between the two phases disappear. This example

illustrates an important point: certain phases of matter can only be distinguished in

the presence of symmetry.

There is obviously more to this story, as alluded to by the title of this Disserta-

tion, “Subdimensional topological quantum phases of matter”. The word “topolog-

ical” here loosely refers to quantum phases of matter which cannot be understood

in terms of spontaneous symmetry breaking. Firstly, we now know that there exist

many different gapped quantum phases with the same unbroken symmetry, known as

symmetry-protected topological (SPT) phases [2] (these include the Haldane chain [3]

and topological insulators [4]). Secondly, we know that even in the absence of any sym-

metry, non-trivial gapped quantum phases of matter known as topological order [5]

can exist (these include quantum Hall states and Chern insulators [6]). 1 These

phases are distinguished by their patterns of quantum entanglement in the ground

state. SPT phases possess only short-ranged entanglement and, if the symmetry is

neglected, may be smoothly deformed to the trivial unentangled phase. Topologically

ordered phases, on the other hand, possess long-range quantum entanglement which

1A third possibility combines topological order with symmetry, resulting in symmetry-enrichedtopological order.

2

cannot be removed by any local deformations. These statements are made precise by

local unitary equivalence [7]. Detailed introduction to the relevant physics of such

phases will be presented later in this Dissertation when they become relevant.

While the standard family of SPT and topologically ordered phases can be de-

scribed without any reference to an underlying lattice structure, there have also been

developments on phases of matter which depend crucially on the details of their

underlying lattice. These include, for example, SPT phases protected by lattice

symmetries [8], higher-order topological insulators [9], and the main topics of this

Dissertation: fracton topologically ordered [10] and subsystem symmetry-protected

topological (SSPT) phases [11]. Fracton and SSPT phases are relatively recent devel-

opments in condensed matter physics, with much of the current understanding being

developed through a concentrated effort in just the past few years.

These phases possess properties which are “subdimensional”, reminiscent of lower-

dimensional physics embedded within the full spatial dimension of the system. Frac-

ton topological phases are 3+1D long-range entangled phases (like standard topologi-

cal order) but exhibit a number of unique features. They can roughly be divided into

two types [10]. Type-I fracton phases are characterized by subdimensional point-like

excitations, meaning they can only be moved within some restricted manifold i.e.

along a plane, a line, or not at all. Such excitations are created at corners of rectan-

gular or other regularly shaped operators. Type-II fracton phases also have immobile

point-like excitations, but in contrast they are created by fractal shaped operators.

Fracton phases were shown to be deeply connected to systems with a special kind of

symmetry known as subsystem symmetry [10]. Systems with subsystem symmetries

exist in two or higher spatial dimensions. A subsystem symmetry is similar to the

Z2 spin-flip symmetry of the Ising chain, but acts only on a subextensive subsystem.

Similar to with fracton phases, such symmetries can be roughly divided into two

types: (1) regular subsystem symmetries which act along regular subsystems such as

3

a line of the square lattice, and (2) fractal subsystem symmetries which act along a

fractal-dimensional subsystem of, say, the square lattice. Like with usual symmetries,

it was quickly realized that subsystem symmetries could also protect non-trivial SPT

phases, known as SSPT phases.

This Dissertation contains a series of works by myself and collaborators in a jour-

ney to better understand these new phases of matter. Part I focuses on fracton, and

other related forms of, topologically ordered phases. Part II focuses on SSPT phases

with regular subsystem symmetries. Finally, Part III focuses on SSPT phases with

fractal subsystem symmetries.

Starting with Part I Chapter 1, I review some basic notions in topological order

and introduce the X-Cube model [10], a canonical model for type-I fracton topological

order, as the plaquette Ising gauge theory (PGT) in analogy with the regular Ising

gauge theory (IGT). In the same way that the IGT is the gauge theory of the Ising

model with the global Z2 symmetry, the PGT is the gauge theory of the 3D plaquette

Ising model, which has a subsystem planar Z2 symmetry. Chapter 2 is based on

the paper [12], which generalizes an order parameter which diagnoses the topological

phase of the IGT, introduced by Gregor et al [13], to the PGT. We supplement

this result with a quantum Monte Carlo determination of the PGT phase diagram.

Chapter 3 is based on the paper [14] in which I discuss resonating plaquette phases,

the natural generalizations of resonating valence bond phases [15] whereby spins in a

plaquette (rather than a bond) are strongly bound into a singlet. The investigation

was motivated by the realization that the X-Cube model may be thought of as a

resonating plaquette phase on a simple cubic lattice (previously studied and shown

to be confined by Pankov, Moessner and Sondhi [16] and Xu and Wu [17]) had the

U(1) hard-core constraint per site only been relaxed to a Z2 constraint. Although

a resonating plaquette phase (with a U(1) constraint) realizing a fracton phase was

not found, one interesting result was that the resonating plaquette phase on the face-

4

centered cubic lattice exhibited a Z3 topological order arising from the geometry of the

lattice. A number of ZN generalizations are then discussed, some of which contain

fracton phases. Finally, Chapter 4 is based on the paper [18] in which I discuss

some features of floating topological phases, which are stacks of lower-dimensional

topologically ordered phases. These may be thought of as being “in between” regular

and fracton topological phases. A modification of the order parameter from Chapter 2

is proposed to diagnose floating topological order, and their stability is discussed in

both gapped and gapless cases.

Part II focuses on regular subsystem symmetries, motivated by their connection

to fracton phases. The observation was first made in Ref [11] that subsystem sym-

metries, like regular symmetries, could protect non-trivial SPT phases. The various

signatures of SSPT phases were analyzed through a number of examples, starting in

2D with linear (line-like) subsystem symmetries. The main example studied was the

2D square lattice cluster model, which was protected by a Z2 × Z2 linear subsystem

symmetry. The physics of such phases are similar to those of lower-dimensional 1D

SPT phases. Some of these basic properties are reviewed in Chapter 5. One major

contribution of Ref [11] was the notion of a “strong” and “weak” SSPT. Weak SSPTs

were those that were equivalent to stacks of lower-dimensional SPT phases: a stack

of 1D SPT chains, for example, is a weak SSPT. However, the square lattice cluster

model could not be written as a stack of 1D phases, and therefore was conjectured to

be a strong SSPT. The distinction between weak and strong was made precise in the

paper [19] in terms of linear-symmetric local unitary (LSLU) transformations, which

is the topic of Chapter 6. A complete classification of 2D linear SSPT phases (using

the LSLU to define strong phase equivalence) was accomplished for an arbitrary finite

Abelian group, and we were able to show that the square lattice cluster model indeed

described a strong SSPT phase. At the same time, there had also been progress on 3D

SSPT phases protected by planar subsystem symmetries, which were dual to various

5

“twisted” fracton phases by the aformentioned gauge duality. Their basic properties

and a number of examples were given in Ref [20], although their classification in

terms of weak or strong remained unclear. At around the same time, there was a

parallel effort to classify 3D fracton phases, most notably by Shirley, Slagle, Chen,

and collaborators [21, 22, 23, 24, 25, 26] who defined the notion of “foliated fracton

phases”, defining equivalence classes of fracton phases based on whether they can be

obtained by stacking 2D topological orders. They were able to show that one of the

main examples in Ref [20] was actually dual to a fracton phase that was in the same

foliated phase as the X-Cube model. Although the connection between the foliated

phase equivalence and strong phase equivalence was not clear, this suggested that

the SSPT example from Ref [20] may be a weak SSPT phase. This was made con-

clusive in the paper [27], the topic of Chapter 7, which generalized the classification

of 2D linear SSPTs to 3D planar SSPTs. We were able to show that, indeed, the

SSPT example from Ref [20], as well as all previously discovered 3D planar SSPTs

(including any dual to any studied fracton phase and any obtainable by proposed

layer constructions at the time such as p-string condensation [28]) were weak SSPT

phases. The first strong SSPT phases and their dual fracton phases are presented

and discussed in detail.

Shortly after linear subsystem SPTs came the notion of fractal subsystem SPTs,

the topic of Part III. Chapter 8 is based on the paper [29]. Fractal symmetries are

defined, and the properties of phases with such symmetries are discussed through a

number of examples. Chapter 9 concerns the classification of fractal SSPT phases and

is based on the paper [30]. The approach to classifying SSPT phases as weak or strong

(which worked well for regular subsystem symmetries) does not apply in the case of

fractal symmetries. The necessity of strong equivalence classes in the classification

of regular SSPTs can be explained by the fact that there are uncountably infinitely

many regular SSPT phases. A sensible classification required the identification of

6

finitely many (or countably infinitely many) equivalence classes. The main result

of this chapter is that fractal SSPT phases are fundamentally different, and are al-

ready countably infinite and can be enumerated by their locality. Finally, Chapter 10

concludes the Dissertation with some brief remarks.

7

Part I

Subdimensional Topological Orders

8

Chapter 1

Preliminaries

1.1 Topological order

1.1.1 The Toric Code

We start by discussing the Hamiltonian for Kitaev’s toric code [31], originally intro-

duced as a model for fault-tolerant quantum computing. The toric code is an exactly

solvable spin model with many interesting properties that we will briefly review here,

one of which is topological order [32].

The model is described in terms of qubit (equivalently, spin-1/2) degrees of free-

dom. The Hilbert space for a single qubit is H = C2. Choosing an orthonormal basis

|0〉 , |1〉, we define the three Pauli matrices

σx =

0 1

1 0

σy =

0 −i

i 0

σz =

1 0

0 −1

(1.1)

That is, any single-qubit operator may be written as O = c01 + c1σx + c2σ

y + c3σz

for complex constants c0, . . . , c3. Some basic properties of Pauli matrices include: (1)

they alll square to identity, (2) any pair of distinct Pauli matrices anticommute with

9

each other, e.g. σxσz = −σzσx, and (3) they obey the algebra σxσy = iσz (and cyclic

permutations of X, Y, Z). Together with the identity 1, the Pauli matrices form a

complete basis for all 2× 2 matrices. We will sometimes refer to these Pauli matrices

simply as X, Y , or Z.

The toric code is defined on a system of qubits living on the bonds ` of a square

lattice. The Hamiltonian describing the toric code is

HTC = −∑

x

Ax −∑

B (1.2)

where x denotes crosses (the four bonds straddling a vertex) and denotes plaquettes.

The term

Ax =∏

`∈xσx` (1.3)

is the cross term, a tensor product of four σx operators on the bonds straddling a

site, and

B =∏

`∈σz` (1.4)

is the plaquette term, a tensor product of four σz operators on the bonds encircling

a square plaquette.

This model is exactly solvable due to the fact that all terms commute. All A

terms commute with one another due to only being a tensor product of Pauli σzs,

and similarly all B terms. All A terms also commute with all B terms since any

cross shares an even number of bonds with any plaquette, so the −1 signs picked up

from commuting σx` with σz` always comes in pairs. The ground state is therefore the

simultaneous +1 eigenstate of all A and B terms.

Working in the σx basis, we see that the A term enforces that, at each site, only

an even number of bonds straddling it can have σx` = −1. This means that basis

states which satisfy Ax |ψ〉 = + |ψ〉 for all Ax are only those where σx` = −1 bonds

10

form closed loops. The B term flips four σx` around a plaquette, thus transitioning

between two configurations of closed loops. Starting with the polarized configuration

with all σx` = 1, |0〉, the ground state can be obtained by projecting to the B = +1

subspace,

|GS〉 =∏

(1 +B

2

)|0〉 (1.5)

and has the interpretation of being the equal amplitude sum of all loop configurations

reachable by applying B on |σx` = 1〉.

Now consider this model defined on a torus. Valid configurations which satisfy

the Ax = 1 constraint include those with loops that wind non-trivially around the

torus. (such configurations are absent in |GS〉). Starting with these states with

non-trivial winding, we can define new ground states as before by projecting to the

B = 1 subspace. There are four possible ways to wind non-trivially around the

torus (no winding, winding around the x direction, winding around the y direction,

or winding around both) which generate the four degenerate ground states for HTC .

On more general manifolds of genus g, the toric code will have a 2g degenerate ground

state manifold. This topology dependent ground state degeneracy is the trademark

of topological order.

These ground states are locally indistinguishable, meaning there is no local oper-

ator that can distinguish between these ground states. The parity of loops winding

around the x direction, for example, is measured by a non-local operator that winds

around the whole system. This is the major difference between the topological ground

state degeneracy in topologically ordered systems and that arising from spontaneous

symmetry breaking (e.g. the Ising ferromagnet). Following from this local indis-

tinguishability is the fact that the ground state degeneracy cannot be split (in the

thermodynamic limit of large system size) by any local perturbation. Gapped topolog-

ically ordered phases such as this are therefore stable to arbitrary small perturbations

to the Hamiltonian [33].

11

The ground state degeneracy may also be obtained by counting stabilizers. The

stabilizer group S is generated by the terms in the Hamiltonian, Ax and B. Although

there are also 2N such terms, they are not all independent: the product∏

xAx =∏B = 1. Thus, there are 2N − 2 independent generators of the stabilizer group.

On the square lattice with N sites, there are 2N qubits for a 22N dimensional Hilbert

space. Restricting to the subspace in which each of the 2N − 2 generators are +1

leaves us with a 22N−(2N−2) = 22 ground state degeneracy.

Let us define the Wilson loop

WC =∏

`∈Cσz` (1.6)

which is a product of σz` on all bonds going along a loop denoted by C, along with

the dual Wilson (or ‘t Hooft) loop

VC =∏

`∈C

σx` (1.7)

which is a product of σx` on all bonds cutting a dual loop C. Both W and V commute

with HTC . Also, since W1 can be multiplied by B without changing its action on

the ground state, the exact shape of the loop C does not matter — only its overall

topology. Let W1, V1 denote the non-contractible Wilson loops going around the torus

in the x direction, and W2, V2 along y. In this case, W1 and V2 anticommute, and

similarly W2 and V1. The operators (W1, V2) and (W2, V1) therefore generate the Pauli

algebra on the four-dimensional ground state manifold.

In the quantum information language, the toric code is a stabilizer code which en-

codes two logical qubits into its ground state manifold. The operators (W1, V1,W2, V2)

are called logical operators, since they act non-trivially on the encoded qubits.

Another important feature of topological order is the emergence of topological

quasiparticle excitations. A quasiparticle excitation is simply a point-like excitation

12

above the ground state, which is topological if it cannot be created by itself from the

vacuum (here, vacuum means the ground state). Instead, they must be created in

multiples (for example, a particle-antiparticle pair may be created from the vacuum).

The toric code has two fundamental types of excitations, which we call the e and the m

excitation (for “electric” and “magnetic” from the analogy to electromagnetism). The

e excitations are created at the ends of an open Wilson loop (aka Wilson line) operator

W . Acting on the ground state, W |GS〉 results in a state with two excitations located

at its endpoints: it has Ax = −1 at the two endpoints, while remaining identical to

the ground state everywhere else. Similarly, the m excitations are excitations of the

B terms in the Hamiltonian, and are created at the ends of an open dual Wilson

line operator V .

The topological nature of the quasiparticle excitations is apparent through their

topological braiding phase. One can imagine a process in which, starting with a

state with spatially separated quasiparticles, one quasiparticle is adiabatically moved

around another and finally returning to the same state up to a phase factor. In

addition to the usual dynamical phase factor e−i∫E(t)dt coming from the usual Schro-

dinager equation, where E(t) is the total energy at time t, there is an additional

topological phase that may arise from the braiding statistics of the quasiparticles

themselves. For the toric code, this braiding phase is straightforward to obtain: e

and m pick up a −1 phase when one is brought around the other, and each e or

m braids trivially with itself. Between two identical particles, one can also define a

topological exchange phase obtained when their two positions are exchanged. Both

e and m are bosons, since they have a trivial self-exchange phase. The composite

excitation of both an e and an m, however, is a fermion: exchanging two ψ ≡ em

particles results in an overall −1 sign.

13

1.1.2 The Ising Gauge Theory

Let us now show how the toric code Hamiltonian, just discussed, emerges natu-

rally from the Ising gauge theory [34]. In particular, this exercise demonstrates an

important connection (a “gauge duality”) between certain non-topologically-ordered

systems with symmetries and topologically ordered systems, which will play an im-

portant role for much of this thesis.

Consider the 2+1D Ising model in a transverse field,

HIM = −J∑

〈i,j〉τ zi τ

zj − ΓM

∑

i

τxi (1.8)

where τx,y,zi are Pauli matrices for the spin-1/2 on site i, and the sum is over nearest-

neighbor pairs 〈i, j〉 on the square lattice.

The Ising model has a global Z2 symmetry, whose action is given by

S =∏

i

τxi (1.9)

which simply flips every spin τ z → −τ z. Indeed, [HIM , S] = 0 since HIM only consists

of pairwise τ z terms, and τx, both of which commute with S.

We may now proceed analogously to the gauging process of the U(1) charge con-

servation symmetry in electromagnetism, except the group here is the finite cyclic

group Z2.

We begin by introducing gauge degrees of freedom, which we will call σ`, along the

bonds ` = 〈i, j〉 of the square lattice. From now on, we will call τ and σ the matter

and gauge degrees of freedom respectively. Like the physical spins, σ` are two level

(qubit) degrees of freedom, so we may define the Pauli matrices σx,y,z` acting on them.

We must then modify the Hamiltonian accordingly by including the matter-gauge

14

coupling, which is done by making the substitution

τ zi τzj → τ zi σ

z〈i,j〉τ

zj (1.10)

in HIM .

The resulting Hamiltonian now has a local gauge symmetry, which is generated

by the local operator

Gi = τxi∏

〈i,j〉σx〈i,j〉 (1.11)

which flips the matter qubit τ zi , along with the four gauge qubits σz〈i,j〉 on the bonds

straddling it. This reflects a redundancy in our description of the system: not all con-

figurations of τ z, σz correspond to different physical states. We define the physical

subspace to be the subspace of states satisfying Gi |ψ〉 = |ψ〉 for all i. The final in-

gredient is a flux-free constraint. In the original theory, the product of the four bond

terms τ zi τzj around a square plaquette is trivially identity, since each τ zi is included

twice. However, in the gauged theory, this results in a non-trivial term consisting

purely of gauge qubits. To ensure that the gauge theory representation is faithful to

the original model, we must therefore enforce the constraint

1 =∏

`∈σz` (1.12)

which we interpret as the Z2 version of a flux-free condition.

The Ising gauge theory (IGT) takes this model as the starting point. The flux-free

condition is relaxed down to an energetic constraint on the ground state by a term in

the Hamiltonian, and the gauge fields are given dynamics in the form of a σx term.

The IGT Hamiltonian is

HIGT = −J∑

〈i,j〉τ zi σ

z〈i,j〉τ

zj − ΓM

∑

i

τxi −K∑

∏

`∈σz` − Γ

∑

`

σx` . (1.13)

15

along with the constraint that physical states lie in the Gi = +1 subspace. It is always

possible to use this constraint to gauge-fix the matter qubits, such that we obtain

a representation of HIGT only in terms of the gauge qubits. For each configuration

|σz, τ z〉, we use the fact that Gi = +1 in the physical subspace and the fact that

Gi flips τ zi → −τ zi , to find a representative state with τ zi = +1∀i. In other words, we

fix the gauge to one in which τ zi = +1∀i. The resulting gauge-fixed Hamiltonian is

H ′IGT = −ΓM∑

x

∏

`∈xσx` −K

∑

∏

`∈σz` − J

∑

`

σz` − Γ∑

`

σx` . (1.14)

This makes the connection to the toric code clear. Setting ΓM = K = 1 and

J = Γ = 0, this is exactly the toric code Hamiltonian. The J and Γ terms act

as σz and σx perturbations to the toric code. There are two phases to HIGT : the

deconfining or confining phases. The deconfined phase hosts deconfined fractional-

ized quasiparticle excitations, meaning finite energy states exists in the spectrum in

which quasiparticle excitations are arbitrarily far apart. This corresponds to the topo-

logically ordered phase, which exists at small J/ΓM , Γ/K. In the confining phase,

however, quasiparticle excitations do not exist in isolation (c.f. quark confinement).

This is the topologically trivial phase, which exists at large J/ΓM or large Γ/K.

These two limits are smoothly connected as shown by Fradkin and Shenker [35]

Note that the topological phase of HIGT corresponds (on the pure-matter Γ = 0

axis) to the paramagnetic phase of the original Ising model. We briefly mention here

that this type of connection between a symmetric non-topologically ordered phase

(the paramagnetic phase of the Ising model) and a topologically ordered phase (the

deconfined phase of the Ising gauge theory) is an example of a gauge duality. This

gauge duality relates a gapped symmetric phase of matter with a unique ground

state to a gapped topologically ordered phase of matter (which no longer has the

original symmetry). In particular, there may be many distinct symmetric phases of

16

matter: these are known as symmetry-protected topological (SPT) phases. Distinct

SPT phases will be dual to distinct topological orders. Exactly how this works will

be reviewed in Chapter 7.

1.1.3 More general topological orders

The toric code is but one simple example of a topologically ordered phase. Specifi-

cally, it is a bosonic gapped Abelian topologically ordered phase. Bosonic referring

to the fact that the fundamental degrees of freedom are spins (bosonic); gapped re-

ferring to the finite energy gap in the thermodynamic limit; and Abelian meaning

that quasiparticle braiding/fusion are Abelian processes. It is also worth mentioning

that there is a rich and active literature surrounding gapped non-Abelian topological

orders as well. Gapped topologically ordered phases up to 2+1D have been fully

classified, including with symmetries, in seminal work by Wen and collaborators [36].

The majority of phases that will be discussed in this thesis fall into this same category

of bosonic gapped and Abelian.

For such phases, there is a simple description in terms of the low energy topological

quantum field theory. The low energy physics of the toric code is described by the

BF theory [37, 38], or equivalently the Chern-Simons theory with a K matrix, with

the Lagrangian density

L =KIJ

4πεµνρaIµ∂νa

Jρ (1.15)

where K is a 2×2 integer matrix K = 2σx, εµνρ the antisymmetric Levi-Civita symbol,

I, J ∈ [1, 2], µ, ν, ρ ∈ [0, 1, 2] corresponding to time and two spatial dimensions, aIµ

are the real valued quantum fields, and summation over repeated indices is implied.

In fact, all Abelian topologically ordered phases may be described by such a theory

with an appropriate integer K matrix [39]. Topological properties may be directly

obtained from this K matrix description. The ground state degeneracy is simply

17

given by

GSD = detK. (1.16)

The matrix elements of the matrix inverse K−1 determine the braiding and exchange

statistics of quasiparticle excitations. Let each I corresponds to a label of some

generating set of quasiparticle excitation, each of which are represented by the unit

vector eI . Composite excitations are a combination of these generating set, and

are represented by a general vector ~v. The topological braiding phase between two

quasiparticles ~v, ~w is

Braid(~v, ~w) = e2π~vTK−1 ~w (1.17)

and the self-exchange phase for a ~v quasiparticle is

Exchange(~v) = eπ~vTK−1~v. (1.18)

In higher dimensions, many more things are possible. In 3+1D, gapped topological

phases similar to those in 2+1D are known to exist. Take for example a 3+1D

generalization of the toric code: the Ax term is modified now to include all 6 bonds

straddling a site, and B is now summed over all three orientations of squares on

the simple cubic lattice. This model has pointlike excitations of the Ax term (the e

particle) as before, which are created at the ends of Wilson line operators. However,

excitations of the B terms are no longer point-like but instead loop-like and are

created at the edges of a dual Wilson surface operator. But what else is possible?

In the search for a fault-tolerant quantum memory, Haah considered a class of

stabilizer codes which lacked string logical operators [40]. The presence of string log-

ical operators means the presence of mobile topological quasiparticle excitations (a

truncated string operator creates two excitations at its endpoints — this may alterna-

tively be interpreted as an operator which moves an excitation from one endpoint to

another). The absence of any string logical operators means that a topological quasi-

18

particle excitation (if it exists) is strictly immobile. The codes that Haah found had a

ground state degeneracy which behaved erratically (but as a overall increasing expo-

nentially) with system size and logical operators that took the shape of complicated

fractal structures. The model had quasiparticle excitations which were immobile in

isolation, but certain configurations of them could be moved together. These are all

trademarks of what are now known as fracton topologically ordered phases.

1.2 Fracton topological order

1.2.1 The X-Cube model

As in the previous section, we start with an example: the X-Cube model [10]. The

X-Cube model is, to fracton topological order, like the toric code is to conventional

topological order. The model is described on the simple cubic lattice with qubit

degrees of freedom on the bonds. The Hamiltonian is

HXC = −∑

c

Ac −∑

x

Bx (1.19)

where the first sum is over crosses x, and the second is over cubes c (hence the name

X-Cube). Each site of the cubic lattice has six outgoing bonds, and is associated with

three crosses of different orientations. Each orientation corresponds to a separate

term in the Hamiltonian

Bx =∏

`∈xσz` (1.20)

which is a product of four σz. For each cube c, we have the term

Ac =∏

`∈cσx` (1.21)

19

which is a product of twelve σx along the edges of the cube c. Like the toric code, all

these terms commute. The ground state is the simultaneous +1 eigenstate of all Ac

and Bx.

There are many ways to compute the topological ground state degeneracy of this

model on the 3-torus. Here we take the approach of counting logical operators. Note

that the operator

Kxy(x, y) =∏

`∈line(x,y)

σx` (1.22)

where line(x, y) is the line going along the z direction at xy coordinate (x, y). This

operator commutes with H, but is also not a product of stabilizers. It is therefore

a logical operator, and operates non-trivially on the ground state manifold. Not all

(x, y) are independent, however. We have

Kxy(x, y)Kxy(x′, y)Kxy(x, y′)Kxy(x′, y′) = Stabilizers (1.23)

for all x, x′, y, y′, where the r.h.s. is a product of stabilizers which, on the ground state

manifold, is equal to one. Picking a reference x0, y0, all Kxy(x0, y) and Kxy(x, y0)

may be found to be ±1 independently in the ground state. Then, all others are

fixed by the relation Eq 1.23, Kxy(x, y) = Kxy(x0, y)Kxy(x, y0)Kxy(x0, y0). There

are therefore 2L − 1 independent logical operators of this kind. Similarly, we have

Kyz and Kzx, leading to a total of 6L − 3 independent operators. There are also

other Pauli Z-type logical operators which complete the Pauli algebra on the ground

state manifold, but we will not go into detail on them here. A single ground state is

specified by a particular choice of ±1 for each of these independent operators, leading

to a total ground state degeneracy of 26L−3 in agreement with Ref 10.

A subextensive topological ground state degeneracy that depends on system size

L is one of the defining features of a fracton topologically ordered phase. The other

20

defining characteristic is the presence of topological quasiparticles that have restricted

mobility.

Consider an excitation of a single Ac term. Such an excitation may be created at

the corners of a dual membrane operator. A rectangular shaped operator will create

four of such excitations. Since they can only be created in groups of four (and in

certain spatial configurations) there is generically no operator which moves a single

quasiparticle from one lattice point to another. Such a quasiparticle is immobile —

such immobile particles are called fractons. A pair of fractons created at the ends of

a ribbon shaped operator, however, can be moved in the plane perpendicular to their

orientation. This composite of two fractons is therefore mobile in two dimensions.

We call such an excitation a planon.

An excitation of a Bx term on a single vertex can be created at the ends of a line

operator (since the three Bx on a vertex are not independent, we must have that two

of the three Bx at a vertex are −1). The line operator must be along a straight line,

if it bends then another excitation will be created at its bending point. Thus, a single

Bx excitation may only move along a one dimensional line. We call this excitation a

lineon excitation.

Despite having restricted mobility, there are various braiding processes between

the lineon and fracton that results in a non-trivial topological phase. These braid-

ing processes either create additional excitations during the braiding process or by

braiding composite planon excitations. In conventional 3+1D systems, two particles

must braid trivially with one another due to the fact that the braiding process can

always be smoothly deformed to a trivial operation, and the self-exchange statis-

tics must be either bosonic or fermionic. In fracton phases, quasiparticle excitations

have restricted mobility. The braiding of two planons, for example, is topologically

non-trivial.

21

1.2.2 The Plaquette Ising Gauge Theory

In the previous section, we made the connection between the toric code model and

the gauge theory of the 2+1D Ising model with a global Z2 symmetry. The same is

true for the X-Cube model but instead to a 3+1D Plaquette Ising model with what

is known as a Z2 planar subsystem symmetry. Starting from the Ising theory, we will

demonstrate how the X-Cube model arises naturally through a generalized gauging

process first introduced by Vijay, Haah, and Fu [10] and Williamson [41].

THe starting point is the Plaquette Ising model [42] on a cubic lattice

HPIM = −J∑

∏

i∈τ zi − ΓM

∑

i

τxi (1.24)

where qubits τ live on the sites, and there is a four-body τ zτ zτ zτ z interaction along

every plaquette.

We first note that this model has some very special symmetries. Like the Ising

model, it has the global Z2 symmetry that involves flipping all spins Sglob =∏

i τxi .

But it also is invariant under a planar Z2 subsystem symmetry

Sss =∏

i∈plane

τxi (1.25)

where the product is over all sites in any xy, yz, or zx plane. Every plaquette always

has an even number of sites within a plane, thus [Sss, HPIM ] = 0 for all planes. This is

the first instance of a subsystem symmetry that we have encountered thus far. They

will be the subject of the next chapter.

Subsystem symmetries are very different from conventional global symmetries. For

one, the total symmetry group of a system grows with its size. For an L×L×L system,

there are 3L planes and the total symmetry group is Gtot = (Z2)3L−2 (the −2 comes

from the fact that the product of all xy planar symmetries is the same as the product

22

of all yz or zx planar symmetries). One consequence of this large symmetry group

is that, for ΓM = 0, the model HPIM has 23L−2 degenerate spontaneous subsystem

symmetry breaking ground states.

All the ground states in the h = 0 limit may be obtained by applying any combi-

nation of subsystem symmetries to the fully polarized state |τ zi = +1〉. For small

h, these states remain degenerate (there is a stable spontaneous subsystem symme-

try breaking phase at zero temperature). At a critical value of field (ΓM,c ≈ J/0.3

according to numerics in Sec 2.4) there is a transition to a paramagnetic phase where

qubits are polarized along τx and there is a unique gapped ground state.

How do we gauge such a symmetry? Although the conventional gauging process

still works when applied to this system, it only affects the global part of the symmetry.

We would ideally like some process which utilizes the full subsystem symmetry group

— this is the generalized gauging procedure [10, 41].

Let us forget what we knew about gauging and simply apply the same operational

steps to HPIM as we did to HIM . In gauging HIM , we introduced gauge qubits σz

along the bonds. We did this so that we could couple the gauge qubit to the two-body

interaction term, τ zτ z → τ zσzτ z, thereby leading to the local gauge transformation.

In HPIM , however, the simplest τ z interaction term possible is the four body term∏

i∈ τzi present in HPIM . Suppose we wish to perform the same replacement, except

with the four-body interaction term:∏τ z → σz

∏τ z. The natural step would be to

introduce “gauge” qubits σ on each plaquette of the cubic lattice, rather than the

bonds, such that one gauge qubit can be associated with each term in the Hamiltonian.

We therefore perform the following replacement of each plaquette term

∏

i∈τ zi → σz

∏

i∈τ zi . (1.26)

23

Proceeding as with the IGT, the local gauge transformation is generated by the

operator

Gi = τxi∏

|i∈σx (1.27)

which involves flipping a spin τ zi , along with the gauge qubits σz on the twelve

plaquettes containing the vertex i. Like before, we define the physical subspace to be

the Gi = +1 subspace. Finally, there is a constraint on certain pure-gauge (purely σz)

operators. The product of four plaquette terms along a matchbox m (four plaquettes

encircling a cube, in one of three possible orientations) is identity, as each τ z is

included twice. We must therefore enforce the constraint

∏

∈mσz = 1 (1.28)

on the gauged theory. This is the analogue of the zero-flux constraint of the IGT.

To obtain the Plaquette Ising Gauge theory (PGT), we enforce zero-flux constraint

energetically by a term in the Hamiltonian, and add dynamics to the gauge qubits,

HPGT = −J∑

σz∏

i∈τ zi − ΓM

∑

i

τxi −K∑

m

∏

∈mσz − Γ

∑

σx. (1.29)

The redundancy in our description can be removed by gauge-fixing to τ zi = +1,

resulting in the gauge-fixed PGT Hamiltonian

H ′PGT = −J∑

σz∏

i∈τ zi − ΓM

∑

i

∏

|i∈σx −K

∑

m

∏

∈mσz − Γ

∑

σx. (1.30)

Let us first examine the limit J = Γ = 0. In this limit, HPGT reduces to the

X-Cube Hamiltonian. To see this, we switch from the original (“direct”) lattice to

the dual lattice: sites of direct lattice are mapped on to cubes on the dual lattice,

and plaquettes are mapped to bonds. Then, the ΓM term becomes exactly the cube

24

term Ac and the K term becomes the cross term B+ from the X-Cube Hamitonian

HXC . The physics this model in the small J , Γ , phase are therefore described by

the X-Cube model. In analogy with the IGT, we shall refer to this phase as the

deconfined (fracton topologically ordered) phase.

As one increases J or Γ , there is once again a phase transition into a confining

phase, in which σ are mostly polarized along some directionk The subject of the

next chapter is the identification of this phase transition by means of a non-local

correlation function, where we map out the phase diagram.

We have therefore shown that fracton physics emerges naturally through a general-

ized gauging duality applied to systems with subsystem symmetries. Haah’s code [40],

a more complicated fracton model, has also been shown [10, 41] to arise from apply-

ing the generalized gauging procedure to an Ising model with a fractal subsystem

symmetry. This will be elaborated on when we discuss fractal symmetries in Part III.

Fracton phases can broadly be categorized as either “Type-I” or “Type-II” [10]

(although there also exist examples that fit into neither or are in between). Type-I

fracton phases are like the X-Cube: the ground state degeneracy scales exponentially

with L and quasiparticles may be fractons, lineons, or planons. These are dual to

systems with regular (planar, for example) subsystem symmetries. Type-II fracton

phases, on the other hand, are like Haah’s code: the ground state degeneracy may be

a complicated function of system size and topologically non-trivial quasiparticles are

generically only fractons. These tend to be dual to systems with fractal symmetries.

25

Chapter 2

Correlation function diagnostics

The goal of this section is construct a correlation function diagnostic which is capable

of diagnosing deconfinement in type-I fracton theories. We will first review how this

is done in the case of conventional topological order using the IGT as an archetypical

example [13]. Then, we will discuss a generalization of this order parameter to the

PGT which is capable of diagnosing deconfined fracton topological order. Finally,

as a demonstration, we use the order parameter to numerically compute the phase

diagram of the PGT using quantum Monte Carlo (more specifically, the stochastic

series expansion method). This chapter is based on the paper

[12] T. Devakul, S. A. Parameswaran, S. L. Sondhi, “Correlation function diagnos-

tics for type-I fracton phases”, Phys. Rev. B 97, 041110(R) (2018).

Despite rapid progress in advancing the theory of these novel 3D topological

phases, there is a paucity of sharp characterizations of fracton deconfinement away

from the stabilizer limit, e.g. when fractons acquire dynamics or are at finite density.

One possible diagnostic is to extract topological contributions to the entanglement

entropy [43, 44, 45], but this requires an exact computation of ground states, typically

challenging in 3D, and does not immediately generalize to T > 0. For topological

orders described by standard lattice gauge theories, a trio of loop observables suitably

26

oriented in Euclidean space-time serves this role, and furthermore may be directly

computed from, e.g. quantum Monte Carlo simulations. Can such diagnostics be

adapted to study these new states in the presence of dynamical fractonic matter?

Here, we answer this in the affirmative for the X-cube model, and argue that our

results may be generalized to all Type-I fracton phases of which it is the paradigmatic

example. We do so by studying the PGT (and another dual Ising model, which we will

introduce). Although quasiparticle excitations of these models are always constrained

to lower-dimensional subspaces and are hence not truly deconfined, they are in a sense

partially deconfined within these subspaces. We show that the standard technology

for diagnosing the deconfined and confined phases [13, 46], reviewed next, can indeed

be generalized in a straightforward manner to detect this partial deconfinement that

can be viewed as a defining property of fractonic matter.

2.1 Ising gauge theory

We first begin with a quick review of the diagnostics in the case of the IGT Hamilto-

nian, reprinted here from Eq 1.13,

HIGT = −J∑

〈i,j〉τ zi σ

z〈i,j〉τ

zj − ΓM

∑

i

τxi −K∑

∏

`∈σz` − Γ

∑

`

σx` . (2.1)

As discussed, this model reduces to Kitaev’s Toric code [31] in the limit J = Γ = 0.

Introducing nonzero J or Γ can then be thought of as perturbations from the Toric

code point. Turning Γ too high will drive the gauge theory into a trivial confined

phase, and turning J too high will result in a Higgs transition into a symmetry broken

phase. These two limits are smoothly connected [35], thus we will refer to both as

the confined limits, and small perturbations of the Toric code point as the deconfined

limit (characterized by Z2 topological order).

27

Let us now consider moving along the “pure gauge theory” axis, Γ > 0, J = 0,

along which the matter is static, τxs = 1 and therefore can be ignored. Here, the

spatial Wilson loop, W =∏

`∈C σz` , where C is a closed loop (taken for simplicity

to be an L × L square), serves as a diagnostic that can distinguish the confined

and deconfined phases. At the Toric code point Γ = 0, we have 〈W 〉 = 1. Small

perturbations in Γ create local fluctuations of pairs of “visons”, plaquettes on which∏

`∈ σz` = −1 (the magnetic flux excitations of the theory). As the Wilson loop

measures the average parity of visons contained within it, these fluctuations will

cause the expectation value to decay proportionally to the perimeter of the loop,

following a perimeter law: log〈W 〉 ∼ −L for large L. In the confined phase at large

Γ , the visons are condensed and so here log〈W 〉 ∼ −L2 follows an area law for large

L. However, as soon as we add dynamical matter J > 0, the Wilson loop follows a

perimeter law everywhere. To see this, notice that in perturbation theory in J about

the J = 0 ground state |ψ0〉, a term matching the Wilson loop operator appears at

O(JL): |ψ〉 = |ψ0〉+αe−βLW |ψ0〉+. . . for some numbers α ∼ O(1) and β ∼ − ln J , so

that there is at least a perimeter law component to 〈W 〉 which dominates as L→∞.

Thus, the Wilson loop fails as a deconfinement diagnostic as soon as J > 0.

Now, consider moving along the “pure matter theory” axis, with J > 0, Γ = 0.

Here, the gauge field exhibits no fluctuations, and it is convenient to work with

σz = 1, and project onto the gauge invariant subspace if needed. In this subspace, the

Hamiltonian is simply the original Ising model in a transverse field. Beyond a critical

J , there is a transition to an ordered phase where 〈τ z〉 gains an expectation value.

However, τ z alone does not correspond to a gauge invariant operator; only pairs of τ z

do. This transition can therefore be diagnosed by a Wilson line W = τ zi τzj

∏`∈Cij σ

z`

where Cij is a path connecting sites i and j, which in this subspace is simply the

spin-spin correlation function 〈τ zi τ zj 〉. As one takes |~ri − ~rj| → ∞, this either goes to

zero in the deconfined (paramagnetic) phase, or approaches a constant in the confined

28

(Higgs ferromagnetic) phase. This can also be understood without referring to the

matter theory as the vanishing of a line-tension in the Euclidean action [13]. Now

consider adding in a small Γ perturbatively: σx anticommutes with the σz chain,

and so 〈τ zs σz . . . σzτ zs′〉 decays to zero exponentially with |~ri − ~rj| in both phases. We

therefore again are in a situation where a diagnostic that works exactly along this

axis fails as soon as Γ > 0.

How then can we distinguish the confined from the deconfined phase away from

these special axes? The answer is to measure an appropriate line tension, using wis-

dom gained from the Euclidean path integral representation which maps the problem

on to an isotropic 3D statistical mechanical problem of edges and surfaces [13, 47].

This can be linked to the expectation value of a “horseshoe operator”, viz. an

L × L Wilson loop cut in half (with τ z inserted at the ends for gauge invariance),

W1/2 = τ zi τzrj

∏`∈C1/2

σz` , where C1/2 defines the half-Wilson loop of dimension L/2×L,

terminating at sites i and j. The ratio of expectation values as L→∞,

R(L) =〈W1/2〉√〈W 〉

L→∞−−−→

0 deconfined

const. confined

(2.2)

can then be understood as measuring the “cost” of opening the Wilson loop. In the

deconfined phase, opening a Wilson loop will cause the expectation value to decay

exponentially with the size of the gap. In the confined phase, the expectation value of

the Wilson loop follows a perimeter law regardless of whether it is opened or closed,

thus the scaling with L is exactly cancelled out by dividing by the square root of the

full Wilson loop.

Since the Euclidean IGT is space-time symmetric, by choosing distinct orientations

and ‘cuts’ of the loop, we can identify three different diagnostics. Besides (1) the

‘spatial loop’ discussed above, the two possible cuts for the orientation extending

along the time direction also have elegant physical interpretations [13]: either (2) as

29

the Fredenhagen-Marcu diagnostic [48, 49], measuring the overlap between the ground

state and the normalized two-spinon state; or (3) as a measure of delocalized spinon

(electric-charge) excitations. By the self-duality of the IGT this exercise could have

been done in the dual model, which defines a different Wilson loop object and exactly

interchanges the role of the gauge (Γ , K) and matter (J , ΓM) sectors [34].

2.2 Euclidean Path Integral and Wilson Loops

We will now proceed with our analysis of the plaquette Ising gauge theory (PGT),

which arises from applying the generalized gauging procedure to the classical plaque-

tte Ising model [42].

From Eq 1.29,

HPGT = −J∑

σz∏

i∈τ zi − ΓM

∑

i

τxi −K∑

m

∏

∈mσz − Γ

∑

σx. (2.3)

where now the σs live at the center of plaquettes , c denotes a cube, and m cor-

respond to one of three distinct combinations of four plaquettes that wrap around a

cube (matchboxes). We further have a constraint defined on each site i,

Gi = τxi∏

|i∈σx = 1, (2.4)

where the product is over the 12 plaquettes touching the site i. This model, as

discussed, is just a perturbed X-cube model for small J and Γ . The deconfined phase

of this model hosts two types of excitations: the “electric” excitations are fractons,

while the “magnetic” excitations are lineons.

In standard gauge theory, one is often only concerned about the deconfinement

of the electric charge excitations. The X-cube model (unlike the Toric code) does

not possess an electro-magnetic (σz ↔ σx) self-duality, so for completeness we also

30

consider the “electromagnetic” dual to the PGT. This dual model arises naturally

from the same generalized gauging procedure on the classical dual of the PIM, which

can be written as an anisotropically coupled Ashkin-Teller model [50, 51]. Note that

the duality discussed here maps between two full gauge-matter theories; the “F-S

duality” between a pure matter theory and pure fracton gauge theory [10] is a limiting

case. We construct deconfinement diagnostics for the electric charge in both the PGT

and its dual, thus providing diagnostics for both fracton and lineon excitations.

For a full space-time discussion of Wilson loop analogues, we construct a discrete-

time Euclidean path integral for the PGT. The gauge constraint will be enforced by

the introduction of auxiliary spin-1/2 degrees of freedom along the time-links of the

4D hypercubic lattice [34, 52], that we will denote λ (in the IGT one has a space-time

symmetric structure so these spins can be thought of as σ spins along the time-links,

but this is not the case here).

The starting ingredient is the PGT Hamiltonian Eq. (2.3) (which we simply refer

to as H) and the constraint Eq. 2.4 that must be satisfied at every site.

We are interested in calculating the partition function Z(β) = Tre−βH for inverse

temperature β (we take β →∞ to access the relevant, zero-temperature limit). To do

this, we employ the usual Suzuki-Trotter decomposition: we divide the interval β into

Lt small steps of size ε, such that β = Ltε. This then allows us to write the partition

function as a path integral in the z-basis. Finally, to enforce the constraint, we insert

the projector into the gauge-invariant subspace at every time step, P =∏

i(1+Gi)/2.

So, we have

Z(β) =∑

σz(t),τz(t)

Lt∏

t=1

〈σz(t+1), τ z(t+1)|Pe−εH |σz(t), τ z(t)〉 (2.5)

= limε→0

∑

σz(t),τz(t)

Lt∏

t=1

〈σz(t+1), τ z(t+1)|Pe−εHxe−εHz |σz(t), τ z(t)〉 (2.6)

31

where in the second step we have performed a Trotter decomposition e−εH ≈

e−εHxe−εHz + O(ε2), separating the parts of the H containing σx, τx and σz, τ z into

Hx and Hz respectively. We have also enforced periodic boundary conditions on the

time direction.

Let us now focus on evaluating a single one of these terms in the product 2.6. The

path integral is performed in the z-basis, thus we can move the state past e−εHz , pick-

ing up only a number e−εHz(σz ,τz) where Hz(σz, τ z) denotes 〈σz, τ z|Hz|σz, τ z〉.

Then, what’s left is to compute 〈σz′, τ z′|Pe−εHx|σz, τ z〉.

For ease of notation, let us define the projector for O, PO ≡ (1−O)/2. In terms

of these operators, we have the following:

〈σx, τx|σz, τ z〉 = eiπ(

∑ Pσx

Pσz

+∑i Pτxi

Pτzi

)(2.7)

e−εHx ∝ e−2εΓ

∑ Pσx

−2εΓM∑i Pτxi (2.8)

where we are ignoring an overall shift in Hx, and finally

P =1

2Ns/2+Np/2

∏

i

(1 + τxi∏

|i∈σx) =

1

2Ns

∑

λi=±1eiπ

∑i Pλi (Pτxi

+∑|i∈ σ

x)

(2.9)

whereNs (Np) is the number of sites (plaquettes), and we introduced the Ising variable

λi to mediate the constraint on site i.

Inserting a resolution of the identity 1 =∑σx,τx |σx, τx〉〈σx, τx|, and using

Eqs. (2.7-2.9), we get

〈σz′, τ z′|Pe−εHx|σz, τ z〉 =

1

22Ns+Np

∑

λi

∑

σxp ,τxs e∑ Pσx

(−2εΓ+iπ[Pσz

+Pσz′

+∑i∈ Pλi ])

× e∑i Pτxi

(−2εΓM+iπ[Pτz′i

+Pτzi

+Pλi ])

32

=1

22Ns+Np

∑

λi

∏

(1 + e−2εΓ+iπ[Pσz

+Pσz′

+∑i∈ Pλi ])

∏

i

(1 + e−2εΓM+iπ[Pτz′

i+Pτz

i+Pλi ])

=1

22Ns+Np

∑

λi

∏

(1 + e−2εΓσzσz′

∏

s∈λi)∏

i

(1 + e−2εΓM τ z′i τzi λi)

∝∑

λieΓ

∑ σ

z′σ

z

∏i∈ λi+ΓM

∑i τz′i τ

zi λi

where Γ = −12

log tanh εΓ and ΓM = −12

log tanh εΓM . Thus, λs can be thought of

as a spin variable located on the bond between site s at time t and t + 1. Labelling

each λ(t)s by the time index and combining all our parts, the total partition function

is given by Z(β) ∝∑σ(t),τ (t),λ(t) e−SPGT(σ(t),τ (t),λ(t)) where we have suppressed the z

label on σ(t), τ (t), with the action

SPGT =− K∑

t,m

∏

∈mσ

(t) − ΓM

∑

t,i

τ(t)i τ

(t+1)i λ

(t)i

− J∑

t,

σ(t)

∏

i∈τ

(t)i − Γ

∑

t,

σ(t) σ

(t+1)

∏

i∈λ

(t)i

(2.10)

where we have defined K = εK and J = εJ , and m are matchboxes as before. Note

that the gauge constraint manifests as a local symmetry in the action: a simultaneous

flip of τ(t)i , σ

(t) for |i ∈ , λ

(t)i , and λ

(t−1)i leaves the action unchanged. Thus, we

have successfully obtained the Euclidean action for the PGT. The zero temperature

limit can be taken by making the time direction infinite.

Finally, due to the Ising nature of these variables, we may now express the partition

function as a sum of products involving every possible combination of terms in the

33

action,

Z ∝ Trσ,τ,λ∏

t,m

(1 + [tanh K]

∏

∈mσ

(t)

)∏

t,i

(1 + [tanh ΓM ]τ

(t)i τ

(t+1)i λ

(t)i

)

×∏

t,

(1 + [tanh J ]σ

(t)

∏

i∈τ

(t)i

)∏

t,

(1 + [tanh Γ ]σ

(t) σ

(t+1)

∏

i∈λ

(t)i

)

(2.11)

Expanding the product, any term that contains an odd number of any σ(t) , τ

(t)i , λ

(t)i

vanish under the trace. Thus, only combinations in which each of these appear an even

number of times contribute to the partition function. This can therefore be thought

of as a statistical mechanical model of edges, and surfaces, where each configuration

appears with its own weights, but with a more complex set of rules for allowed shapes

than in the edge-surface statistical mechanical interpretation that can be given to the

Euclideanized partition function of a conventional gauge theory. Nevertheless, it is

still possible to assign an interpretation of the confinement/deconfinement transition

in terms of vanishing string and surface tensions: K (Γ ) play the role of a surface

cost in the space (time) directions, and J (ΓM) play the role of the edge cost in the

space (time) directions. In this language, the deconfined phase corrsponds to a phase

with zero (macroscopic) surface tension and high line tension, and the confined phase

to one where either surface tension is nonzero or line tension is zero.

To summarize, we have ZPGT = Trτ,σ,λe−SPGT , with SPGT in Eq 2.10. This can

be viewed as a statistical mechanical model of edges, surfaces, and volumes in 4D,

but with a more subtle set of rules for how to build allowed objects from these.

Proceeding by analogy with the IGT, we now construct the Wilson loops for the

PGT and its dual (Fig. 2.1). Spatial loops are constructed by choosing a set of cubes

c whose centers lie in a plane and taking the product of their matchbox terms (terms

multiplying K in the action) such that the vacant squares of each matchbox lie par-

34

cb

a

Plaquette Ising Plaquette Ising Dual

Spatial Loop

Temporal Loop

Horseshoes

Figure 2.1: The Euclidean time representation of the Wilson loop and horseshoe gen-eralizations for the PGT and its dual, which realize the X-cube topological phase.Blue circles represents τ (which lie on vertices), red represent σ (which lie on thespatial plaquettes in the PGT, but on spatial links in its dual), and green lines repre-sent the auxiliary spin λ (which lie on the links along the imaginary time direction).Non-equal time operators are shown projected to a 2+1D subspace, with the timedirection pointing “up” in the page. The three possible cut orientations are labeledby a,b, and c.

allel to the plane, resulting in a ‘ribbon loop’ encircling it. This can equivalently

can be thought of as the dynamical process of moving a two-dimensionally mobile

combination of charges around in a loop lying in a plane, via applications of the term

multiplying J in the action. For the PGT, this is a pair of fractons, while for the dual

it is a pair of parallel-moving lineons. Temporal Wilson loops are constructed in a

similar fashion, by taking the product of the six-spin terms (that multiply Γ ) corre-

sponding to each space-time cube in an L×Lτ spacetime sheet, leaving open spatial

ribbons at the initial and final slices, whose corners are linked by strings of λs. This

can equivalently be constructed by moving a one-dimensionally mobile combination

of charges a distance L apart, evolving both for Lτ in imaginary time, and bringing

35

them back together again. The combination again consists of a fracton-pair in the

PGT, but now only a single lineon in the dual. The corresponding horseshoes (or cut

Wilson loop) operators are then obtained by cutting open the loop and terminating it

with appropriate combination of τs, with three distinct possible orientations labeled

a, b, and c in Fig. 2.1.

2.3 Diagnostic behaviors

We now consider the expectation value of these operators at various points in the

phase diagram. First, note that the spatial Wilson loop alone functions as a diagnostic

only in the pure gauge theory. When J = 0, for small Γ , vison-pair fluctuations

occur only on small length scales, so that only pairs along the perimeter of the loop

will affect the expectation value. In contrast, flux excitations are condensed in the

confined phase at large Γ , so that the loop now exhibits an area law. As in the IGT,

for any J > 0 the loop obeys a perimeter law in both phases.

Next, notice also that the spatial horseshoe alone serves as a diagnostic only

along the Γ = 0 axis, where it can be understood as measuring the vanishing of a

macroscopic string tension. To understand why this expectation value is nonzero in

the Higgs/confined phase, we draw on known results for the PIM [42]. Early work on

the “fuki-nuke” model [53], which may be thought of as an anisotropic limit of the

CPIM with J = 0 for the plaquettes in the xy plane, reveals that this model maps on

to a stack of decoupled 2D (xy-planar) Ising models. In terms of the original spins,

the local observable 〈τ zi τ zi+z〉 gains a nonzero expectation value in the ordered phase,

but is free to spontaneously break the symmetry in different directions for each xy

plane. Now, the horseshoe operator (a) obtained by cutting open a xy Wilson loop

is exactly the correlation function of this observable: 〈τ zi τ zi+zτ zj τ zj+z〉 for i,j which

are constrained to be in the same xy plane, which therefore approaches a constant

36

as |~ri − ~rj| → ∞ in the ordered phase. This correlator continues to function as a

diagnostic even for the isotropic model, where we are free to choose planes oriented

in any direction [54, 55, 56].

Away from the J = 0 or Γ = 0 cases, we must rely on the ratios R(L) (Eq. (2.2))

to distinguish between the confined and (partially) deconfined phases. The ratio for

the spatial cut (a in Fig. 2.1) as before measures of the cost of opening up a gap in the

loop, which depends exponentially on the size of the gap in the deconfined phase, but

not in the confined phase. This behavior will be verified numerically using quantum

Monte Carlo soon.. At Γ = 0, R(L) reduces to the “fuki-nuke” correlation function

above.

Next, we examine the temporal loops. Consider the cut b of the PGT, W1/2 =

τ zi τzi+uτ

zj τ

zi′+u

∏∈Cuij σ

z(−T/2), where i,j are two sites on the same plane orthogonal

to u = x, y, z, and Cuij defines the set of plaquettes forming a path between them (as in

Figure 2.1). We have also defined σz(T ) = eHTσze−HT , and T = L/c for a velocity c

in the continuum time limit ε→ 0. Calling our candidate two-fracton-pair (4 fractons

in total) state |χ〉 = W1/2|G〉, created from the ground state |G〉, we see that R(L) =

〈G|χ〉/√〈χ|χ〉 measures the overlap between the ground state and our candidate

state. This is a generalization of the Fredenhagen-Marcu diagnostic [48, 49] measuring

the deconfinement of fracton-pairs, with the constraint that the two fracton-pairs

must be in the same plane of movement. The final orientation of the horseshoe (cut

c) probes the existence of delocalized fracton-pair states in the spectrum, in exactly

the same way as the delocalized spinons are probed the IGT [13].

Thus, rather than measuring the deconfinement of single spinons as in the IGT, our

Wilson loop and horseshoe generalizations instead measure the same quantities but

for the smallest mobile combinations of quasiparticles in their subspace of allowed

movement. For the PGT, this is a fracton-pair. As stated, these diagnostics only

probe the deconfinement properties of fracton-pairs, and not single fractons. To

37

identify the deconfinement of individual fractons one can do the same calculation

but using Wilson loops and horseshoes with a finite width that also scale with L.

This distinction can be important, for example, in an anisotropic version of the PGT

which exhibits an intermediate phase in which single fractons are confined into pairs,

while pairs remain deconfined (reminiscent of quark confinement into mesons) (see

supplemental material of Ref 12).

2.4 Phase Diagram and Quantum Monte Carlo

Here, we perform quantum Monte Carlo calculations to verify the behavior of the

correlation function R(L) in the different phases, as well as to map out the phase

diagram of HPGT.

2.4.1 Stochastic series expansion

We perform these simulations using the stochastic series expansion (SSE) formal-

ism [57, 58] for simplicity. For the purpose of the calculation, we gauge fix τ z = 1 and

move to the dual lattice. In the dual lattice, σ degrees of freedom live on the links `.

The Hamiltonian HPGT describes the X-cube model with σx and σz perturbations,

HPGT = −K∑

x

∏

`∈xσz` − ΓM

∑

c

∏

`∈cσx` − J

∑

`

σz` − Γ∑

`

σx` (2.12)

where x represents crosses (of which there are three per vertex), c represents cubes,

` ∈ x represents the four links taking part in the cross, and ` ∈ c the 12 links along

the edges of the cube. We also assume all parameters are positive.

38

For the purpose of the calculation, we introduce the operators

H0,0 = 1 (2.13)

Hl,0 = C` + Jσz` (2.14)

Hl,1 = Γσx` (2.15)

Hc,0 = Cc (2.16)

Hc,1 = ΓM∏

`∈cσx` (2.17)

Hx,0 = Cx +K∏

`∈xσz` (2.18)

for each link `, cube c, and cross x, such that

HPGT = −∑

`,j

H`,j −∑

c,j

Hc,j −∑

x,j

Hx,j (2.19)

up to a constant (and j = 0, 1 represents diagonal or offdiagonal terms, in the σz-

basis). The constants C`, Cc, and Cx must be chosen such that all these terms are

positive. Here, we choose C` = max(J, Γ ) + 0.5, Cc = ΓM , and Cx = K + 0.5.

In the SSE approach, we expand the partition function

Z = e−βHPGT =∑

α

∞∑

n=0

βn

n!〈α|(−HPGT)n|α〉 (2.20)

=∑

α

∞∑

n=0

∑

Sn

βn

n!〈α|

n∏

i=1

Hs(i),j(i)|α〉 (2.21)

where Sn designates a particular sequence of operators by their label,

Sn = [s(1), j(1)], [s(2), j(2)], . . . [s(n), j(n)] (2.22)

39

s(i) designates a link, cube, or cross, and j(i) = 0, 1 (except when s(i) is a cross,

in which case we only have j(i) = 0). The sum over α is over all product states

|α〉 = | σz`〉.

To construct an efficient sampling scheme, the expansion is truncated at some

power n = M sufficiently high that the cutoff error is negligible (in practice M is

increased dynamically as necessary until there is no cutoff error). A further simplifi-

cation is obtained by keeping the length of the operator string Sn fixed, and allowing

M − n unit operators H0,0 to be present in the operator list. Correcting for the(Mn

)

possible ways that H0,0 may be inserted in the list gives

Z =∑

α

∑

SM

βn(M − n)!

M !〈α|

M∏

i=1

Hs(i),j(i)|α〉 (2.23)

where now [s(i), j(i)] = [0, 0] is a valid entry in SM , and n only counts the number

of non-identity operators. For convenience, we define the state |α(i)〉 = | σz` (i)〉 ob-

tained by propagating |α〉 by the first i operators of SM . We then need to step through

the space of possible configurations SM and states α, with probability proportional

to the weight in the partition function sum Eq. 2.23.

Update procedure Due to the four and twelve-spin interactions, along with the

arbitrary transverse and longitudinal fields, naively we cannot efficiently apply non-

local update techniques such as loop or cluster algorithms [59, 60]. Notice that along

the particular axes J = 0 or Γ = 0, the model can be mapped on to Ising models

with a transverse field, for which more efficient algorithms can surely be devised.

Here, we apply a simple spatially local Metropolis update procedure. A Monte carlo

step consists of a diagonal update step, followed by a number of off-diagonal updates,

which we will detail below.

40

Diagonal update The diagonal update consists of stepping through the M ele-

ments of SM . If an off-diagonal operator [s(i), 1] is encountered, we continue on to

the next operator in the list. If a diagonal ([s(i), 0]i) or identity ([0, 0]i) operator is

encountered, we propose to replace it with an identity or diagonal operator, respec-

tively (the subscript i indicates the position of the operator in SM). If a diagonal

operator [s(i), 0]i is encountered, we remove it with probability

P ([s(i), 0]i → [0, 0]i) =M − n+ 1

β [N`(C` + J) +Nx(Cx +K) +NcCc](2.24)

where N`, Nx, Nc is the total number of links, crosses, and cubes.

If an identity operator is encountered, we propose to add a diagonal operator with

probability

P ([0, 0]i → [s(i), 0]) =β [N`(C` + J) +Nx(Cx +K) +NcCc]

M − n (2.25)

If we have decided to add an operator, we must further decide the type of operator.

The type of operator to add chosen with probabilities

P (link) =N`(C` + J)

N`(C` + J) +Nx(Cx +K) +NcCc(2.26)

P (cross) =Nx(Cx +K)

N`(C` + J) +Nx(Cx +K) +NcCc(2.27)

P (cube) =NcCc

N`(C` + J) +Nx(Cx +K) +NcCc(2.28)

(2.29)

If the type chosen is a link, we randomly pick a link ` and insert a diagonal bond

operator [`, 0] with probability

P (add link `) =C` + Jσz` (i)

C` + J(2.30)

41

otherwise, if the type chosen is a cross, we randomly pick a cross x and insert [x, 0]

with probability

P (add cross x) =Cx +K

∏`∈∂x σ

z` (i)

Cx +K(2.31)

and finally, if a cube is chosen, we choose a random cube c and insert the operator

[c, 0] with probability 1. If we fail any of these probability checks, we simply consider

the move failed and continue on to the next element i+ 1 in SM . This concludes the

diagonal update step.

Offdiagonal update We perform simple local offdiagonal updates. These come in

two types, link operator flips and cube operator flips.

The link operator flip consists of picking a link operator (diagonal or offdiagonal)

[`, j(i)]i randomly in SM . We then find the next operator acting on the same link,

[`, j(i′)]i′ , i 6= i′, and propose to flip the spin state between the two, which we accept

with a Metropolis probability

P ([`, j(i)]i[`, j(i′)]i′ → [`, j(i)]i[`, j(i

′)]i′) = min

(1,Wnew

Wold

)(2.32)

where Wnew/Wold is the ratio of the weights after and before the flip, and j(i) =

1− j(i). The weight difference depends only on the difference between the number of

satisfied and dissatisfied cross operators acting on the link ` between i and i′. Letting

n± be the number of∏

`∈x σz` = ±1 cross operators acting on site ` between i and i′

before the flip. The weight ratio before and after the flip is simply given by

Wnew

Wold

=

(Cx −KCx +K

)n+−n−(2.33)

The cube operator flip is similarly a flip of two consecutive cube operators acting

on the same cube c. We randomly pick an operator [c, j(i)]i and its next [c, j(i′)]i′ ,

and propose to flip the state of all 12 spins between the two, which we again accept

42

with probability

P ([c, j(i)]i[c, j(i′)]i′ → [c, j(i)]i[c, j(i

′)]i′) = min

(1,Wnew

Wold

)(2.34)

Notice that since the cube shares two links with any cross, there is no weight change

due to cross operators between i and i′. The only weight change due to this flip comes

from diagonal link operators [`, 0]. Similarly to before, letting n± be the number of

[`, 0] operators acting on the involved links between i and i′ with σz` = ±1, we have

the weight ratio

Wnew

Wold

=

(C` − JC` + J

)n+−n−(2.35)

Finally, we note that including only these moves is not sufficient for ergodicity,

as the total number of offdiagonal link operators acting on link l is always even, and

the total number of offdiagonal cube operators acting on a cube c is also always even.

We can presumably make the algorithm ergodic by allowing moves in which one cube

operator is flipped along with 12 link operators. We do not consider such moves, as

the parity of such operators is a non-local measurement (along the time direction),

and should be locally indistinguishable. We have verified that including such moves do

not make a discernible difference. Also, since we have periodicity along the expansion

direction, the offdiagonal flips that cross the boundary also sample through states

|α〉.

A full Monte carlo step then consists of the diagonal update step, followed by a

number of link and cube offdiagonal updates. We begin the system with some M and

SM consisting of only identity operators. As the number of non-identity operators n

increases, we increase M such that M > (3/2)n at all times, so that the truncation

error is completely negligible.

43

2.4.2 Results

Finally, we present modest numerical results using the above local update procedure.

We perform simulations on a 10 × 10 × 10 periodic lattice with K = ΓM = 1 at

β = 8, and consider the breakdown of the deconfined phase as we introduce J and

Γ . We have verified that our choice of β is high enough that we are essentially seeing

only ground state behavior. The present algorithm is also prone to getting “stuck”

in suboptimal configurations, but manages to find the correct ground states in the

various (J = Γ = 0, large-J , or large-Γ ) limits following a slow ramp from β = 0

(infinite temperature).

Figure 2.2 shows the internal energy 〈E〉 per site (with the additional constants

introduced for the QMC calculation in Eq 2.13-2.18 subtracted out), as a function of

J and Γ perturbations, where the QMC system is swept along both increasing and

decreasing J and Γ . Looking at the energy allows one to identify the confinement

transition, which appears to be strongly first order everywhere, as evidenced by the

strong hysteresis which appears to be independent of sweeping rate. The confinement

transition occurs at roughly J/ΓM ≈ 0.3 or Γ/K ≈ 0.9. Finally, we note that akin to

the phase diagram of the Ising gauge theory [35, 61, 62, 63, 64, 65] there appears to

be a line of first order transition extending from the corner of the deconfined phase,

as shown in Figure 2.2(right) (which are smoothly connected in the large-J, Γ limit,

where the Hamiltonian becomes simply a rotation of a field). These result in the

phase diagram shown in Figure 2.3(right). Note that this line of first order transition

must terminate at a critical point, where one can perform scaling analysis. We leave

a more complete analysis of the phase diagram to future work.

In Figure 2.3(left), we show the behavior of the diagnostic R(L) introduced in the

main text for length L = 2 and L = 4 loops, across the confinement transition as we

increase J keeping Γ = 0.8. These small loops are already enough for convergence, as

R(L) is already independent of L in the confining (high-J) phase, and very close to

44

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

J/ΓM

−4.8

−4.6

−4.4

−4.2

−4.0

−3.8

〈E〉

Γ/K = 0.2

Γ/K = 0.5

Γ/K = 0.8

Sweep rightSweep left

0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20

Γ/K

−5.0

−4.8

−4.6

−4.4

−4.2

−4.0

〈E〉

J/ΓM = 0.2

Sweep rightSweep left

0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Γ/K

−6.0

−5.5

−5.0

−4.5

−4.0

〈E〉

J/ΓM = 0.5

Sweep rightSweep left

Figure 2.2: Plots of the energy 〈E〉 as a function of J and Γ , for a 10×10×10 systemat β = 8 with K = ΓM = 1 (the energy shift due to the QMC constants (Eq 2.13-2.18)are subtracted out). We only show data until the QMC state becomes unstable andtransitions into a lower-energy state. (upper left) Sweeping J at various values of Γ ,sweeping right from the X-cube limit and left from the trivial σz = 1 large-J limit,showing strong first order transitions at J/ΓM ≈ 0.3. (upper right) Sweeping of Γwith J = 0.2, showing a first-order transition at around Γ/K ≈ 0.93 (other values ofJ < 0.3 look very similar). (bottom) Sweeping Γ with J = 0.5 (which is confining),showing a first-order transition between the two confined phases.

45

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

J/ΓM

0.0

0.2

0.4

0.6

0.8

1.0

R(L

)

Γ/K = 0.8

L = 2

L = 4

J/ΓM

Γ/K

Confined(Trivial)

Dec

onfi

ned

(Fra

cton

)

0.3

0.9

Figure 2.3: (left) The expectation value of the diagnostic R(L) defined in the maintext, which approaches zero (a constant) in the deconfined (confined) phase as L →∞. Here, the loop is taken to be an L×L square, and the horseshoe has dimensionsL/2×L. We look at the transition induced by increasing J at fixed Γ/K = 0.8. Thecorrelation lengths are very short near the first order transition and already L = 2 isindistinguishable from L = 4, thus we are already in the large-L limit and R(L) showsthe expected behavior. Note that we only show the lower-energy state at the firstorder transition. (right) A schematic phase diagram summarizing the sweep resultsfrom Figure 2.2. All transitions are first-order.

0 already in the deconfined phase. Identifying the transition along the increasing-Γ

direction using R(L) is difficult as the expectation value for both the Wilson loop

and the horseshoe are exponentially small in L and close to 0, thus leading to large

statistical errors in their ratio. For practical purposes, one should instead use the

dual Wilson loop and horseshoes (defined as products of σx in our model) to diagnose

the transition along this direction.

Finally, these results have since been compared with series expansion [66] to ex-

cellent agreement.

46

Chapter 3

Resonating Plaquette Phases

The topologically ordered phases discussed thus far, the deconfined phases of the

IGT and PGT, have been exactly solvable commuting spin Hamiltonians (in some

limit). They are both examples of quantum spin liquid phases [67]. In this section,

we discuss some examples which are not commuting, in the form of resonating singlet

models. Such models potentially describe topologically ordered spin liquid phases,

complete with a topological ground state degeneracy [68, 69], non-trivial quasiparticle

statistics [70], edge states [71], and topological entanglement entropy [72, 73, 74]. This

chapter is based on the paper

[14] T. Devakul, “Z3 topological order in the face-centered-cubic quantum plaquette

model”, Phys. Rev. B 97, 155111 (2018).

As the classic example of a gapped quantum spin liquid, we have short-ranged

resonating valence bond (RVB) states originally introduced by Anderson [15, 75, 76,

77], where pairs of electrons form local singlet bonds and the state is a superposition

of such configurations. Rather than independently fluctuating spins, we can instead

simply consider the dynamics of such valence bonds. The low energy physics are well

captured by quantum dimer models [78] (QDM) originally introduced by Rokhsar and

Kivelson [79], where the presence or absence of a dimer is indicated by an Ising degree

47

of freedom living on the links between two sites. The key difference between the dimer

and valence bond representation being that the states corresponding to two different

dimer configurations in the QDM are orthogonal by definition, but have non-zero

overlap in the valence bond representation [79]. These models have the nice feature

that at a special point, called the Rokhsar-Kivelson (RK) point, the ground state can

be solved for exactly and is an equal amplitude superposition of all possible dimer

configurations, allowing expectation values of diagonal observables to be computed

from the classical equal probability ensemble. The ability to describe such phases by

bond variables in conjunction with a site constraint hints at a connection between such

models and gauge theories. Indeed, at the microscopic level they can be formulated

as hybrid lattice gauge theories with a local U(1) gauge invariance [80] due to the

fixed number constraint at each site but with Ising valued electric fields [81, 82] which

reflect the binary character of dimer occupations. The challenge in this language is to

show that the gauge theory exhibits a deconfined phase which can be identified with

the RVB phase. As it turns out, the QDM on the square (or any bipartite) lattice in

d = 2 is gapless with power law decaying dimer-dimer correlations at the RK point,

which sits at the boundary between a resonating plaquette [83, 84, 85] and a staggered

phase, and so does not host an RVB phase (upon general perturbation, one can have

more complex phenomena such as Cantor deconfinement [86]). This lack of an RVB

phase is due to the fact that the square lattice QDM maps on to a U(1) gauge theory

at long wavelengths [80], which is only gapless at one particular point (the RK point)

in 2D (while there exists an extended gapless U(1) RVB Coulomb phase in 3D [87,

88, 89]). The triangular lattice QDM, however, does exhibit exponentially decaying

correlations at the RK point and hosts a fully fledged Z2 topologically ordered RVB

liquid phase [90] characterized by a long wavelength Z2 gauge field. It is also useful to

note that one can also deform QDMs by loosening the fixed dimer number constraint

to variable numbers. Specifically we can loosen the constraint to allow for all odd

48

or even numbers of dimers per site—the latter now yields a microscopic Ising gauge

theory (IGT) and the former its “odd” cousin. In this limit one can find a deconfined

phase on any lattice although the connection to the original RVB picture is less clear.

As a natural extension of the RVB idea, the resonating singlet valence plaque-

tte [16, 17] (RSVP) generalizes from the two spin-1/2 SU(2) singlet to SU(n) sin-

glets formed by n spins in the fundamental representation of SU(n) (note that the

plaquette structure is not necessary, we could form SU(n) singlets of n spins from

simplices of any form). Following the RVB discussion, it is natural to ask whether

one can find a liquid phase in these models, and if so, what is the character of this

liquid? In Ref 16, this idea was investigated first for n = 4 on the simple cubic

lattice, where spins formed tetramers along the square plaquettes, with a hard-core

constraint (each site was only allowed to be included in one tetramer), but was shown

to exhibit a weak crystalline order (which would lead to a confining phase) at the

RK point, rather than a gapped liquid [16, 17]. In this chapter, we are motivated by

the observation that had the hard-core constraint been “loosened” to an even or odd

constraint (that each site had to be a part of an even or odd number of tetramers),

one exactly obtains the Ising plaquette gauge theory in the X-Cube limit. In this

context, the crystalline order at the RK point can be explained as an instability of

the U(1) X-cube phase to crystalline order [17]. Notice how the connection between

this model and the plaquette gauge theory [12] parallels that of the QDM and the

IGT.

This suggests that there is potentially much of interest to be found in RSVP can-

didates. In this section, we investigate another model looked at in Ref 16, for which

Monte Carlo results show, in contrast to the cubic model, exponentially decaying cor-

relations at the RK point indicative of a gapped RSVP phase whose character was left

undetermined. The model is inspired from an SU(3) version of the above on the face

centered cubic (FCC) lattice, where three mutually nearest neighbor spins (which sit

49

at the corners of equilateral triangular plaquettes as can be seen in Figure 3.1) form an

SU(3) singlet. Consequently, we may examine the quantum plaquette model (QPM)

whereby each plaquette is associated with it an Ising degree of freedom representing

the presence or absence of such a singlet (a trimer) in combination with a hard-core

constraint on each site. We describe such models in more detail in Section 3.1.

Notice that had a liquid phase existed in the cubic QPM, that phase would have

been characterized by fracton order. One might then consider the possibility that

the liquid phase in the FCC QPM might also be a realization of fracton order. Alas,

this is not the case, and we show that it instead has (somewhat surprisingly) Z3

topological order in its liquid phase. This order emerges naturally from the geom-

etry of the FCC lattice (despite the trimer degrees of freedom still being Ising), as

detailed in Section 3.2. Inspired by the connection between the IGT and the QDM,

we examine in Section 3.3 a ZN commuting-projector generalization of this model.

This model exhibits Z3 order when N is divisible by 3, and is trivial otherwise —

making explicitly clear the origin of the Z3 order in the hard-core limit. Finally, we

also consider similar generalizations for plaquette models on other lattices (some of

which show ZN fracton order). In a sense, we make a connection between the classic

ideas of RVB and RSVP and more modern concepts of topological order. Models

with plaquette degrees of freedom have the potential to describe fracton phases (as

in the simple cubic or corner-sharing octahedra lattices discussed in the appendix),

or they may alternatively describe a conventional non-fracton topologically ordered

phases (of which the FCC model to be discussed is an example of).

Before continuing with the discussion of the FCC QPM, we first review the key

features of ZN topological order in 3 + 1D [67]. The theory hosts two fundamental

types of excitations: a point-like quasiparticle (called the charge or “electric” excita-

tion) and loop-like excitations with a finite energy per length (which we call vortex

loops [91] or “magnetic” flux excitations). The charge quasiparticles are self-bosons

50

(the wavefunction does not pick up a sign upon interchanging two), but picks up a

non-trivial phase when brought around a path that links with one vortex loop. More

generally, bringing n charge particles around a loop linked with m vortex loops re-

sult in an e2πinm/N phase factor. The main identifying feature of such a phase is the

topological ground-state degeneracy: a system defined on a manifold with genus g

has an N g-fold degenerate ground state that cannot be broken by local perturbations.

The different states in the ground-state manifold can be connected by the non-local

action of creating a charge-anticharge pair, bringing one around the system along a

non-contractible loop, and finally annihilating the pair. We verify all these features

in our model system.

3.1 FCC Plaquette model

We begin by defining a generalized plaquette model (GPM). To clarify our nomen-

clature, we use “generalized” in the parlance of Ref 52 to mean that we have not yet

specified a site constraint. The quantum plaquette model (QPM) will refer specifi-

cally to the GPM with the hard-core site constraint. The ZN generalized plaquette

model examined in Section 3.3 and the Appendix will be referred to as N -GPM.

The model of interest is defined on the FCC lattice, a unit cell of which is shown

in Figure 3.1, with sites at each of the lattice points. We will take the system defined

on the 3-torus (periodic in all three directions) for simplicity. A trimer is defined as

some bound state of three mutually nearest neighbor sites, which form equilateral

triangles on the FCC lattice. We assign an Ising (Z2) variable σx to each triangle,

and define σx = 1(−1) as the presence (absence) of a trimer on that triangle, and take

directly the set of all trimer configurations as an orthonormal basis for our Hilbert

space.

51

Figure 3.1: A unit cell of the face centered cubic lattice. Nearest neighbor pairs areconnected by gray lines. Triangles on which trimers may occupy are formed by threemutually nearest neighbor sites. Regular polyhedra formed by the triangular facesinclude octahedra (one shown in red) and tetrahedra (one shown in green).

We may now begin to discuss Hamiltonians on this Hilbert space. These will

consist generically of three parts: a site constraint, a kinetic term, and a potential

term. The site constraint is a local constraint diagonal in the trimer basis, which

is defined for each site and must be satisfied, thus permitting only a subset of the

Hilbert space. This constraint may be enforced externally, or energetically on the

ground state by attaching a large energy penalty to violating states. For example,

the QPM will be obtained by enforcing that each site is only allowed to be a part

of exactly one trimer, but one can also write down a theory where each site is only

allowed to be a part of an odd (even) number of trimers (thus producing an Ising

plaquette gauge theory on the FCC lattice). The kinetic term is a sum of purely

off-diagonal local terms that transition between trimer configurations respecting by

the site constraint. Finally, the potential term is a sum of diagonal local terms, which

may be used to tune the Hamiltonian to the RK point — where the ground state can

be solved for exactly.

Before jumping straight to the hard-core QPM, one might expect that there may

be something to learn first from the GPM with the odd/even constraint. This ex-

52

pectation turns out to be wrong: the exactly solvable even/odd models are actually

non-topologically ordered liquids. First, note that the even and odd models are uni-

tarily related, thus it is only necessary to examine the even case. Let us write this

down explicitly for the even model. The Hamiltonian is given by 1

Heven = −∑

Ce

∏

t∈Ceσzt −

∑

s

∏

t∈sσxt (3.1)

where t refers to triangles, and σz,xt are Pauli matrices acting on the trimer degree

of freedom on each triangle. The second sum is over sites s, and t ∈ s corresponds

to the triangles containing the site s (of which there are 24 of). The set Ce refers to

a set of triangles for which each site on the lattice is shared by an even number of

triangles in Ce (thus guaranteeing the term commutes with the site constraint), and

Ce does not consist of multiple disjoint sets of triangles (the subscript e stands for

even). The first sum is over all such sets Ce up to a certain size |Ce|max, which we will

assume is large enough for ergodicity (within a topological sector, should they exist).

We will return to the discussion of what these terms look like in more detail in the

context of the (hard-core) QPM in Sec 3.2. The first term is the kinetic term, and

the second term enforces the constraint that every site must have an even number

of trimers connected to it (there is no potential term needed here). By construction,

this Hamiltonian consists of mutually commuting terms and one can deduce that

an equal amplitude superposition of all constraint-satisfying configurations within a

topological sector (should they exist) is the exact ground state.

In fact, no such topological sector exists. An easy way to see this is by examining

the excitation structure. In the gauge theory language, consider creating a “charge”

excitation: an excitation of the second term in the Hamiltonian, where a site par-

ticipates in an odd number of trimers. It is in fact possible to create a single such

1Note that this even model is also what one would have found starting from an FCC Ising modelwith triangular plaquette interactions and proceeded with the generalized gauging procedure [10, 41].

53

an excitation locally at site s by applying an operator σzt1σzt2σzt3 on the ground state,

where t1, t2, t3 are the three triangles around a tetrahedron that contain the site s.

These overlap the site s three times, and the three other sites in the tetrahedron

twice, thus it anticommutes with the site term only on site s. We have therefore cre-

ated a single charge excitation using only local operators acting on the ground state,

thus implying that a single charge excitation does not carry any topological charge.

By topological charge, we refer to charge that can be measured by a membrane-like

operator akin to Gauss’ law in standard U(1) electromagnetism. As the action of

moving a charge excitation around a non-contractable loop plays a key role in diag-

nosing topological order, and such an action is topologically trivial in this case, we are

forced to conclude that this Hamiltonian does not possess the other key features of

topological order such as topological degeneracy and quasiparticle/loop excitations

with non-trivial statistics. Nevertheless, as we will show in the next section, the

QPM (specified by a number site constraint) at the RK point does exhibit the signs

of topological order, more specifically, Z3 topological order. The reason the above

construction fails is that we have implicitly tried to force a Z2 order by using an even

constraint, while the geometry of the model favors a Z3 order.

3.2 The Hard-Core constraint

We now examine the FCC QPM: the model of trimers with the hard-core constraint

that each site must participate in only one trimer. The allowed Hilbert space now

consists of the set of hard-core trimer coverings of the FCC lattice. The set of local

trimer moves are now more restricted than in the even theory. Any local trimer move

can be represented by a non-disjoint bipartite set of triangles C = CA ∪ CB, with the

constraint that every site in the lattice must be included in exactly one triangle from

CA and one from CB, or none at all. By non-disjoint, we mean that one cannot express

54

C as C = C1 ∪ C2 for C1,2 both being valid bipartite sets as previously defined. The

trimer move then consists of taking all trimers that were originally on all the triangles

in CA and moving them to CB, or vice versa. Let us represent the local state in which

all triangles in CA are occupied with trimers as |CA〉, and similarly |CB〉. We can then

define a RK type model as

HRK = −t∑

C(|CA〉〈CB|+ |CB〉〈CA|) + V

∑

C(|CA〉〈CA|+ |CB〉〈CB|) (3.2)

where the sum is over all C as previously described up to some |C|max. We further

have the site constraint of one trimer per site:∑

t∈s(σxt + 1)/2 = 1 for every site s.

This can be expressed as enforcing the constraint Gs|ψ〉 = |ψ〉 for all s with

Gs = e−iα[1−∑t∈s(σ

xt +1)/2] (3.3)

for any α. Note that this Hamiltonian, written in terms of Pauli matrices, has a U(1)

symmetry σ±t → e±iασ±t , where σ± = σy±iσz are σx raising/lowering operators. This

U(1) symmetry corresponds to the conservation of total trimer number, as every such

bipartite path satisfies |CA| = |CB|.

Exactly at t = V , the RK point, the Hamiltonian is a sum of projectors,

Ht=V=1RK = 2

∑

C(|CA〉 − |CB〉)(〈CA| − 〈CB|) (3.4)

whose exact ground state is an equal amplitude sum of all constraint-obeying trimer

configurations that can be reached by the local flips C.

At the RK point, which will be the focus of our discussion, expectation values of

diagonal operators are exactly that of the equal probability classical ensemble. The

trimer-trimer correlation function at the RK point was calculated via Monte Carlo

simulation in Ref 16, and was found to decay exponentially with a small correlation

55

(a)

(b)

(c)

(d)

Figure 3.2: Illustration of a few terms in the Hamiltonian, which we describe by setsof triangles C = CA∪CB, where the orange and blue triangles indicate CA and CB. All|C| = 4 terms are loop terms of the form (a) or (b). (c) and (d) shows terms involvinga larger number of triangles. The term (c) involves flipping between configurationswith local “divergence” ±3 (as described in the text), and (d) is an example of a|C| = 8 length loop term.

length. This indicates that should a suitable RK type Hamiltonian be defined, the

RK point sits within (or at the boundary of) a gapped RSVP phase — if the RK

point were a critical point between two phases or part of a gapless phase, one would

expect power law decaying correlations.

Let us now discuss what possible terms, denoted by the set of flipped triangles C,

arise in our model and how large clusters |C|max one should include for ergodicity. The

simplest types of moves are loop moves, where C consists of a loop of an even number

of triangles joined in alternating orientation (each triangle shares sites with only two

other triangles, as shown in Figure 3.3a). The smallest moves are |C| = 4 terms of

this type, which come in two flavors: a loop of four triangles around an octahedron,

and a loop of four triangles around two edge-sharing tetrahedra, shown in Figure 3.2a

and 3.2b. Finally, we note that this model differs from the QPM proposed in Ref 16

in that larger cluster flips are included which are necessary for ergodicity.

56

(a)(b)

(c)

Figure 3.3: The convention for assigning directions to trimer configurations. The toprow shows the configuration (for example) in state |CA〉, and the bottom shows theflipped state |CB〉; the red arrows indicate the direction assignment. Configurationsalong loop-like paths are assigned a direction as shown in (a). Terms which involveflips along non-loop paths include triangles with local “divergence” ±3, as shown in(b). Finally, (c) shows how a monomer (an untrimerized site) may be moved along apath via trimer flips.

To more effectively visualize the action of these loop terms, we can unambiguously

assign a directionality to the loop configurations |CA〉 and |CB〉. To set a convention,

imagine the triangles in CA as arrowheads which all point in one direction around

the loop, which we define to be the direction of the configuration |CA〉, as shown in

Figure 3.3a. Similarly, we may look at the configuration |CB〉, which always points in

the opposite direction. Pictorially, the kinetic term then looks like −t(|〉〈 | + |

〉〈|) in this language. In this description, the loop terms are always flipping between

“divergenceless” configurations. A flip is characterized as a loop if every triangle is

only in contact with two other triangles. However, a triangle may also be in contact

with three other triangles. In our picture, such triangles have a “divergence” of ±3, as

shown in Figure 3.3b. Terms involving such triangles first appear in the Hamiltonian

at |C| = 6, one such example is shown in Figure 3.2c.

57

Figure 3.4: A sample trimer configuration in an xy plane specified by z-coordinatez0, which includes triangles spanning the site-layers z0 and z0 + 1/2. Upwards facingtrimers are shown in orange, while downwards facing trimers are shown in blue. Thetopologically conserved “winding number” is the difference between the number ofupwards facing trimers (N4) and downwards facing trimers (NO) modulo 3 (Eq 3.5).

As we will show, there exists a conserved number that is left invariant under local

trimer manipulations, modulo 3. However, the loop terms with |C| = 4 leave this

number unchanged not modulo 3 and we have an extra unwanted conservation law

that we can get rid of by including larger terms. At |C| = 6, the term in Figure 3.2c

is sufficient to accomplish this, and at |C| = 8, there are larger loop terms such as the

one shown in Figure 3.2d that also accomplish this. Thus, we need at least |C|max = 6

to achieve ergodicity. We do not investigate this question of ergodicity further here,

and assume that there is a small finite value of |C|max (which may just be 6) for which

the Hamiltonian is ergodic enough within each topological sector.

We can now proceed to discuss conserved quantities that remain invariant under

such local flips. Consider two adjacent xy-plane of sites defined by the z-coordinate

z0 and z0 +1/2 of the FCC lattice, as shown in Figure 3.4 (where the linear dimension

of the cubic unit cell is taken to be 1). All the triangles with all three sites within

these two planes are oriented with either: two sites on the lower and one on the upper,

which we call “upwards pointing” (4), or the opposite, which we call “downwards

pointing” triangles (O). We claim that the “winding number” for this xy plane,

W (z0)xy = N

(z0)4 −N (z0)

O mod 3 (3.5)

58

is conserved by arbitrary local trimer moves, where N(z0)4 (N

(z0)O ) is the number of

upwards (downwards) pointing trimers between layers z0 and z0 + 1/2.

Furthermore, knowing W(z0)xy for one z0 determines the value for all other xy planes.

We can show this using a simple counting argument. The number of sites on layer

z0 + 1/2 that are included in the trimers spanning z0, z0 + 1/2, is N(z0)4 + 2N

(z0)O . Let

Nxy be the total number of sites an xy layer. This leaves Nxy − (N(z0)4 + 2N

(z0)O ) free

sites in layer z0 + 1/2 that must be used in the trimers spanning z0 + 1/2, z0 + 1, as

there are no untrimerized (monomer) sites. Therefore, we must have

2N(z0+1/2)4 +N

(z0+1/2)O = Nxy − (N

(z0)4 + 2N

(z0)O ), (3.6)

and taking both sides modulo 3, we find

W (z0+1/2)xy = W (z0)

xy −Nxy mod 3. (3.7)

Therefore, knowing W(z0)xy for z0 fixes its value for every z. This alone is proof that

W(z0)xy cannot be modified by any local trimer move: to modify one we must simul-

taneously change this value for every value of z, which requires a non-local trimer

move. The same argument holds for the yz and zx planes, which therefore give us

access to three independent conserved winding numbers. Measuring these winding

numbers requires counting the number of triangles within an entire plane: a non-local

measurement. At the RK point (and the RSVP liquid phase), this leads to a locally

indistinguishable 33-fold degenerate ground state manifold on a 3-torus. Thus, we

have already uncovered the topological ground state degeneracy — a key features of

a Z3 topologically ordered phase.

Next, we observe that the non-local trimer shift needed to change these winding

numbers correspond to flips on paths C that are equivalent to non-contractible loops.

Consider the non-local trimer loop move C which runs along a non-contractible loop

59

wrapping once around the z direction. Let |CA〉 be the configuration where the “di-

rection” of the loop as previously discussed points along the positive z direction, and

|CB〉 along the negative direction. Then, flipping |CB〉 → |CA〉 will increment W(z0)xy

by 1. Since W(z0)xy for every slice must be changed identically, we further see that any

further local manipulations one makes to the details of C will not change its effect on

W(z0)xy .

To complete the picture of the Z3 topological order, we next consider the form of

the excitations. At the RK point, we only have the ground state that can be solved

for exactly — and while we can write down variation states with localized excitations,

these will not be exact (they must be locally “dressed” and the true eigenstates will be

a definite momentum superposition) [79]. We examine two types of excitations in this

model: point-like monomer (“charge”) excitations and loop-like vortex (“magnetic”)

excitations.

Monomer excitations are sites which do not participate in any trimer. To include

these, we must relax our constraint in Eq. 3.3 to allow states with Gs|ψ〉 = eiα|ψ〉

at some energy cost. A single monomer can be moved from site s to s′ by a trimer

flip along a path, as shown in Figure 3.3c. Adding a two-triangle hopping term gives

monomer excitations a finite mass and dispersion. We can now identify the non-local

flip that increments the winding number by one as corresponding to the action of

bringing a monomer excitation around along a non-contractible loop in the negative

z direction once.

To create vortex excitations, consider a loop L, and let WL count the winding

number as previously defined in Eq 3.5 but for an open surface with boundary at L.

We then define the “vortex operator” as vL = e2πiWL/3. Our cartoon state containing

a vortex loop along L will then look like

|vL〉 ≈ |WL = 0〉+ e2πi/3|WL = 1〉+ e−2πi/3|WL = 2〉 (3.8)

60

where |WL = k〉 is the component of the ground state wavefunction with WL = k.

Any term in the Hamiltonian far away from the loop L does not change the value of

WL, and so this state remains a local eigenstate of those terms. This is not true for

terms near the loop which do change the value of WL, and so this state will have a

finite energy density along L (but will not be an eigenstate of those terms). In this

cartoon picture, one can imagine threading n monomer excitations through m vortex

loops before returning to its original position, resulting in an overall phase e2πinm/3

(of course, actually rigorously defining such a process requires more care).

Thus, we have argued that the QPM in its RSVP phase does indeed possess Z3

topological order, with all of its important features. In the next section, we will

examine a ZN generalization of the FCC QPM in an exactly solvable limit, which

shares much of the properties of the hard-core model just discussed, including a Z3

order for all N divisible by 3. The properties of these models generically depend

strongly on N and the details of the lattice, and we will also cover a few more

characteristic examples.

3.3 ZN Generalization

To motivate the study of the ZN generalization, we observe that by doing a simple

operator substitution on the hard-core Hamiltonian, one can get a Hamiltonian of

mutually commuting projectors which can be solved exactly.

The first step is to enlarge the Z2 degree of freedom on each plaquette to a ZN

degree of freedom. Acting on each of these degrees of freedom, we have the operators

X,Z, for each bond obeying algebra

ZN = XN = 1

XZ = ωZX (3.9)

61

where ω = e2πi/N . Thus, the eigenvalues of X are ωn for n = 1 . . . N , and Z acts as a

raising operator in the X eigenbasis. Interpreting the X eigenvalue ωn as the presence

of n trimers on a bond, we can then enforce a site constraint that the sum of trimers

connected to a site always be zero mod N . For large N , these can be interpreted as

bosonic or quantum rotor degrees of freedom. Note that we could have equally chosen

the site constraint to be any number without changing the physics, as the resulting

Hamiltonians can be shown to be unitarily related to each other. Quantum dynamics

that respect this constraint can then be represented by substituting σ+ → Z, σ− →

Z† in the kinetic term of the RK Hamiltonian Eq 3.4 when expressed in terms of

raising/lowering operators. Since the kinetic term does not annihilate any state, the

potential term is not needed.

Thus, we have

HN = −∑

C

(∏

t∈CAZt∏

t∈CBZ†t + h.c.

)−∑

s

(∏

t∈sXt + h.c.

)(3.10)

where the first sum is over all bipartite connected sets of triangles C = CA ∪ CB such

that every site contains an equal number of triangles from CA and CB. Note that this

is a looser constraint than in the hard-core case (where each site had to have one

from each, or none).

We can motivate that this model will have Z3 order only if N is a multiple of 3,

and trivial otherwise, by just looking at the quasiparticle structure. We may define

the charge as Qs =∏

t∈sXt, where the product is over the 24 triangles touching a site.

However, acting with Zt creates a set of three charges ω each, and so we are therefore

forced to conclude that three charges combined carries no topological charge (note

that if the lattice were tripartite, then a different charge definition could be used on

each sublattice and this conclusion would not hold — some examples of this happening

are discussed later). If N is not a multiple of three, then one can create a single ω

62

charge via local operations, and we are left with a trivial liquid. On the other hand,

if N is a multiple of three, there is the possibility for a Z3 topological order. In this

situation, the correct definition of the topological charge operator should be

Qtops = QN/3

s . (3.11)

We assume that N is a multiple of three moving forwards.

First, note that there may be non-topological degeneracies that exist due to com-

muting terms which are not included in the Hamiltonian because they cannot be

expressed as products of terms on bipartite C. The product (Zt1Zt2Zt3Zt4)N/3 around

the four faces of a tetrahedron is such an example, which leads to an extra 3-fold

non-topological degeneracy. We will ignore non-topological degeneracies as they can

be broken by local perturbations.

To count the topological degeneracy, consider the operator that counts N(z0)4 −

N(z0)O for an xy-plane of triangles (as considered earlier for QPM),

e2πi(N(z0)4 −N(z0)

O )/N =∏

t∈4Xt

∏

t∈OX†t (3.12)

where the product t ∈ 4 (t ∈ O) is over all upwards (downwards) pointing triangles

in the xy-plane spanning z0,z0 + 1/2. While this commutes with all |C| = 4 terms in

the Hamiltonian, it fails to do so with some |C| = 6 terms, (such as the one shown

in Figure 3.2 for the QPM), and general local perturbations. Instead, like in the

QPM, this number is only conserved mod 3 under local operations, and so the correct

operator is

Wxy =

(∏

t∈4Xt

∏

t∈OX†t

)N/3

(3.13)

which does commute with every term in the Hamiltonian. We have suppressed the

z0 label, as it is possible to relate W(z0)xy for different z by terms present in the Hamil-

63

tonian. To see this, observe that multiplying W(z0)xy by (Qtop

s )†

on every site s in the

z0 + 1/2 layer results in W(z0+1/2)xy , and so therefore W

(z0)xy = W

(z0+1/2)xy in the ground

state where Qtops = 1. We have W 3

xy = 1 and so Wxy can take on one of three values,

and since there are three independent planes one could have defined this for, this

leads to a 33 topological degeneracy. Notice the remarkable similarity to the QPM

discussion in Section 3.2.

The advantage of this model over the QPM at the RK point is that the excitations

are static can be solved exactly. A monomer excitation from the QPM correspond to

a Qs = ω charge sitting on a site s, which carries topological charge Qtops = e2πi/3.

By application of a chain operator Z†t1Zt2 . . . Z†tL−1

ZtL , a monomer can be moved from

one site to another, and moving one monomer around a non-contractible loop in the

z direction will modify the value of the conserved winding number Wxy by e±2πi/3

depending on which direction the monomer goes around the loop.

The vortex (magnetic) excitations of this model are loop-like, and are created at

the boundary of a membrane operator,

WL =

∏

t∈4LXt

∏

t∈OLX†t

N/3

(3.14)

where 4L (OL) are all the upwards (downwards) oriented triangles along an open

surface with boundary along the loop L (which we may take to be a flat loop in an

xy plane, where this operator can be thought of as a truncated version of the W(z0)xy

operator). Acting with this operator on the ground state creates an excited eigenstate

of the Hamiltonian, which is locally the ground state away from L, but an excited

eigenstate with gap ∆E = 2(1 − cos 2π/3) = 3 for each term near the loop L that

doesn’t commute with WL.

We can now also explicitly verify the statistical phase obtained by bringing charge

excitations through vortex loops. Consider the action of bringing n charge excitations

64

ZN model Lattice Phase

DimerSquare ZNTriangular Zgcd(2,N)

Trimer

Triangular ZN × ZNCorner-sharing ZN fractonoctahedra (X-cube phenomenology)Face centered cubic Zgcd(3,N)

Tetramer Simple Cubic ZN fracton (X-cube)

Table 3.1: Table summarizing the topological phases found for the ZN generalizeddimer models (first two rows) and ZN generalized plaquette models (remaining rows).Zgcd(p,N) for p = 2, 3 simply means Zp order if N is a multiple of p, and trivialotherwise. The FCC QPM is discussed in Section 3.2 of the main text.

around in a circle linking with m vortex loops, bringing us back to the same state but

with a overall phase. In the simplest case, computing this phase involves commuting

a Zn with (X†)Nm/3, which results in a ωNnm/3 = e2πinm/3 phase factor overall, in

agreement with what one expects from a Z3 phase.

3.4 Generalized Models on other lattices

In this section, we example ZN generalized models on various other lattices. We mo-

tivate the study of these ZN generalized models from an observation that by doing a

simple operator substitution on the hard-core Hamiltonian for QDMs or QPMs, one

gets a Hamiltonian of mutually commuting projectors which can be solved exactly.

Some possible phases found in these exactly solvable models are summarized in Ta-

ble 3.1. We will refer to such models as N -GDM (specifically for dimer models), and

N -GPM for the plaquette models (which include trimer models and a tetramer model

which we also discuss).

To illustrate the construction for a general lattice model, we first consider the

Rokhsar-Kivelson QDM on the square lattice. Letting σx = 1(−1) on a bond signify

65

the presence (absence) of a dimer, we can write the Hamiltonian as

HRK = −t∑

σ+l1σ−l2σ

+l3σ−l4 + h.c.

−V∑

Pσxl1Pσxl3

+ Pσxl2Pσxl4

−Γ∑

s

e−iα

[1−∑l∈s Pσxl

]+ h.c. (3.15)

where we have defined the projection operator PO = (1 + O)/2 for an operator O

with eigenvalues ±1, Γ = ∞ enforces the hard-core constraint, and α can be any

number (except for some special choices, such as π, for example). The first sum

is over square plaquettes on the lattice, and l1...4 are the four links going around

clockwise or counterclockwise around it, and the second sum is over all sites which

touch four links in a cross.

To arrive at the N -GDM on the square lattice, we first enlarge the Z2 degree

of freedom on each bond to a ZN degree of freedom, with operators X,Z acting on

them with algebra given in Eq 3.9. We can then substitute σ+ → Z, σ− → Z† in the

kinetic term of the RK Hamiltonian 3.15. Since the kinetic term does not annihilate

any state, again the potential term is not needed. We then have (schematically)

HSquareN -GDM = −

∑

(ZZ†ZZ† + h.c.)−∑

+

(∏

l∈+

Xl + h.c.) (3.16)

where we have suppressed the l subscripts on the kinetic term which act on the

four bonds around a square as illustrated in Figure 3.5a. The second term is the

site constraint, which is a product over all four bonds emanating from a site. This

Hamiltonian is composed to mutually commuting terms (so we have set t = Γ = 1)

and can be solved exactly. On the square lattice, this model is a ZN generalization

of the toric code, which exhibits ZN topological order as we will show.

66

Z

Z†

X(a)

(b) (c)

Figure 3.5: Pictorial representation of the terms in the Hamiltonian for (a) the squarelattice N -GDM, (b) the triangular lattice N -GDM, and (c) the triangular lattice N -GPM. Blue and orange bonds/triangles indicate operators involved in the kineticterms in the Hamiltonian (Z and Z†), and red indicates those involved in the site-constraint (X). Only one of three possible rhombus orientations is shown for thekinetic term in the triangular lattice N -GDM (b).

For plaquette models, there is an additional difference between the N -GPM and

the (hard-core) QPM in which kinetic terms are allowed. In the QPM, the allowed

flips C = CA ∪ CB may only have each site being included in zero or two plaquettes,

one from CA and one from CB. In the N -GPM, the constraint is instead that each

site only be a part of an equal number of triangles from CA and CB. Thus, there are

terms involving configurations where a site is included in more than two triangles

total, that were not allowed in the QPM.

We shall now examine the properties of the N -GDM and N -GPM on a few char-

acteristic lattices, starting with the square lattice N -GDM we just derived.

3.4.1 N-GDM on Square Lattice

On this (bipartite) lattice, the N -GDM is equivalent to a ZN lattice gauge theory. The

Hamiltonian is given by Eq 3.16, and we take the system on a torus which respects

the bipartiteness of the square lattice.

67

The ground state degeneracy can be found by noting that for a non-contractible

loop, the product W = Zl1Z†l2. . . ZlL−1

Z†lL along that loop commutes with and is

independent of any of the terms in the Hamiltonian. Furthermore, powers of W

are also independent of terms in the Hamiltonian. Since WN = 1, eigenstates may

take on any eigenvalue ωn for n = 1 . . . N . As there are two such independent loop

operators, the ground state sector is N2-fold degenerate.

We can define the charge operator on site s as

Qs =

∏l∈sXl s ∈ A

∏l∈sX

†l s ∈ B

(3.17)

where A and B correspond to the two sublattices of the square lattice. We then

see that acting on the ground state with with Zl creates the exact eigenstate with

two oppositely-charged excitations of charge ω and ω−1 on the two sites touching

l. Therefore, total charge is preserved under any local operation modulo N . Notice

crucially that this construction works only due to the bipartite nature of the lattice.

Finally, we note that by doing a transformation Zl, Xl → Z†l , X†l on a subset of

the links, one can recover the usual form of the ZN Toric code on the square lattice.

3.4.2 N-GDM on Triangular Lattice

On non-bipartite lattices, the N -GDM describes a Z2 ordered phase for even N , and

a topologically trivial liquid otherwise. The Hamiltonian is

HTriN -GDM = −

∑

rhombus

(ZZ†ZZ† + h.c.)−∑

s

(∏

l∈sXl + h.c.) (3.18)

where the first sum is now over length-4 loops on the triangular lattice which are

rhombuses, and the second term is now a product over 6 links touching a site, which

are illustrated in Figure 3.5b.

68

We first consider the case of even N . The first thing to note is that there is

now an additional two-fold non-topological ground state degeneracy. We can write

down the local operation Tt = (Zl1Zl2Zl3)N/2 where l1...3 are three links to go around

a triangle t, which is independent of and commutes with the Hamiltonian. Such

triangle operators on different triangles can be related to each other via applications

of terms in the Hamiltonian, and since T 2t = 1, there are degenerate ground states

with Tt = ±1. This is non-topological, as one can simply add a term −hTt to the

Hamiltonian for just a single triangle, which would break the degeneracy. We will

ignore this degeneracy moving forwards.

Because the lattice is no longer bipartite, we cannot use the definition of charge

from Eq. 3.17. Instead, the best we can do is simply

Qs =∏

l∈sXl. (3.19)

The action of applying Zl to a link l creates two charges ω on each of the two sites

it connects. As it is possible to locally create two charges ω2, a pair of such charges

must be topologically indistinguishable from the vacuum. In this case, we must make

a distinction from the charge in Eq 3.19 and the topological charge operator, which

should be

Qtops = QN/2

s , (3.20)

and can only take two values. This is already an indication of the Z2 order to come,

which we show by observing the 22-fold topological degeneracy.

As before, consider the product W = Zl1Z†l2. . . ZlL−1

Z†lL along a non-contractible

loop of length L. Again, W is independent of and commutes with the Hamiltonian, so

one might be tempted to say it can take on any of N values. However, this turns out

not to be true, as W 2 can be written as a product of terms in the Hamiltonian. This

is consistent with our previous finding that two charges are topologically identical to

69

the vacuum: W can be thought of as the process of moving a charge around the non-

contractible loop, W 2 would correspond to moving two charges along the loop, which

must therefore be trivial. Since W 2 = 1 we are left with only a choice of W = ±1.

There are two independent non-contractible loops, and so we are left with a 22-fold

topological degeneracy, for any even N .

For odd N , even a single charge must be topologically identical to the vacuum.

To see this, observe that the local operator (Zl1Z†l2Zl3)

(N+1)/2 for l1...3 going around

a triangle creates a total charge ω on a single site, which therefore cannot carry any

topological charge.

3.4.3 N-GPM on Triangular Lattice

We next consider ZN generalized plaquette models (N -GPM). Similar to how the

properties of the N -GDM depended heavily on the bipartiteness of the lattice, we

will find that the properties of the N -GPM with triangular plaquettes will depend

heavily on the tripartiteness of the lattice.

For this reason, we first examine the N -GPM on the triangular lattice, which has

triangular plaquettes and is tripartite. On this lattice, the N -GPM maps to a ZN

bosonic ring-exchange model on the (dual) honeycomb lattice originally studied by

Motrunich [92] the strong coupling limit, which was found to have a fully deconfined

ZN × ZN phase, which we will find here as well.

The Hamiltonian is

HTriN -GPM = −

∑

s

(ZZ†ZZ†ZZ† + h.c.) (3.21)

−∑

s

(XXXXXX + h.c.)

70

where each term involves the product of operators over 6 triangles touching a site,

as illustrated in Figure 3.5c. We again assume the system to be defined on a torus

which respects the tripartiteness of the lattice.

Again, a simple method of analysis is by examining the quasiparticle structure.

Acting with Zt on a triangle creates three charge excitations, one on each sublattice

which we label A, B, and C. This leads to the “fusion rule” a×b×c = 1, where a, b, c

are charge excitations on each of the three sublattices. Thus, we can represent c as a

bound state of an a and b antiparticle, and define the charge operators accordingly:

Qas =

∏t∈sXt s ∈ A

∏t∈s 1 s ∈ B

∏t∈sX

†t s ∈ C

Qbs =

∏t∈s 1 s ∈ A

∏t∈sXt s ∈ B

∏t∈sX

†t s ∈ C

(3.22)

both of which are conserved under local operations. Going through a similar exercise

as before, one can readily verify the existence of four independent non-contractible

loop operators, which leads to the N2 × N2 topological ground state degeneracy.

These loop operators correspond to bringing an a or b particle around along a non-

contractible loop. For a more detailed analysis of this ZN × ZN phase, we direct the

reader to Ref [92], which discusses the model on the dual (honeycomb) lattice.

3.4.4 N-GPM on Corner-Sharing Octahedra Lattice

Here we highlight yet another interesting case: the N -GPM on the lattice defined

by corner-sharing octahedra (a tripartite lattice with triangular plaquettes). The

71

(a) (b)

(c)

(d)Z

Z†

Figure 3.6: The corner-sharing octahedra lattice, on which the N -GPM shows aZN X-cube fracton phase. (a) shows of the corner-sharing octahedra lattice, wheresites from the three sublattices are colored red, green, and blue. The centers of theoctahedra form into a simple cubic lattice, with lattice constant taken to be 1. Sitesfrom each sublattice themselves also sit an offset simple cubic lattice. (b) shows twotype of octahedron flips |Coct| = 4, and (c) shows a cuboctahedron flip |Ccuboct| = 8.(d) shows a portion of the Wz(x0, y0) operator, which measures a ZN topologicallyconserved quantity. Blue triangles indicates Z operators and orange indicates Z†

operators.

lattice can be understood as an underlying simple cubic lattice where each vertex is

the center of an octahedron and the sites lie on the bonds of the underlying simple

cubic lattice. A portion of this lattice is shown in Figure 3.6a, which also illustrates

the tripartiteness of the lattice. The N -GPM on this lattice will turn out to exhibit

ZN fracton topological order, which appears to be in the same phase as the ZN X-

cube model. We will show that this model exhibits the key features of this phase:

quasiparticle excitations which exhibit restricted movement and the characteristic

subextensive topological ground state degeneracy. Fundamental quasiparticle excita-

tions of this (and the X-cube) model are the one-dimensionally mobile quasiparticle

(lineons) and zero-dimensional immobile fractons, which are created at the corners of

membrane operators.

72

The Hamiltonian describing this model is

HC-S OctN -GPM = −

∑

Coct

(ZZ†ZZ† + h.c.

)

−∑

Ccuboct

(ZZ†ZZ†ZZ†ZZ† + h.c.

)

−∑

s

(∏

t∈sXt + h.c.

)(3.23)

The first sum is over all bipartite sets of triangles Coct = CA ∪ CB that go around

an octahedron, such that each site is a part of an equal number of triangles in CAand CB, of size |Coct| = 4 These come in two main types, as shown in Figure 3.6b

(the rest are obtained by symmetry relations on the octahedron of these two). The

second sum is over all such sets on cuboctahedra (the 14-faced polyhedron with 8

triangular faces and 6 square faces), and involve all |Ccuboct| = 8 triangles, as shown

in Figure 3.6c. Finally, the third term is the usual site constraint, with the product

going over 8 triangles touching a site.

Again, we may begin our analysis by examining the quasiparticle structure. Apply

a Zt to a triangle creates three charge excitations, one on each sublattice. Let A,B,

and C correspond to the three sublattices, and a,b,and c a single charge excitation

on the respective sublattice. We can apply the charge definition from Eq 3.22 and

treat the c charge as a bound state of an a and b anticharge. However, there is an

additional conservation law here arising from the geometry of the lattice.

Consider what happens when we have a single a charge sitting on a site s in the

A sublattice. The simplest way it can be moved from s to some other site s′ is by

applying the operator Z†t1Zt2 , where t1 must touch the site s and share two sites with

t2, who must then touch another site s′. The geometry of the lattice allows only for

s′ to be one of two choices, which are both along one axis. Thus, this a charge is

confined to move along only one axis: it is the one-dimensional lineon of the X-cube

73

model! The a, b, and c charges then correspond to lineons confined to move along x,

y, and z directions respectively.

The vortex excitations can come in two forms: either as violations of the octahe-

dron terms or as violations of the cuboctahedron terms. We first examine excitations

of the cuboctahedron term: consider the operator Xt1Xt2Xt3Xt4 around the four tri-

angles around a square-based pyramid (which comprises half of an octahedron). This

operator commutes with every octahedron term, but creates four cuboctahedron ex-

citations. Thus, cuboctahedron excitations can only be created in groups of four,

and one can confirm that by repeated applications of this operator along a mem-

brane, these excitations can be moved further apart and appear at the corners of the

membrane operator. Alone, one such excitation cannot be moved without creating

additional excitations. The cuboctahedron vortex excitations are therefore fractons!

Various combinations of octahedron excitations can then be interpreted as bound

states of fracton excitations.

Finally, we can compute the ground state degeneracy. Consider the operator that

corresponds to creating a z-moving lineon-antilineon pair at coordinates (x0, y0), mov-

ing the lineon around in the positive z direction, and then annihilating them again.

This is done by a ZZ† chain as shown in Figure 3.6d, which we call Wz(x0, y0) and

commutes with the Hamiltonian. Note that the details of how the z-lineon goes along

each octahedron can be related to each other by octohedron terms in the Hamilto-

nian, and so are not independent. We can henceforth freely choose Wz(x0, y0) = ωn

for n = 1 . . . N . Furthermore, by application of the cuboctohedron term, we can show

that in the ground state

Wz(x0, y0)W †z (x0 + 1, y0)W †

z (x0, y0 + 1)Wz(x0 + 1, y0 + 1) = 1 (3.24)

74

where we have taken the length of the cubic unit cell to be 1, and so not all of these

Wz(x, y) are independent. In fact, there are 2L− 1 independent Wz(x, y)’s, where L

is the linear dimension of the system. To see this, let us define for convenience

Wz(x, y) =

Wz(x, y) if x+ y even

W †z (x, y) if x+ y odd

(3.25)

Then, we can specify 2L− 1 of Wz(x, y0) and Wz(x0, y), and then obtain the rest via

the relation

Wz(x, y) = W †z (x, y0)W †

z (x0, y)W †z (x0, y0). (3.26)

Therefore, we have 2L − 1 independent choices to make for the z direction, and

similarly along x and y. This leads to a topological ground state degeneracy of

N6L−3, which for N = 2 exactly matches with that of the X-cube model, despite

being microscopically appearing very different. Thus, the N -GPM on the corner-

sharing octahedra lattice results in ZN fracton topological order, which appears to

describe the same phase as the X-cube model.

3.4.5 N-GPM on Simple Cubic Lattice

Here, we briefly show how the N -GPM on the simple cubic lattice maps on to the ZN

X-cube model. First, notice that this model has square plaquettes (thus describes a

square tetramer model, rather than a trimer model). The Hamiltonian is given by

HSCN -GPM = −

∑

matchboxes

(ZZ†ZZ† + h.c.)−∑

s

∏

p∈sXp (3.27)

where the first sum is over four plaquettes going around a cube, which we refer to as

“matchboxes”. There are three distinct orientations per cube. To map the model on

to the X-cube model, we transform to the dual lattice: cubic volumes are replaced

75

by vertices, and plaquette faces are replaced by bonds. The first sum then becomes

the cross-term, and the second sum becomes the cube term. Finally, after mapping

Z → Z† and X → X† for all operators on bonds going from A to B sublattices of the

dual cubic lattice in the positive x,y, and z directions, one obtains the ZN X-cube

generalization obtained in Ref [93] from a layered construction.

3.5 Conclusions

To conclude this section, we have investigated in detail the topological properties of

the FCC QPM, a prime candidate for an RSVP phase. In doing so, we discovered

the presence of a Z3 topological conserved quantity that leads to a 33-fold topological

ground state degeneracy at the RK point on a 3-torus, where this model was shown

to have exponentially decaying trimer-trimer correlations [16] indicating the presence

of a gapped liquid RSVP phase. Our result would then imply that this topological

degeneracy is a feature of the whole phase, and we show that it also shares the

features one expects of a phase can be described by a Z3 gauge theory, such as Z3

quasiparticle excitations and loop-like vortex excitations. This Z3 emerges naturally

from the geometry of the FCC lattice, in the same way that a Z2 order emerges in

the triangular lattice QDM.

76

Chapter 4

Floating topological phases

Having discussed a number of both conventional and fracton topologically ordered

phases, we now turn to floating topological phases. A floating topological phase is a

3+1D phase which is smoothly connected to a state of decoupled 2+1D topologically

ordered layers. These phases are “in between” conventional and fracton topologically

ordered phases. In some ways, they are the simplest (almost trivial) examples of

fracton topological order. Nevertheless, many material candidates for spin liquids are

quasi-2D, and so topological order (if it were to exist in such a material) will be of

this layered type. In this section, we define floating topological phases and discuss

their stability in both gapped and gapless cases. Then, we discuss its diagnosis via a

correlation function order parameter (as in Sec 2). This chapter is based on parts of

the paper

[18] T. Devakul, S. L. Sondhi, E. Berg, S. A. Kivelson, “Floating topological phases”,

Phys. Rev. B 102, 125136 (2020) [Editor’s Suggestion].

77

4.1 Gapped topological floating phases

We begin with the simplest possible case: a floating phase of gapped topological orders

in two spatial dimensions (i.e. 2+1D including the time dimension). In such models,

topological properties such as the ground state degeneracy and statistical properties

of fractionalized quasiparticle excitations are stable to arbitrary interlayer couplings.

As the models we will consider later take the form of stacked gauge theories coupled

to matter, we will first begin with a simplest such example: the stack of 2+1D Ising

gauge theories coupled to Ising matter. This model is equivalent to a stack of 2+1D

Kitaev toric codes.[31]

The 2+1D Ising gauge theory (IGT) on the square lattice is described [34, 35] by

a model with matter degrees of freedom τr living on the sites r, and gauge degrees

of freedom σ` living on the links ` = (rr′) connecting nearest neighbor sites r and r′.

The IGT Hamiltonian is

HIGT =−K∑

∏

`∈σz` − Γ

∑

`

σx`

− J∑

(rr′)

τ zr σz(rr′)τ

zr′ − ΓM

∑

r

τxr

(4.1)

where the first sum is over square plaquettes, and σx,y,z and τx,y,z are Pauli matrices

acting on the σ and τ degrees of freedom respectively. Here the first two terms are the

“Maxwell” terms, and the remaining terms are the gauge-invariant matter terms. This

Hamiltonian is invariant under local gauge transformations, G†rHGr = H, generated

by

Gr = τxr∏

r′

σx(rr′) (4.2)

on each site r, where the product is over the four nearest neighbors. We take the

physical subspace to be the one with Gr = 1 on every site. K > 0 favors a zero-flux

ground state, and Γ makes the gauge field dynamical.

78

This model describes a deconfined Z2 gauge theory when K/Γ 1 and ΓM/J

1. In this limit, HIGT describes a perturbed version of Kitaev’s toric-code model [31]

— this can be seen by working in the gauge τ zr = 1, in which HIGT can be expressed

entirely in terms of the gauge-fields

HIGT =−K∑

∏σz − ΓM

∑

+

∏σx

− J∑

`

σz` − Γ∑

`

σx`

=HTC − J∑

`

σz` − Γ∑

`

σx`

(4.3)

where the K and ΓM terms are the stabilizers of the toric code Hamiltonian HTC ,

and J and Γ are small σz and σx perturbations that make the model dynamical.

The Toric Code Hamiltonian HTC possesses non-trivial topological order which is

stable to arbitrary local perturbations [33]. When placed on a torus, HTC has four

exactly degenerate ground states in the thermodynamic limit. These ground states

may be distinguished via non-local Wilson and ’t Hooft operators. Define the Wilson

loop operator

WC =∏

l∈Cσzl (4.4)

where C denotes a closed loop on the square lattice, and l ∈ C are all the links

involved. We may also define the dual Wilson loop (or ’t Hooft operator),

VC =∏

l∈C

σxl (4.5)

where C denotes a loop on the dual square lattice, and l ∈ C are all the links

cut by C. These operators commute with HTC . Let W1 and V1 denote the non-

contractible Wilson loops going around the torus in the x direction, and similarly W2

and V2 along y. The operators (W1, V2) and (W2, V1) generate Pauli algebras on the

79

4-dimensional ground state manifold. The ground state degeneracy is stable to J and

Γ perturbations due to the fact that these non-contractible Wilson loop operators

only appear at high order O(L) in perturbation theory where L is the circumference

of the torus; any lifting of the degeneracy is thus exponentially suppressed at large

L, going as ∼ (J/ΓM)L or ∼ (Γ/K)L.

Next, consider the bilayer of two such systems, with some weak coupling between

them

H = H(1)IGT +H

(2)IGT + λHinter (4.6)

where Hinter contains local terms coupling the two layers. On a torus, this bilayer now

has a 42-fold degenerate ground state manifold, which is only split perturbatively by

the interlayer couplings at order λL as before. This simply describes a new topological

order, which inherits all its topological properties from the stack of two decoupled

toric codes. Indeed, this is simply the Z2×Z2 generalization of the toric code (which

describes the gauge theory of a bilayer Ising model in which each layer has a separate

Z2 symmetry).

Now, let us consider a 3+1D system on a 3-torus obtained by stacking L such

models in the xy plane along the z direction, allowing for small (but arbitrary) local

perturbations. This model will have a robust 4L ground state degeneracy which is

stable to any arbitrary small interlayer interactions. We define a D+ 1 dimensional

system to be in a non-trivial gapped floating topological phase if it can be smoothly

connected (via a finite depth local unitary transformation) to a decoupled stack of

d + 1 dimensional topologically ordered systems, where 0 < d < D. In the cases we

consider, D = 3 and d = 2. The stack of toric code models, with weak interlayer

coupling terms, realizes a non-trivial floating topological phase by this definition.

Quasiparticle excitations of this model are constrained to move within a single

2+1D xy plane. As an aside, we note that decoupled stacks of topologically ordered

planes, exactly as we have formulated, have appeared multiple times [94, 24] in the

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literature of fracton topological order [95, 96]. If we take the definition of a fracton

topological order to be a subextensive ground state degeneracy lnGSD ∼ L on a

3-torus and subdimensional quasiparticle excitations, then the stack of 2+1D topo-

logical orders are indeed (very simple versions of) fracton models. These may also be

obtained as gauge theories of models with planar subsystem symmetries along each

xy plane (a 1-foliated planar subsystem symmetry [27]).

The low energy field theory description of the single-layer toric code is given by

the 2+1D BF theory, or equivalently the Chern-Simons theory [37, 38]

LCS =KIJ

4πεµνρaIµ∂νa

Jρ (4.7)

with the 2× 2 K matrix KTC = 2σx. The ground state degeneracy of such a model

on a manifold of genus g is then given by GSD = | det K|g. The bilayer toric code

described by the Hamiltonian Eq 4.6 then admits a similar low-energy description,

except with the 4 × 4 K matrix given by the direct sum K2TC = KTC ⊕KTC . The

floating topological phase of toric codes discussed above may then be characterized

by the extensively large K matrix Kfloat = KTC⊕· · ·⊕KTC (such “giant” K matrices

also appear in the classification of fracton phases [24]). This can be generalized to

stacks of general Abelian topological phases characterized by the matrix K [39].

We also note the interesting possibility of offdiagonal elements in the large K

matrix which couples different layers. These types of systems have been studied [97,

98, 99, 100] and found to exhibit interesting behavior (such as an irrational braiding

statistic) which cannot be found in 2+1D systems. Such systems are not floating

topological phases by our definition as they cannot be deformed to the decoupled

limit, but nevertheless have an emergent decoupled gauge symmetry.

81

Finally, most of our discussion can also be extended to stacks of non-Abelian

topological orders 1. Such a phase can be characterized by the topological properties of

the quasiparticle excitations, such as the fusion coefficients and topological spin [103].

Stacking two layers results in a new phase whose quasiparticles are directly inherited

from the individual layers (see Ref [103]), and survive interlayer couplings as long as

the gap is not closed. The floating phase of gapped non-Abelian topological orders

are therefore also stable.

4.2 Floating phases via the Fredenhagen-Marcu

order parameter

The goal of this section is to differentiate a floating topological phase from either the

3+1D topological order or the trivial phase by means of a correlation function.

4.2.1 Usual deconfinement diagnostic

We first review how this is done in the usual case of diagnosing deconfinement in

the Ising gauge theory [13]. Let us take W (L) to be the Wilson loop defined in

Eq. 4.4 along the contour C taken to be an L × L square. For a pure gauge theory

without dynamical matter (J = 0 in Eq 4.3), the scaling of the expectation value of

the Wilson loop is sufficient to diagnose deconfinement: for large L, ln〈W (L)〉 ∼ −L

scales linearly with the perimeter of the loop in the deconfined phase, but in the

confined phase scales with the area, ln〈W (L)〉 ∼ −L2. However, as soon as J 6= 0,

the Wilson loop scales with the perimeter in both the deconfined and confined phases,

and therefore fails to distinguish between the two.

1More specifically, we concern ourselves with only non-invertible topological orders, which possessfractionalized quasiparticles. Stacks of invertible topological phases [101, 102] need not form stablefloating phases.

82

To correct this shortcoming, consider (an equal time formulation of) the

Fredenhagen-Marcu order parameter [13, 48, 49] (FMOP). Define

W 12(L) ≡ τ zr τ

zr′

∏

l∈C 12

σzl (4.8)

to be a gauge-invariant open Wilson line (or horseshoe) operator, where C 12

is the

L×L/2 horseshoe terminated at sites r, r′, obtained by cutting C in half. Similarly,

let W− 12

be the other half of the Wilson loop. We note that a more general geometry

is possible, we simply choose to horseshoe shape for simplicity. It can be shown (see

below) that the ratio

R(L) = 〈W 12(L)〉〈W− 1

2〉/〈W (L)〉 (4.9)

goes to 0 in the deconfined phase, while limL→∞R(L) = R0 > 0 in the confined phase.

Thus, other than in the special case J = 0, this provides a suitable signature of a

deconfined phase.

This behavior can be understood in many ways. If we adopt the same gauge choice

as in Eq. 4.3, we can express the ground-state of the Hamiltonian in the σx basis as

|ψ〉 =∑

c

αc |c〉 , (4.10)

where c = σxl label all the configurations. The ground-state of the toric code is the

equal amplitude superposition of all configurations of closed σx = −1 loops, with zero

amplitude for all other c; it is a loop condensate. With perturbations, the weights of

each configuration in the ground state are no longer exactly equal and configurations

with open strings now exist, ableit with weights that are exponentially small in the

separation between the two endpoints ` going as ∼ (J/ΓM)`.

83

The expectation value of the horseshoe operator (Eq. 4.8)

〈W 12〉 =

∑

c

α∗cαc′ (4.11)

where |c′〉 = W 12|c〉 is the configuration c with σx flipped along the support of W 1

2.

(The terminal factors of τ zr in Eq. 4.8 are set equal to 1 by the choice of gauge.)

There is an analogous expression for the expectation value of W . To see how these

considerations distinguish the two phases, we use this expression to compute R(L) at

points deep inside the respective phases.

Manifestly, 〈W 12(L)〉 vanishes in the toric-code ground-state, since W1/2 generates

a string from r to r′, meaning that for any c such that αc 6= 0, αc′ = 0. In the

perturbed problem, 〈W 12〉 is not identically zero, but decays exponentially with L.

The form of its decay can be derived using perturbation theory in both J and Γ :

〈W 12(L)〉 ∼ e−(2a+b)L. Here, a ∼ (Γ/K) comes from the sides of the horseshoe and

b ∼ − ln(J/ΓM) comes from the string from r to r′. The Wilson loop scales as

〈W (L)〉 ∼ e−4aL. Thus R(L) ∼ e−2bL → 0 in the deconfined phase.

To characterize the confined phase, consider the ground state of Eq. 4.3 in the

large Γ limit. Here, the significant configuration are those which are mostly polarized

with σx = +1, plus small loop fluctuations (suppressed by factors of (K/Γ )A where

A is the enclosed area) and open line fluctuations (exponentially suppressed in their

length as (J/Γ )`) For large L, 〈W1/2(L)〉 ∼ (J/Γ )2L and 〈W (L)〉 ∼ (J/Γ )4L. Hence,

R(L)→ const in the large L limit in the confined phase. Some intuition can be gained

along the special Γ = 0 axis, where R(L) is (the square of) the original Ising σzσz

correlation function, and the approach to a constant can be understood in terms of

spontaneous symmetry breaking.

The generalization of this FMOP construction beyond Z2 is possible and discussed

by Gregor et al [13].

84

4.2.2 Floating deconfinement diagnostic

We now turn our discussion to the problem of diagnosing a floating phase of 2+1D

topological orders from either a confining phase or a fully 3+1D topological ordered

phase. To do this we consider an anisotropic 3D generalization of the gauge-fixed

version of the Ising gauge theory (Eq. 4.3) with qubits defined on the nearest-neighbor

bonds of a tetragonal lattice, with couplings K, J, Γ for the in-plane terms, and

K⊥, J⊥, Γ⊥ for those involving bonds in the interplane direction. This system now

has the same Hilbert space as that of the 3+1D toric code. Indeed, for J = J⊥ =

Γ = Γ⊥ = 0 and K = K⊥, this model reduces to the 3+1D toric code, and to stacks

of 2+1D toric codes if we then take the limit Γ⊥ →∞, K⊥ → 0.

It is thus clear that in different limits, this one model can support all three of the

possible phases in question. Our diagnostic for floating topological order is inspired

by the usual deconfinement diagnostic just discussed.

Let W (z;L) be the Wilson loop on a L×L loop in the zth plane. We further split

the loop into two equal horseshoes such that W (z;L) = W 12(z;L) W− 1

2(z;L). where

W 12

is defined on the left horseshoe and W− 12

on the right. We then consider the ratio

R2(L) =W 1

2(z;L) W− 1

2(z + 1;L)

√W (z;L) W (z + 1;L)

. (4.12)

As we will see, in analogy with the previous analysis,

limL→∞

R2(L) =

0 deconfined floating

const confined or 3+1D deconfined

(4.13)

distinguishes between the deconfined floating topological phase from a confined or

fully 3+1D deconfined topological phase.

85

First, consider the decoupled limit with all the interplane couplings set equal to

0: now R2(L) factors as R2(L) = R(L), where R(L) is the FMOP for a single 2+1

d plane defined in Eq. 4.9. Thus in this case, for the deconfined phase, R2(L) → 0,

while for the confining phase R2(L)→ const > 0. It is easy to see that these results

remain true even in the presence of arbitrary small perturbations.

Finally, consider the case of an isotropic 3+1D phase. The numerator is now

the expectation value of a Wilson loop minus only the two vertical bonds - the full

Wilson loop thus differs from this by a factor which is order O(1) (i.e. independent

of L). Moreover, this Wilson loop can be viewed as a slightly distorted relative

of the the Wilson loop in the denominator other except for two “kinks” where it

changes over between planes z and z + 1. These kinks also make an O(1) correction

to the total expectation value of the Wilson loop. The scaling of numerator and

denominator again cancel, so independent of whether the 3+1D system is confining

or not, R2(L) → const. In short, the vanishing of R2(L) at large L is a signature of

a topological floating phase.

Owing to the emergent subdimensional Lorentz symmetry of the floating topo-

logical phase, the order parameter may be oriented in various space-time directions

(within the x, y, τ subspace), each of which have a physical interpretation [13]. The

order parameter Eq 4.12 corresponds to an equal time-slice orientation. We will

now discuss another orientation of this order parameter (which in the usual case cor-

responded to the order parameter discovered by Fredenhagen and Marcu [48, 49]).

Let us denote by |ψz〉 a trial state with two (spinon) excitations located at ~r and

~r′ = ~r + Lx on plane rz = z, defined by

|ψz〉 = τ zr τzr′Vrr′(−T/2) |GS〉 (4.14)

86

where Vrr′(−T/2) ≡ e−HT/2Vrr′eHT/2, Vrr′ =∏

l σzl is the (non gauge-invariant) Wil-

son line operator connecting the points r and r′, and |GS〉 is the ground state.

V (−T/2) acts on the ground state by creating two “defects” at r and r′ where

Gr = −1, and (in the limit of large T ) projects to the lowest energy state with

such defects.

The order parameter is given by

R2(L, T ) =〈ψz〉ψz+1√

〈ψz〉ψz 〈ψz+1〉ψz+1

(4.15)

which will exhibit the same asymptotic behavior as L, T → ∞ as F (L). In this

picture, we see that R2 is probing the orthogonality of the trial spinon states on plane

z and z+1. In the deconfined floating phase, |ψz〉 and |ψz+1〉 will be orthogonal, since

spinons cannot tunnel between planes. However, in a fully 3+1D deconfined phase

a spinon may move between the two planes and so 〈ψz〉ψz+1 6= 0. While R is useful

conceptually, in condensed matter systems where the gauge symmetry is emergent,

the equal-time formulation R2 (Eq 4.12) should be used [13].

Finally, all of this discussion can be extended beyond Z2. One simply replaces V

with the appropriate Wilson line operator and τ by the appropriate charged matter

operator for gauge invariance [13].

4.3 Gapless floating topological phases

In this section, we analyze the stability of gapless floating topological phases. We first

consider the floating phase of gapless Dirac matter coupled to a gapped Z2 gauge field.

We then go on to consider the floating phase of Dirac fermions coupled to a U(1) gauge

field, in which both the matter and gauge sectors are gapless. We show that these

gapless floating phases are stable to interlayer couplings in the renormalization group

87

(RG) sense. That is, the interlayer couplings are irrelevant perturbations: at long

distances and low energies, the system flows back to the decoupled limit.

4.3.1 Gapless matter, gapped gauge

Let us first consider the case with gapless Dirac matter coupled to a gapped gauge

field. At low energies, a single 2+1D layer is described by Dirac fermions hopping

in a background static gauge field. This describes, for example, the gapless phase of

Kitaev’s Honeycomb model [104], in which case the minimum energy configuration

for the gauge field is equivalent to the trivial (flux-free) configuration.

The low-energy continuum Hamiltonian for a single layer l is

Hl =

∫d2rψ†l (~r)

[−iv~σ · ~∂

]ψl(~r) (4.16)

where ψ†l (~r) (ψl(~r)) is a 2-component spinor which creates (annihilates) a complex

fermion at position ~r = (x, y) on layer l, v is the Fermi velocity, ~∂ = (∂x, ∂y), and

~σ = (σx, σy) is a vector of Pauli matrices acting on the spinor indices 2.

The corresponding Euclidean action for the single layer is

Sl =

∫d2rdτψl(~r, τ)γµ∂µψl(~r, τ) (4.17)

where summation over µ = τ, x, y is implied, γµ = (σz, σy,−σx), ψ = ψ†σz, and we

have rescaled coordinates to set v = 1.

The gauge fields are fully gapped out, and hence do not appear in the low-energy

description. Nevertheless, they are important as they restrict the terms which are

allowed to appear to only those which are gauge invariant. When we have multiple

layers, gauge invariance within each layer implies that any interlayer term must consist

2This low-energy form is obtained from Kitaev’s Honeycomb model by first combining the twoMajorana cones into a single Dirac cone.

88

of operators which are individually gauge invariant on each layer. Crucially, this

forbids quadratic interlayer hopping terms, which are not gauge invariant within a

single layer. The simplest gauge-invariant terms coupling two layers are four-body

interaction terms such as (ψlψl)(ψl′ψl′) between two layers l, l′. However, all such

quartic terms are irrelevant at the 2+1D Dirac fermion Gaussian fixed point with S =∑

l Sl. This can be seen by simple power counting: The field ψ has length dimensions

[ψ] = L−1, and so the quartic term has dimensions [(ψψ)2] = L−4. Under an RG

transformation in which we rescale time and the two continuous spatial dimensions,

the quartic term therefore flows to zero. If the system were to flow to a fully 3 + 1D

phase, we would instead expect that the interlayer couplings would increase under this

RG flow. In this case, however, the system flows back to the decoupled layers limit

at large distances. This system is therefore an example of a stable gapless floating

phase.

Finally, we note that there are single-layer terms which are relevant. These include

quadratic terms of the form ψγµψ, which may open up a gap or create a Fermi surface.

In assessing the stability of these phases, we have implicitly assumed that these terms

are forbidden by symmetries of the microscopic Hamiltonian, and that the microscopic

interlayer couplings also respect such symmetries.

4.3.2 Gapless matter, gapless gauge

Let us now consider the case of Dirac fermions coupled to a gapless U(1) gauge field.

The pure U(1) gauge theory is confining in 2 + 1D, but a stable deconfined phase can

exist when coupled to a large number of Dirac fermions [105]. In this situation, the

low energy continuum description of each 2 + 1D layer l is simply large-N quantum

electrodynamics: N Dirac fermion flavors ψi,l (i = 1, . . . , N) coupled to an emergent

89

gauge field aµ,l. The Euclidean Lagrangian is

LQED3l =

N∑

i=1

ψi,lγµ(∂µ + iaµ,l)ψi,l +

1

4e2fµν,lf

µνl (4.18)

where fµν,l = ∂µaν,l − ∂νaµ,l is the field strength. The gauge transformation sends

ψl → eiαlψl and aµ,l → aµ,l−∂µαl for an arbitrary spacetime function αl(~r, τ) on each

layer l.

The Maxwell term, although included, is irrelevant at large N . This is exemplified

by the fact that with a clever choice of non-local gauge fixing term, the gauge photon

propagator can be written in such a way that the e2 → ∞ limit can be taken at

the beginning of a calculation [106]. Indeed, as written, we have [a] = L−1 and so

[fµνfµν ] = L−4 is irrelevant, while the coupling [ψaψ] = L−3 is marginal.

As in the previous case, the simplest interlayer coupling terms that are gauge-

invariant are either quartic in the fermion operators or pure-gauge (of the form

fµν,lfµνl′ ), both of which are strictly irrelevant perturbations at large N . However,

as before, we have implicitly assumed that, for reasons of symmetry, relevant single-

layer terms are not present. These now include, for example, a Chern-Simons term.

Like in the gapped case, rather than the low energy model with a separate gauge

symmetry on each individual layer, a more natural starting point is a single anisotropic

emergent 3+1D gauge field with interlayer couplings much weaker than intralayer. In

this case, one has a gauge fields az,l, and gauge-invariant interlayer hopping terms

of the form ψl+1eiaz,lψl are allowed. In this limit, az,l are strongly fluctuating (and

therefore gapped), so they can be integrated out resulting in a local effective action

with a separate gauge symmetry on each individual layer. Such a system is partially

confined: fractionalized quasiparticles are confined along z, but deconfined within

each layer.

90

Such “layered” phases of the U(1) gauge theory have been studied previously [107,

108, 109] and found to be stable in higher than 3 + 1D. These are examples of stable

higher-dimensional gapless floating topological phases.

91

Part II

Regular Subsystem Symmetric

Phases

92

Chapter 5

Preliminaries

5.1 Symmetry-protected topological phases

The subject of this chapter are symmetry-protected topological (SPT) phases [110,

111, 112, 113, 114, 2, 115].

The topologically ordered phases of the previous chapter are said to have long-

range entanglement — meaning that there is no finite-depth local unitary circuit

which transforms its ground state into a disentangled product state [7]. In the absence

of symmetry, phases of matter may be classified as either short-range or long-range

entangled, and all short-range entangled states are adiabatically connected to the

trivial state.

With symmetry, however, there are many more possibilities. Suppose we have

an on-site symmetry group G. By this, we mean that each site degree of freedom

transforms under a unitary representation of G, and the total action of the symme-

try element g ∈ G on the system is a tensor product of the on-site representation.

Multiple physical spins may to be grouped into a single site if necessary for the rep-

resentation to be on-site. In the case of the Ising model, for example, G = Z2 and

the on-site representation is σxi . One possibility is Landau spontaneous symmetry

93

breaking [1], in which the ground state may only be symmetric under some (possi-

bly trivial) subgroup of G. The ferromagnetic state of the Ising model, for example,

spontaneously breaks the Z2 symmetry down to its trivial subgroup.

However, there may also be distinct phases of matter even when none of the

symmetries are broken spontaneously. In the presence of symmetry, two gapped

systems are said to belong to the same phase if their ground states are be connected

by a finite-depth symmetric local unitary (SLU) circuit [7]: a local unitary circuit in

which each gate must individually commute with the symmetry operator. A system

belongs to a non-trivial SPT phase if its unique gapped ground state cannot be

connected to the trivial product state by an SLU (but if the symmetry is not enforced,

then it can be). Such phases with a global on-site symmetry group G may be classified

by group cohomology [2], which we will briefly review when needed.

The classification of phases may be understood intuitively as some non-trivial

action of the symmetry at the edges of the system. Take for example a 1D chain. A

very simply exactly solvable 1D SPT is the 1D cluster model [116] which is described

by the Hamiltonian

H = −∑

i

Zi−1XiZi+1 (5.1)

where Xi,Yi,Zi, denote Pauli matrices acting on the ith qubit in the chain. Each term

is commuting and so the ground state is the simultaneous +1 eigenstate of every term.

There is a unique ground state since each term is independent and there are exactly

as many terms as qubits. This model has a G = Z2 × Z2 global symmetry, which is

generated by

Se =∏

i∈even

Xi

So =∏

i∈odd

Xi

(5.2)

94

and involves flipping all spins on the even or odd sites. To see how this describes a

non-trivial SPT phase, take the system on an open chain of length L, and consider the

way the symmetry acts on the edge. Terms near the edges that are not fully contained

in the system are neglected. Let |ψ〉 be a state in the ground state manifold (on an

open chain, the ground state may no longer be unique). Since the bulk is invariant

under the symmetry, we must have that, effectively, on the ground state manifold,

the symmetry acts as

Se |ψ〉 = V Le V

Re |ψ〉

So |ψ〉 = V Lo V

Ro |ψ〉

(5.3)

where VL/Re/o is some unitary acting only near the left (L) or right (R) edge. To find

out what they are in this case, we need to find operators such that V LV RS |ψ〉 = |ψ〉,

which is satisfied if V L†V R†S belongs to the stabilizer group. We find, when L is

even,

V Le =Z1

V Re =ZL−1XL

V Lo =X1Z2

V Ro =ZL

(5.4)

which tells us how each symmetry generator Se/o acts on the edges of the system.

Focusing on only the left edge, we have V Le = Z1 and V L

o = X1Z2, which do not

commute: V Le V

Re = −V R

e VLe . Although Se and So must commute with each other as

a whole, it is OK for the edge actions to commute up to a phase, as long as the phase

from the left edge and right edge cancel with one another. The fact that there is a −1

sign from the left edge which must cancel with the −1 from the right means that we

are in a non-trivial phase, as any symmetry-respecting perturbation can only change

95

things locally. Hence, we see that the symmetry group G = Z2×Z2 is realized on the

edge up to a phase.

Such a representation is known as a projective representation, and 1D SPT phases

are in one-to-one correspondence with classes of projective representations of G [2],

which is classified according to the second group cohomology H2[G,U(1)]. A projec-

tive representation u(g) satisfies the group algebra up to a phase,

u(g1)u(g2) = ω(g1, g2)u(g1g2) (5.5)

where g1, g2 ∈ G, ω(g1, g2) ∈ U(1). The representation must be associative, meaning

u(g1)u(g2)u(g3) =ω(g1, g2)u(g1g2)u(g3) = ω(g1g2)ω(g1g2, g3)u(g1g2g3)

=ω(g2, g3)u(g1)u(g2g3) = ω(g1, g2g3)ω(g2g3)u(g1g2g3)

(5.6)

implying that ω must satisfy

ω(g1g2)ω(g1g2, g3) = ω(g1, g2g3)ω(g2g3) (5.7)

for all g1, g2, g3. Finally, we are always free to redefine u(g) → u(g)α(g) for some

phase α(g) ∈ U(1), which we will refer to as a “gauge” transformation. This gauge

transformation transforms ω according to

ω(g1, g2)→ ω′(g1, g2) =α(g1g2)

α(g1)α(g2)ω(g1, g2), (5.8)

which represents the same projective representation. The set of function ω(g1, g2)

satisfying Eq 5.6 modulo transformation Eq 5.7 is precisely the definition of the

second group cohomology, H2[G,U(1)]. We will come back to this in Chapter 6.

Thus, 1+1D SPTs are characterized by the edge representation of the symmetry

group failing to be commutative. They are classified according to projective represen-

96

tations. In 2+1D or higher, SPTs may still be classified by an anomalous action of the

symmetry on the edge [117], albeit it is no longer as simple as a failure to commute.

For example, 2+1D SPTs are classified by the third group cohomology H3[G,U(1)],

which may be interpreted as a failure of a representation to be associative. We will

come back to this in Chapter 7

In 2+1D, there is an intimate connection between the classification of SPT phases

and a family of topologically ordered phases known as quantum double models which

are obtained by gauging of a global symmetry. The simplest quantum double model

is the toric code, whose quasiparticle excitations are the e, m, and ψ = e × m

particles. e and m self-bosons, and ψ is a self-fermion (meaning they have +1 or −1

self exchange statistic, respectively). Meanwhile, e and m are mutual semions. A

semion is a quasiparticle that is “one-half” of a fermion, meaning they have a i self

exchange statistic and a −1 self-braiding statistic (a braid, one particle going fully

clockwise around the other and returning to its original position, may be thought of

as two clockwise self-exchange processes). The statement that e and m are mutual

semions means that they have a −1 mutual braiding statistic. As we showed in

Sec 1.1.2, the toric code is obtained from the paramagnetic phase of the Ising model

after gauging its global Z2 symmetry. We therefore say that the toric code is dual to

the trivial Z2 symmetric phase: the trivial paramagnet.

For G = Z2 in 2+1D, there is a single non-trivial SPT [118, 119]. The Levin-

Gu model [119] is an exactly solvable model belonging to this non-trivial SPT phase

The gauging procedure, applied to the Levin-Gu model, results in a topologically

ordered phase known as the double semion model. Like the toric code, we can label

to quasiparticle excitations as 1, e,m, ψ. Although the quasiparticle content is

the same, the topological order is different due to the self and mutual statistics of

these quasiparticles. First, e and m are still mutual semions (−1 braiding statistic).

However, m is now a semion: exchanging two m particles results in an overall i phase

97

in the wavefunction. As a result, ψ = e×m is now an (anti-) semion, as interchanging

two results in a −i phase. Hence, the name double semion.

The double semion is a “twisted” version of the toric code, meaning that it has the

same quasiparticle content, but self/mutual statistics may differ. The more general

observation is that non-trivial SPT phases will be dual to various twisted topological

ordered phases, with the same particle content but whose statistics are determined

by the specific SPT phase [119]. A physical interpretation of the classification of

2+1D SPT phases is therefore through the quasiparticle statistics of their gauged

topological orders.

This chapter will be focused on systems with subsystem symmetries. Recall that

the study of subsystem symmetries was initially motivated by their connection to

fracton topological order (Sec 1.2.2). In a system with subsystem symmetries, beyond

the trivial symmetric phase and the spontaneously symmetry broken phases (which we

saw in the plaquette Ising model), there is also the possibility of non-trivial subsystem

SPT (SSPT) phases. What do such phases look like? How can we characterize

them? Can we classify all the possible phases? In this chapter, we will answer

these questions for systems with regular symmetries. Specifically, for 2+1D systems

with linear subsystem symmetries [11, 19], and 3+1D systems with planar subsystem

symmetries [20, 27]. The more complicated case of fractal subsystem symmetries will

be covered in the next chapter.

As we will see, SSPTs may also be understood in terms of some anomalous action

of the subsystem symmetries on the edge. The dimensionality of the edge action

will be determined by the dimensionality of the subsystem: for example, a linear

subsystem symmetry acts on a single zero-dimensional point at the edges. A planar

subsystem symmetry, on the other hand, acts along a one-dimensional line along the

edges. Thus, while projective representations are useful for understanding the physics

of 1+1D global SPTs, they will also be useful in understanding 2+1D SSPTs with

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linear subsystems. Similarly, the tools for understanding 2+1D SPT phases will also

be useful in understanding 3+1D SSPTs with planar subsystem symmetries.

5.2 Linear subsystem symmetries

5.2.1 The square lattice cluster model

Before stating things more generally, it is instructive to begin with a simple exactly

solvable model.

Consider the cluster model on the square lattice, given by the Hamiltonian

Hclus =∑

~r

X~rZ~r+xZ~r+xZ~r+yZ~r−y (5.9)

where the sum is over all sites at ~r, and each term involves X~r and the product of four

Z on the four neighboring site. This model has a subsystem symmetry that involves

flipping every spin along a diagonal of the square lattice:

S(x0, y0) =∏

n

Xx0+n,y0+n

S(x0, y0) =∏

n

Xx0+n,y0r−n

(5.10)

for any choice of (x0, y0). The fact that all symmetry operators commutes with Hclus

is easily verified. Absent boundaries, there is a unique ground state, which can be

verified by the fact that all terms are independent, and there is one term per qubit.

Thus model therefore describes a gapped symmetric phase with a unique ground

state.

To see that this describes a non-trivial SPT phase, we can identify some anomalous

action of the symmetry along the edges. Let us take the system on an L× L square,

x, y ∈ [1, L], and neglect any terms not fully contained in the system. Define a set of

99

Pauli operators along the top edge

πxi = Xi,LZi−1,LZi,L−1Zi+1,Lπzi = Zi,L (5.11)

and πzi πxi = iπyi . It is straightforward to verify that πx,y,zi all commute with Hclus, and

they generate the Pauli algebra for L qubits along the top edge (ignoring subtleties

near the corners). The same can be done along all four edges. Hence, this model has

a ground state degeneracy that scales as 2Ledge .

To see that this degeneracy is non-trivial and protected by symmetry, we can

observe the way a symmetry acts along the edge. As in 1D, a subsystem symmetry

acts in some way on the ground state manifold which is localized to two points along

the edge where the subsystem terminates. Take S(1, y), which terminates on the

left edge at the point (1, y) and on the top edge at (1 + L− y, L) ≡ (x, L). Working

out the edge action on the ground state manifold, we find

S(1, y) |ψ〉 = VLVT (5.12)

where

VL =X1,yZ1,y+1Z2,y

VT =Xx,LZx−1,LZx,L−1 = πxxπzx+1

(5.13)

we see that this symmetry acts on two neighboring π qubits as πxi πzi+1. If we look

at the action of two neighboring parallel subsystem symmetries, for example S(1, y)

and S(1, y + 1), their action along the edge anticommute: πxi , πzi+1, πxi+1π

zi+2 = 0.

The minus sign obtained from the top edge is cancelled out by one from the left edge,

as all symmetries have to commute as a whole. This signifies a non-trivial SPT phase,

as any symmetry-respecting perturbation cannot remove the minus sign arising from

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faraway edges. The fact that the edge representation of the symmetry is non-trivial

also requires the large ground state degeneracy along the edge GSD ∼ 2Ledge .

The cluster model on the square lattice is an example of a strong subsystem

symmetry-protected topological (SSPT) phase. Roughly, a weak SSPT is one that is

connected to decoupled 1D SPT chains going along the subsystems, while a strong

SSPT cannot [11]. This distinction will be made precise shortly.

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Chapter 6

Classifying 2D linear subsystem

SPTs

In this section, we are interested in the question of classification for 2D linear subsys-

tem symmetry-protected topological (SSPT) phases. What SSPT phases can exist,

and can we classify them group cohomology classifies 1D and 2D (global) SPTs? The

example above of the square lattice cluster model demonstrates that such non-trivial

SSPT phases exist, and that they are non-trivial in a similar way as 1D global SPTs:

through a non-trivial projective representation realized at the edge. In particular, the

classification will make the distinction between strong and weak SSPT phases.

This chapter is based on the paper

[19] T. Devakul, D. J. Williamson, Yizhi You, “Classification of subsystem

symmetry-protected topological phases”, Phys. Rev. B 98, 235121 (2018).

We propose a natural definition of a strong equivalence relation for two-

dimensional SSPTs protected by line-like symmetries, whereby two phases are in the

same equivalence class if they can be connected to each other via a linearly-symmetric

local unitary (LSLU) evolution, which we will define. By construction, the weak

SSPT composed of decoupled 1D SPT chains may be transformed into a trivial

102

product state via an LSLU evolution. Importantly, we find that the square lattice

cluster model cannot be transformed to the trivial state. We may therefore take this

equivalence relation to define a strong SSPT phase as one that cannot be connected

to the trivial product state via an LSLU evolution. Moreover, we find that there

are several distinct equivalence classes of strong SSPTs, which are in one-to-one

correspondence with the non-trivial elements of the group

C[Gs] = H2[G2s, U(1)]/(H2[Gs, U(1)])3 (6.1)

where Gs is the finite abelian onsite symmetry group characterizing the subsys-

tem symmetries (to be defined), and H2[G,U(1)] is the second cohomology group

which classifies the projective representations of G. We have utilized the fact that

(H2[Gs, U(1)])3 always appears as a (normal) subgroup of H2[G2s, U(1)] for abelian

Gs (for details see section 6.3.4). This therefore presents a classification for strong

SSPT phases, according to our strong phase equivalence. Finally, we note that the

equivalence class defined by LSLU is the same as that defined by standard symmet-

ric local unitaries in combination with stacking with 1D SPT chains (See Sec 6.5.4),

which indeed has a natural interpretation of being a 2D equivalence class “modulo”

1D physics.

In an appendix of Ref. [11], it was argued that strong SSPT phases did not exist

for conventional continuous symmetry groups such as U(1) or SU(2), while the effect

of an additional global ZT2 time reversal symmetry does not lead to new strong phases,

as diagnosed by the projective representation at the edge. Furthermore, a non-abelian

Gs implies the existence of a local symmetry, as we will show. We therefore focus

on unitary representations of finite abelian groups Gs, which encompass most known

examples of strong SSPTs (e.g. Z2 or Zn × Zm).

103

2

decoupled 1D SPT chains. It was noted42,43 that this ex-ample of a weak SSPT could not serve as a resource foruniversal MBQC using only single-spin measurements.However, it is not clear how general this statement is asthere is currently no clear definition for what specifies astrong or weak SSPT. In this paper, we hope to tacklethe question of what constitutes a weak or strong SSPT,and whether such SSPTs may be classified in a naturalway.

Our work draws inspiration from a series of recentworks on fracton topological orders, where the conceptof a foliated fracton phase has been introduced36,48–51 toclassify non-fractal (Type-I24) fracton orders. A foliatedfracton phase is an equivalence class of fracton topolog-ical orders, whereby two fracton phases are consideredequivalent if one can be brought to the other via a com-bination of local unitary52 (LU) evolution and the addi-tion and removal of 2D topologically ordered phases. Weremark that standard phase equivalence only allows theaddition of trivial product states along with LU evolu-tion. Therefore, foliated fracton phases present a drasticdeparture from the norm. Foliated fracton phases may bethought of as a 3D phase equivalence “modulo” any 2Dphysics: this motivates a similar construction for SSPTs.

We propose a natural definition of a strong equiva-lence relation for two-dimensional SSPTs protected byline-like symmetries, whereby two phases are in the sameequivalence class if they can be connected to each othervia a linearly-symmetric local unitary (LSLU) evolution,which we will define. By construction, the weak SSPTcomposed of decoupled 1D SPT chains may be trans-formed into a trivial product state via an LSLU evolu-tion. Importantly, we find that the square lattice clustermodel cannot be transformed to the trivial state. Wemay therefore take this equivalence relation to define astrong SSPT phase as one that cannot be connected tothe trivial product state via an LSLU evolution. More-over, we find that there are several distinct equivalenceclasses of strong SSPTs, which are in one-to-one corre-spondence with the non-trivial elements of the group

C[Gs] = H2[G2s, U(1)]/(H2[Gs, U(1)])3 (1)

where Gs is the finite abelian onsite symmetry groupcharacterizing the subsystem symmetries (to be defined),and H2[G, U(1)] is the second cohomology group whichclassifies the projective representations of G. We haveutilized the fact that (H2[Gs, U(1)])3 always appears asa (normal) subgroup of H2[G2

s, U(1)] for abelian Gs (fordetails see section IV D). This therefore presents a classi-fication for strong SSPT phases, according to our strongphase equivalence. Finally, we note that the equivalenceclass defined by LSLU is the same as that defined bystandard symmetric local unitaries in combination withstacking with 1D SPT chains (See Sec VID), which in-deed has a natural interpretation of being a 2D equiva-lence class “modulo” 1D physics.

In an appendix of Ref. 38, it was argued that strongSSPT phases did not exist for conventional continuous

Shy (g)

Svx(g)

FIG. 1: A depiction of the types of models we consider here.Each site transforms as a linear representation of some onsitesymmetry group Gs. For any g 2 Gs, there is a verticalsymmetry Sv

x(g) on every column, and a horizontal symmetrySh

y (g) on every row, which apply the onsite representation ofg to every site in a column or row, respectively . We requirea Hamiltonian to commute with all Sv

x(g) and Shy (g).

symmetry groups such as U(1) or SU(2), while the e↵ectof an additional global ZT

2 time reversal symmetry doesnot lead to new strong phases, as diagnosed by the pro-jective representation at the edge. Furthermore, a non-abelian Gs implies the existence of a local symmetry, aswe will show. We therefore focus on unitary representa-tions of finite abelian groups Gs, which encompass mostknown examples of strong SSPTs (e.g. Z2 or Zn Zm).

In Sec. II, we define the class of models we are in-terested in, and what we mean by subsystem symme-tries. Sec. III contains a review of standard 1D SPTphase equivalence and classification, in addition to a re-view of various useful tools such as projective represen-tations. Then, in Sec. IV we present our strong phaseequivalence and classification of strong SSPTs, for ourgeneral class of models. Sec. V then walks through theresults of the previous section with an example at hand,the square lattice cluster model. Finally, we finish witha few additional comments and conclusions in Sec. VIand VII. These include some straightforward general-izations as well as a connection to spurious topologicalentanglement entropy53,54 observed in non-topologicallyordered states on a cylinder55.

II. SETTING

Let us consider 2D models protected by line-like sub-system symmetries of a specific form. Let bosonic degreesof freedom live at the sites s of a square lattice, with lo-cal Hilbert space Hxy at site (x, y), such that the totalHilbert space is H =

Nxy Hxy. Each site transforms as

a unitary linear representation of some onsite symme-try group, Gs, which we take to be finite and abelian.For each element g 2 Gs, we demand that the system

Figure 6.1: A depiction of the types of models we consider here. Each site transformsas a linear representation of some onsite symmetry group Gs. For any g ∈ Gs, thereis a vertical symmetry Svx(g) on every column, and a horizontal symmetry Shy (g) onevery row, which apply the onsite representation of g to every site in a column orrow, respectively. We require a Hamiltonian to commute with all Svx(g) and Shy (g).

In Sec. 6.1, we define the more generally the class of models we are interested in,

and what we mean by subsystem symmetries. Sec. 6.2 contains a review of standard

1D SPT phase equivalence and classification, in addition to a review of various useful

tools such as projective representations. Then, in Sec. 6.3 we present our strong phase

equivalence and classification of strong SSPTs, for our general class of models. Sec. 6.4

then walks through the results of the previous section with an example at hand, the

square lattice cluster model. Finally, a few additional comments and conclusions are

included in Sec. 6.5 and 6.6. These include some straightforward generalizations as

well as a connection to spurious topological entanglement entropy [73, 74] observed

in non-topologically ordered states on a cylinder [120].

104

6.1 Setting

We consider 2D models protected by line-like subsystem symmetries of a specific form.

Let bosonic degrees of freedom live at the sites s of a square lattice, with local Hilbert

space Hxy at site (x, y), such that the total Hilbert space is H =⊗

xyHxy. Each site

transforms as a unitary linear representation of some onsite symmetry group, Gs,

which we take to be finite and abelian. For each element g ∈ Gs, we demand that

the system respects the following subsystem symmetries,

Svx(g) =∞∏

y=−∞uxy(g)

Shy (g) =∞∏

x=−∞uxy(g)

(6.2)

for every x, y ∈ Z, where uxy(g) is the on-site unitary (faithful) representation trans-

forming the site (x, y) by g. Sv and Sh act along vertical and horizontal rows, re-

spectively, as illustrated in Figure 6.1. The total symmetry group is therefore a

(sub)extensively large group (which should not be confused with the finite onsite

symmetry group Gs). We consider local short-range entangled Hamiltonians which

respect all these symmetries.

We remark that if Gs were a non-abelian group, then the symmetry

Shy (g−12 )Svx(g−1

1 )Shy (g2)Svx(g1) = uxy(g−12 g−1

1 g2g1) (6.3)

may act non-trivially on only a single site if g−12 g−1

1 g2g1 6= 1. This implies the existence

of a local [Gs, Gs] symmetry on every site, and an effective abelian G′s = Gs/[Gs, Gs]

subsystem symmetry. We therefore focus our attention on abelian groups Gs from

the beginning. We also note that Eq. (6.2) induces an identification of the group

elements g ∈ Gs across all sites of the system.

105

These symmetries present a drastic change from the now well-understood phases

protected by a global on-site symmetry group Gs in 2D, which are classified by the

3rd cohomology group H3[Gs, U(1)]. What distinct phases are possible under such

subsystem symmetries?

Consider the following scenario: suppose we construct a 2D phase by aligning

1D SPT chains horizontally, in such a way that all the vertical symmetries are still

respected (in this process a single SPT chain may span multiple rows in order to

respect all the vertical symmetries). We call such a phase a “weak” SSPT [11]. Under

the standard SPT phase equivalence, which we will review briefly in Sec. 6.2, two

states are in the same phase if they can be adiabatically transformed to one another

while respecting the symmetry, via a symmetric local unitary (SLU) evolution. In

our weak SSPT, each 1D SPT chain could be in any allowed 1D SPT phase, and by

this definition these are all distinct phases. The number of distinct phases therefore

grows exponentially with the system size. Note that we never assume any translational

invariance in any of our discussion. Nevertheless, we would like to be able to make a

clear distinction between these weak SSPT phases and a “strong” SSPT phase, which

cannot be written as a product of 1D SPT phases.

To this end, the main result of this section is a definition of a strong equivalence

relation for SSPT phases in Sec. 6.3, under which all weak SSPT phases are equiv-

alent to the trivial phase. This defines the meaning of a strong SSPT phase. The

secondary result is a classification of strong SSPT phases under this equivalence rela-

tion: strong SSPT phases may be classified according to the group C[Gs] in Eq.(6.1).

As an example, in Sec. 6.4 we show that the SSPT phase of the square lattice cluster

model, which has the onsite symmetry group Gs = Z2 × Z2, is non-trivial under this

equivalence relation. We further show that these equivalence classes of strong SSPT

phases are in one-to-one correspondence with elements of the group C[(Z2)2] = (Z2)6,

and exhibit the group structure under stacking.

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6.2 Standard SPT phase equivalence

To set the stage for our discussion of 2D SSPT phases, we first present a review of

the relevant standard concepts coming from the study of 1D SPT phases.

6.2.1 Symmetric local unitary transformations

Let |ψ〉 be the unique ground state of a gapped local Hamiltonian H, with symmetry

group G with an onsite representation ux(g) on site x. The symmetries are S(g) =∏

x ux(g) for g ∈ G, and the Hamiltonian respects [S(g), H] = 0. Two states |ψ〉 and

|ψ〉′ are said to be in the same SPT phase if there exists a symmetric local unitary

(SLU) evolution, USLU, that connects the two: |ψ〉 = USLU |ψ〉′. A state is in the

trivial phase if it can be connected via an SLU to a product state. For convenience,

we may always express an SLU evolution as a symmetric finite-depth quantum circuit,

which we now define.

A quantum circuit of depth d representing an SLU evolution, USLU, may be rep-

resented as

USLU = U (d)pwU

(d−1)pw . . . U (1)

pw . (6.4)

Here, each Upw is a piecewise local unitary operator,

Upw =⊗

i

U (i)s (6.5)

where U (i)s are local symmetric unitary operators which all act on local disjoint

regions. Importantly, the radius of support for each Us must be bounded by some

finite length. Finally, to represent an SLU evolution, we require that [S(g), Us] = 0

for all Us. Without this symmetry restriction, all short-range entangled phases can

be connected to a product state (via an LU). Such a quantum circuit is shown in

107

Fig. 6.2 for a 1D chain. Two quantum states are in the same SPT phase if and only if

there exists a quantum circuit USLU connecting the two, where d is a finite constant.

6.2.2 Projective Representations

In 1D, SPT phases with symmetry group G are in one-to-one correspondence with the

projective representations of the group G [2]. Projective representations will also play

a key role in our classification of strong SSPT phases, so we present an introduction

here. A non-projective or linear representation of a group G is a mapping from group

elements g ∈ G to unitary matrices V (g), such that V (g1)V (g2) = V (g1g2), for all

g1, g2 ∈ G. A representation V is projective if it instead satisfies

V (g1)V (g2) = ω(g1, g2)V (g1g2), (6.6)

where ω(g1, g2) ∈ U(1) is a phase, referred to as the factor system of a particular

projective representation. The factor system must satisfy

ω(g1, g2)ω(g1g2, g3) = ω(g1, g2g3)ω(g2, g3)

ω(1, g1) = ω(g1, 1) = 1

(6.7)

for all g1, g2, g3 ∈ G, where 1 is the identity element.

A different choice of prefactors, V ′(g) = α(g)V (g), leads to the factor system

ω′(g1, g2) =α(g1g2)

α(g1)α(g2)ω(g1, g2) (6.8)

Two factor systems ω(g1, g2) and ω′(g1, g2) related in this way are said to be equivalent,

and both belong to the same equivalence class ω.

Given two projective representations V1(g) and V2(g) with factor systems ω1(g1, g2)

of equivalence class ω1, and ω2(g1, g2) of equivalence class ω2, we may define the

108

projective representation

V (g) = V1(g)⊗ V2(g) (6.9)

with factor system

ω(g1, g2) = ω1(g1, g2)ω2(g1, g2) (6.10)

which now belongs to the class ω, defining a group operation ω1ω2 = ω. Under this

operation, the equivalences classes form an abelian group which is given by the second

cohomology group H2[G,U(1)]. The identity element of H2[G,U(1)] corresponds to

the linear representations, while other elements correspond to non-trivial projective

representations.

We consider cases where G is a finite abelian group. In this case, projective

representations simply allow for non-trivial commutation relations of the form

V (g1)V (g2) = φ(g1, g2)V (g2)V (g1) (6.11)

where

φ(g1, g2) = ω(g1, g2)/ω(g2, g1) (6.12)

is explicitly invariant under equivalence transformations of the form in Eq. 6.8, and

can therefore be regarded as a signature of the class ω. Under the group operation

on two classes, ω = ω1ω2, we have that φ(g1, g2) = φ1(g1, g2)φ2(g1, g2).

As an example, consider the group

G = Z2 × Z2 = 1, ga, gb, gagb (6.13)

where ga and gb are defined to be the two generators for G. In this case, there

are two classes of projective representations: the trivial linear representation where

φ(ga, gb) = 1, and the non-trivial projective representation with φ(ga, gb) = −1. An

109

example of the latter is given by the Pauli representation,

V (ga) = X, V (gb) = Z, V (gagb) = XZ (6.14)

where X,Z, are the Pauli matrices, with non-trivial ω(g1, g2) given by

ω(gb, ga) = ω(gagb, ga) = ω(gagb, gagb) = −1 (6.15)

In this case φ is a complete invariant and the projective representations of Z2 × Z2

are therefore in one-to-one correspondence with elements in H2[Z2 × Z2, U(1)] = Z2.

6.2.3 1D classification

Non-trivial SPT phases in 1D may be identified by their non-trivial edges, where the

symmetry group G is realized projectively leading to a symmetry-protected degener-

acy at the edge. We motivate this classification in a way that will prove useful for

our classification of strong SSPT phases to follow.

Let |ψ〉 be the ground state of a gapped symmetric local Hamiltonian in the

absence of a boundary, with symmetry group G which we take to be finite and abelian

(as these are the ones relevant for the case of SSPTs). As the ground state is unique,

we must have |ψ〉 = S(g) |ψ〉 up to a phase which can be absorbed into S(g). Now,

consider the truncated symmetry operator

U[x0,x1)(g) =

x1−1∏

i=x0

ui(g) (6.16)

where x0 < x1 are the endpoints, which we take to be separated by much further

than the correlation length. Acting on |ψ〉, this may create two local excitations in

the neighborhood of x0 and x1. These excitations may be locally annihilated by some

unitary operators V Lx0

(g) and V Rx1

(g) with support size on the order of the correlation

110

length about x0 and x1, such that

V Lx0

(g)V Rx1

(g)U[x0,x1)(g) |ψ〉 = |ψ〉 . (6.17)

Note that in writing this, we have assumed that |ψ〉 is short-range entangled and not

spontaneous symmetry breaking, as is the case for SPT phases.

We may also simplify this picture by not distinguishing between local excitations

created at the left/right endpoints of U(g), as they can be related to each other by a

symmetry operation:

U[x0,x1)(g)S(g−1) = U[−∞,x0)(g−1)U[x1,∞)(g

−1) (6.18)

Thus, the local excitation created at the right (left) end of U(g) is the same as

the excitation created at the left (right) end of U(g−1). We may therefore simply

substitute Vx(g) ≡ V Lx (g) on the left endpoint and Vx(g

−1) on the right endpoint, and

choose a U(1) phase prefactor such that Eq. 6.17 is still satisfied.

Observe the commutation relation,

[S(g1), Vx0(g2)Vx1(g

−12 )U[x0,x1)(g2)

]|ψ〉 = 0 (6.19)

While [S(g1), U[x0,x1)(g2)] = 0 as the onsite representation ui(g) is linear, S(g1) need

not commute with Vx0(g2) and Vx1(g−12 ) individually. Indeed, we may have that when

acting on |ψ〉,

S(g1)Vx0(g2) |ψ〉 = φ∗(g1, g2)Vx0(g2)S(g1) |ψ〉

S(g1)Vx1(g−12 ) |ψ〉 = φ(g1, g2)Vx1(g

−12 )S(g1) |ψ〉

(6.20)

111

which still satisfies Eq. 6.19, where φ(g1, g2) is a U(1) phase. Note that this phase

cannot depend on x0, x1, nor on choice of V s, as we may change each independently

of the others — it is therefore a global property of the bulk.

Now suppose we introduce edges into the system at x = 1 and x = `. As the

ground state need not be unique in the presence of an edge, we no longer require that

S(g) |ψ〉 = |ψ〉 (it may move |ψ〉 around in the ground state manifold). Instead, we

may find local operators V1(g) and V`(g−1) on the edges such that

V1(g)V`(g−1)S(g) |ψ〉 = |ψ〉 (6.21)

for any |ψ〉 in the ground state manifold. Put differently, this means that

S(g) |ψ〉 = V †1 (g)V †` (g−1) |ψ〉 (6.22)

and we may decompose S(g) into separated operations with support at the left and

right edges separately, which act nontrivially only within the ground state manifold.

Repeating our previous analysis with x0 or x1 at an edge, we find that the represen-

tation of the symmetry on the left edge, Ve(g) ≡ V †1 (g), may be projective, and the

class is completely determined by the previously discovered bulk φ,

Ve(g1)Ve(g2) = φ(g1, g2)Ve(g2)Ve(g1) (6.23)

Similarly, operators at the right edge must exhibit the same projective representation.

A non-trivial projective representation requires a degenerate ground space manifold

on which the matrices Ve may act, thus leading to the protected edge modes of non-

trivial SPTs.

Phases with projective representation from different classes cannot be adiabati-

cally transformed into one another via a SLU evolution. While such an evolution may

112

5

We may also simplify this picture by not distinguishingbetween local excitations created at the left/right end-points of U(g), as they can be related to each other by asymmetry operation:

U[x0,x1)(g)S(g1) = U[1,x0)(g1)U[x1,1)(g

1) (19)

Thus, the local excitation created at the right (left) endof U(g) is the same as the excitation created at the left(right) end of U(g1). We may therefore simply substi-tute Vx(g) V L

x (g) on the left endpoint and Vx(g1) onthe right endpoint, and choose a U(1) phase prefactorsuch that Eq. 18 is still satisfied.

Observe the commutation relation,

S(g1), Vx0

(g2)Vx1(g1

2 )U[x0,x1)(g2)| i = 0 (20)

While [S(g1), U[x0,x1)(g2)] = 0 as the onsite representa-tion ui(g) is linear, S(g1) need not commute with Vx0

(g2)and Vx1

(g12 ) individually. Indeed, we may have that

when acting on | i,

S(g1)Vx0(g2) | i = (g1, g2)Vx0

(g2)S(g1) | iS(g1)Vx1

(g12 ) | i = (g1, g2)Vx1

(g12 )S(g1) | i

(21)

which still satisfies Eq. 20, where (g1, g2) is a U(1)phase. Note that this phase cannot depend on x0, x1,nor on choice of V s, as we may change each indepen-dently of the others — it is therefore a global propertyof the bulk.

Now suppose we introduce edges into the system atx = 1 and x = `. As the ground state need not be uniquein the presence of an edge, we no longer require thatS(g) | i = | i (it may move | i around in the groundstate manifold). Instead, we may find local operatorsV1(g) and V`(g

1) on the edges such that

V1(g)V`(g1)S(g) | i = | i (22)

for any | i in the ground state manifold. Put di↵erently,this means that

S(g) | i = V †1 (g)V †

` (g1) | i (23)

and we may decompose S(g) into separated operationswith support at the left and right edges separately, whichact nontrivially only within the ground state manifold.Repeating our previous analysis with x0 or x1 at an edge,we find that the representation of the symmetry on the

left edge, Ve(g) V †1 (g), may be projective, and the class

is completely determined by the previously discoveredbulk ,

Ve(g1)Ve(g2) = (g1, g2)Ve(g2)Ve(g1) (24)

Similarly, operators at the right edge must exhibit thesame projective representation. A non-trivial projectiverepresentation requires a degenerate ground space man-ifold on which the matrices Ve may act, thus leading tothe protected edge modes of non-trivial SPTs.

SLU

LSLU

FIG. 2: A symmetric local unitary (SLU) evolution expressedas a quantum circuit defines phase equivalence in SPTs pro-tected by global symmetries. Each green gate represents asymmetry-respecting unitary evolution. We propose to de-fine a strong equivalence relation for 2D SSPT phases usinglinearly-symmetry local unitary evolution, in which each in-dividual local gate (shown in red) need not be symmetric, butmust be grouped into gates acting along straight lines whichare, as a whole, symmetric.

Phases with projective representation from di↵erentclasses cannot be adiabatically transformed into one an-other via a SLU evolution. While such an evolution maychange V in Eq. (21), it must leave invariant. Mean-while, if two phases are of the same class, then there isno obstacle to connecting the two adiabatically. By thephase equivalence relation given in Sec. III A, distinctSPT phases are in one-to-one correspondence with theprojective representations of G, and can be diagnosed bythe projective representation observed at the edges2,56.

Now consider two states, | 1i and | 2i, characterizedby the projective classes !1 and !2. Consider the 1Dsystem obtained by stacking the two chains on top of eachother, such that | i = | 1i | 2i, with the symmetryacting onsite as ui(g) = ui,1(g) ui,2(g). Following theabove, the projective action of the symmetry at the leftedge is given by Ve(g) = Ve,1(g)Ve,2(g), which is of theclass ! = !1!2. Therefore, under stacking, SPT phasesform a group structure given by the second cohomologygroup H2[G, U(1)].

IV. STRONG EQUIVALENCE OF SSPT PHASES

Having set the stage with a review of 1D SPT phaseequivalence, we now turn to our main topic of interest:strong SSPTs in 2D. Recall that we take such a model tobe defined with respect to an onsite finite abelian symme-try group Gs, which in turn defines the total symmetrygroup generated by the set of Sv

x(g) operators, which actalong vertical columns with fixed x, and the set of Sh

y (g)operators, which act along horizontal rows with fixed y,for g 2 Gs.

Figure 6.2: A symmetric local unitary (SLU) evolution expressed as a quantum circuitdefines phase equivalence in SPTs protected by global symmetries. Each green gaterepresents a symmetry-respecting unitary evolution. We propose to define a strongequivalence relation for 2D SSPT phases using linearly-symmetry local unitary evo-lution, in which each individual local gate (shown in red) need not be symmetric,but must be grouped into gates acting along straight lines which are, as a whole,symmetric.

change V in Eq. (6.20), it must leave φ invariant. Meanwhile, if two phases are of

the same class, then there is no obstacle to connecting the two adiabatically. By the

phase equivalence relation given in Sec. 6.2.1, distinct SPT phases are in one-to-one

correspondence with the projective representations of G, and can be diagnosed by

the projective representation observed at the edges [110, 117].

Now consider two states, |ψ1〉 and |ψ2〉, characterized by the projective classes

ω1 and ω2. Consider the 1D system obtained by stacking the two chains on top

of each other, such that |ψ〉 = |ψ1〉 ⊗ |ψ2〉, with the symmetry acting onsite as

ui(g) = ui,1(g)⊗ ui,2(g). Following the above, the projective action of the symmetry

at the left edge is given by Ve(g) = Ve,1(g) ⊗ Ve,2(g), which is of the class ω = ω1ω2.

Therefore, under stacking, SPT phases form a group structure given by the second

cohomology group H2[G,U(1)].

6.3 Strong equivalence of SSPT phases

Having set the stage with a review of 1D SPT phase equivalence, we now turn to our

main topic of interest: strong SSPTs in 2D. Recall that we take such a model to be

defined with respect to an onsite finite abelian symmetry group Gs, which in turn

113

defines the total symmetry group generated by the set of Svx(g) operators, which act

along vertical columns with fixed x, and the set of Shy (g) operators, which act along

horizontal rows with fixed y, for g ∈ Gs.

6.3.1 Linearly symmetric local unitary transformations

To proceed, we introduce the concept of a linearly symmetric local unitary (LSLU)

evolution, which are a generalization of the previously defined SLU evolution, and

take the form shown in Fig. 6.2 (bottom). Such an evolution may be constructed as

a finite-depth quantum circuit ULSLU ,

ULSLU = U(d)lpwU

(d−1)lpw . . . U

(1)lpw (6.24)

where each Ulpw is a linearly piecewise unitary, taking the form

Ulpw =⊗

i

U(i)ls (6.25)

where U (i)ls are linearly-supported symmetric local unitaries with disjoint support.

By linearly-supported, we mean that the support of Uls may extend indefinitely in

either the x or y direction, but only a small finite range in the other. A single

green rectangle in Fig. 6.2 (bottom) represents one Uls. We also require that they all

commute with all symmetries,

[Uls, Svx(g)] = [Uls, S

hy (g)] = 0 (6.26)

for all x, y, and g ∈ Gs.

114

The only restriction on Uls beyond this is that it must be a local unitary transfor-

mation. For completeness, we may express Uls as a finite depth δ quantum circuit.

Uls = U (δ)pwU

(δ−1)pw . . . U (1)

pw (6.27)

where each Upw is a piecewise local unitary operator, given by

Upw =⊗

i

U(i)loc (6.28)

where U (i)loc are disjoint unitary operators with a finite radius of support. Crucially,

neither any Uloc nor Upw need respect any symmetries — only the final product, Uls,

need respect all subsystem symmetries. The total depth of the circuit is given by dδ,

and must be a constant independent of system size for it to represent an LSLU.

Conceptually, an SLU may be represented as a quantum circuit where each gate

must respect all symmetries. In an LSLU, each gate need not individually respect the

symmetries, but there must be a way of grouping the gates into disjoint operations

acting along vertical or horizontal lines such that the combined action along the line

as a whole respects the symmetries.

The first main result of this chapter is the proposal of the following equivalence

relation: Two SSPTs, with unique ground states |ψ〉 and |ψ′〉, are in the same strong

SSPT equivalence class if there exists a finite-depth LSLU circuit ULSLU connecting

the two, such that |ψ〉 = ULSLU |ψ′〉.

The motivation for this definition comes from the observation that any 1D SPT

may be deformed to a product state via a (non-symmetric) LU evolution. If both

the initial and final states are symmetric, we may take this LU to be, as a whole,

symmetric. Consider a weak SSPT phase consisting of 1D chains aligned horizon-

tally. An LU along the horizontal direction is able to disentangle a 1D chain, while

remaining symmetric as a whole. Such an operation is allowed in an LSLU, and are

115

represented by Uls above. Thus, by allowing an LSLU in our equivalence relation,

we are essentially “modding out” 1D chains. Whatever remains must contain some

fundamentally two-dimensional physics. This is similar in spirit to the definition of

foliated [21, 22, 23, 24, 25] fracton phases, where the equivalence relation for 3D

foliated fracton phases is defined modulo the addition or removal of 2D topological

orders.

An example of an LSLU is shown in Figure 6.7, for the explicit case of the square

lattice cluster model. This LSLU consists of a product of controlled-Z gates which as

a whole commutes with all subsystem symmetries.

The goal now is to show that there indeed exists non-trivial equivalence classes

under this definition, which leads to a classification of such phases.

6.3.2 Bulk Invariants

In this section, we derive the existence of bulk properties (much like the projective

phases φ(g1, g2) in the 1D SPT classification) that are invariant under LSLU trans-

formations. Later, in Section 6.4, we give an explicit example which makes this

construction clear.

First, let us introduce the truncated symmetry operation,

Uy0y1x0x1

(g) =

x1−1∏

x=x0

y1−1∏

y=y0

uxy(g) (6.29)

where x0 < x1, y0 < y1, for any g ∈ Gs, which represents the application of the

symmetry g to a rectangular region of the system. Let us take |x1− x0| and |y1− y0|

to be much larger than any correlation length in the system.

We may think of Uy0y1x0x1

(g) as the application of Svx(g) truncated to [y0, y1), for x ∈

[x0, x1). We may also alternatively think of it as the application of Shy (g) truncated

to [x0, x1), for y ∈ [y0, y1). Therefore, the only place where Uy0y1x0x1

(g) does not look

116

7

V (g)

V (g-1)

V (g-1)

V (g)

U(g)

U(g) U(g1)

U(g1) U(g)

SB-

ST-

SL 6? SR6

?UQ2(g1)

UQ1(g)

FIG. 3: A rectangular truncated symmetry operationUy0y1

x0x1(g) creates local excitations only at its corners. (left)

Corners of di↵erent orientations that can be related to oneanother other by the action of symmetries, as shown here, byapplying S(g) or S(g1) of di↵erent orientations. (top right)The four excitations created by Uy0y1

x0x1(g) may be locally anni-

hilated by operators Vxy(g) or Vxy(g1) at its corners. (lowerright) We show the symmetry operation SQ

xy(g) discussed inthe text.

within a correlation-length sized region near (x, y). EachVxy(g) must form some (possibly projective) representa-tion of Gs, which may depend on our choice of V s. InFigure 3 (left), we show how di↵erent types of cornersmay be related to each other by application of subsys-tem symmetries: we therefore do not need to distinguishbetween them and we may simply define

Vxy(g) V BLxy (g) = V TL

xy (g1) = V TRxy (g) = V BR

xy (g1)(32)

with a choice of overall phase prefactors such that Eq. 31becomes

Vx0y0(g)Vx0y1(g1)Vx1y1(g)Vx1y0(g

1)Uy0y1x0x1

(g) | i = | i(33)

as illustrated in Figure 3 (right). The operators Vxy(g)contain all the information we need about the system.

Let us define the symmetry operations,

SRx (g) =

1Y

x0=x

Svx0(g), SL

x (g) =x1Y

x0=1Sv

x0(g) (34)

STy (g) =

1Y

y0=y

Shy0(g), SB

y (g) =

y1Y

y0=1Sh

y0(g) (35)

where the superscript denotes that we are applying thesymmetry g to all sites to the right, left, top, or bottomof the coordinate x or y. We will mainly use SR and ST ,although we have defined them all for completeness.

We now proceed to prove that the U(1) phase Rxy(g),

given by

Rxy(g) = h | S†R

x (g)V †xy(g)SR

x (g)Vxy(g) | i (36)

is independent of x, y, and therefore cannot be changed byan LSLU evolution. We could have also chosen any ori-entation, R, T , L, or B, which would give the same value.We may therefore simply refer to (g) = R

xy(g). Notethat (g) is nothing but the phase obtained from com-muting SR

xy(g) with Vxy(g), when acting on the groundstate.

The proof consists of three steps: First, we prove thatR

xy(g) is the same for all y, and then that Txy(g) is the

same for all x. Then, we show that for a given (x, y),R

xy(g) = Txy(g). It then follows that (g) = R

xy(g) isindependent of x,y.

The first two steps can be accomplished by lookingat our rectangle operator Uy0y1

x0x1(g) from earlier. Since

the combination in Eq. 33 acts trivially on | i, as does

SR/Tx1 (g), they must commute when acting on | i. Let

us first deal with R. SRx1

(g) only overlaps with V s from

the top right and bottom right corners: Vx1y0(g1) and

Vx1y1(g). Therefore,

SR

x1(g), Vx1y0

(g1)Vx1y1(g)| i = 0 (37)

but we also have that, from our definition of Rxy(g),

SRx1

(g)Vx1y1(g) | i = R

x1y1(g)Vx1y1

(g)SRx1

(g) | iSR

x1(g)Vx1y0

(g1) | i = Rx1y0

(g)Vx1y0(g1)SR

x1(g) | i

(38)

(note that Vxy(g1) = !xy(g, g1)V †xy(g), where

!xy(g, g0) is the factor system of the representationVxy(g)). Therefore, we must have that

Rx1y1

= Rx1y0

(39)

However, we could have chosen y0 or y1 independently.This phase R

xy(g) is therefore independent of y. Simi-

larly, from T , we discover that Txy(g) must be indepen-

dent of x.The final step requires another ingredient. Consider

the symmetry operation

SQxy(g) = SR

x (g)[STy (g)]† = SR

x (g)STy (g1) (40)

which consists of applying g in the bottom right quad-rant, and g1 in the top left quadrant (it acts as identityon the top right quadrant), as shown in Figure 3 (lowerright).

We now show that SQxy(g) commutes with Vxy(g). To

do this, notice that we may split SQxy as

SQxy(g) =

24

1Y

x0=x

y1Y

y0=1u(g)

3524

x1Y

x0=1

1Y

y0=y

u(g1)

35 (41)

UQ2

xy (g)

UQ1xy (g1)

(42)

where UQ1xy (g1) only has support in the top left quad-

rant, and UQ2xy (g) in the bottom right quadrant, about

Figure 6.3: A rectangular truncated symmetry operation Uy0y1x0x1

(g) creates localexcitations only at its corners. (left) Corners of different orientations that can berelated to one another other by the action of symmetries, as shown here, by applyingS(g) or S(g−1) of different orientations. (top right) The four excitations created byUy0y1x0x1

(g) may be locally annihilated by operators Vxy(g) or Vxy(g−1) at its corners.

(lower right) We show the symmetry operation SQxy(g) discussed in the text.

like the application of a symmetry is near its corners, and it may therefore create four

local excitations at each of the corners. As the ground state is unique, short-range

entangled, and symmetric, these local excitations may be locally annihilated via a

unitary operator at each of the corners, such that

V BLx0y0

(g)V TLx0y1

(g)V TRx1y0

(g)V BRx1y1

(g)Uy0y1x0x1

(g) |ψ〉 = |ψ〉 (6.30)

Here, B(T )L(R) indicates the bottom (top) left (right) corner of the rectangle, and

Vxy(g) only has support within a correlation-length sized region near (x, y). In Fig-

ure 6.3 (left), we show how different types of corners may be related to each other by

application of subsystem symmetries: we therefore do not need to distinguish between

117

them and we may simply define

Vxy(g) ≡ V BLxy (g) = V TL

xy (g−1) = V TRxy (g) = V BR

xy (g−1) (6.31)

with a choice of overall phase prefactors such that Eq. 6.30 becomes

Vx0y0(g)Vx0y1(g−1)Vx1y1(g)Vx1y0(g

−1)Uy0y1x0x1

(g) |ψ〉 = |ψ〉 (6.32)

as illustrated in Figure 6.3 (right). The operators Vxy(g) contain all the information

we need about the system. It is also always possible to choose Vxy(g) such that it com-

mutes with all Uy0yx0x

(g′) (for example, by letting Vxy(g) extend the symmetry operator

slightly), in which case Vxy(g) must form some (possibly projective) representation of

Gs, which we will assume for convenience (note that this assumption is not strictly

necessary).

Let us define the symmetry operations,

SRx (g) =∞∏

x′=x

Svx′(g), SLx (g) =x−1∏

x′=−∞Svx′(g)

STy (g) =∞∏

y′=y

Shy′(g), SBy (g) =

y−1∏

y′=−∞Shy′(g)

(6.33)

where the superscript denotes that we are applying the symmetry g to all sites to the

right, left, top, or bottom of the coordinate x or y. We will mainly use SR and ST ,

although we have defined them all for completeness.

We now proceed to prove that the U(1) phase βRxy(g), given by

βRxy(g) = 〈ψ| S†Rx (g)V †xy(g)SRx (g)Vxy(g) |ψ〉 (6.34)

118

is independent of x, y, and therefore cannot be changed by an LSLU evolution. We

could have also chosen any orientation, R, T , L, or B, which would give the same

value. We may therefore simply refer to β(g) = βRxy(g). Note that β(g) is nothing but

the phase obtained from commuting SRxy(g) with Vxy(g), when acting on the ground

state.

The proof consists of three steps: First, we prove that βRxy(g) is the same for all

y, and then that βTxy(g) is the same for all x. Then, we show that for a given (x, y),

βRxy(g) = βTxy(g). It then follows that β(g) = βRxy(g) is independent of x,y.

The first two steps can be accomplished by looking at our rectangle operator

Uy0y1x0x1

(g) from earlier. Since the combination in Eq. 6.32 acts trivially on |ψ〉, as does

SR/Tx1 (g), they must commute when acting on |ψ〉. Let us first deal with R. SRx1(g)

only overlaps with V s from the top right and bottom right corners: Vx1y0(g−1) and

Vx1y1(g). Therefore,[SRx1(g), Vx1y0(g

−1)Vx1y1(g)]|ψ〉 = 0 (6.35)

but we also have that, from our definition of βRxy(g),

SRx1(g)Vx1y1(g) |ψ〉 = βRx1y1(g)Vx1y1(g)SRx1(g) |ψ〉

SRx1(g)Vx1y0(g−1) |ψ〉 = β∗Rx1y0(g)Vx1y0(g

−1)SRx1(g) |ψ〉(6.36)

(note that Vxy(g−1) = ωxy(g, g

−1)V †xy(g), where ωxy(g, g′) is the factor system of the

representation Vxy(g)). Therefore, we must have that

βRx1y1 = βRx1y0 (6.37)

However, we could have chosen y0 or y1 independently. This phase βRxy(g) is therefore

independent of y. Similarly, from T , we discover that βTxy(g) must be independent of

x.

119

The final step requires another ingredient. Consider the symmetry operation

SQxy(g) = SRx (g)[STy (g)]† = SRx (g)STy (g−1) (6.38)

which consists of applying g in the bottom right quadrant, and g−1 in the top left

quadrant (it acts as identity on the top right quadrant), as shown in Figure 6.3 (lower

right).

We now show that SQxy(g) commutes with Vxy(g). To do this, notice that we may

split SQxy as

SQxy(g) =

[ ∞∏

x′=x

y−1∏

y′=−∞u(g)

][x−1∏

x′=−∞

∞∏

y′=y

u(g−1)

]

≡[UQ2xy (g)

] [UQ1xy (g−1)

](6.39)

where UQ1xy (g−1) only has support in the top left quadrant, and UQ2

xy (g) in the bottom

right quadrant, about (x, y). Importantly, they only touch each other at the point

(x, y), as shown in Figure 6.3 (lower right). Then, supposing that

Vxy(g−1)UQ1

xy (g) |ψ〉 = Vxy(g−1)UQ2

xy (g) |ψ〉 = |ψ〉 (6.40)

and using UQ1xy (g) = [UQ1

xy (g−1)]†, we have that

V †xy(g)SQxy(g)Vxy(g) |ψ〉

= V †xy(g)SQxy(g)Vxy(g)Vxy(g−1)UQ1

xy (g) |ψ〉

= ωxy(g, g−1)V †xy(g)SQxy(g)[UQ1

xy (g−1)]† |ψ〉

= ωxy(g, g−1)V †xy(g)UQ2

xy (g) |ψ〉

= Vxy(g−1)UQ2

xy (g) |ψ〉 = |ψ〉

(6.41)

120

where we have used Vxy(g)Vxy(g−1) = ωxy(g, g

−1), and V †xy(g) = ω∗xy(g, g−1)Vxy(g

−1).

We remark that statements such as Eq. 6.40 are dangerous, as they deal with operators

of infinite support acting on |ψ〉 (for which the overall phase factor is not-so-well-

defined). Instead of using these infinite operators, we may instead replace them with

finite rectangular operators with appropriately dressed corners,

UQ1xy (g) = Vx0y(g)Vx0y1(g

−1)Vxy1(g)Uyy1x0x

(g)

UQ2xy (g) = Vxy0(g)Vx1y0(g

−1)Vx1y(g)Uy0yxx1

(g)

(6.42)

for some x0 x x1 and y0 y y1. These satisfy Eq. 6.40 exactly, and

SQxy(g) ≡ UQ1xy (g−1)UQ2

xy (g) acts in the same way as SQxy(g) near Vxy(g). The important

fact is that these operators only touch at (x, y), so the other corners may effectively

be ignored and we arrive at the same result. From this, we conclude that SQxy(g)

commutes with Vxy(g) when acting on |ψ〉.

Finally, since SQxy(g) = SRx (g)[STy (g)]†, we have from the definition of βR/Txy that

βRxy(g)β∗Txy (g) = 1 (6.43)

With all these parts combined, we may conclude that β(g) = βRxy(g) = βTxy(g) does not

depend on x, y, or T/R. Analogous arguments also show that βLxy(g) = βBxy(g) = β(g).

It then follows that β(g) cannot be changed by an LSLU evolution. A local

symmetric unitary cannot transform β(g) throughout the entire system at once, for

the same reason it could not change φ for a 1D SPT. Similarly, a linearly-symmetric

local unitary may make changes to quantities defined along whole lines but cannot

make a global change that would affect β(g).

We remark that such a result does not hold for other similar quantities. For

example, the phase obtained from commuting a single line symmetry, Svx′(g′), with

Vxy(g), may be non-trivial if x′ is near x. This phase is independent of y and therefore

121

cannot be changed by a SLU evolution. However, it can be changed by an LSLU

evolution, which acts along the entire column at once. Also, the phase obtained

from commuting SRx (g′) with Vxy(g), for g′ 6= g, need not be the same as for STy (g′).

This is therefore again only a property of a line, and can be changed by an LSLU

evolution. Only those phases β(g) coming from g′ = g are bulk properties and

therefore conserved under LSLU evolution. Note that this procedure is isomorphic

to observing the charge response of the symmetry SRx (g) to a twist of the symmetry

SLx (g) [119, 36].

A question still remains as to what consistent choices are possible for β(g). This

will lead to a classification of all strong equivalence classes of SSPT phases with onsite

symmetry Gs.

6.3.3 At the edge

At this point, it is convenient to introduce an edge into our system at y = 1 and y = `y.

This allows us to present an alternate view of our findings in the previous section.

We proceed to derive some of the same results, but from a different perspective.

After introducing edges, the ground state manifold becomes massively degenerate,

with degeneracy growing exponentially as exp(O(Ledge)), where Ledge is length of the

edge [11]. Similar to the case of 1D SPTs, a vertical subsystem symmetry may be

decomposed into two operations acting on the top/bottom edge of the system,

Svx(g) |ψ〉 = V topx (g)V bot

x (g) |ψ〉 (6.44)

which operate within the ground state manifold. We focus on the group of vertical

symmetries, an extensively large group Gv = (Gs)Ledge , with a linear representation

generated by Svx(g). In analogy to 1D, the representation of the symmetry group

Gv on the top edge, V topy (g), may be a projective representation. Note that, unlike

122

for 2D SPTs under global symmetries, the symmetries act locally at the edge and do

not give rise to non-trivial 3-cocycles.

Let hgx ∈ Gv be the group element represented by Svx(g), ωtop(h, h′) be the factor

system of V topx (g), and define φtop(h, h′) = ωtop(h, h′)/ωtop(h′, h) the phase obtained

from commuting h, h′. As the Hamiltonian is local, we may assume that ωtop is a

local projective representation, which we define to be one such that φtop(hgx, hg′

x′) = 1

if x and x′ are separated by a distance much larger than the correlation length.

Equivalently, this means they can be brought into a form where ωtop(hgx, hg′

x′) = 1 for

far separated x, x′.

Under LSLU evolution, the class of this projective representation may be changed

“locally”, subject to certain extra constraints. By a “local” change in projective

representation, we mean modifications to ωtop that can be made up of consecutive

single local changes, where a single local change is one in which ωtop → ωtopωloc for

ωloc satisfying

φloc(hgx, h

g′

x′) ≡ωloc(h

gx, h

g′

x′)

ωloc(hg′x′ , h

gx)

= 1

if x /∈ [x0, x1] or x′ /∈ [x0, x1]

(6.45)

for some finite range [x0, x1] on the order of the correlation length. Note that a single

local change is accomplished by a long vertical 1D unitary evolution that only respects

the symmetries as a whole (a Uls from earlier). Any change that can be made up of

consecutive local changes is itself a local change, which can be implemented by an

LSLU evolution. However, as alluded to earlier, there are some extra constraints that

ωtop must satisfy, which arise due to the requirement that the orthogonal horizontal

symmetries Shy (g)must also be respected. Thus, we are interested in the equivalence

class of local projective representations satisfying these constraints, modulo local

changes.

123

The extra constraints may be thought of as the following: V topx (g) must commute

with all Shy (g′), since the overall representation V topx (g)V bot

x (g) must be linear, and

the horizontal symmetries may only overlap with one of them at most. At the same

time, we have the identity

`y∏

y=1

Shy (g′) =∞∏

x=−∞Svx(g′) (6.46)

which implies that [V topx (g),

∞∏

x′=−∞V topx′ (g′)

]|ψ〉 = 0 (6.47)

placing a constraint on possible classes of projective representations ωtop. In terms of

φtop, this implies∞∏

x′=−∞φtop(hgx, h

g′

x′) = 1. (6.48)

We remark that there are no issues with the ∞, as the representation is local and we

may simply restrict the product over x′ to some finite range about x.

All single local changes ωloc, φloc, must also satisfy this constraint. Take φloc to be

non-trivial only within the range [x0, x1], and let x 12

lie within this interval. Then,

observe the phase resulting from commuting hgleft =∏

x<x 12

hgx with hg′

right =∏

x≥x 12

hg′x ,

φloc(hgleft, h

g′

right) =∏

x<x 12

∏

x′≥x 12

φloc(hgx, h

g′

x′) (6.49)

using the fact that φloc must satisfy the same constraints as φtop, multiplying by the

conjugate of Eq. 6.48 we get

φloc(hgleft, h

g′

right) =∏

x0≤x<x 12

∏

x0≤x′<x 12

φ∗loc(hgx, h

g′

x′) (6.50)

124

Since φloc is only non-trivial with [x0, x1], we have explicitly restricted x and x′ to

this interval. In the case where g = g′, φloc(hgx, hgx′) = φ∗loc(h

gx′ , h

gx), and so

φloc(hgleft, h

gright) = 1 (6.51)

Hence, a local modification φtop → φtopφloc cannot changed the value of φtop(hgleft, hgright).

It is possible that this value will be non-trivial in φtop. Consider putting periodic

boundary conditions along the x direction, identifying x = 0 and x = `x, such that the

overall topology is a cylinder. Let x 12

be, say, near `x/2. If we define hgleft and hgright to

be products from 0 to x 12

and from x 12

to `x, respectively, we would similarly find that

φtop(hgleft, hgright) = 1. However, as φtop is local, we may decompose φtop(hgleft, h

gright)

into a contribution coming from near x 12

and coming from the boundaries 0 and `x.

To isolate the contribution coming from x 12, let us define ξ `x to be some length

for which φtop(hgx, hgx′) is trivial if |x− x′| > ξ. Then, redefining

hgleft =∏

(x 12−ξ)≤x<x 1

2

hgx, hgright =∏

x 12≤x<(x 1

2+ξ)

hgx (6.52)

we find that φtop(hgleft, hgright) need not be 1. In fact,

φtop(hgleft, hgtop) = β(g) (6.53)

is exactly our bulk invariant from earlier. This can be seen (similar to in 1D) by

placing a side of the truncated symmetry operator Uy0y1x0x1

(g) along an edge. It then

follows from our previous proof that this phase is independent of where the cut x 12

is made. Furthermore, the phase is insensitive to the orientation of the cylinder and

cut.

125

6.3.4 Classification

Let us now discuss the possible consistent choices for β(g), and in this way classify

all strong equivalence classes of SSPT phases.

In the previous section we reduced the 2D bulk physics down to the 1D prob-

lem of local projective representations along an edge, and finally down to a 0D

problem involving hgleft/right about a single cut in the edge. In this final picture,

we are essentially examining properties of the projective representation ωtop, φtop,

of the group G2s = Gleft

s × Grights . Certain parts of this representation, namely

β(g) = φtop(hgleft, hgright), are universal throughout the system and invariant under

LSLU transformations, and hence define the equivalence class. The different equiv-

alence classes are therefore in one-to-one correspondence with projective representa-

tions of G2s, modulo changes that leave β(g) invariant.

Let us denote by the superscript gL(R) the element g from Gleft(right)s , and ω a factor

system of Glefts × Gright

s . Consider modifying ω → ωω. There are three classes of ω

that leave β(g) = ω(gL, gR)/ω(gR, gL) unchanged. Let ω0 be a factor system for any

projective representation of Gs,

1. We may define ω(gL1 gR2 , g

L3 g

R4 ) = ω0(g1, g3). This trivially leaves β(g) unchanged,

as ω(gL, gR) = ω(gR, gL) = 1.

2. We may also use ω(gL1 gR2 , g

L3 g

R4 ) = ω0(g2, g4).

3. Finally, we may again use ω0 to define

ω(gL1 gR2 , g

L3 g

R4 ) =

ω0(g1g2, g3g4)

ω0(g1, g3)ω0(g2, g4)(6.54)

This is independent of the previous two, satisfies the factor system condition

(Eq. 6.7), and leaves β(g) invariant, as ω(gL, gR) = ω(gR, gL) = ω0(g, g).

126

Therefore, we want the projective representations of G2s, classified as H2[G2

s, U(1)],

modulo these three types of transformations, each of which are classified according to

H2[Gs, U(1)]. This is shown graphically for the explicit example of Gs = Z2 × Z2 in

the next section. We remark that Eq. (6.54) is unambiguous since Eq. (6.2) specifies

an isomorphism between any pair of onsite groups Gs.

This leads us to the second main result of this chapter: The possible choices of

β(g), and therefore distinct strong equivalence classes of SSPT phases, are in one-to-

one correspondence with elements of the group

C[Gs] = H2[G2s, U(1)]/(H2[Gs, U(1)])3. (6.55)

The group structure is induced by a stacking operation. Consider two strong

SSPT phases with onsite symmetry Gs, characterized by β1(g), β2(g), corresponding

to two elements c1, c2 ∈ C[Gs]. Let us stack these two SSPTs, such that the local

Hilbert space at site (x, y) is Hxy = Hxy,1 ⊗ Hxy,2 and the onsite symmetry acts

as uxy(g) = uxy,1(g) ⊗ uxy,1(g). The number of rows or columns, and therefore the

number of symmetries, is unchanged in this process. For the resulting stacked system,

β(g) = β1(g)β2(g), which corresponds to the element c = c1c2 following the group

structure of C[Gs].

We note that there is an alternate (perhaps more intuitive) formulation 1 of the

classification C[Gs]. Let A[G] be the Abelian group of all bilinear functions G×G→

U(1), meaning functions satisfying a(g1g2, g3) = a(g1, g3)a(g2, g3) and a(g1, g2g3) =

a(g1, g2)a(g1, g3), for gi ∈ G. Then, let Aanti[G] be the subgroup of A[G] consisting

of functions a which satisfy a(g, g) = 1 (or, equivalently, a(g1, g2) = a(g2, g1)−1).

Then, the classification is given by C[Gs] = A[Gs]/Aanti[Gs], which one can verify is

equivalent to Eq 6.55.

1This was pointed out by a referee, which we are thankful for.

127

Actually computing C[Gs] for a particular group Gs is straightforward. By the

fundamental theorem of finite abelian groups, a general finite abelian group G may

be written as

G =∏

i

Zni (6.56)

where ni are prime powers, and i = 1, . . . , N for some finite N . The second coho-

mology group for G is obtained by applying the Kunneth formula (for this particular

case, see for example the Appendix of Ref. [121]),

H2[G,U(1)] =∏

i<j

Zgcd(ni,nj) (6.57)

Applying Eq. 6.57 to the group G2 instead, we get

H2[G2, U(1)] =

(∏

i<j

[Zgcd(ni,nj)]4

)(∏

i

Zni

)(6.58)

Finally, we wish to compute

C[G] = H2[G2, U(1)]/(H2[G,U(1)])3 (6.59)

which is easily obtained from Eq. 6.57 and Eq. 6.58,

C[G] =

(∏

i<j

Zgcd(ni,nj)

)(∏

i

Zni

)

=∏

i≤jZgcd(ni,nj)

(6.60)

6.4 Example: 2D cluster model

Our statements thus far have been quite general. Let us now focus on the canonical

example of an SSPT [11], the 2D cluster model on the square lattice.

128

10

Consider modifying ! ! !!. There are three classes of! that leave (g) = !(gL, gR)/!(gR, gL) unchanged. Let!0 be a factor system for any projective representationof Gs,

1. We may define !(gL1 gR

2 , gL3 gR

4 ) = !0(g1, g3). Thistrivially leaves (g) unchanged, as !(gL, gR) =!(gR, gL) = 1.

2. We may also use !(gL1 gR

2 , gL3 gR

4 ) = !0(g2, g4).

3. Finally, we may again use !0 to define

!(gL1 gR

2 , gL3 gR

4 ) =!0(g1g2, g3g4)

!0(g1, g3)!0(g2, g4)(58)

This is independent of the previous two, satisfiesthe factor system condition (Eq. 7), and leaves (g)invariant, as !(gL, gR) = !(gR, gL) = !0(g, g).

Therefore, we want the projective representations of G2s,

classified as H2[G2s, U(1)], modulo these three types of

transformations, each of which are classified according toH2[Gs, U(1)]. This is shown graphically for the explicitexample of Gs = Z2Z2 in the next section. We remarkthat Eq. (58) is unambiguous since Eq. (2) specifies anisomorphism between any pair of onsite groups Gs.

This leads us to the second main result in this paper:The possible choices of (g), and therefore distinct strongequivalence classes of SSPT phases, are in one-to-onecorrespondence with elements of the group

C[Gs] = H2[G2s, U(1)]/(H2[Gs, U(1)])3. (59)

The group structure is induced by a stacking opera-tion. Consider two strong SSPT phases with onsite sym-metry Gs, characterized by 1(g), 2(g), correspondingto two elements c1, c2 2 C[Gs]. Let us stack these twoSSPTs, such that the local Hilbert space at site (x, y)is Hxy = Hxy,1 Hxy,2 and the onsite symmetry actsas uxy(g) = uxy,1(g) uxy,1(g). The number of rowsor columns, and therefore the number of symmetries, isunchanged in this process. For the resulting stacked sys-tem, (g) = 1(g)2(g), which corresponds to the ele-ment c = c1c2 following the group structure of C[Gs].

We note that there is an alternate (perhaps more in-tuitive) formulation66 of the classification C[Gs]. LetA[G] be the Abelian group of all bilinear functionsGG ! U(1), meaning functions satisfying a(g1g2, g3) =a(g1, g3)a(g2, g3) and a(g1, g2g3) = a(g1, g2)a(g1, g3), forgi 2 G. Then, let Aanti[G] be the subgroup of A[G]consisting of functions a which satisfy a(g, g) = 1 (or,equivalently, a(g1, g2) = a(g2, g1)

1). Then, the classi-fication is given by C[Gs] = A[Gs]/Aanti[Gs], which onecan verify is equivalent to Eq 59.

Actually computing C[Gs] for a particular group Gs isstraightforward, and done in Appendix A.

-

FIG. 4: Illustration of the rotated square lattice on whichthe cluster model is originally defined. In order to bring thesymmetries into the form considered here (Eq. 2), we combinetwo qubits into a single site, which we label as a (blue) and b(red), forming the square lattice shown on the right.

V. EXAMPLE: 2D CLUSTER MODEL

Our statements thus far have been quite general. Letus now focus on the canonical example of an SSPT38, the2D cluster model on the square lattice.

While the symmetries of this model act along the di-agonals of the original square lattice, we can easily bringthem into the form considered here by rotating 45 andforming a unit cell of two qubits, which we label by a andb. The Hamiltonian is given by

Hclus = X

xy

X(b)xy Z(a)

xy Z(a)x+1,yZ

(a)x,y+1Z

(a)x+1,y+1 (60)

X

xy

X(a)xy Z(b)

xy Z(b)x1,yZ

(b)x,y1Z

(b)x1,y1 (61)

where X(↵)xy is the X Pauli matrix acting on the ↵ spin on

site (x, y), and similarly for Z. This model is composedof mutually commuting terms, and thus can be solvedexactly.

The onsite symmetry group Gs is

Gs = Z2 Z2 = 1, ga, gb, gagb (62)

where ga, gb, are defined as the two generators of Gs. Thequbit degrees of freedom transform under this symmetryas

uxy(ga) = X(a)xy , uxy(gb) = X(b)

xy (63)

and therefore lead to the generators of our subsystemsymmetries

Svx(g↵) =

1Y

y=1X(↵)

xy , Shy (g↵) =

1Y

x=1X(↵)

xy (64)

for ↵ 2 a, b, which one can readily verify all commutewith Hclus. Each row/column is therefore associated withtwo symmetry generators.

Suppose we take the system on a torus of dimensionsLx Ly, in which case there are 2(Lx + Ly) symmetry

Figure 6.4: Illustration of the rotated square lattice on which the cluster model isoriginally defined. In order to bring the symmetries into the form considered here(Eq. 6.2), we combine two qubits into a single site, which we label as a (blue) and b(red), forming the square lattice shown on the right.

While the symmetries of this model act along the diagonals of the original square

lattice, we can easily bring them into the form considered here by rotating 45 and

forming a unit cell of two qubits, which we label by a and b. The Hamiltonian is

given by

Hclus = −∑

xy

X(b)xy Z

(a)xy Z

(a)x+1,yZ

(a)x,y+1Z

(a)x+1,y+1

−∑

xy

X(a)xy Z

(b)xy Z

(b)x−1,yZ

(b)x,y−1Z

(b)x−1,y−1

(6.61)

where X(α)xy is the X Pauli matrix acting on the α spin on site (x, y), and similarly

for Z. This model is composed of mutually commuting terms, and thus can be solved

exactly.

The onsite symmetry group Gs is

Gs = Z2 × Z2 = 1, ga, gb, gagb (6.62)

129

where ga, gb, are defined as the two generators of Gs. The qubit degrees of freedom

transform under this symmetry as

uxy(ga) = X(a)xy , uxy(gb) = X(b)

xy (6.63)

and therefore lead to the generators of our subsystem symmetries

Svx(gα) =∞∏

y=−∞X(α)xy , S

hy (gα) =

∞∏

x=−∞X(α)xy (6.64)

for α ∈ a, b, which one can readily verify all commute with Hclus. Each row/column

is therefore associated with two symmetry generators.

Suppose we take the system on a torus of dimensions Lx×Ly, in which case there

are 2(Lx + Ly) symmetry generators from Eq. 6.64. However, not all symmetries are

unique, as we haveLx∏

x0=1

Svx0(gα) =

Ly∏

y0=1

Shy0(gα) =∏

xy

X(α)xy (6.65)

for each α ∈ a, b. The total symmetry group is therefore only G = (Z2 ×

Z2)2(Lx+Ly−1).

Let us first probe the nontriviality of this phase in the bulk according to the

procedure in Sec. 6.3.2. Construct the rectangular truncated symmetry operator,

Uy0y1x0x1

(g) =

x1−1∏

x=x0

y1−1∏

y=y0

uxy(g) (6.66)

which creates excitations at the corners. These excitations may be locally annihilated

by operators Vxy(g) at the bottom left and top right corners, and Vxy(g−1) on the

130

remaining two, given by

Vxy(ga) = Z(b)x−1,y−1, Vxy(gb) = Z(a)

xy

Vxy(gagb) = Z(b)x−1,y−1Z

(a)xy

(6.67)

and in the case of Z2, g = g−1. Note that there is some freedom in choosing V , and

we have made a choice in this definition. Calculating the invariants β(g) (which are

independent of our choice of V ) using Eq. 6.34, we find

β(1) = β(ga) = β(gb) = 1, β(gagb) = −1 (6.68)

Since β(gagb) 6= 1, this phase is indeed a non-trivial strong SSPT. Utilizing Eq. 6.55,

the classification of strong SSPTs with this symmetry group is given by

C[Z2 × Z2] = Z2 × Z2 × Z2 . (6.69)

This calculation may be understood graphically as described in Figure 6.5. In this

case, each of β(ga), β(gb), and β(gagb) may be chosen independently, giving rise to a

total of eight possible equivalence classes.

We may also arrive at this conclusion by examining the edge. Consider a top edge

at y = `y. A symmetry Svx(g) acts on the top edge as

V topx (ga) = Z

(b)x−1,`y

Z(b)x,`y

V topx (gb) = Z

(a)x,`y

X(b)x,`y

Z(a)x+1,`y

V topx (gagb) = Z

(b)x−1,`y

Z(a)x,`y

Z(b)x,`y

X(b)x,`y

Z(a)x+1,`y

(6.70)

which forms a projective representation for large total vertical symmetry group Gv =

GLedges . As before, let hgx ∈ Gv be the element represented by V top

x (g). This projective

representation is characterized by the non-commuting pairs of generators, i.e. those

131

11

generators from Eq. 64. However, not all symmetries areunique, as we have

LxY

x0=1

Svx0

(g↵) =

LyY

y0=1

Shy0

(g↵) =Y

xy

X(↵)xy (65)

for each ↵ 2 a, b. The total symmetry group is there-fore only G = (Z2 Z2)

2(Lx+Ly1).Let us first probe the nontriviality of this phase in the

bulk according to the procedure in Sec. IVB. Constructthe rectangular truncated symmetry operator,

Uy0y1x0x1

(g) =

x11Y

x=x0

y11Y

y=y0

uxy(g) (66)

which creates excitations at the corners. These excita-tions may be locally annihilated by operators Vxy(g) atthe bottom left and top right corners, and Vxy(g1) onthe remaining two, given by

Vxy(ga) = Z(b)x1,y1, Vxy(gb) = Z(a)

xy (67)

Vxy(gagb) = Z(b)x1,y1Z

(a)xy (68)

and in the case of Z2, g = g1. Note that there is somefreedom in choosing V , and we have made a choice inthis definition. Calculating the invariants (g) (whichare independent of our choice of V ) using Eq. 36, we find

(1) = (ga) = (gb) = 1, (gagb) = 1 (69)

Since (gagb) 6= 1, this phase is indeed a non-trivialstrong SSPT. Utilizing Eq. 59, the classification of strongSSPTs with this symmetry group is given by

C[Z2 Z2] = Z2 Z2 Z2 . (70)

This calculation may be understood graphically as de-scribed in Figure 5. In this case, each of (ga), (gb),and (gagb) may be chosen independently, giving rise toa total of eight possible equivalence classes.

We may also arrive at this conclusion by examining theedge. Consider a top edge at y = `y. A symmetry Sv

x(g)acts on the top edge as

V topx (ga) = Z

(b)x1,`y

Z(b)x,`y

(71)

V topx (gb) = Z

(a)x,`y

X(b)x,`y

Z(a)x+1,`y

(72)

V topx (gagb) = Z

(b)x1,`y

Z(a)x,`y

Z(b)x,`y

X(b)x,`y

Z(a)x+1,`y

(73)

which forms a projective representation for large total

vertical symmetry group Gv = GLedges . As before, let

hgx 2 Gv be the element represented by V top

x (g). Thisprojective representation is characterized by the non-commuting pairs of generators, i.e. those with the non-trivial tops,

top(hgax , hgb

x ) = top(hgbx , hgb

x+1) = 1 (74)

C[Z2 Z2] = (Z2)6/(Z2 Z2 Z2)

gLa

gLb

gRa

gRb

GLs GR

s

FIG. 5: A graphical representation of the strong classifica-tion of the Gs = Z2 Z2 model. The four generators ofG2

s = Glefts Gright

s are indicated by gLa , gL

b , gRa , gR

b . Follow-ing Eq. 59, we first compute H2[G2

s, U(1)] = (Z2)6, corre-

sponding to a freedom to choose (g, g0) = (g0, g) = ±1for each pair of generators. The LSLU invariant quantitiesare (g) = (gL, gR) between the same element g from GL

s

and from GRs (for any g 2 Gs, not just generators). The

three types of non-trivial changes that leave all (g) invariant(given by near Eq. 58) act on these generator (g, g0) as (1)(gL

a , gLb ) ! (gL

a , gLb ), (2) (gR

a , gRb ) ! (gR

a , gRb ), or (3)

(gLa , gR

b ),(gLb , gR

a ) ! (gLa , gR

b ),(gLb , gR

a ) (illustratedby blue, red, and green). The overall classification is thereforeobtained by modding out three copies of H2[Gs, U(1)] = Z2

from (Z2)6, resulting in C[(Z2)

2] = (Z2)3.

for all x, and the property top(h, h0) = top(h0, h). No-tice that the set of top(h, h0) between all generators ofGv provides a complete description of the projective rep-resentation, as (see Appendix of Ref. 38 for details) theelements of

H2[Zn2 , U(1)] = Z

n2n2

2 (75)

are in one-to-one correspondence with di↵erent choices oftop(h, h0) for the n generators of Zn

2 , where n = 2Ledge.A graphical understanding of the projective represen-

tation may be obtained by representing each generator,hga

x and hgbx , as vertices along a line ordered by x. This is

demonstrated in Fig. 6. Two points h and h0 are con-nected by a link if top(h, h0) = 1. The constraint(Eq. 52) means that each point must be connected bya link to an even number of hga

x , and an even number ofhgb

x . The invariant (ga) is obtained by cutting the linesomewhere, and counting the parity of links between twohga vertices that were cut, and similarly for (gb). Inthis case, both are zero and so (ga) = (gb) = 1. Thefinal invariant, (gagb), is simply the total parity of linkscrossing the cut. In this case, there is one link, and so(gagb) = 1.

As an instructive example, consider the case of Gs =ZN = 1, ga, g2

a, . . . , gN1a . We draw a chain of hga

x or-dered by x. In this case, top(h, h0) may be any Nth rootsof unity. In the graphical representation, we may drawa directed link going from hga

x to hga

x0 , and associate with

Figure 6.5: A graphical representation of the strong classification of the Gs = Z2×Z2

model. The four generators of G2s = Gleft

s × Grights are indicated by gLa , g

Lb , g

Ra , g

Rb .

Following Eq. 6.55, we first compute H2[G2s, U(1)] = (Z2)6, corresponding to a free-

dom to choose φ(g, g′) = φ(g′, g) = ±1 for each pair of generators. The LSLUinvariant quantities are β(g) = φ(gL, gR) between the same element g from GL

s

and from GRs (for any g ∈ Gs, not just generators). The three types of non-

trivial changes that leave all β(g) invariant (given by near Eq. 6.54) act on thesegenerator φ(g, g′) as (1) φ(gLa , g

Lb ) → −φ(gLa , g

Lb ), (2) φ(gRa , g

Rb ) → −φ(gRa , g

Rb ),

or (3) φ(gLa , gRb ), φ(gLb , g

Ra ) → −φ(gLa , g

Rb ),−φ(gLb , g

Ra ) (illustrated by blue, red, and

green). The overall classification is therefore obtained by modding out three copiesof H2[Gs, U(1)] = Z2 from (Z2)6, resulting in C[(Z2)2] = (Z2)3.

with the non-trivial φtops,

φtop(hgax , hgbx ) = φtop(hgbx , h

gbx+1) = −1 (6.71)

for all x, and the property φtop(h, h′) = φtop(h′, h). Notice that the set of φtop(h, h′)

between all generators of Gv provides a complete description of the projective repre-

sentation, as (see Appendix of Ref. [11] for details) the elements of

H2[Zn2 , U(1)] = Zn2−n

22 (6.72)

are in one-to-one correspondence with different choices of φtop(h, h′) for the n gener-

ators of Zn2 , where n = 2Ledge.

132

12

hgax1 h

gbx2

-x

[1, 1, 1]

[1,1, 1]

[1, 1,1]

hgax

hgbx

top(h, h0)= 1

:

:

:

| i

U(1)lpw

U(2)lpw

| 0i[1, 1, 1] (Trivial)

FIG. 6: A graphical representation of a particular state maybe obtained by looking at the projective representation of ver-tical symmetries terminating at the top edge (bottom left).The examples here are for the square lattice cluster model

Gs = Z2 Z2. (top) The generators of Gv = GLedges , hga

x

and hgbx , are denoted by the blue and red markers respec-

tively, and ordered by increasing x. A link is drawn betweengenerators if they have non-trivial commutation relations,top(h, h0) = 1. The graphical representation of three rep-resentative states from distinct non-trivial equivalence classesare shown, labeled by [(ga),(gb),(gagb)]. Note that therepresentative states here have a translation-invariant projec-tive representation — this need not be the case for a generalstate. (bottom right) We show a state, | i, with a non-trivialprojective representation on the edge, that nevertheless be-longs to the weak (trivial) [1, 1, 1] equivalence class, and howit may be transformed into the trivial projective representa-tion via an LSLU composed of two linearly piecewise sym-

metric unitaries U(1)lpw and U

(2)lpw. The graphical representation

shown for U(i)lpw is that of the state U

(i)lpw |+i, where |+i is the

trivial symmetric state (see Sec VI D).

each link a value (hgax , hga

x0 ). The constraint (Eq. 52) de-mands that the total flux going in to a vertex is equal tothe outgoing flux mod N , i.e. the flux flow is divergence-less. The invariant (ga) gives the net flux flow goingalong the length of the line. Indeed, C[ZN ] = ZN , andthere are N total strong equivalence classes correspond-ing to the N values the flux can take. In this picture, itis immediately clear that (ga) cannot be modified viaa local change in the projective representation, as only aglobal action can change the total flux flow.

Going back to the cluster model with Gs = Z2 Z2,we found that there were a total of eight strong equiv-alence classes. What do the states in thes equivalenceclasses look like? We may represent an equivalence classby [(ga),(gb),(gagb)]. The square lattice cluster statehere corresponds to the [1, 1,1] equivalence class. Thegraphical representation for representative states fromthe other equivalence classes, [1, 1, 1] and [1,1, 1], areshown in Figure 6. All these strong equivalence classesmay also be realized by commuting projector Hamiltoni-ans. For instance, the [1, 1, 1] equivalence class is real-

=)

FIG. 7: An explicit example of an LSLU composed of a singleUls =

Qhiji CZij , acting on the square lattice cluster model.

The product is over all green links hiji connecting qubits iand j and CZij is the controlled-Z gate acting on the twoqubits. One can verify that Uls commutes with all subsys-tem symmetries (except at edges) as a whole, but cannot bewritten as a product of individually symmetric local gates(otherwise it would also be an SLU). The resulting state af-ter applying Uls can be thought of as a cluster state on amodified lattice, shown on the right. As a result of Ulpw, twocolumns have e↵ectively been disentangled from the rest ofthe system. However, because the square lattice cluster staterepresents a non-trivial strong SSPT phase, the two subsys-tems on the left and right of Ulpw are still entangled. Above,we show the graphical picture of the projective representation(see Fig 6 for legend). One can verify that before and after,[(ga),(gb),(gagb)] = [1, 1,1] remains invariant.

ized by the Hamiltonian

H[1,1,1] = X

xy

X(b)xy Z(a)

xy Z(a)x+1,yZ

(a)x,y+1Z

(a)x+1,y+1

X

xy

hX(a)

xy Z(b)xy Z

(b)x1,yZ

(b)x,y1Z

(b)x1,y1

Z(a)x+1,y+1Z

(a)x+1,y1Z

(a)x1,y+1Z

(a)x1,y1

i(76)

We show in Figure 6 (bottom right) an example of astate | i in the trivial equivalence class [1, 1, 1] that nev-ertheless realizes a seemingly non-trivial projective repre-sentation at the edge. This projective representation maybe trivialized by two linearly piecewise unitaries Ulpw,which modify the top(h, h0) of the state as shown. Theaction of Ulpw may be thought of as trivializing a stackof non-trivial (Z2)

4 1D SPT chains in the SSPT. Thus

| i and a trivial state | 0i = U(2)lpwU

(1)lpw | i are connected

via an LSLU evolution and therefore belong to the same(trivial) strong equivalence class.

Figure 7 shows an explicit example of an LSLU actingon the square lattice cluster model. The LSLU shownmanages to disentangle two columns of qubits, but sincethe model represents a non-trivial strong SSPT, the twosubsystems on the left and right of these columns are stillentangled.

Finally, a Gs = Zn Zm generalization of the squarelattice cluster model SSPT was introduced in Ref. 38.It was found that such a construction could give rise to

Figure 6.6: A graphical representation of a particular state may be obtained bylooking at the projective representation of vertical symmetries terminating at thetop edge (bottom left). The examples here are for the square lattice cluster model

Gs = Z2 × Z2. (top) The generators of Gv = GLedges , hgax and hgbx , are denoted by

the blue and red markers respectively, and ordered by increasing x. A link is drawnbetween generators if they have non-trivial commutation relations, φtop(h, h′) = −1.The graphical representation of three representative states from distinct non-trivialequivalence classes are shown, labeled by [β(ga), β(gb), β(gagb)]. Note that the rep-resentative states here have a translation-invariant projective representation — thisneed not be the case for a general state. (bottom right) We show a state, |ψ〉, witha non-trivial projective representation on the edge, that nevertheless belongs to theweak (trivial) [1, 1, 1] equivalence class, and how it may be transformed into the trivialprojective representation via an LSLU composed of two linearly piecewise symmetricunitaries U

(1)lpw and U

(2)lpw. The graphical representation shown for U

(i)lpw is that of the

state U(i)lpw |+〉, where |+〉 is the trivial symmetric state (see Sec 6.5.4).

133

12

hgax1 h

gbx2

-x

[1, 1, 1]

[1,1, 1]

[1, 1,1]

hgax

hgbx

top(h, h0)= 1

:

:

:

| i

U(1)lpw

U(2)lpw

| 0i[1, 1, 1] (Trivial)

FIG. 6: A graphical representation of a particular state maybe obtained by looking at the projective representation of ver-tical symmetries terminating at the top edge (bottom left).The examples here are for the square lattice cluster model

Gs = Z2 Z2. (top) The generators of Gv = GLedges , hga

x

and hgbx , are denoted by the blue and red markers respec-

tively, and ordered by increasing x. A link is drawn betweengenerators if they have non-trivial commutation relations,top(h, h0) = 1. The graphical representation of three rep-resentative states from distinct non-trivial equivalence classesare shown, labeled by [(ga),(gb),(gagb)]. Note that therepresentative states here have a translation-invariant projec-tive representation — this need not be the case for a generalstate. (bottom right) We show a state, | i, with a non-trivialprojective representation on the edge, that nevertheless be-longs to the weak (trivial) [1, 1, 1] equivalence class, and howit may be transformed into the trivial projective representa-tion via an LSLU composed of two linearly piecewise sym-

metric unitaries U(1)lpw and U

(2)lpw. The graphical representation

shown for U(i)lpw is that of the state U

(i)lpw |+i, where |+i is the

trivial symmetric state (see Sec VI D).

each link a value (hgax , hga

x0 ). The constraint (Eq. 52) de-mands that the total flux going in to a vertex is equal tothe outgoing flux mod N , i.e. the flux flow is divergence-less. The invariant (ga) gives the net flux flow goingalong the length of the line. Indeed, C[ZN ] = ZN , andthere are N total strong equivalence classes correspond-ing to the N values the flux can take. In this picture, itis immediately clear that (ga) cannot be modified viaa local change in the projective representation, as only aglobal action can change the total flux flow.

Going back to the cluster model with Gs = Z2 Z2,we found that there were a total of eight strong equiv-alence classes. What do the states in thes equivalenceclasses look like? We may represent an equivalence classby [(ga),(gb),(gagb)]. The square lattice cluster statehere corresponds to the [1, 1,1] equivalence class. Thegraphical representation for representative states fromthe other equivalence classes, [1, 1, 1] and [1,1, 1], areshown in Figure 6. All these strong equivalence classesmay also be realized by commuting projector Hamiltoni-ans. For instance, the [1, 1, 1] equivalence class is real-

=)

FIG. 7: An explicit example of an LSLU composed of a singleUls =

Qhiji CZij , acting on the square lattice cluster model.

The product is over all green links hiji connecting qubits iand j and CZij is the controlled-Z gate acting on the twoqubits. One can verify that Uls commutes with all subsys-tem symmetries (except at edges) as a whole, but cannot bewritten as a product of individually symmetric local gates(otherwise it would also be an SLU). The resulting state af-ter applying Uls can be thought of as a cluster state on amodified lattice, shown on the right. As a result of Ulpw, twocolumns have e↵ectively been disentangled from the rest ofthe system. However, because the square lattice cluster staterepresents a non-trivial strong SSPT phase, the two subsys-tems on the left and right of Ulpw are still entangled. Above,we show the graphical picture of the projective representation(see Fig 6 for legend). One can verify that before and after,[(ga),(gb),(gagb)] = [1, 1,1] remains invariant.

ized by the Hamiltonian

H[1,1,1] = X

xy

X(b)xy Z(a)

xy Z(a)x+1,yZ

(a)x,y+1Z

(a)x+1,y+1

X

xy

hX(a)

xy Z(b)xy Z

(b)x1,yZ

(b)x,y1Z

(b)x1,y1

Z(a)x+1,y+1Z

(a)x+1,y1Z

(a)x1,y+1Z

(a)x1,y1

i(76)

We show in Figure 6 (bottom right) an example of astate | i in the trivial equivalence class [1, 1, 1] that nev-ertheless realizes a seemingly non-trivial projective repre-sentation at the edge. This projective representation maybe trivialized by two linearly piecewise unitaries Ulpw,which modify the top(h, h0) of the state as shown. Theaction of Ulpw may be thought of as trivializing a stackof non-trivial (Z2)

4 1D SPT chains in the SSPT. Thus

| i and a trivial state | 0i = U(2)lpwU

(1)lpw | i are connected

via an LSLU evolution and therefore belong to the same(trivial) strong equivalence class.

Figure 7 shows an explicit example of an LSLU actingon the square lattice cluster model. The LSLU shownmanages to disentangle two columns of qubits, but sincethe model represents a non-trivial strong SSPT, the twosubsystems on the left and right of these columns are stillentangled.

Finally, a Gs = Zn Zm generalization of the squarelattice cluster model SSPT was introduced in Ref. 38.It was found that such a construction could give rise to

Figure 6.7: An explicit example of an LSLU composed of a single Uls =∏〈ij〉CZij,

acting on the square lattice cluster model. The product is over all green links 〈ij〉connecting qubits i and j and CZij is the controlled-Z gate acting on the two qubits.One can verify that Uls commutes with all subsystem symmetries (except at edges)as a whole, but cannot be written as a product of individually symmetric local gates(otherwise it would also be an SLU). The resulting state after applying Uls can bethought of as a cluster state on a modified lattice, shown on the right. As a resultof Ulpw, two columns have effectively been disentangled from the rest of the system.However, because the square lattice cluster state represents a non-trivial strong SSPTphase, the two subsystems on the left and right of Ulpw are still entangled. Above,we show the graphical picture of the projective representation (see Fig 6.6 for leg-end). One can verify that before and after, [β(ga), β(gb), β(gagb)] = [1, 1,−1] remainsinvariant.

A graphical understanding of the projective representation may be obtained by

representing each generator, hgax and hgbx , as vertices along a line ordered by x. This is

demonstrated in Fig. 6.6. Two points h and h′ are connected by a link if φtop(h, h′) =

−1. The constraint (Eq. 6.48) means that each point must be connected by a link to

an even number of hgax , and an even number of hgbx . The invariant β(ga) is obtained

by cutting the line somewhere, and counting the parity of links between two hga

vertices that were cut, and similarly for β(gb). In this case, both are zero and so

β(ga) = β(gb) = 1. The final invariant, β(gagb), is simply the total parity of links

crossing the cut. In this case, there is one link, and so β(gagb) = −1.

134

As an instructive example, consider the case of Gs = ZN = 1, ga, g2a, . . . , g

N−1a .

We draw a chain of hgax ordered by x. In this case, φtop(h, h′) may be any Nth roots

of unity. In the graphical representation, we may draw a directed link going from

hgax to hgax′ , and associate with each link a value φ(hgax , hgax′ ). The constraint (Eq. 6.48)

demands that the total flux going in to a vertex is equal to the outgoing flux mod

N , i.e. the flux flow is divergenceless. The invariant β(ga) gives the net flux flow

going along the length of the line. Indeed, C[ZN ] = ZN , and there are N total strong

equivalence classes corresponding to the N values the flux can take. In this picture, it

is immediately clear that β(ga) cannot be modified via a local change in the projective

representation, as only a global action can change the total flux flow.

Going back to the cluster model with Gs = Z2 × Z2, we found that there were a

total of eight strong equivalence classes. What do the states in thes equivalence classes

look like? We may represent an equivalence class by [β(ga), β(gb), β(gagb)]. The square

lattice cluster state here corresponds to the [1, 1,−1] equivalence class. The graphical

representation for representative states from the other equivalence classes, [−1, 1, 1]

and [1,−1, 1], are shown in Figure 6.6. All these strong equivalence classes may

also be realized by commuting projector Hamiltonians. For instance, the [−1, 1, 1]

equivalence class is realized by the Hamiltonian

H[−1,1,1] = −∑

xy

X(b)xy Z

(a)xy Z

(a)x+1,yZ

(a)x,y+1Z

(a)x+1,y+1

−∑

xy

[X(a)xy Z

(b)xy Z

(b)x−1,yZ

(b)x,y−1Z

(b)x−1,y−1

× Z(a)x+1,y+1Z

(a)x+1,y−1Z

(a)x−1,y+1Z

(a)x−1,y−1

]

(6.73)

We show in Figure 6.6 (bottom right) an example of a state |ψ〉 in the trivial

equivalence class [1, 1, 1] that nevertheless realizes a seemingly non-trivial projective

representation at the edge. This projective representation may be trivialized by two

linearly piecewise unitaries Ulpw, which modify the φtop(h, h′) of the state as shown.

135

The action of Ulpw may be thought of as trivializing a stack of non-trivial (Z2)4

1D SPT chains in the SSPT. Thus |ψ〉 and a trivial state |ψ′〉 = U(2)lpwU

(1)lpw |ψ〉 are

connected via an LSLU evolution and therefore belong to the same (trivial) strong

equivalence class.

Figure 6.7 shows an explicit example of an LSLU acting on the square lattice

cluster model. The LSLU shown manages to disentangle two columns of qubits, but

since the model represents a non-trivial strong SSPT, the two subsystems on the left

and right of these columns are still entangled.

Finally, a Gs = Zn × Zm generalization of the square lattice cluster model SSPT

was introduced in Ref. [11]. It was found that such a construction could give rise to

q ≡ gcd(n,m) different phases (if q = 1, the model was always trivial). We now know

that this model may be classified according to

C[Zn × Zm] = Zn × Zm × Zq (6.74)

Each of the q phases constructed in Ref. [11] lie in distinct strong equivalence classes,

and live within the final Zq factor. Thus, there are many more strong SSPT phases

involving the Zn or Zm factor that were missed in the construction of Ref. [11].

6.5 Other Aspects

6.5.1 Additional line-like subsystem

We may also consider systems with additional line-like subsystem symmetries.

For example, consider the cluster model on the triangular lattice. We may redefine

the unit vectors such that the triangular lattice is mapped on to the square lattice

with additional connections going along the x + y direction. This Hamiltonian then

136

takes the form

Htri = −∑

xy

[XxyZx−1,y−1Zx−1,yZx,y−1

×Zx,y+1Zx+1,yZx+1,y+1]

(6.75)

which has the onsite symmetry group Gs = Z2, but now with three directions of

subsystem symmetries: horizontal, vertical, and diagonal. The diagonal symmetries

are given by

Sdq (g) =∞∏

x=−∞ux,x−q(g) (6.76)

where q ∈ Z corresponds to the different diagonals.

We must modify our definition of strong phase equivalence in this case. It is

natural to extend the definition of LSLU to allow for unitaries Uls along the diagonal

x+ y direction. In general, we should allow for Uls to extend along any direction for

which subsystem symmetries exist (in the case of line-like symmetries).

It is convenient to think in terms of edge projective representations. Only symme-

tries going along the same direction may have non-trivial projective representations.

One can define a β(g) from this projective representation, one for each of the three

directions. It can then be shown like before that these three directions are not inde-

pendent (and must be the same), and we are left with the same classification of β(g)

as before. Thus, strong SSPTs with these extra subsystems also have a C[Z2] = Z2

classification, and this model lies in the non-trivial phase.

6.5.2 Adding or removing degrees of freedom

In standard SPT phases protected by global symmetries, we are allowed to add or

remove degrees of freedom. This is necessary to compare SPT phases on different

system sizes. However, we are only allowed to add or remove degrees of freedom that

transform as a linear representation of the symmetry. For example, the edge modes

137

of the AKLT chain protected by time reversal symmetry can be gapped out if we add

a spin-1/2 degree of freedom to the edges.

In the case of SSPTs comparing phases on different system sizes is more subtle, as

the total symmetry group increases with the system size. Consequently, it is neccesary

to consider adding and removing degrees of freedom in several different ways. We may

locally add unentangled degrees of freedom to a site, as long as they transform linearly

under Gs, which does not change anything. However, we may also add an entire row

or column at once, which actually increases the size of the total symmetry group. To

achieve this we may add an unentangled symmetric row of sites, for example, and each

site should transform as a linear representation of Gs. This defines a new horizontal

symmetry acting on the new row, and existing vertical symmetries should be modified

to act on this new row at their intersection. Similarly, we can allow the removal of

entire rows or columns, along with their symmetry, that are unentangled from the

rest of the system. This allows us to meaningfully compare SSPTs on different lattice

sizes, which lie in distinct conventional phases and have different total symmetry

groups. Our strong equivalence relation successfully identifies SSPT models defined

by the same local rule on different system sizes as belonging to the same equivalence

class.

6.5.3 Blocking changes the symmetry structure

With global SPTs, we may block multiple existing sites together to define a new site,

without changing the structure of the symmetries. We note here that the same is not

true for SSPTs. In particular, the choice of what defines a site is important.

For example, consider a model with a single spin on each site and a Gs onsite

symmetry group. Suppose we take every odd row and combine it with the row below

it, and combine each odd column with the one to its left. Each unit cell now contains

four spins. However, each row should now be associated with two types of horizontal

138

symmetries, one which acts on the lower two qubits and another which acts on the

upper two, and similarly for each column. This does not take the form of an SSPT

as defined in Sec. 6.1, even if we allow for a larger onsite symmetry group.

This motivates a more general definition of subsystem symmetries than discussed

previously. For the above example, a natural generalization is to allow a triple of on-

site abelian groups Gboth×Ghorz×Gvert, where Gboth×Ghorz participate in horizontal

symmetries and Gboth × Gvert participate in the vertical symmetries. In the above

example Gboth = Gs is given by the diagonal Gs subgroup of the total onsite (Gs)4

symmetry, of the form (g, g, g, g) (labeling the onsite spins counterclockwise from the

top right), where g ∈ Gs. Similarly, Ghorz is given by another Gs subgroup of the form

(g, g, 1, 1), and Gvert is given by the Gs subgroup of the form (1, g, g, 1). As the LSLU

equivalence relation does not care about choice of unit cell or specific symmetry struc-

ture, such a coarse-graining cannot affect the overall classification which is therefore

still given by C[Gs]. We claim that the classification for this generalized symmetry

structure is given by simply C[Gboth], and is independent of Gvert or Ghorz. Indeed,

the projective representations at an edge involving Ghorz or Gvert are not subject to as

strong constraints as those placed on Gboth, and can always be trivialized via LSLU

(specifically, the constraint in Eq. 6.48 only has to hold for g′ ∈ Gboth × Gvert which

have a non-trivial Gboth component). This means, for example, that models with

subsystem symmetries only along only one direction (e.g. Gboth = Ghorz = Z1, but

Gvert is non-trivial) are always weak SSPTs.

In the above coarse-graining example we could instead chose to preserve only the

Gboth = Gs onsite symmetry group. This achieves a mapping between SSPTs on

lattices of different scales with the same onsite group, but different representations.

This highlights that symmetry-respecting real-space renormalization of SSPTs is a

subtle issue.

139

6.5.4 Equivalence of LSLU and stacking with weak SSPTs

One useful perspective on the effect of allowing LSLUs, as opposed to simply SLUs, is

that equivalence under LSLUs may be thought of as equivalence under a combination

of SLUs and stacking with weak SSPTs. Take the linearly supported symmetric

unitary Uls, which acts upon a horizontal or vertical line (which may encompass

multiple rows or columns), and commutes with all symmetries as a whole. Acting

on the trivial symmetric product state, denoted by |+〉, Uls |+〉 may describe a non-

trivial 1D SPT state running along this line. From the perspective of the symmetry

action at the edge, there is no difference between acting with the unitary on the

state, |ψ〉 → Uls |ψ〉, versus stacking with this 1D SPT state, |ψ〉 → |ψ〉 ⊗ Uls |+〉

(and extending the onsite representation appropriately). Stacking with a disjoint set

of such 1D SPT chains is then identical to a linearly piecewise unitary Ulpw, and

allowing for multiple layers of such stacks captures the effect of an arbitrary LSLU

evolution. One should also allow for local unitaries and isometries that can reduce

the local Hilbert space dimension in this picture. Note that we have assumed here

that any 1D SPT may be be created by a local unitary circuit acting on the product

state — this is true in 1D (but not in higher dimensions where other types of SPTs

exist [122]).

For example, consider the weak phase with a highly non-trivial projective rep-

resentation at the edge, as in Fig. 6.6 (bottom right). The action of U(1)lpw may be

thought of as stacking with a weak SSPT (the fact that it is weak is clear from the

disjointness of the edge projective representation), and similarly U(2)lpw. Stacking these

two phases on top of the initial phase produces one with a trivial linear representation,

which may then be brought to a trivial product state via SLUs (once the horizontal

symmetries have similarly been brought to a linear representation at the edge). For

a strong SSPT phase, this is not possible.

140

This shows that the equivalence relation defined by LSLUs indeed coincides with

the intuitive defition of a strong or weak SSPT. We may define a disjoint SSPT as a

subclass of weak SSPTs, which is one such that the projective representation along

the edge may be separated into those coming from disjoint sets of rows or columns.

This is the intuitive definition of a weak SSPT employed in Ref. [11]. Stacking two

disjoint SSPTs will generally result in a weak (but not necessarily disjoint) SSPT. If

one wishes for disjoint SSPTs to be weak, and for weak SSPTs to be closed under

stacking, then one is lead to precisely the equivalence relation proposed in this paper.

6.5.5 Spurious topological entanglement entropy

Recently, a connection was made [123] between SSPTs and spurious values of topo-

logical entanglement entropy [73, 74] (TEE) found in the bulk of certain short-range

entangled 2D phases [120]. Here we show that strong SSPT phases always lead to

spurious values of TEE.

One of the examples given in Ref. [120] is that of the triangular lattice cluster

state (which we have shown in Sec 6.5.1 belongs to a non-trivial strong SSPT phase

protected by three directions of linear symmetries). It was noted that using the cylin-

der extrapolation method [124] for this state leads to a spurious non-zero value of the

TEE, despite the lack of topological order (Ref. [120] only found spurious contribu-

tions via the cylinder extrapolation method, but they have since also been noted to

occur for SSPTs via the general A,B,C partitioning methods [73, 74] when the bound-

aries of the chosen partitions run along the directions of subsystem symmetry [123]).

In the cylinder extrapolation method, the 2D system is taken on a cylinder of circum-

ference L along (say) the vertical direction, and bipartitioned into a left and right

half as in Fig. 6.8. The TEE γ is obtained from the limit γ = −S(L = 0). Ref. [120]

reduced the calculation of the entanglement entropy of the 2D system down to that

of 1D system going along the cut, but with an extensive bipartitioning. It was found

141

15

x0

FIG. 8: The cylindrical setup used in the cylinder extrap-olation method to calculation the TEE. The system is puton a cylinder, and the entanglement entropy across the cutdividing the system into two halves is calculated. Via uni-tary evolution Uleft(right) with support only on the left (right)half, all the sites far away from the cut may be disentangledinto the trivial product state. The remaining entangled sitesare near the cut and shown as blue/red circles. The calcula-tion of entanglement entropy is therefore reduced to that of a1D system but with an extensive bipartitioning into left/rightpartitions.

where ! is the factor system characterizing the 1D SPTphase. It was shown generally that the 1D system ap-pearing at the cut being non-trivial under the productgroup G1 G2, where G1 acts only on the left and G2

only on the right of the cut, is a sucient condition for anon-zero spurious TEE. It is no coincidence that this isreminiscent of our strong classification, which relied ona particular non-trivial projective representation of theproduct group Gleft

s Grights .

Quite generally, consider an SSPT state | i with onsitesymmetry group Gs, on a cylinder with circumference Ly

along the vertical (y) direction, and infinite extent in thehorizontal (x) direction. Consider calculating the entan-glement entropy S(| i) across a cut at x = x0 dividingthe system into a left and right half. Let us define thesymmetry operations

Sleft(g) =Y

x<x0

Svx(g), Sright(g) =

Y

xx0

Svx(g) (81)

for g 2 Gs, which act on the left/right side of thecut. Since the entanglement entropy is invariant un-der unitary operations localized on either side, S(| i) =S(UleftUright | i), we may simply choose Uleft(right) suchthat the system becomes the trivial product state awayfrom the cut (here Uleft(right) is only supported on theleft(right) half). Let us also enforce that Uleft(right) com-

mute with all Sleft(right)(g) as a whole (note that this doesnot interfere with the ability to disentangle spins awayfrom the cut, as they are being brought to the trivial sym-metric product state). This reduces the calculation downto that of a 1D system going along the cut, as shown inFig. 8, with an extensive bipartitioning. As Uleft(right)

commutes with Sleft(right)(g), this remaining 1D systemis itself a 1D SPT and is symmetric under the symme-try group Gleft

s Grights . The first factor (Gleft

s ) acts onthe left (blue) sites, and the second (Gright

s ) acts on theright (red) sites in Fig. 8. The result in Ref. 55 impliesthat if this 1D SPT is non-trivial under the product group

Glefts Gright

s , then there will be a non-zero spurious con-tribution to the calculated TEE.

By our classification, an SSPT is strong if there exists

(g) =!(gL, gR)

!(gR, gL)6= 1, (82)

where gL and gR correspond to the same group elementg 2 Gs from Gleft

s and Grights , respectively. We therefore

see immediately from Eq. 80, with G1 = Glefts and G2 =

Grights that a non-trivial (g) necessarily implies that the

1D SPT is non-trivial under the product group, whichtherefore implies a non-zero spurious contribution to theTEE. That is,

Strong SSPT =) Spurious TEE (83)

when measured using the cylinder extrapolation method,if the cut lies parallel to a subsystem direction. However,the converse implication is not true: a non-zero spuriousTEE in an SSPT does not imply that the SSPT is strong.A zero TEE implies that the projective representation isdisjoint across the cut x0 (there are no lines crossing thecut in the graphical representation), which implies thatthe SSPT is weak (this is simply the transposition ofEq. 83).

VII. CONCLUSION

We have proposed a natural equivalence relation forstrong SSPT phases, and correspondingly a classificationof a particular class of strong SSPT phases. Phases withsubsystem symmetries di↵er in key ways from those withtraditional global symmetries, necessitating these addi-tional tools. There are various pertinent directions forfuture work.

The 2D models with linear (1D) subsystem symme-tries studied here are the simplest cases of SSPTs. Var-ious other SSPTs exist for which it is not obvious howour construction generalizes. For example, it remains un-clear whether there is a meaningful distinction betweenstrong and weak fractal SSPTs13,26 in 2D. One also has3D SSPTs with linear subsystem symmetries38, which aremore complicated and for which one must again specifythe meaning of a weak SSPT.

Perhaps of more interest are 3D SSPTs protected byplanar (2D) subsystem symmetries, which are mappedto fracton topological orders38 after gauging the symme-tries. An appealing equivalence relation for strong SSPTsof this type would, on the gauge dual, coincide with thefoliated fracton phases36,48–51 (although the connectionis not clear at the present). However, in Ref. 51, thesemionic X-cube was shown to be in the same foliatedfracton phase as the regular X-cube, which therefore im-plies that, on the dual side, the SSPT presented in Ref. 38should be categorized as weak (if the defintions of strongSSPT and foliated fracton phase indeed coincide). Thereare also certain fracton models63,64 for which it is not

Figure 6.8: The cylindrical setup used in the cylinder extrapolation method tocalculation the TEE. The system is put on a cylinder, and the entanglement entropyacross the cut dividing the system into two halves is calculated. Via unitary evolutionUleft(right) with support only on the left (right) half, all the sites far away from the cutmay be disentangled into the trivial product state. The remaining entangled sites arenear the cut and shown as blue/red circles. The calculation of entanglement entropyis therefore reduced to that of a 1D system but with an extensive bipartitioning intoleft/right partitions.

that this 1D system exhibited an additional Z2×Z2 symmetry, and was a non-trivial

1D SPT under the product group. A 1D SPT with symmetry G = G1×G2 is defined

to be non-trivial under the product group if there exists

φ(g, h) ≡ ω(g, h)

ω(h, g)6= 1, g ∈ G1, h ∈ G2 (6.77)

where ω is the factor system characterizing the 1D SPT phase. It was shown generally

that the 1D system appearing at the cut being non-trivial under the product group

G1 × G2, where G1 acts only on the left and G2 only on the right of the cut, is

a sufficient condition for a non-zero spurious TEE. It is no coincidence that this

is reminiscent of our strong classification, which relied on a particular non-trivial

projective representation of the product group Glefts ×Gright

s .

Quite generally, consider an SSPT state |ψ〉 with onsite symmetry group Gs, on a

cylinder with circumference Ly along the vertical (y) direction, and infinite extent in

the horizontal (x) direction. Consider calculating the entanglement entropy S(|ψ〉)

across a cut at x = x0 dividing the system into a left and right half. Let us define

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the symmetry operations

Sleft(g) =∏

x<x0

Svx(g), Sright(g) =∏

x≥x0Svx(g) (6.78)

for g ∈ Gs, which act on the left/right side of the cut. Since the entanglement

entropy is invariant under unitary operations localized on either side, S(|ψ〉) =

S(UleftUright |ψ〉), we may simply choose Uleft(right) such that the system becomes the

trivial product state away from the cut (here Uleft(right) is only supported on the

left(right) half). Let us also enforce that Uleft(right) commute with all Sleft(right)(g) as

a whole (note that this does not interfere with the ability to disentangle spins away

from the cut, as they are being brought to the trivial symmetric product state). This

reduces the calculation down to that of a 1D system going along the cut, as shown in

Fig. 6.8, with an extensive bipartitioning. As Uleft(right) commutes with Sleft(right)(g),

this remaining 1D system is itself a 1D SPT and is symmetric under the symmetry

group Glefts ×Gright

s . The first factor (Glefts ) acts on the left (blue) sites, and the second

(Grights ) acts on the right (red) sites in Fig. 6.8. The result in Ref. [120] implies that

if this 1D SPT is non-trivial under the product group Glefts ×Gright

s , then there will be

a non-zero spurious contribution to the calculated TEE.

By our classification, an SSPT is strong if there exists

β(g) =ω(gL, gR)

ω(gR, gL)6= 1, (6.79)

where gL and gR correspond to the same group element g ∈ Gs from Glefts and Gright

s ,

respectively. We therefore see immediately from Eq. 6.77, with G1 = Glefts and G2 =

Grights that a non-trivial β(g) necessarily implies that the 1D SPT is non-trivial under

the product group, which therefore implies a non-zero spurious contribution to the

TEE. That is,

Strong SSPT =⇒ Spurious TEE (6.80)

143

when measured using the cylinder extrapolation method, if the cut lies parallel to

a subsystem direction. However, the converse implication is not true: a non-zero

spurious TEE in an SSPT does not imply that the SSPT is strong. A zero TEE

implies that the projective representation is disjoint across the cut x0 (there are no

lines crossing the cut in the graphical representation), which implies that the SSPT

is weak (this is simply the transposition of Eq. 6.80).

6.6 Conclusion

In this chapter, we have proposed a natural equivalence relation for strong SSPT

phases, and correspondingly a classification of a particular class of strong SSPT

phases. Phases with subsystem symmetries differ in key ways from those with tra-

ditional global symmetries, necessitating these additional tools. There are various

pertinent directions for future work.

The 2D models with linear (1D) subsystem symmetries studied here are the sim-

plest cases of SSPTs. Various other SSPTs exist, such as those with fractal subsystem

symmetries [29]. These will be discussed along with fractal subsystem symmetries in

Part III. One also has 3D SSPTs with linear subsystem symmetries [11], which are

more complicated and for which one must again specify the meaning of a weak SSPT.

Perhaps of more interest are 3D SSPTs protected by planar (2D) subsystem sym-

metries [20], which are mapped to fracton topological orders after gauging the sym-

metries. An appealing equivalence relation for strong SSPTs of this type would, on

the gauge dual, coincide with the foliated fracton phases [21, 22, 23, 24, 25]. As it will

turn out, this is not the case. Generalizing our definition of strong equivalence 3D

with planar symmetries, by means of a planar-symmetric local unitary transforma-

tion, we find that the foliated fracton phase definition (on the gauged side) is weaker

than the strong phase equivalence (on the ungauged side). Similarly to how strong

144

linear SSPT phases are characterized by non-trivial 2-cocycles between symmetries

from different rows, a strong 3D planar SSPT will be characterized by non-trivial

3-cocycles between symmetries from different planes. This will be the topic of the

next section.

145

Chapter 7

Classifying 3D planar subsystem

SPTs

We will begin this section by a review of the group cohomological classification of

SPT phases in 2D, before discussing 3D planar SSPTs. Recall that 3D planar SSPTs

are dual to type-I fracton topological order, as we have seen with the plaquette Ising

model and X-Cube. We will then generalize the classification of strong 2D linear

SSPTs from the previous section to 3D planar SSPTs with Abelian fracton duals.

This chapter is based on the paper

[27] T. Devakul, W. Shirley, J. Wang, “Strong planar subsystem symmetry-

protected topological phases and their dual fracton orders”, Phys. Rev.

Research 2, 012059 (2020).

7.1 Review of 2D SPTs

We begin this section with a review the group cohomological classification of SPTs

in 2D, as well as some additional aspects which will prove useful for our arguments

related to the SSPT. These include the interpretation of SPT phases as an anomalous

146

action of the symmetries on the edge, and the connection to the braiding and exchange

statistics of quasiparticle excitations in the dual gauge theories.

7.1.1 Group cohomological classification of 2D SPTs

In the presence of symmetry, the unique ground states of two gapped Hamiltonians

belong to the the same phase if they can be transformed into each other via a sym-

metric local unitary (SLU) transformation.[7] That is, a finite depth quantum circuit

in which each gate commutes with the symmetry operation. A state describes a non-

trivial 2D SPT phase if it cannot be connected to the trivial product state via an

SLU, but can be trivialized if the symmetry restriction is removed. Two dimensional

bosonic SPTs with on-site symmetry G, under this phase equivalence relation, are

known [2] to be classified according to the third cohomology group H3[G,U(1)]. For

the finite abelian group G =∏

i ZNi , this can be written out explicitly as

H3[G,U(1)] =∏

i

ZNi∏

i<j

Zgcd(Ni,Nj)

∏

i<j<k

Zgcd(Ni,Nj ,Nk) (7.1)

where gcd denotes the greatest common denominator. The three factors are com-

monly referred to as type-I, type-II, and type-III cocycles. Type-III cocycles cor-

respond to a gauge dual with non-abelian quasiparticle excitations; as our focus is

on SSPTs with abelian fracton duals, we will discuss only on type-I and II cocycles

(the duals of 2D SPTs with non-trivial Type-III cocycles correspond to non-Abelian

topological orders).

The Else-Nayak procedure

Let us derive the group cohomological classification via a series of dimensional re-

duction procedures, introduced by Else and Nayak. [117] which will prove useful in

our discussion of SSPTs. Although the original procedure observed a system with a

147

physical edge, here we prefer to deal with a “virtual” edge, meaning: the full system

has no edges, but we will consider applying the symmetry only to a finite region M of

the system. At the edges of M , this symmetry will act non-trivially as if at a physical

edge. The advantage of this approach is that it removes any ambiguity related to

choice of how the model is defined at the physical edges (and will be useful in the

case of SSPTs).

Let |ψ〉 be the unique gapped ground state of our Hamiltonian H with on-site

symmetry group G, and S(g) be the symmetry operation realizing the symmetry

element g ∈ G. We have that [H,S(g)] = 0 and, without loss of generality, take the

ground state to be uncharged under the symmetry S(g) |ψ〉 = |ψ〉. Now, let SM(g)

be the symmetry operation S(g), but restricted to a region M . SM(g) acting on the

ground state will no longer leave it invariant, but will create some excitation along

the boundary of this region, ∂M . Since |ψ〉 is the unique ground state of a gapped

Hamiltonian, this excitation may always be locally annihilated by some symmetric

unitary transformation U∂M(g)†, which only has support near ∂M . That is,

SM(g) |ψ〉 = U∂M(g) |ψ〉 (7.2)

It is straightforward to show that the matrices U∂M(g) form a twisted representation

of G, satisfying

SM (g2)U∂M(g1)U∂M(g2) |ψ〉 = U∂M(g1g2) |ψ〉 (7.3)

where BA ≡ BAB† denotes conjugation of A by B, and that they must commute

with any global symmetry operation, [U∂M(g), S(g′)] = 0.

We now perform a further restriction: from ∂M down to a segment C, UC(g).

This is always possible. UC(g) need only satisfy Eq 7.3 up to some unitary operator

148

V∂C(g1, g2) at the two endpoints of C,

SM (g2)UC(g1)UC(g2) |ψ〉 = V∂C(g1, g2)UC(g1g2) |ψ〉 (7.4)

By associativity, V∂C must satisfy

SM (g3)V ∂C(g1, g2)V∂C(g1g2, g3) =

SM (g2g3)UC(g1)V ∂C(g2, g3)V∂C(g1, g2g3)

(7.5)

when acting on |ψ〉. The final restriction is from ∂C, which consists of two disjoint

regions a and b, down to simply a: V∂C(g) = Va(g)Vb(g) → Va(g). Va(g) need

only satisfy Eq. 7.5 up to a U(1) phase factor, which can be cancelled out by the

contribution from Vb(g).

SM (g3)V a(g1, g2)Va(g1g2, g3) =

ω(g1, g2, g3)SM (g2g3)UC(g1)V a(g2, g3)Va(g1, g2g3)

(7.6)

where ω : G3 → U(1). This entire dimensional reduction process is shown in Fig-

ure 7.1.

One can further show that ω(g1, g2, g3) satisfies the 3-cocycle condition [117]

1 =ω(g1, g2, g3)ω(g1, g2g3, g4)ω(g2, g3, g4)

ω(g1g2, g3, g4)ω(g1, g2, g3g4)(7.7)

and since Va(g1, g2) is only defined up to a phase factor β(g1, g2), we must identify

ω(g1, g2, g3) ∼ b(g1, g2, g3)ω(g1, g2, g3) (7.8)

where

b(g1, g2, g3) =β(g1, g2)β(g1g2, g3)

β(g2, g3)β(g1, g2g3)(7.9)

149

is called a coboundary. The classification of functions satisfying Eq. 7.7, modulo

transformations Eq. 7.8, is exactly the definition of the third cohomology group

H3[G,U(1)]. The class of ω is the element of H3[G,U(1)] to which it corresponds.

Invariant combinations in H3

Suppose we have followed the Else-Nayak procedure on a system and obtained the

cocycle function ω(g1, g2, g3). How do we identify which class in Eq. 7.1 it belongs

to? One way to do so is to identify combinations of ω which are invariant under the

transformation Eq. 7.8, whose value can tell us about the class.

For simplicity, we focus first on G = (ZN)M . Let us first write down an explicit

form [125, 126] for ω,

ω(g1, g2, g3) = exp

∑

i≤j

2πipij

N2gi1(gj2 + gj3 − [gj2 + gj3])

(7.10)

where gi is an integer modulo N denoting the component of g in the ith ZN factor,

g = (g1, g2, . . . , gM), [·] denotes the interior modulo N , and pij are integers mod N . It

is straightforward to confirm that ω satisfies the 3-cocycle condition. As we will show,

the different choices of pij for i ≤ j correspond to different classes in H3[G,U(1)].

From Eq. 7.1, piI ≡ pii specify the value of the type-I cocycles and pijII ≡ pij specify

the type-II cocycles.

Define

Ω(g) =N∏

n=1

ω(g, gn, g) (7.11)

and

ΩII(g, h) =Ω(gh)

Ω(h)Ω(h)(7.12)

both of which one can readily verify are invariant under transformations of the type

Eq. 7.8. Given a choice of generators, G = 〈a1, . . . , aM〉, an explicit calculation shows

150

that

Ω(ai) = e2πiNpiI (7.13)

and

ΩII(ai, aj) ≡Ω(aiaj)

Ω(ai)Ω(aj)= e

2πiNpijII (7.14)

thus correctly identifying the value of the type-I and type-II cocycles. Thus, if we are

given an unknown ω, we may simply compute Ω(ai) and ΩII(ai, aj) for all i and j to

identify its class.

We may define the symmetric matrix Mij = pijII and Mii = 2piI . Then, we have

Ω(g) = eπiN~gTM~g (7.15)

and

ΩII(g, h) = e2πiN~gTM~h (7.16)

for arbitrary elements g and h, where ~g = (g1, . . . , gM).

Group cohomology models

The group cohomology models are a powerful construction that allows us to explic-

itly write down models realizing SPT phases corresponding to an arbitrary cocy-

cle [125, 126]. Although these models have an elegant interpretation in terms of a

path integral on arbitrary triangulations of space-time, we will use them to simply

define Hamiltonian models on a square lattice.

We first define the homogenous cocycle ν : G4 → U(1),

ν(g1, g2, g3, g4) = ω(g−11 g2, g

−12 g3, g

−13 g4) (7.17)

151

which satisfies ν(gg1, gg2, gg3, gg4) = ν(g1, g2, g3, g4). In terms of ν, the cocycle con-

dition (Eq. 7.7) is

1 =ν(g1, g2, g3, g4)ν(g1, g2, g4, g5)ν(g2, g3, g4, g5)

ν(g1, g2, g3, g5)ν(g1, g3, g4, g5)(7.18)

We will use ν to define our ground state wavefunction.

Take G-valued degrees of freedom on each site r, |gr〉. The ground state of our

model |ψ〉 is an equal amplitude sum of all possible configurations

|ψ〉 =∑

grf(gr) |gr〉 (7.19)

where f(gr) is a U(1) phase for each configuration. The group cohomology model

is defined by the choice

f(gr) =∏

r

ν(gr, gr+x, gr+x+y, g∗)

ν(gr, gr+y, gr+x+y, g∗)≡∏

r

fr(gr) (7.20)

where x,y are the two unit vectors, g∗ ∈ G is an arbitrary element which we can

simply take to be the identity g∗ = 1, and we have defined a phase contribution fr

for each plaquette. This arises from a triangulation of each square plaquette into

two triangles, each of which contribute a phase; those interested in the details of the

construction are directed to Ref [125].

Performing the Else-Nayak procedure outlined in Section 7.1.1 on this ground

state results in exactly the cocycle ω used to construct the state, up to a coboundary

(Eq. 7.8).

To obtain a gapped local Hamiltonian realizing this state as its ground state, we

simply consider a set of local ergodic transitions 〈gr → g′r〉, multiplied by an

152

Figure 7.1: The dimensional reduction procedure in the Else-Nayak procedure. Westart with a truncated global symmetry operator, SM(g). This acts on the groundstate as a unitary U∂M(g) along the edge of M . We further restrict this unitarydown to a line segment C, UC(g). Restricted to C, UC(g) behaves as a representationof G only up to unitaries V∂C(g) at its endpoints. Finally, we restrict to a singleendpoint Va(g), where associativity of the representation is only satisfied up to aphase ω(g1, g2, g3), defining our 3-cocycle.

appropriate phase factor,

H = −∑

〈g′r→gr〉

f(g′r)f(gr)

|g′r〉 〈gr| (7.21)

which by construction has |ψ〉 as its unique ground state. We can simply choose

g′r to differ from gr by the action of a generator ai of G on a single site r. The

Hamiltonian will then be a sum of mutually commuting terms consisting of a “flip”

operator |aigr〉 〈gr| on each site, multiplied by an appropriate phase factor depending

on the state gr near that site.

153

Gauge duality

The group cohomological classification of an SPT has an elegant interpretation in

terms of braiding statistics of its gauge dual. [119] We will briefly outline the gaug-

ing process (as applied to the group cohomology models), and discuss the relevant

statistical processes.

Consider the group cohomology SPT model on a square lattice given by Eq. 7.21.

To gauge the global symmetry, we define gauge degrees of freedom gr,r′ = g−1r′,r for

each nearest neighbor pair (r, r′), and enforce a Gauss’s law constraint at each vertex

r which involves the matter degree of freedom gr and the adjacent gauge degrees of

freedom gr,r′ . Then, we minimally couple the symmetric Hamiltonian to the gauge

degrees of freedom by replacing the operators gr′g−1r with gr′gr,r′g

−1r throughout. In

addition, we energetically enforce the zero-flux constraint gr1r2gr2r3gr3r4gr4r1 = 1 for

the square plaquette with corners r1...4 (labeled going clockwise or counterclockwise),

by adding an appropriate projection term to the Hamiltonian.

The resulting model describes a topologically ordered phase, with characteristic

properties such as a topological ground state degeneracy on a torus and quasiparticle

excitations with anyonic braiding statistics. There are two types of excitations: gauge

charge, denoted by eg, and gauge flux, denoted by mg, for each g ∈ G. The former

are created by gauged versions of operators of the form

Z†g(r1)Zg(r2) =∑

gre

2πiN

(gir2−gir1

) |gr〉 〈gr| (7.22)

which creates a charge-anticharge pair, eg and e−1g , at positions r2 and r1. To create

gauge flux excitations, instead consider the gauged version of the operator

L(g) |ψ〉 ≡ U †∂M(g)SM(g) |ψ〉 = |ψ〉 (7.23)

154

where SM(g) is a symmetry operator restricted to a region M and U∂M(g) is the action

on the boundary ∂M , as in the dimensional reduction procedure of Section 7.1.1. The

gauged version of SM(g) only flips gr,r′ near at the boundary, and so the gauged L(g)

operator has support only on ∂M . Now, if we further restrict L(g) → LC(g) to an

open segment C, LC(g) creates two quasiparticle excitations at the two endpoints,

which we identify as the gauge flux-antiflux pair mg and m−1g . Note that there is

an ambiguity in defining the gauged version of L(g), which may result in a different

definition of the gauge flux excitation, mg ∼ mgeg′ . Thus, gauge fluxes are only well

defined modulo attachment of charges.

The group cohomological classification of the ungauged SPT manifests in the self

and mutual statistics of gauge fluxes in the gauged theory. Let ai be the generator of

the ith factor of ZN in G, and ei and mi be its gauge charge and flux excitations. For

two identical excitations, we can define an exchange phase via a process in which their

two positions are exchanged. For two different excitations, we may instead define the

full braiding phase, which is accumulated when one particle encircles another. In the

gauge theory, ei all have trivial exchange and only braid non-trivially with its own

gauge flux mi. Meanwhile, the gauge flux mi has an exchange statistic e2πipiIN2 with

itself, and a mutual braid e2πip

ijII

N2 with mj. Notice that the exchange and mutual braid

of mi is only well defined modulo e2πiN , since mi is only well defined modulo charge

attachment. For a general gauge flux mg, its exchange phase is given by an Nth root

of Ω(g), which can be straightforwardly calculated from the M matrix (Eq. 7.15).

In the K matrix formulation of Abelian topological orders (Sec 1.1.3), the K

matrix characterizing the gauged theory is given by

K =

−M N1

N1 0

(7.24)

155

such that

K−1 =

0 1N

1

1N

1 1N2 M

(7.25)

where the indices represent quasiparticle excitations ordered as e1, e2, . . . ,m1,m2, . . . .

The exchange statistic of a quasiparticle ~l written in this basis is given by eπi~lTK−1~l,

and the mutual braiding statistic between ~l1 and ~l2 is e2πi~lT1 K−1~l2 .

7.2 3D Planar SSPTs

The brief history of 3D planar SSPT phases begins with Ref. [20], which constructed

a non-trivial 3D planar SSPT model. However, it was later discovered that its fracton

dual belonged to the same foliated fracton phase as the X-cube model [24], imply-

ing that it is weak. More recently, fracton phases were constructed in Ref. [26]

which possess ‘twisted’ foliated fracton orders, raising the question as to the nature

of their SSPT duals. We will find that these phases, too, are weak. This prompts

the question: do any strong planar SSPTs exist? The answer to this question is

yes, and we explicitly construct strong SSPT phases. Such strong phases are dual to

novel strong fracton phases with unusual braiding statistics that cannot be obtained

by coupling 2D theories. In a sense these statistical interactions are “intrinsically”

three-dimensional.

We will first show how to construct weak 3D planar SSPT phases via a stacking

process of 2D SPTs. We then ask whether there are SSPT phases which cannot

be realized by this process. We identify mechanisms by which an SSPT may be

strong, leading to a classification of such phases, and construct exactly solvable, zero-

correlation length models realizing these phases. In the fracton dual picture, this

construction corresponds to one in which 2D topological orders are stacked on to and

strongly coupled to an existing fracton model [26]. The duals of our strong SSPTs are

156

novel fracton phases which cannot be attained via such a procedure, also implying

that they cannot be realized by a p-string condensation transition [28, 93], as we will

show.

7.2.1 Planar subsystem symmetries

Throughout we will consider a system with degrees of freedom on each site of a cubic

lattice. Each site r transforms under the finite abelian on-site symmetry group G

under a unitary representation ur(g), where g ∈ G. An xy planar symmetry acting

on plane z acts as Sxy(z; g) =∏

x,y ur=(x,y,z)(g) for g ∈ G. Similarly, we may define

Syz(x; g) and Szx(y; g), which act on yz and zx planes respectively. Importantly,

individual sites transform under the same on-site representation regardless of the

orientation of the planar symmetry — there is therefore a redundancy: the product

of all xy symmetries is identical to the product of all yz or all zx symmetries. We will

refer to models which respect only one orientation of planar symmetry as 1-foliated,

those with two as 2-foliated, and those with all three as 3-foliated. To construct

explicit models, we choose the on-site degrees of freedom to be G-valued, |gr〉, which

transform under the on-site symmetry as ur(g) |gr〉 = |ggr〉.

7.2.2 Construction of weak SSPT phases

It is possible to construct non-trivial SSPT phases from known 2D global SPTs, as

we will show in this section. Phases obtained in this way are ‘weak’, by definition,

whose nontrivial properties are in some sense a manifestation of lower-dimensional

physics. We emphasize here that we do not assume any translation invariance in our

system. Hence, our definition is different (but similar in spirit) to weak crystalline

SPTs with global symmetries, which are stacks of lower dimensional SPTs protected

by translation symmetry.

157

As reviewed, the classification of such phases [2] is given by the third cohomology

group H3[G,U(1)]. For simplicity, we may consider G = (ZN)n, in which case an

element of H3[G,U(1)] is specified by integers, piI (i ∈ [1, n]), pijII (i < j), and pijkIII

(i < j < k), all modulo N , called type-I, II, and III cocycles respectively. We

will specify piI and pijII compactly in a single symmetric n × n integer matrix M

with Mii = 2piI and Mij = Mji = pijII . Upon gauging the global symmetries of a

2D SPT, one obtains a topologically ordered system with fractional quasiparticles

carrying gauge charge or flux (or both). Nontrivial type-III cocycles give rise to non-

abelian topological order, [127, 125] which we will not consider here. A generating set

of quasiparticles are the “electric” excitations (gauge charges) ei and “magnetic”

excitations (gauge fluxes) mi. Each ei has a e2πi/N braiding statistic with mi and

trivial statistics with all other generators. The elements of M characterize the self and

mutual statistics of gauge flux excitations. In particular, the type-I cocycles give rise

to a self exchange statistic eπiMii/N2

of the gauge flux mi, and type-II cocycles lead to

a mutual braiding statistic of e2πiMij/N2

between mi and mj. Note that these phases

are only well defined modulo e2πi/N , since flux is only well defined up to attachment

of charge, mi → miej. Finally, we note that abelian topological orders in 2D can all

be described by K matrix Chern-Simons theories. [39, 128] The topological orders we

have discussed have a 2n× 2n K matrix description with

K = N

− 1N

M 1

1 0

, K−1 =

1

N

0 1

1 1N

M

(7.26)

where the indices labeling quasiparticles are ordered as e1, . . . , en,m1, . . . ,mn.

Quasiparticles are described by an integer vector ` in this basis, and have self-

exchange statistic eπi`T ·K−1·` and mutual braiding statistics e2πi`T1 ·K−1·`2 .

It is always possible to view a 3D planar SSPT as a quasi-2D system in the xy plane

with a subextensively large symmetry group GL by compactifying the z direction.

158

Figure 7.2: (Left) Examples of our construction of 1-foliated or weak 2 or 3-foliatedmodels, for G = ZN × ZN , in the graphical notation. 2D SPTs to be stacked, areshown in the blue boxes, and the large arrow points to the resulting SSPT afterstacking. The color of the edges connecting two vertices indicate its weight moduloN . (Right) Examples of M matrices that cannot be obtained by stacking 2D phasesonto 2 or 3-foliated models. The Type 1 phase is only strong for even N , and Type2 strong phases can only be realized for 2-foliated symmetries.

We may then proceed to compute its classification in terms of H3[GL, U(1)], which

is characterized by a subextensively large M matrix. We note that it is possible to

define M matrices corresponding to yz or zx as well, but for reasons that will become

clear we will always consider the xy symmetries only. It is useful to introduce a

graphical notation for M, which is used in Fig. 7.2. The αth generator of G in a

plane z is denoted by a vertex ai=(α,z). Two vertices i and j are connected by an

undirected edge with weight Mij, and a vertex i is connected to itself via a self-loop

with weight Mii/2, where weights are defined modulo N .

Consider the 2D global symmetry group G2D = GK for an integer K. For

appropriate choice of the pure phase function f2D, the wavefunction |ψ〉2D =∑gr f2D(gr) |gr〉2D on a 2D square lattice is a zero-correlation length ground

state of a commuting Hamiltonian with SPT order (see the group cohomology models

in Sec 7.1.1). All phases in the group cohomology classification can be realized in

this way [125, 126, 129].

159

Suppose we start with the trivial disordered wavefunction |ψ0〉 =∑gr |gr〉 on

the 3D cubic lattice. We can construct a nontrivial 1-foliated SSPT by identifying each

factor of G in G2D in the function f2D(gr) with a planar G symmetry in an arbitrary

collection of planes z1, . . . , zK (where zk are all within some finite range to ensure local-

ity). The wavefunction |ψ〉1-fol = U |ψ0〉 with U =∑gr f2D(grrz∈zk) |gr〉 〈gr|

is the ground state of a 1-foliated 3D SSPT, which is nontrivial only near the planes

zk. We may then repeat this procedure arbitrarily many times, each time acting on

the previous state with U for different choices of f2D and zk. We will call this

procedure “stacking” the 2D SPT |ψ〉2D onto the planes zk of a 1-foliated SSPT.

More generally, we may define a stacking operation between two SSPTs in which

the two systems, with on-site symmetry representations u(1)r (g) and u

(2)r (g), are placed

on top of each other to create a new SSPT with on-site representation ur(g) = u(1)r (g)⊗

u(2)r (g). The group structure of the standard SPT classification is realized under such

a stacking operation. Stacking a 2D SPT onto a 3D SSPT can be viewed as stacking

two 3D SSPTs, in which the first is only nontrivial in the vicinity of a number of

planes zk. We define any phase realizable by stacking 2D SPTs in this way to be

weak. In the case of our 1-foliated SSPT construction, each additional stacked 2D

SPT simply adds to the corresponding elements of M, shown graphically in Fig. 7.2.

For 1-foliated symmetries, it is thus possible to realize any M by stacking 2D SPTs;

hence all phases are weak.

On the other hand, for 2- or 3-foliated models, this procedure may not work be-

cause |ψ〉1-fol is not guaranteed to be symmetric under the orthogonal planar symme-

tries (if it is, we can simply follow the same procedure). Instead, let us define variables

dr = gr+zg−1r , which transform under xy planar symmetries but are invariant under

all orthogonal symmetries. We may then define non-trivial SSPT wavefunctions as

160

before, but in terms of dr instead using the unitary

U =∑

grf2D(drrz∈zk) |gr〉 〈gr| , (7.27)

which is explicitly invariant under the orthogonal symmetries. However, in this case

the M matrix of the 2D SPT does not map directly onto that of the SSPT — instead

one should view the 2D SPT as living “in between” the planes of the SSPT, at

zk + 1/2. To obtain the M matrix of the SSPT, one can compute the appropriate

type-I and II cocycles of the 2D SPT in the basis of the xy planar symmetries. This

simply involves a change of basis.The process is shown in Fig. 7.2. As will be discussed

in the next section, unlike for 1-foliated symmetries, there are now allowable phases

which cannot be realized by stacking any number of 2D SPT.

Note that in this discussion we have implicitly ignored nontrivial SSPTs that

have trivial M matrices. Such phases do exist, however, we conjecture that all such

phases are weak (they can be realized by stacking 2D linear SSPTs [19]) and therefore

irrelevant in the classification of strong phases.

7.2.3 General constraints and invariants

In the presence of orthogonal symmetries, there are general constraints that must be

satisfied by M. Conceptually, these arise due to the aforementioned redundancy: the

global symmetry Sglob(g) =∏

z S(xy)(z; g) =

∏x S

(yz)(x; g). Since yz symmetries do

not contribute to M, the generator Sglob(g) must therefore manifest trivially in M.

This leads to two types of constraints on the elements of M: the global symmetry

must have trivial type-II cocycle with any other symmetry and trivial type-I cocycle

with itself. We will prove that these constraints must hold generally in the next

section. Let us label the αth generator of G on the zth plane by i = (α, z). Then,

161

the two constraints are expressed as

∑

z′

M(α,z),(β,z′) ≡ 0 mod N, ∀α, z, β (7.28)

and

1

2

∑

z,z′

M(α,z),(α,z′) ≡ 0 mod N, ∀α (7.29)

These constraints define a restricted subgroup of H3[GL, U(1)] in which 2- or 3-

foliated SSPTs must reside. As we will show, there are now allowed phases which

cannot be realized by stacking any number of 2D SPTs — these are precisely the

strong phases we are searching for. This motivates us to define two types of strong

invariants, F1 and F2, which cannot be changed by stacking with 2D SPTs.

Strong SSPTs: Type 1 Consider G = Z2N . Then Mzz′ is an L× L matrix. Pick

an arbitrary cut that divides the system into two halves z < z0 and z ≥ z0. Then,

F1 ≡∑

z<z0

∑

z′≥z0Mzz′ mod 2 (7.30)

is a Z2-valued global invariant. To see why, view Mzz′ mod 2 as a Z2 “flux” flowing

from vertex z to z′ in the graphical representation. Then, Eq. 7.28 is a divergence-free

constraint at each vertex. The invariant F1 is simply the total Z2 flux flowing through

a cut at z0. It is therefore clear that F1 does not depend on the choice of cut z0, nor

can it be modified by stacking a 2D SPT which amounts to adding closed flux loops

locally.

162

Type 2 Consider G = ZN × ZN , so that M(α,z),(β,z′) is a 2L × 2L matrix. Again

pick a cut z0. Then,

F2 ≡∑

z<z0

∑

z′≥z0

(M(1,z),(2,z′) −M(2,z),(1,z′)

)mod N (7.31)

is a ZN -valued global invariant. To see how this arises, interpret M(1,z),(2,z′) as a ZN

“flux” flowing from vertex (1, z) to (2, z′). Like before, Eq. 7.28 is a divergence-free

constraint on this flux and F2 measures the total flux flowing across a cut, which

therefore does not depend on z0 nor can it be modified by stacking with 2D SPTs.

We also prove three additional statements (the proofs are technical and can be

found in the Supplemental material of Ref [27]). First, that the invariant F1 or

F2 is the same regardless of whether we consider the M matrix obtained from xy

symmetries or that obtained from yz (or zx) symmetries. Secondly, 3-foliated systems

must have trivial F2 = 0. Thirdly, the set of F1 and F2 (which we also define for

general G) completely classify M modulo stacking with 2D SPTs. Sec 7.3 provides an

explicit construction of a 3-foliated model which realizes a non-trivial type 1 strong

phase F1 = 1, and a 2-foliated model which realizes arbitrary F1 and F2, thereby

demonstrating the existence of such strong phases. Examples of M matrices with

non-trivial F1 and F2 are shown in Fig. 7.2 (right).

Let us define a ‘strong’ equivalence relation between SSPTs, under which two

phases belong to the same equivalence class if they can be connected with one an-

other by stacking of 2D phases (along with, of course, symmetric local unitary trans-

formations and addition/removal of disentangled degrees of freedom transforming as

an on-site linear representation of G [7]). For an arbitrary finite abelian group G, the

163

set of equivalence classes is given by

C3-fol[G] =∏

i

Zgcd(2,Ni)

C2-fol[G] =∏

i

Zgcd(2,Ni) ×∏

i<j

Zgcd(Ni,Nj)

(7.32)

for 3-foliated and 2-foliated models respectively. The group structure is realized via

the stacking operation between two SSPTs. We note that this equivalence relation

can be naturally formulated in terms of planar-symmetric local unitary circuits, gen-

eralizing the LSLU [19]. Indeed the unitaries U used to construct weak SSPTs are

examples of such circuits.

7.3 Strong models

In this section, we introduce two exactly solvable models of strong planar SSPT

phases. The first is the 3-foliated Type 1 strong phase with G = Z2, which we write

down in the form of a Hamiltonian. The fracton dual is a novel fracton model which

we explicitly write down. The second is the 2-foliated Type 1 and Type 2 strong

phase with G = ZN × ZN , for which we write down the ground state wavefunction

|ψ〉. We may consider the 2-foliated model as part of a model with two sets of 2-

foliated symmetries, in which case the fracton dual is again a novel model with unusual

braiding statistics between fractons. Alternatively, we may examine the fracton dual

of a single 2-foliated model by itself, which results in a 2-foliated fracton phase, with

non-trivial braiding statistics between gauge fluxes. To obtain models for strong

phases for more general groups G, one may simply identify Z2 or ZN ×ZN subgroups

of G, and define the model in terms of those degrees of freedom.

164

7.3.1 3-foliated Type 1 strong model

The G = Z2 strong 3-foliated model is defined on the square lattice with qubit degrees

of freedom on each site. Define the Pauli matrices Z and X,

Z =

1 0

0 −1

X =

0 1

1 0

(7.33)

as well as the S =√Z matrix and the controlled-Z (CZ) matrix

S = i(1−Z)/2 =

1 0

0 i

CZ12 = (−1)(1−Z1)(1−Z2)/4 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 −1

(7.34)

The Hamiltonian will be written as a sum of terms of the form

H = −∑

r

XrFr(Zp) ≡ −∑

r

Br (7.35)

where Zp are products of Z on the four corners of a plaquette p, and Fr(Zp) is some

function of these variables near the site r. The planar symmetries will act as products

of Xs along xy, yz, or zx planes. As Fr(Zp) only depends on the combinations Zp

which commutes with all planar symmetries, this Hamiltonian is explicitly symmetry

respecting.

The function Fr(Zp) consists of 6 Zp, 12 Sp, and 12 CZp1p2 operators on various

plaquettes, and an overall factor of i. Fig. 7.3 shows the model on the dual lattice,

where plaquettes are represented by bonds, and the site r is mapped on to the red

165

Figure 7.3: The operator Bc for each cube in the gauged Type 1 strong model. Theaction of the Z, S, and CZ operators precede the action of the X. The CZ operatorsare always between two bonds oriented in different directions, and are denoted by aline connecting the two bonds. For ease of viewing, CZ operators between bonds ofvarious pairs of orientations are shown in a different color. The model is symmetricunder three-fold rotation about the (111) axis.

Figure 7.4: Taking a product of the cubic terms Bc (Fig. 7.3) results in a wireframeoperator with support along the hinges of the cube. This wireframe operator is shownhere for a 7 × 7 × 7 cube, where the action of the Z, S, and CZ operators precedethe X (which acts along the red cube).

166

cube. Careful calculation will show that [Br, Br′ ] = 0 and B2r = 1. This Hamilto-

nian is is therefore simply a commuting projector Hamiltonian, and as every term

is independent (only Br can act as Xr) and there are the same number of terms as

sites, H has a unique gapped group state |ψ〉 and describes a valid SSPT. We found

it simplest to write a small computer script to confirm these commutation relations

(and to compute the wireframe operator later), rather than doing so by hand.

The wireframe operator (Fig. 7.4) obtained as a product of Br over a large cube,

when ungauged, gives the action of the symmetry on the hinges of the cube. One

may confirm using the Else-Nayak procedure that this model has M matrix

Mz,z = 2

Mz,z+1 = Mz,z−1 = 1

(7.36)

and all other elements zero. This therefore realizes the Type 1 strong phase shown

in Fig. 1 for G = Z2.

The fracton dual of this model is defined on the square lattice with qubit degrees

of freedom on the bonds. The Hamiltonian is given by

H = −∑

v

(Axyv + Ayzv + Azxv )−∑

c

Bc (7.37)

where c represents cubes, Bc is the operator shown in Fig. 7.3, v represents vertices,

and Aµνv is the product of Zs along the four bonds touching v in the µν plane (the

usual cross term from the X-cube model). Bc consists of Xs along the cube (the

cube term from the X-cube model) but with an additional phase factor depending on

the Z state around it in the form of S, Z, and CZ operators. Note that while the

ungauged operator Br squares to 1, the gauged operator Bc does not square to 1, it

instead squares to a product of Av operators.

167

This model has the same fracton charge excitations as the usual X-cube model.

However, the lineon excitations are modified. To find out what they are, consider the

product of Bc over a large cube,∏

cBc, shown in Fig. 7.4. This results in an operator

with support only along the hinges of the cube. This operator, when truncated, is

the operator which creates lineon excitations at its ends.

From this, the crossing (braiding) statistic of two lineon can be readily extracted.

Reading off of Fig. 7.4, a pair of x-moving lineons on line (y1, z1) is constructed by

the operator

Lx ≡x2∏

x=x0

X(x)x,y1,z1

S(z)x,y1,z1

CZ(x↔y)x,y1,z1

(7.38)

where X(x)x,y,z is an X on the bond originating from the vertex at (x, y, z) going in the

positive x direction, and similarly for S(z)x,y,z, and CZ

(x↔y)x,y,z is a CZ between Z

(x)x,y,z and

Z(y)x,y,z. Lx creates two lineons at x0 and x2. Meanwhile, a pair of y-moving lineons is

constructed by

Ly ≡y2∏

y=y0

X(y)x1,y,z1

S(x)x1,y,z1

CZ(y↔z)x1,y,z1

(7.39)

which creates two lineons at y0 and y2. Note that depending on which hinge of the

wireframe we obtain Lx and Ly from, there may be additional Z operators, which

correspond to a choice of lineon or antilineon (and will affect the braiding phase by a

±1). It can be readily verified that when these two operators cross (i.e. y0 < y1 < y2

and x0 < x1 < x2), they only commute up to a factor of i,

LyLx = iLyLx (7.40)

using the relations XSX = iZS and X1CZ12X1 = Z2CZ12. Thus, the braiding phase

of any two lineons in this model is ±i.

168

7.3.2 2-foliated strong model

In this section, we describe a 2-foliated model which realizes both Type 1 and/or

Type 2 strong phases.

A group cohomology model on the square lattice

First, let us explicitly construct a group cohomology model on the square lattice, for

G = ZMN . Recall that the ground state of such models are an equal amplitude sum of

all configurations

|ψ〉 =∑

grf(gr) |gr〉 (7.41)

where f(gr) is a pure phase up to an overall normalization, which we ignore.

From Eq 7.20, f(gr) is a product of terms fr coming from each square plaquette

at r, given by

fr(gr) =ν(gr, gr+x, gr+x+y, 1)

ν(gr, gr+y, gr+x+y, 1)(7.42)

Alternatively, we may choose to defined the same wavefunction using instead

f ′r(gr) =ν(1, gr, gr+y, gr+x+y)

ν(1, gr, gr+x, gr+x+y)(7.43)

which one can verify using the cocycle condition (Eq 7.18) differs from fr only by

terms along the edges of the plaquette which are cancelled out by the same terms

from neighboring plaquettes. Plugging the explicit form for the cocycles, we get

f (2D)r (gr) = exp

∑

i≤j

2πipij

N2gir ( [gjr+y − gjr] + [gjr+x+y − gjr+y]

−[gjr+x − gjr]− [gjr+x+y − gjr+x])

(7.44)

which we have called f(2D)r .

169

The strong SSPT

Let us take G = ZN × ZN , with subsystem symmetries along xy and yz planes. The

ground state of our strong SSPT is again described by a function f(gr), which can

be written as a product of fr(gr), which are now associated with the cube at r.

The function fr is given by

f (SSPT )r (gr) = exp

∑

α≤β

2πiqαβ

N2

((gαr+z − gαr )([gβr+y+z − gβr+z] + [gβr+x+y+z − gβr+y+z]

− [gβr+x+z − gβr+z]− [gβr+x+y+z − gβr+x+z])

−(gαr+y − gαr )([gβr+x+y+z − gβr+y+z]− [gβr+x+y − gβr+y]

+ [gβr+y+z − gβr+x+y+z + gβr+x+y − gβr+y])

)

(7.45)

for qαβ integers mod N . Here, α, β ∈ 1, 2 for each factor of ZN in G.

We claim that f(SSPT )r describes an SSPT phase which is Type 1 strong if q11 or

q22 are odd (and N is even), and Type 2 strong if q12 6= 0.

First, let us examine the state as a quasi-2D SPT along the xy plane, with a

GL symmetry group. Let us label each generator of GL by (α, z), for α ∈ 1, 2

and z ∈ [1, L]. The second term in the exponent (the term multiplying (gir+y − gir))

is completely invariant under an xy planar symmetry. This second term therefore

cannot affect the xy cocycle class, as it can be removed by a symmetric local unitary

transformation respecting all xy planar symmetries (but will break the yz planar

symmetries). Thus, the xy cocycle class is determined simply by the first term.

However, this term is exactly of the form f(2D)r (gr) for GL, with the mapping

p(α,z),(β,z) = qαβ

p(α,z),(β,z+1) = −qαβ(7.46)

170

and other elements zero. In terms of the M matrix,

M(α,z),(β,z) = (1 + δαβ)qαβ

M(α,z),(β,z+1) = −qαβ(7.47)

and all other elements (except those related by symmetry) are zero.

The F1 invariants are therefore simply q11 and q22 modulo 2, and the F2 invariant

is −q12. By the proof from our previous section, the invariants will also the same for

the yz symmetries.

But before we can conclude that we have constructed a strong phase, we must

show that this state is symmetric under yz symmetries. The purpose of the second

term in f(SSPT )r is to ensure that this is the case. Let us examine how fr(gr)

transforms under a yz planar symmetry which sends gr → g(yz)gr, or, on the

relevant degrees of freedom,

(gr, gr+y, gr+z, gr+y+z)→ (ggr, ggr+y, ggr+z, ggr+y+z)

(gr+x, gr+x+y, gr+x+z, gr+x+y+z) unchanged

(7.48)

A calculation shows that

fr(g(yz)gr)fr(gr)

= exp

∑

α≤β

2πiqαβ

N2

((gαr+z − gαr )([gβr+x+y+z − gβr+y+z + gβ]− [gβr+x+y+z − gβr+y+z]

− [gβr+x+z − gβr+z + gβ] + [gβr+x+z − gβr+z])

−(gαr+y − gαr )([gβr+x+y+z − gβr+y+z + gβ]− [gβr+x+y+z − gβr+y+z]

− [gβr+x+y − gβr+y + gβ] + [gβr+x+y − gβr+y])

)

(7.49)

171

which simplifies to

fr(g(yz)gr)fr(gr)

=P (gr+z, gr+x+y+z, gr+y+z, g)

P (gr, gr+x+y, gr+y, g)

P (gr, gr+x+z, gr+z, g)

P (gr+y, gr+x+y+z, gr+y+z, g)

P (gr+y, gr+x+y, gr+y, g)

P (gr+z, gr+x+z, gr+z, g)

(7.50)

where

P (g1, g2, g3, g) = exp

∑

α≤β

2πiqαβ

N2

(gα1 ([gβ2 − gβ3 + gβ]− [gβ2 − gβ3 ])

)(7.51)

If one considers the contribution from neighboring cubes, one finds that the factors of

P (. . . ) exactly cancel out between neighboring cubes. Repeating this calculation for

a yz-planar symmetry which transforms the other four sites in Eq 7.48, one finds the

same result. Thus, the wavefunction is indeed symmetric under yz planar symmetries

and describes a strong SSPT phase for a 2-foliated model. If one wished, one could

confirm that the matrix M(yz) obtained from yz planar is also strong with the same

F1 and F2 invariants, by following the Else-Nayak procedure. Obtaining a gapped

local Hamiltonian corresponding to this ground state is straightforward, and is done

in the same way as for the standard group cohomology models, Eq 7.21.

7.4 Fracton duals

Let us finally discuss the fracton phases, which are gauge dual [10, 41, 25, 20, 26] to

the SSPT phases we have been discussing. The simplest and most well-studied fracton

model is the X-cube model [10], which is obtained by gauging the planar symmetries

of the plaquette Ising paramagnet, and hosts fractional quasiparticle excitations with

limited mobility including immobile fractons, lineons mobile along lines, and planons

mobile within planes (which are either fracton dipoles or lineon dipoles).

Let us begin with 3-foliated SSPTs, which are dual to ‘twisted’ X-cube fracton

topological orders with fractonic charge [26]. The gauge flux m(g,z) of an element g

172

on the plane z is a planon: a composite excitation composed of a lineon anti-lineon

pair on the planes z + 1/2 and z − 1/2, i.e. a lineon dipole. A single lineon can be

regarded as a semi-infinite stack of lineon dipoles mobile in the x and y directions.

The constraints on the matrix M have a simple interpretation in this language:

the infinite stack of lineon dipoles, which belongs to the vacuum superselection sec-

tor, must have trivial braiding statistics with all other lineon dipoles, and a trivial

exchange statistic with itself. The invariant F1 also has a simple interpretation in this

picture: the quantity e2πiF1/N2corresponds to the braiding (or crossing [28]) statistic

of a lineon and its anti-lineon on the same plane, modulo e4πi/N2.

It is possible to construct fracton topological orders by strongly coupling intersect-

ing stacks of topologically ordered 2D discrete gauge theories oriented along the xy,

yz, and zx planes, inducing a type of transition called p-string condensation [28, 93].

More generally, these stacks of 2D gauge theories can be replaced by arbitrary 1-

foliated gauge theories [26]. The twisted X-cube models that emerge from this con-

struction are dual to weak 3-foliated SSPTs constructed via the planar-symmetric

local unitaries U in Eq. 7.27. (See the supplementary material of Ref [27]).

Equivalently, twisted X-Cube models dual to weak SSPTs may be obtained by

effectively “binding” 2D anyons to existing planons in the fracton model. As an

example, consider placing one layer of the doubled semion topological order (with

bosonic e and semionic m) onto a plane z0 of the X-Cube model, and condensing

pairs of e and fracton dipoles in the plane z0. The end result is that x or y mobile

lineons on plane z0 and m become confined, but the bound state of the two remain

deconfined and form the new lineon excitations. Since m is a semion, the new lineons

now also inherit their semionic statistics. This procedure can be extended to general

twisted quantum doubles living on multiple planes zk, thereby binding more general

2D anyons to the lineons; this process is exactly dual to stacking a 2D SPT according

to Eq. 7.27.

173

Conversely, strong 3-foliated SSPTs are dual to fracton models that cannot be

realized through such a construction. This correspondence sheds light on the F1

strong invariant — in p-string condensation, lineon crossing statistics are inherited

from the self-braiding statistics of fluxes in the 1-foliated gauge theories, and are

therefore the square of a flux exchange statistic, i.e. a multiple of e4πi/N2for G = ZN

with N even. In a strong phase, F1 = 1 implies that this statistic is offset by e2πi/N2.

The fracton dual of the Type 1 strong G = Z2 model is an example of a novel such

fracton order in which lineons satisfying a triple fusion rule have ±i mutual crossing

statistic, and therefore cannot be realized via p-string condensation. A Hamiltonian

realizing this phase is shown in Fig 7.3.

One can also consider the fracton duals of 2-foliated SSPTs, which are novel

‘twisted’ versions of the 2-foliated lineon-planon model introduced in Ref. [24]. Fur-

thermore, the X-cube model may be ungauged in two different ways, by regarding

either the fracton sector or the lineon sector as gauge charge. The former procedure

results in a paramagnet with G-valued degrees of freedom transforming under all 3

sets of planar symmetries as before, whereas the latter yields a model with two G-

valued degrees of freedom per site, the first transforming under xy and yz planar

symmetries, and the second under yz and zx planar symmetries. The classification of

the latter system is given by (C2-fol)2. Thus, both Type 1 and Type 2 strong SSPTs,

as well as arbitrary weak SSPTs, may be constructed. Their fracton duals are novel

variants of the X-cube model whose fracton dipoles exhibit non-trivial braiding and

exchange statistics.

A Type 2 strong SSPT can also be diagnosed through the statistical phases of

quasiparticles of the gauged dual. Although fractons are immobile particles, we may

still define a braiding statistic between two fractons by regarding a single fracton

as a semi-infinite stack of fracton dipoles mobile in the xy plane. Consider a G =

ZN × ZN model which has two flavors of fractons. Then, let eiθab be the statistical

174

phase obtained by braiding two such fractons of flavors a and b on plane z0, where

the first argument is a semi-infinite stack in the z → ∞ direction, and the second

argument in the z → −∞ direction. The Type 2 strong invariant is then obtained by

eiF2/N = eiN(θab−θba). This makes it clear why this strong phase with F2 6= 0 cannot

be obtained by binding 2D anyons to the fractons, since braiding of 2D anyons is

manifestly symmetric with respect to its two arguments.

7.5 Conclusions

We have therefore formulated a classification of strong 3D planar SSPTs, generalizing

the result from Chapter 6. Each phase falls into one of a finite set of equivalence classes

modulo stacking with 2D phases, which we have fully enumerated. For 1-foliated

systems, all SSPT phases are weak. For 2-foliated systems, there are two mechanisms

by which a phase may be strong, characterized by Type 1 and Type 2 invariants. For

3-foliated systems, only Type 1 strong phases exist. Under a generalized gauge duality,

our classification has a natural interpretation in terms of p-string condensation [28],

and we have explicitly constructed strong SSPT models which are dual to fracton

phases that cannot be realized via this mechanism. More discussion of the connection

to p-string condensation is available in the supplemental material of Ref [27]. The

fractional quasiparticles in these strong phases thus have novel statistical interactions

which cannot be interpreted as the statistics of 2D anyonic bound states.

There are various natural extensions. A relevant open question regards the struc-

ture of entanglement in strong SSPT phases [123, 130, 131, 120, 132]. Another is the

addition of non-trivial type-III cocycles, which leads to non-abelian fracton topolog-

ical orders. Finally, it would be interesting to study the foliation structure of the

fracton duals.

175

Part III

Fractal Subsystem Symmetric

Phases

176

Chapter 8

Fractal symmetric phases

Having thoroughly discussed regular subsystem symmetries (linear and planar sym-

metries), we now turn to another, more exotic, type of subsystem symmetry: fractal

subsystem symmetries. We will start in this chapter by defining these symmetry

operations, and then go on to discuss the possible SPT phases protected by such

symmetries. In the next chapter, Chapter 9, we will exhaustively classify fractal SPT

phases in 2D.

There are a couple additional works that are relevant but not included in this Dis-

sertation: these include an application of fractal SPT phases for measurement-based

quantum computation [133] and a more general view of fractal models (both SPT and

type-II fracton topological phases) through a process called “fractalization” [134].

This chapter is based on the paper

[29] T. Devakul, Y. You, F. J. Burnell, S. L. Sondhi, “Fractal Symmetric Phases of

Matter”, Scipost Phys. 6, 007 (2019).

We first present in Sec 8.1 a brief introduction to CA, and how fractal structures

emerge naturally from them. In Sec 8.2, we take these fractal structures to define

symmetries on a lattice in 2D. These symmetries are most naturally defined on a

semi-infinite lattice; here, symmetries flip spins along fractal structures (e.g. trans-

177

lations of the Sierpinski triangle). We describe in detail how such symmetries should

be defined on various other lattice topologies, including the infinite plane. Simple

Ising models obeying these symmetries are constructed in Sec 8.3, which demonstrate

a spontaneously fractal symmetry broken phase at zero temperature, and undergoes

a quantum phase transition to a trivial paramagnetic phase. In Sec 8.4 we use a

decorated defect approach to construct fractal SPT (FSPT) phases. The nontrivial-

ity of these phases are probed by symmetry twisting experiments and the existence

of symmetry protected ungappable degeneracies along the edge, due to a locally

projective representation of the symmetries. Such phases have symmetry protected

fracton excitations that are immobile and cannot be moved without breaking the

symmetries or creating additional excitations. Finally, we discuss 3D extensions in

Sec 8.5, these include models similar to the 2D models discussed earlier, but also

novel FSPT phases protected by a combination of regular fractal symmetries and

a set of symmetries which are analogous to higher form fractal symmetries. These

FSPT models with higher form fractal symmetries, in one limit, transition into a frac-

ton topologically ordered phase while still maintaining the fractal symmetry. Such

a phase describes a topologically ordered phase enriched by the fractal symmetry,

thus resulting in a fractal symmetry enriched (fracton) topologically ordered (fractal

SET [135, 136, 137, 138, 139, 140, 141, 142, 143], or FSET) phase.

8.1 Cellular Automata Generate Fractals

We first set the stage with a brief introduction to a class of one-dimensional CA, from

which it is well known that a wide variety of self-similar fractal structures emerge. In

latter sections, these fractal structures will define symmetries which we will demand

of Hamiltonians.

178

Consider sites along a one-dimensional chain or ring, each site i associated with a

p-state variable ai ∈ 0, 1, . . . , p − 1 taken to be elements of the finite field Fp. We

define the state of the CA at time t as the set of a(t)i . We will typically take p = 2,

although our discussion may be easily generalized to higher primes. We consider CA

defined by a set of translationally-invariant local linear update rules which determine

the state a(t+1)i given the state at the previous time a(t)

i . Linearity here means

that the future state of the ith cell, a(t+1)i , may be written as a linear sum of a

(t)j

for j within some small local neighborhood of i. Throughout this paper, all such

arithmetic is integer arithmetic modulo p, following the algebraic structure of Fp.

Figure 8.1 shows two sets of linear rules which we will often refer to:

1. The Sierpinski rule, given by a(t+1)i = a

(t)i−1 + a

(t)i with p = 2, so called because

starting from the state a(0)i = δi,0 one obtains Pascal’s triangle modulo 2, who’s

nonzero elements generate the Sierpinski triangle with fractal Hausdorff dimen-

sion d = ln 3/ ln 2 ≈ 1.58. In the polynomial representation (to be introduced

shortly), this rule is given by f(x) = 1 + x.

2. The Fibonacci rule, a(t+1)i = a

(t)i−1 + a

(t)i + a

(t)i+1 also with p = 2, so called because

starting from a(0)i = δi,0 it generates a fractal structure with Hausdorff dimen-

sion d = 1 + log2(ϕ) ≈ 1.69 with ϕ the golden mean [144]. The polynomial

representation is given by f(x) = x−1 + 1 + x.

Fractal dimensions for CA with linear update rules may be computed efficiently [145].

To see why such linear update rules always generate self-similar structures, it is

convenient to pass to a polynomial representation. We may represent the state a(t)i

as a polynomial st(x) on Fp over x as

st(x) =∞∑

i=−∞a

(t)i x

i (8.1)

179

?

-

t

i

Figure 8.1: Fractal structures generated by (left,blue) the Sierpinski rule a(t+1)i =

a(t)i−1 + a

(t)i and (right,red) the Fibonacci rule a

(t+1)i = a

(t)i−1 + a

(t)i + a

(t)i+1, starting from

the initial state a(0)i = δi,0. In the polynomial representation, the row t is given by

f(x)t, with (blue) f(x) = 1 + x and (red) f(x) = x−1 + 1 + x over F2. Notice thatself-similarity at every row t = 2l (here, we show evolution up to t = 40).

for an infinite chain (we allow polynomials to have both positive and negative powers).

Alternatively, periodic boundary conditions may be enforced by setting xL = 1. In

this language, these update rules take the form

st+1(x) = f(x)st(x) (8.2)

for some polynomial f(x) containing only small finite powers (both positive or nega-

tive) of x. For the Sierpinski rule we have f(x) = 1 + x, and for the Fibonacci rule

we have f(x) = x−1 + 1 + x.

Then, given an initial state s0(x), we have that

st(x) = f(x)ts0(x) (8.3)

A neat fact about polynomials in Fp is that they obey what is known as the Freshman’s

Dream,

f(x) =∑

i

cixi =⇒ f(x)p

k

=∑

i

cixipk (8.4)

whenever t is a power of p. This can be shown straightforwardly by noting that the

binomial coefficient(pk

n

)is always divisible by p unless n = 0 or n = pk.

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It thus follows that such CA generate fractal structures. Let us illustrate for the

Fibonacci rule starting from the initial configuration s0(x) = 1, i.e. the state where

all ai = 0 except for a0 = 1. Looking at time t = 2l, the state is st(x) = x−2l +1+x2l .

In the following evolution, each of the non-zero cells a−2l = a0 = a2l = 1 each look

locally like the initial configuration s0, and thus the consequent evolution results in

three shifted structures identical to the initial evolution of s0 (up until they interfere),

as can be seen in Figure 8.1. At time t = 2k+1, this process repeats but at a larger

scale. Thus, we can see that any linear update rule of this kind will result in self-

similar fractal structures when given the initial state s0(x) = 1. As the rules are

linear, all valid configurations correspond to superpositions of this shifted fractal.

The entire time evolution of the CA may be described at once by a single poly-

nomial F (x, y) over two variables x and y,

F (x, y) =∞∑

t=0

f(x)tyt (8.5)

and we have that the coefficient of yt in F (x, y)s0(x) is exactly st(x) = f(x)ts0(x).

The two-dimensional fractal structures in Figure 8.1 generated by these CA emerge

naturally due to a set of simple local constraints given by the update rules. In the

next section, we will describe 2D classical spin Hamiltonians which energetically en-

force these local constraints. The ground state manifold of these classical models is

described exactly by a valid CA evolution, which we will then take to define symme-

tries.

8.2 Fractal Symmetries

To discuss physical spin Hamiltonians and symmetries, it is useful to also use a poly-

nomial representation of operators. Such polynomial representations are commonly

used in classical coding theory [146], and refined in the context of translationally

181

invariant commuting projector Hamiltonians by Haah [147]. We will utilize only the

basic tools (following much of Ref [144]), and specialize to Pauli operators (p = 2

from the previous discussion), although a generalization to p-state Potts spins is

straightforward.

Let us consider in 2D a square lattice with one qubit (spin-1/2) degree of freedom

per unit cell. Acting on the qubit at site (i, j) ∈ Z2, we have the three anticommuting

Pauli matrices Zij, Xij, and Yij. We define the function Z(·) from polynomials in x and

y over F2 to products of Pauli operators, such that acting on an arbitrary polynomial

we have

Z

(∑

ij

cijxiyj

)=∏

ij

(Zij)cij (8.6)

and similarly for X(·) and Y (·). For example, we have Z(1 + x+ xy) = Z0,0Z1,0Z1,1.

Some useful properties are that the product of two operators is given by the sum of

the two polynomials, Z(α)Z(β) = Z(α + β), and a translation of Z(α) by (i, j) is

given by Z(xiyjα).

Perhaps the most useful property of this notation is that two operators Z(α) and

X(β) anticommute if and only if [αβ]x0y0 = 1, where [·]xiyj denotes the coefficient of

xiyj in the polynomial, and we have introduced the dual,

p(x, y) =∑

ij

cijxiyj ↔ p(x, y) =

∑

ij

cijx−iy−j (8.7)

which may be thought of as the spatial inversion about the point (0, 0). We will

also often use x to represent x−1 for convenience. More usefully, we may express the

commutation relation between Z(α) and translations of X(β) (given by X(xiyjβ)) as

Z(α)X(xiyjβ) = (−1)dijX(xiyjβ)Z(α) (8.8)

182

where dij may be computed directly from the commutation polynomial of α and β,

P (α, β) =∑

ij

dijxiyj = αβ (8.9)

which may easily be computed directly given α and β. In particular, P = 0 would

imply that every possible translations of the two operators commute.

8.2.1 Semi-infinite plane

We may now transfer our discussion of the previous section here. Let us consider

a semi-infinite plane, such that we only have sites (i, j) with xiyj≥0. We may then

interpret the jth row as the state of a CA at time j, starting from some initial state

at row j = 0. Consider the linear CA with update rule given by the polynomial f(x),

as defined in Eq 8.2. The classical Hamiltonian which energetically enforces the CA’s

update rules is given by

Hclassical = −∞∑

i=−∞

∞∑

j=1

Z(xiyj[1 + f y]) (8.10)

where we have excluded terms that aren’t fully inside the system.

As an example, consider the Sierpinski rule f = 1 + x (f will always refer to a

polynomial in only x). Equation 8.10 for this rule gives,

HSierpinski = −∑

ij

ZijZi,j−1Zi−1,j−1 (8.11)

which is exactly the Newman-Moore (NM) model originally of interest due to being an

exactly-solvable translationally invariant model with glassy relaxation dynamics [148].

The NM model was originally described in a more symmetric way on the triangular

lattice as the sum of three-body interactions on all downwards facing triangles,HNM =

−∑O ZZZ. This model does not exhibit a thermodynamic phase transition (similar

183

to the 1D Ising chain). Fractal codes based on higher-spin generalizations of this

model have also been shown to saturate the theoretical information storage limit

asymptotically [149].

We will be interested in the symmetries of such a model that involve flipping

subsets of spins. Due to the deterministic nature of the CA, such operation must

involve flipping some subset of spins on the first row, along with an appropriate set

of spins on other rows such that the total configuration remains a valid CA evolution.

Operationally, symmetry operations are given by various combinations of F (x, y)

(Eq 8.5). That is, for any polynomial q(x), we have a symmetry

S(q(x)) = X(q(x)F (x, y)) (8.12)

Here, q(x) has the interpretation of being an initial state s0, and S(q(x)) flips spins

on all the sites corresponding to the time evolution of s0. As the update rules are

linear, this operation always flips between valid CA evolutions. For example, S(1)

will correspond to flipping spins along the fractals shown in Fig 8.1.

To confirm that this symmetry indeed commutes with the Hamiltonian, we may

use the previously discussed technology (Eq 8.8 and 8.9) to compute the commutation

polynomial between S(q(x)) and translations of the Hamiltonian term Z(1 + f y),

P = q(x)F (x, y)(1 + fy) = q(x)(1 + fy)∞∑

l=0

(fy)l

= q(x)

( ∞∑

l=0

(fy)l +∞∑

l=1

(fy)l

)= q(x) (8.13)

Since terms which have shift y0 are not included in the Hamiltonian (Eq 8.10), this

operator therefore fully commutes with the Hamiltonian. We may pick as a basis set

of independent symmetries, S(qα), for α ∈ Z with qα(x) = xα. These operators corre-

spond to flipping spins corresponding to the colored pixels in Fig 8.1, and horizontal

184

shifts thereof. Each of these symmetries act on a fractal subset of sites, with volume

scaling as the Hausdorff dimension of the resulting fractal.

8.2.2 Cylinder

Rather than a semi-infinite plane, let’s consider making the x direction periodic with

period L, such that xL = 1, while the y direction is either semi-infinite or finite. In

this case, there are a few interesting possibilities.

Reversible case In the case that there exists some ` such that f ` = 1, then the

CA is reversible. That is, for each state st, there exists a unique state st−1 such that

st = fst−1, given by st−1 = f `−1st.

A proof of this is straightforward, suppose there exists two distinct previous states

st−1, s′t−1, such that fst−1 = fs′t−1 = st. As they are distinct, st−1 + s′t−1 6= 0.

However,

0 = st + st = f(st−1 + s′t−1) = f `−1f(st−1 + s′t−1) = st−1 + s′t−1 6= 0 (8.14)

there is a contradiction. Hence, the state st−1 must be unique. The inverse statement,

that a reversible CA must have some ` such that f ` = 1, is also true.

In this case, all non-trivial symmetries extend throughout the cylinder, and their

patterns are periodic in space with period dividing `. An example of this is the

Fibonacci rule with L = 2m, for which fL/2 = 1. There are L independent symmetries

on either the infinite or semi-infinite cylinder. The symmetries on an infinite cylinder

are given by S(q) = X(q(x)

∑∞l=−∞(fy)l

), where f−1 ≡ f `−1.

Trivial case If there exists ` such that f ` = 0, then the model is effectively trivial.

All initial states s0 will eventually flow to the trivial state s` = 0. On a semi-infinite

cylinder, possible “symmetries” will involve sites at the edge of the cylinder, but will

185

not extend past ` into the bulk of the cylinder. On an infinite cylinder, there are no

symmetries at all. An example of this is the Sierpinski rule with L = 2m, for which

fL = 0.

Neither reversible nor trivial If the CA on a cylinder is neither reversible nor

trivial, then every initial state s0 must eventually evolve into some periodic pattern,

such that st = st+T for some period T at large enough t (this follows from the

fact that there are only finitely many states). Thus, there will be symmetries that

extend throughout the cylinder, like in the reversible case. Interestingly, however,

irreversibility also implies the existence of symmetries whose action is restricted only

to the edge of the cylinder, much like the trivial case.

Let us take two distinct initial states s0, s′0 that eventually converge on to the

same state at time `. Then, let s0 = s0 + s′0 6= 0 be another starting state. After time

`, s` = s` + s′` = 0, this state will have converged on to the trivial state. Thus, the

symmetry corresponding to the starting state s0 will be restricted only to within a

distance ` of the edge on a semi-infinite cylinder.

On an infinite cylinder, only the purely periodic symmetries will be allowed, so

the total number of independent symmetries is reduced to between 0 and L.

8.2.3 On a torus

Let us next consider the case of an Lx × Ly torus. Symmetries on a torus must take

the form of valid CA cycles on a ring of length Lx with period Ly. The total number

of symmetries is the total number of distinct cycles commensurate with the torus size,

which in general does not admit a nice closed-form solution, but has been studied in

Ref [150] Equivalently, there are as many symmetries as there are solutions to the

equation

q(x)f(x)Ly = q(x) (8.15)

186

L k(L) q(L)α (x)

2m 0 -2m − 1 L− 1 xα(1 + x)

2m + 2n − 1 gcd(L, 2n+1 − 1)− 1 xα(1 + x)∑ L

k+1−1

l=0 (x2m−2n)l

2m 2k(m) xα mod 2[q(m)bα/2c(x)]2

Figure 8.2: The number of independent symmetries k(L) and a choice of qα(x) forthe Sierpinski model on an L× L torus for few particular L. Here, m,n are positiveintegers, m > n, and 0 ≤ α < k labels the symmetry polynomials q

(L)α (x), and b·c

denotes the floor function.

with xLx = 1. This may be expressed as a system of linear equations over F2, and can

be solved efficiently using Gaussian elimination. For each solution q(x) of the above

equation, the corresponding symmetry operator is given by

S(q) = X

(q(x)

Ly−1∑

l=0

(fy)l

)(8.16)

As an example, consider the Sierpinski model on an L × L torus. Let k(L) =

log2(Nsym(L)) be the number of independent symmetry operators, where Nsym(L) is

the total number of symmetries. We are free to pick some set of k(L) independent

symmetry operators as a basis set (there is no most natural choice for basis), which

we label by qα(x) with 0 ≤ α < k. To illustrate that k(L) is in general a complicated

function of L, we show in Table 8.2 k(L) and a choice of q(L)α (x) for the few cases of

L where the number of cycles can be solved for exactly. An interesting point is that

for the Sierpinski rule, f(x)2l = 0, thus for L = 2l, there are no non-trivial solutions

to Eq 8.15 and so k(2l) = 0. To contrast, the Fibonacci rule has f(x)2l = 1, and so

k(L = 2l) = L.

8.2.4 Infinite plane

Now, let us consider defining such symmetries directly on an infinite plane, where we

allow all xiyj. In the CA language, we are still free to pick the CA state at time, say

187

t = 0, s0(x), which completely determines the CA states at times t > 0. However,

we run into the issue of reversibility — how do we determine the history of the CA

for times t < 0 which lead up to s0? For general CA, there may be zero or multiple

states s−1 which lead to the same final state s0. For a linear CA on an infinite plane,

however, there is always at least one s−1. We give an algorithm for picking out a

particular history for s0, and discuss the sense in which it is a complete description

of all possible symmetries, despite this reversibility issue. For this discussion it is

convenient to, without loss of generality, assume f contains at least a positive power

of x (we may always perform a coordinate transformation to get f into such a form).

The basic idea is as follows: we have written the Hamiltonian (Eq 8.10) in a form

that explicitly picks out a direction (y) to be interpreted as the time direction of the

CA. However, we may always write the same term as a higher-order linear CA that

propagates in the x direction,

1 + f y = xay

[1 +

nmax∑

n=1

gn(y)xn

]≡ xay [1 + g(x, y)x] (8.17)

where a > 0 is the highest power of x in f , nmax is finite, and gn(y) is a polynomial

containing only non-negative powers of y. This describes an nmax-order linear CA.

For the Sierpinski rule, we have only g1(y) = 1+y, and for the Fibonacci rule we have

both g1(y) = 1 + y and g2(y) = 1. We then further define g(x, y) for convenience,

which only contains non-negative powers of x and y. Now, consider the fractal pattern

generated by

xay[1 + g(x, y)x+ g(x, y)2x2 + . . .

](8.18)

which describes a higher-order CA evolving in the x direction. Note that powers of

g no longer have the nice interpretation of representing an equal time state in terms

of this CA, due to it containing both powers of y as well as x (but evaluating the

series up to the gnxn does give the correct configuration up to xn). As g contains

188

?

-

yj

xi

Figure 8.3: A valid history for the state s0 = 1 for the Fibonacci rule CA. Theforward evolution (red) is fully deterministic, and here an unambiguous choice hasbeen made for states leading up to it (orange). Lattice points are labeled by (i, j)corresponding to xiyj in the polynomial representation.

only negative powers of y, this fractal pattern is restricted only to the half-plane with

yj<0. It thus lives entirely in the “past”, t < 0, of our initial CA.

The full fractal given by

F(x, y) =

[ ∞∑

l=0

(fy)l

]+ xay

[ ∞∑

l=0

(gx)l

](8.19)

unambiguously describes a history of the CA with the t = 0 state s0 = 1. This is

shown in Figure 8.3 for the Fibonacci model, with the forward propagation of f in

red and the propagation of g in orange.

Going back to operator language, it can be shown straightforwardly that the

symmetry

S(q) = X(q(x)F(x, y)) (8.20)

for arbitrary q(x) commutes with the Hamiltonian (Eq 8.10 but with all ij included in

the sum) everywhere. The only term with y0 in F is 1, so this operator only flips the

spins q(x) on row y0. Furthermore, the choice of choosing the y0 row for defining this

symmetry does not affect which operators can be generated, as it is easy to show that

f(x)F(x, y) = yF(x, y), so that S(q(x)f(x)) flips any set of spins q(x) on the row y

189

instead. Simple counting would then suggest that the total number of independent

symmetries thus scales linearly with the size of the system, like on the semi-infinite

cylinder.

This result seems to contradict the irreversibility of the CA. It would suggest that

one can fully determine st at time t < 0 by choosing the state s0 appropriately, which

would seemingly imply that the evolution is always reversible. The resolution to this

paradox lies in the fact that we are on an infinite lattice, and in this procedure we

have chosen the particular f−1 such that it only contains finitely positive powers of

x (there are in general multiple inverses f−1). Defining h(x) = [g(x, y)]y0 such that

f = xa(1 + hx), then we are choosing the inverse

f−1(x) = xa(1 + hx+ (hx)2 + . . . ) (8.21)

from which it can be readily verified that f−1f = 1. In this language, F(x, y) looks

like

F(x, y) = · · ·+ (f−1y)2 + (f−1y) + 1 + (fy) + (fy)2 + . . . (8.22)

which obviously commutes with the Hamiltonian. As an example, with the Sierpinski

rule, the two possible histories for the state s0 = 1 are s(−)−1 =

∑−∞l=−1 x

l and s(+)−1 =

∑∞l=0 x

l. By this inverse, we would only get s(−)−1 . However, if we wanted to generate

the state with history s(+)−1 , we would instead find that the t = 0 state should be the

limit s0 = 1 + x∞. If we were just interested in any finite portion of the infinite

lattice, for example, we may get any history by simply pushing this x∞ beyond the

boundaries.

8.2.5 Open slab

Finally, consider the system on an open slab with dimensions Lx×Ly. Counting the

number of symmetries is the same as counting the number of valid CA configurations

190

on this geometry. The state at time t = 0 may be chosen arbitrarily, giving us Lx

degrees of freedom. Furthermore, at each time step the state of the cells near the edge

may not be fully specified by the CA rules. Hence, each of these adds an additional

degree of freedom. Let x−pmin , xpmax , be the smallest and largest powers of x in f

(if pmin/max would be negative, then set set it to 0). Then, we are free to choose the

cell states in a band pmax × Ly along the left (xi=0) edge, and pmin × Ly along the

right edge as well. Thus, the total number of choices, and therefore symmetries, is

Nsym = 2Lx+(pmin+pmax)(Ly−1) (there are log2Nsym independent symmetries). Note that

some of these symmetries may be localized to the corners.

One may be tempted to pick a certain boundary condition for the CA, for example,

by taking the state of cells outside to be 0, which eliminates the freedom to choose

spin states along the edge and reduces the number symmetries down to simply 2Lx .

What will happen in this case is that there will be symmetries for the full infinite

lattice which, when restricted to an Lx × Ly slab, will not look like any of these

2Lx symmetries. With the first choice, we are guaranteed that any symmetry of the

infinite lattice, restricted to this slab, will look like one of our Nsym symmetries. This

is a far more natural definition, and will be important in our future discussion of edge

modes in Sec 8.4.3.

8.3 Spontaneous fractal symmetry breaking

At T = 0, the ground state of Hclassical is 2k-degenerate and spontaneously breaks the

fractal symmetries, where k is the number of independent symmetry generators (which

will depend on system size and choice of boundary conditions). Note that k will scale

at most linearly with system size, so it represents a subextensive contribution of the

thermodynamic entropy at T = 0. As a diagnosis for long range order, one has the

191

many-body correlation function C(`) given by

C(`) = Z

((1 + f y)

`−1∑

i=0

(f y)i

)= Z(1 + (f y)`) (8.23)

which has C(`) = 1 in the ground states of Hclassical as can be seen by the fact

that Eq 8.23 is a product of terms in the Hamiltonian. If M is the number of

terms in f , then this becomes an M + 1-body correlation function when ` = 2l

is a power of 2. Long range order is diagnosed by lim`→∞C(`) = const. At any

finite temperature, however, these models are disordered and have C(`) vanishing

asymptotically as C(`) ∼ p−Ld, where d is the Hausdorff dimension of the generated

fractal, and p = 1/(1 + e−2β). Thus, there is no thermodynamic phase transition in

any of these models, although the correlation length defined through C(`) diverges

as T → 0.

Even without a thermodynamic phase transition, much like in the standard Ising

chain, there is the possibility of a quantum phase transition at T = 0. We may

include quantum fluctuations via the addition of a transverse field h,

HQuantum = −∑

ij

Z(xiyj[1 + f y])− h∑

ij

X(xiyj) (8.24)

One can confirm that a small h will indeed correspond to a finite correction

liml→∞C(2l) = 1 − const(h), and so does not destroy long range order. This

model now exhibits a zero-temperature quantum phase transition at h = 1, which

is exactly pinpointed by a Kramers-Wannier type self-duality transformation which

exchanges the strong and weak-coupling limits. This self-duality is readily apparent

by examining the model in terms of defect variables, which interchanges the role of

the coupling and field terms. This should be viewed in exact analogy with the 1D

Ising chain, which similarly exhibits a T = 0 quantum phase transition but fails to

have a thermodynamic phase transition.

192

The transition at h = 1 is a spontaneous symmetry breaking transition in which

all 2k fractal symmetries are spontaneously broken at once (although under general

perturbations they do not have to all be broken at the same time). Numerical ev-

idence [151] suggests a first order transition. If one were to allow explicitly fractal

symmetry breaking terms in the Hamiltonian (Z-fields, for example) then it is possible

to go between these two phases adiabatically. Thus, as long as the fractal symmetries

are not explicitly broken in the Hamiltonian, these two phases are properly distinct

in the usual picture of spontaneously broken symmetries. In the following, we will

only be discussing ground state (T = 0) physics.

8.4 Fractal symmetry protected topological phases

Rather than the trivial paramagnet and spontaneously symmetry broken phases, we

may also generate cluster states [116] which are symmetric yet distinct from the trivial

paramagnetic phase. These cluster states have the interpretation of being “decorated

defect” states, in the spirit of Ref [152], as we will demonstrate. These fractal symme-

try protected topological phases (FSPT) are similar to recently introduced subsystem

SPTs [11], and were hinted at in Ref [41]. In contrast to the subsystem SPTs, however,

there is nothing here analogous to a “global” symmetry — the fractal symmetries are

the only ones present!

8.4.1 Decorated defect construction

To describe these cluster Hamiltonians, we require a two-site unit cell, which we will

refer to as sublattice a and b. For the unit cell (i, j) we have two sets of Pauli operators

Z(a)ij , Z

(b)ij , and similarly X

(a/b)ij and Y

(a/b)ij . Our previous polynomial representation is

193

extended as

Z

α

β

= Z

∑

ij c(a)ij x

iyj

∑ij c

(b)ij x

iyj

=

∏

ij

(Z

(a)ij

)c(a)ij(Z

(b)ij

)c(b)ij(8.25)

and similarly for X(·) and Y (·). This notation is easily generalized to n spins per

unit cell, represented by n component vectors.

Our cluster FSPT Hamiltonian is then given by

HFSPT = −∑

ij

Z

xiyj(1 + f y)

xiyj

−

∑

ij

X

xiyj

xiyj(1 + fy)

−hx∑

ij

X

xiyj

0

− hz

∑

ij

Z

0

xiyj

(8.26)

which consists of commuting terms and is exactly solvable at h = hx = hz = 0, which

we will assume for now. There is a unique ground state on a torus (regardless of

the number of symmetries). The ground state is short range entangled, and may

be completely disentangled by applications of controlled-Z (CZ) gates at every bond

between two different-sublattice sites that share an interaction, as per the usual cluster

states — however, this transformation does not respect the fractal symmetries of this

model. These fractal symmetries come in two flavors, one for each sublattice:

Z(a)2 : S(a)(q(x)) = X

q(x)F(x, y)

0

Z(b)2 : S(b)(q(x)) = Z

0

q(x)F(x, y)

(8.27)

where we have assumed an infinite plane with F(x, y) as in Eq 8.22, and q(x) may

be any polynomial.

194

X

X

X

X

Z

Z

ZZ

i

j

(a) (b)X

X X

XX

X X X X

Z Z Z Z Z

Z Z

Z Z

ZZZZ

Z Z Z Z

Z

ZZ

Z

Z

Z

Z

Z

X

X

(c)g1 :

g2 :

g1,<:

j0

Figure 8.4: In (a), we show how to place the Sierpinski FSPT on to the honeycomblattice naturally. The orange circle is the unit cell, and blue/red sites correspond tothe a/b sublattice sites. The interactions involve four spins on the highlighted trianglestriangles. In (b), we show the sites affected by a choice of symmetry operations on

an infinite plane. The large circles are those affected by a particular Z(a/b)2 type

symmetry (Eq 8.27). In (c), we perform a symmetry twist on the Sierpinski FSPTon a 7 × 7 torus. The chosen symmetries g1 (g2) corresponds to operations on allspins highlighted by a large blue (red) circle. The green triangles correspond to termsin the twisted Hamiltonian Htwist(g1) that have flipped sign. The charge responseT (g1, g2) = −1 is given by the parity of red circles that also lie in the green triangles,and is independent of where we make the cut j0.

The picture of the ground state is as follows. Working in the Z(a), Z(b) basis, notice

that if Z(b)ij = 1, the first term in the Hamiltonian simply enforces the Z

(a)ij spins to

follow the standard CA evolution. At locations where Z(b)ij = −1, there is an “error”,

or defect, of the CA, where the opposite of the CA rule is followed. The second

term in the Hamiltonian transitions between states with different configurations of

such defects. The ground state is therefore an equal superposition of all possible

configurations. The same picture can also be obtained from the X(a), X(b) basis, in

terms of the CA rules acting on the X(b)ij spins.

Sierpinski FSPT As a particularly illustrative example, let us consider the FSPT

generated from the Sierpinski rule. The resulting model is the “decorated defect” NM

195

paramagnet, which we refer to as the Sierpinski FSPT. The Hamiltonian is given by

HSier-FSPT = −∑

ij

Z(a)ij Z

(a)i,j−1Z

(a)i−1,j−1Z

(b)ij −

∑

ij

X(b)ij X

(b)i,j+1X

(b)i+1,j+1X

(a)ij (8.28)

It is particularly enlightening to place this model on a honeycomb lattice, as shown

in Fig 8.4a. Fig 8.4b shows the action of two symmetries as an example.

We may then redefine Z(b)ij ↔ X

(b)ij , after which the Hamiltonian takes the partic-

ularly simple form of a cluster model

Hcluster = −∑

s

Xs

∏

s′∈Γ (s)

Zs′ (8.29)

where s = (i, j, a/b) labels a site on the honeycomb lattice and Γ (s) is the set of its

nearest neighbors. However, we will generally not use such a representation. Note

that this model is isomorphic to the 2D fractal SPT obtained in Ref [153].

Fibonacci FSPT Our other example is the Fibonacci FSPT. The Hamiltonian

takes the form

HFib-FSPT = −∑

ij

Z(a)ij Z

(a)i−1,j−1Z

(a)i,j−1Z

(a)i+1,j−1Z

(b)ij (8.30)

−∑

ij

X(b)ij X

(b)i+1,j+1X

(b)i,j+1X

(b)i−1,j+1X

(a)ij (8.31)

which we illustrate in Fig 8.5a. Unlike with the Sierpinski FSPT, this model does not

have as nice of an interpretation of being a cluster model with interactions among

sets of nearest neighbors on some simple 2D lattice.

8.4.2 Symmetry Twisting

To probe the nontriviality of the FSPT symmetric ground state, we may place it on

a torus and apply a symmetry twist to the Hamiltonian, and observe the effect in the

196

charge of another symmetry [154, 155, 156, 157]. To be concrete, let Htwist(g1) be the

g1 symmetry twisted Hamiltonian. The g2 charge of the ground state of Htwist(g1)

relative to its original value tells us about the nontriviality of the phase under these

symmetries. That is, let

〈g2〉g1 = limβ→∞

1

ZTr[g2e−βHtwist(g1)

](8.32)

with Z the partition function, then, we define the charge response

T (g1, g2) = 〈g2〉g1/〈g2〉1 (8.33)

where 〈g2〉1 is simply the g2 charge of the ground state of the untwisted Hamiltonian.

On a torus, we may twist along either the horizontal or vertical direction — here we

first consider twisting along the vertical direction.

Let us be more concrete. Take the FSPT Hamiltonian (Eq 8.26) on an Lx × Lytorus, and let k be the number of independent symmetries of the type Z(a)

2 (which is

also the same as for Z(b)2 ). We assume Lx, Ly have been chosen such that k > 0. The

total symmetry group of our Hamiltonian is therefore(Z(a)

2 × Z(b)2

)k. Let us label the

2k generators for this group

S(a)α = X

q

(a)α (x)

∑Ly−1l=0 (fy)l

0

; S(b)

α ) = Z

0

q(b)α (x)

∑Ly−1l=0 (f y)l

(8.34)

where 0 ≤ α < k and q(a/b)α (x) have been chosen such that the set of S

(a/b)α are all

independent. Recall from Section 8.2.3 that only certain such polynomials q(x) are

allowed on a torus.

To apply a g-twist, we first express the Hamiltonian as a sum of local terms

HFSPT =∑

ij Hij. We then pick a horizontal cut j = j0, dividing the system between

197

j < j0 and j ≥ j0. For each term that crosses the cut, we conjugate Hij → g<Hijg−1< ,

where g< is the symmetry action of g restricted to j < j0. For an Ising system, this

will simply have the effect of flipping the sign of some terms in the Hamiltonian. The

resulting Hamiltonian is Htwist(g).

To understand which terms in the Hamiltonian change sign under conjugation,

consider the choice of symmetry g1 in Fig 8.4c, which consists of flipping all spins in

the large blue (dark and transparent) circles. Restricting g1 to j < j0 leaves g1,<,

flipping only spins in the dark circles. Conjugating by g1,< results in the terms in the

green triangles appearing in Htwist(g1) with a relative minus sign.

Doing this explicitly for a symmetry S(a)α , we find that the incomplete symmetry

restricted to j < j0 is given by

S(a)α,< = X

q

(a)α (x)

∑j0−1l=0 (fy)l

0

(8.35)

The terms in the Hamiltonian that pick up a minus sign when conjugated with S(a)α,<

are exactly translations of the first term in HFSPT (Eq 8.26) given by the non-zero

coefficients of the commutation polynomial along j0: P = q(a)α (x)(fy)j0 . However, the

same twisted Hamiltonian may also be obtained by conjugating the entire HFSPT by

K(a)α = X

0

q(a)α (x)(fy)j0

(8.36)

such that Htwist(S(a)α ) = K

(a)α HFSPTK

(a)†α .

Next, we can compute the charge of another symmetry S(a/b)β in the ground state

of Htwist(S(a)α ). Without any twisting, the ground state is uncharged under all sym-

metries, 〈S(a/b)α 〉1 = 1. After the twist, none of S

(a)β will have picked up a charge

(as they commute with K(a)α ), but some S

(b)β may pick up a nontrivial charge if they

198

anticommute with K(a)α . Letting T (S

(a)α , S

(b)β ) = (−1)Tαβ , we have

Tαβ =

[q(a)α (x)(fy)j0 × q(b)

β (x)

Ly−1∑

l=0

(fy)l

]

x0y0

=[q(a)α (x)q

(b)β (x)

]x0

(8.37)

where we have used yLy = 1 and the definition of a symmetry on the torus, Eq 8.15.

As expected, the result is independent of our choice of j0, and it is also apparent that

T (g1, g2) = T (g2, g1) for any g1,g2. If we choose the same symmetry basis for both

sublattices, q(a)α (x) = q

(b)α (x), then we additionally get that Tαβ = Tβα.

Figure 8.4c is an illustration of this twisting calculation for the Sierpinski FSPT

on a 7 × 7 torus. Letting x0y0 label the unit cell in the top left of the figure, g1 is

an (a) type symmetry with q(a)(x) = x3 + x4 and g2 is a (b) type symmetry with

q(b)(x) = x4 +x5. Then, Eq 8.37 gives T (g1, g2) = −1, which can be confirmed by eye

in the figure.

The exact same procedure may also be applied for twists across the horizontal

direction, which will provide yet another set of independent relations between the

symmetries (but will not have as nice of a form).

8.4.3 Degenerate edge modes

Upon opening boundaries, the ground state manifold becomes massively degenerate.

Away from a corner, we will show that these degeneracies cannot be broken by local

perturbations as long as the fractal symmetries are all respected, much like in the

case of SPTs with one-dimensional subsystem symmetries [11].

Let us review the open slab geometry from Sec 8.2.5 for the FSPT. We take

the system to be a rectangle with Lx × Ly unit cells, such that we are restricted to

x0≤i<Lxy0≤j<Ly . as before, let x−pmin , xpmax , be the smallest and largest powers of x

in f (and let pmin/max = 0 if they would be negative). The total symmetry group is

199

(Z(a)

2 × Z(b)2

)kwith

k = Lx +R(Ly − 1); R = pmin + pmax (8.38)

and we assume Lx > R (otherwise there are no allowed terms in the Hamiltonian at

all!). A Z(a)2 type symmetry acts as

∏X

(a)ij on a subset of unit cells, and a particular

symmetry is fully specified by how it acts on the top row xiy0, the band xi<pmaxyj (on

the left side), and the band xi≥Lx−pminyj (on the right side). A Z(b)2 type symmetry

acts as∏Z

(b)ij and a particular one is fully specified in a similar manner, but spatially

inverted (top↔bottom, left↔right).

On the open slab, we take our Hamiltonian (Eq 8.26) with h = 0 on the infinite

plane and simply exclude terms that contain sites outside of the sample. For each

term with shift xiyj that are excluded, but for which the unit cell xiyj is still in the

system, we lose a constraint on the ground state manifold and hence gain a two-fold

degeneracy. The number of terms excluded is given by exactly the same counting as

before. Along the top (bottom) edge, there is one excluded Z (X) term per unit cell.

Along the left edge, there are pmax Z terms excluded and pmin X terms, for a total of

R excluded terms per unit cell, and similarly for the right edge. Hence, there are a

total of 22k ground states, coming from a 2R-fold degeneracy per unit cell along the

left/right edges, and 2-fold degeneracy per unit cell along the top/bottom (with some

correction for overcounting).

For each excluded Z term at xiyj, we may define a set of three Pauli operators,

X (a)ij = X

0

xiyj

; Z(a)

ij = Z

xiyj(1 + f y)

xiyj

Y(a)ij = Z

xiyj(1 + f y)

0

Y

0

xiyj

(8.39)

200

and for each excluded X term at xiyj we may similarly define

X (b)ij = X

xiyj

xiyj(1 + fy)

; Z(b)

ij = Z

xiyj

0

Y(b)ij = X

0

xiyj(1 + fy)

Y

xiyj

0

(8.40)

which are then truncated to remove operator acting on sites outside the system. We

will call such operators “edge” Pauli operators. There are 2k such sets of edge Pauli

operators, one for each excluded term. It may readily be verified that X (a/b)ij , Y(a/b)

ij ,

and Z(a/b)ij satisfy the Pauli algebra while being independent of and commuting with

every term in the Hamiltonian and each other at different sites. They therefore form

a Pauli basis for operators which act purely within the 22k dimensional ground state

manifold.

In principle, any local perturbation, projected on to the ground state manifold,

will have the form of being some local effective Hamiltonian in terms of these edge

Pauli operators, and may break the exact degeneracy. However, we wish to consider

only perturbations commuting with all fractal symmetries. To deduce what type of

edge Hamiltonian is allowed, we must find out how our many symmetries act in terms

of these edge operators.

A Z(a)2 type symmetry acts as

∏X

(a)ij (Eq 8.27) — these therefore have the po-

tential to anticommute with some of Z(a/b)ij and Y(a/b)

ij . Thus, any Z(a)2 symmetry,

expressed in terms of these edge Pauli operators, will consist of a product of X (a/b)ij

edge operators. Similarly, one may deduce that any Z(b)2 symmetry will act as a

product of Z(a/b)ij edge operators.

We claim that it is always possible to find a particular fractal symmetry that acts

locally on one edge in any way (but it may extend non-trivially into the bulk and act

in some way on the other boundaries). For example, for any (i0, j0) on the left edge,

201

there exists a Z(a)2 symmetry which acts only as X (a/b)

i0j0on the left edge, and there is

also a Z(b)2 symmetry which acts only as Z(a/b)

i0j0on the left edge (although their action

on the other edges may be complicated). There is no non-trivial operator acting on a

single edge that commutes with both X and Z, and therefore we are prohibited from

adding anything non-trivial to the effective Hamiltonian on this edge which therefore

guarantees that no degeneracy can be broken while respecting all fractal symmetries.

Note that we don’t even have the possibility of spontaneous symmetry breaking at the

surface, even simple ZZ couplings along the edge violate the symmetries. The only

way the ground state degeneracy may be broken without breaking the symmetry is by

terms which couple edge Paulis along different edges; these terms are either non-local,

or located at a corner of the system.

A Z(a)2 symmetry g1 and Z(b)

2 symmetry g2 which acts as X and Z respectively

on the same site of a particular edge is said to form a projective representation of

Z(a)2 ×Z(b)

2 on that edge. That is, a faithful representation of Z(a)2 ×Z(b)

2 with generators

g1, g2, would have (g1g2)2 = 1. However, if we look at the action on this particular

edge, then we have that (gedge1 gedge

2 )2 = (X Z)2 = −1. Since we know that as a whole

g1 and g2 must commute, the action of g1 and g2 on some other far away edge must

again anticommute (to cancel out the −1 from this edge). Small manipulations of

the edges (such as adding or removing sites) therefore cannot change the fact that

the actions of g1 and g2 are realized projectively on this edge.

Near particular corners, some symmetries may act essentially locally. As the sym-

metry as a whole must commute with all other symmetries, nothing prevents the

addition of the full symmetry itself as a term in the effective Hamiltonian when it is

local. For example, when h 6= 0 there will be terms appearing in the effective Hamil-

tonian at finite order in perturbation theory near such corners, which commute with

every symmetry (as the term is a symmetry itself, and all symmetries commute). The

202

Z

X

Z

Z

Z

Z Z

Z

Y

X

X

X

X X

Z

X

X

Y

X

X

XX

X

Z

Z

Z ZZ

X X

X

X

Z Z

Z Z

Z Z Z Z

Z

X

X

X

X

X

X

X

X X

X X

X

X X

X (a)

Z(a)

Y(a)

X (b) Z(b) Y(b)

i

j

(a) (b) (c)

Figure 8.5: (a) We illustrate the terms in the Hamiltonian for the Fibonacci FSPT(Eq 8.26 with f = x−1 + 1 + x). The model is defined on a square lattice, with atwo-site unit cell (circled), a (blue) and b (red). The two terms in the Hamiltonianat h = 0 are illustrated in the two triangles. Also shown are the edge Pauli operatorsalong the left edge. (b) We show a family of symmetries on a 10× 10 slab. The blackoutlined circles represent the band of R = 2 unit cells on which we fix the action ofthe symmetry so that it acts only as X (b)

0,7 on the left edge in this case (with (0, 0)being the top left unit cell). This fixes how the symmetry must act on the top andsome of the right edge (gray outlined circles), but there is still some freedom alongthe remaining sites on the right edge (yellow question marks), which will determinehow it acts on the remaining sites (transparent orange circles). There are 2Lx−R = 28

distinct symmetries (corresponding to the 8 question marks) satisfying our constraint.

(c) We also show the family of symmetries which act as Z(b)0,7, and therefore forms a

projective representation with the symmetry shown in (b) on the left edge. Note thatthese symmetries generally have some non-trivial action along the other edges.

magnitude of such terms will decay exponentially away from a corner, however, and

therefore we still have an effective degeneracy per unit length along the boundaries.

Local action of symmetries on edges

To prove our claim that there is always a symmetry which acts locally along an edge,

let us first consider finding a Z(a)2 symmetry which acts locally on an edge as X (a/b)

i0j0.

The ability to find a Z(b)2 symmetry acting locally as well then follows by symmetry.

Such a symmetry will act locally in some way on the edge, but extend into the bulk

in a non-trivial way. Note that there is no “most natural basis” for these symmetries,

unlike in the case of integer d subsystem symmetries [11].

203

Top edge Localizing on the top edge is simple. The only possibilities on the top

edge are X (a)i,0 operators (away from the corners). Any symmetry containing only X

(a)i0,0

along y0 will act only as X (a)i0,0

. As we are free to choose how the symmetry acts on

the top edge, finding a symmetry which does this is straightforward.

Bottom edge Along the bottom edge (again, away from the corners), the only

possibility is for a symmetry to act as X (b)i,Ly−1. Any symmetry containing only X

(a)i0,Ly−1

along xiyLy−1 will act as only X (b)i0,Ly−1 on the bottom edge. There is always such a

symmetry that does this (as we showed for the infinite plane (Sec 8.2.4) one can

always find a history for any CA state).

Left/right edge Along the left/right edges, things are slightly trickier. Let us look

at only the left edge for now. A symmetry may act as X (a)ij for 0 ≤ j < pmax, or as

X (b)ij for 0 ≤ j < pmin. Per unit cell along the left edge, there are 2pmin+pmax = 2R

possible actions for our symmetry. To fully isolate a single edge Pauli operator, we

thus need at least 2R degrees of freedom in our symmetry, but as described earlier

we are only free to choose the action of the symmetry within a band pmax along the

left edge, which is generally less than R. As it turns out, however, we are actually

free to specify the action of the symmetry within the whole band of width R along

the left edge, but at the cost of some of our freedom to choose how it acts along

the top/right edges. One may verify that there are 2RLy choices to make, which is

certainly less than the 2k Z(a)2 symmetries. As each term in the Hamiltonian is R+ 1

unit cells wide, this choice does not inherently force a violation of any terms in the

Hamiltonian, and a family of symmetries with this action on the left band can be

constructed. There are 2k−RLy = 2Lx−R such symmetries which are localized along

the left edge. A similar argument holds for the right edge.

Figure 8.5(right) shows the family of Z(a)2 symmetries localized to act as a X (b) on

the left edge, for the Fibonacci FSPT (Eq 8.31), whos terms are shown in Fig 8.5(left).

204

The freedom to choose how the symmetry acts on the right edge here exactly corre-

sponds to the 2Lx−R distinct symmetries with the specified action on the left edge.

We choose to show the Fibonacci FSPT here rather than the Sierpinski FSPT, as the

latter has R = 1 and is straightforward.

8.4.4 Excitations

On the infinite plane, the lowest lying excitations are strictly immobile. They are

therefore fractons protected by the set of fractal symmetries.

Take h = 0, the lowest lying excited states consist of excitations of a single term

in the Hamiltonian, say the Z term at site x0y0. This excited state can be obtained

by acting on the ground state with X(b)0,0. One may alternatively think in terms of

symmetries. Take an independent set of symmetries g(a/b)α of the form Eq 8.27 with

the basis choice q(a/b)α = xα. We find that this excited state is uncharged, 〈g(a/b)

α 〉 = 1,

with respect to all symmetries except g(b)0 , for which it has −1 charge. In fact, the

only state with a single excitation with 〈g(b)0 〉 = −1 is this one with the excitation at

the origin.

Let us consider the block of the Hamiltonian with symmetry charges 〈g(b)α 〉 =

(−1)dα . The blocks containing states with single fractons will have

∞∑

α=−∞dαx

α = xif j (8.41)

for which the excitation is strictly localized at site xiyj. The excitation may move

away from xiyj, but at the cost of creating additional excitations as well, such that all

symmetries maintain the same charge. If one allows breaking of the fractal symme-

tries, then these charges are no longer conserved and nothing prevents the excitation

from moving to a different site.

205

On lattices with different topology, these fractons may not be strictly immobile.

For example, on a torus, depending on the symmetries, a fracton may be able to

move to some subset of other sites (or all other sites, if there are no symmetries at

all). However, such hopping terms are exponentially suppressed with system size.

In fact, for the Sierpinski FSPT on a torus with no symmetries, it is actually easier

perturbatively to hop a fracton a large power of 2 away than it is to hop a short

distance (mimicking some form of p-adic geometry with p = 2).

On an open slab, the ground state manifold is degenerate and all charge assign-

ments are possible in the ground state, protected by the symmetries. Therefore, a

fracton may be created, or moved, in any way. However, the amplitude for doing

so will decay exponentially away from the edges, and certain processes may only be

possible near certain types of edges or corners. The possibilities will depend on the

details of the model.

8.4.5 Duality

Here we outline a duality that exist generally for these models, which maps the

FSPT phase to two copies of the spontaneous symmetry broken phase of the quantum

Hamiltonian in Sec 8.3. This duality involves non-local transformations and maps the

22k ground states of the FSPT on the open slab to the 22k symmetry breaking ground

states of the dual model.

206

This duality is most naturally described on an Lx×Ly cylinder (with xLx = 1) or

slab. Let us define new Pauli operators Z(·) and X(·) as

Z

0

1

= Z

0

1

; X

1

0

= X

1

0

Z

1

0

= Z

1

1 + f y + (f y)2 + . . .

X

0

1

= X

1 + fy + (fy)2 + . . .

1

(8.42)

and translations thereof. It can be readily verified that the latter two commute, and

as a whole the set of these operators satisfy the correct Pauli algebra. The fractal

symmetries only involve operators in line 8.42, and so are unchanged. In terms of

these operators, we have

Z

1 + f y

0

= Z

1 + f y

1

; X

0

1 + fy

= X

1

1 + fy

(8.43)

so the Hamiltonian HFSPT (Eq 8.26) becomes two decoupled copies of HQuantum

(Eq 8.24) with their own set of symmetries.

From this, it follows that the order parameter measuring long-range order in

HQuantum, C(`) (Eq 8.23), maps on to a fractal order parameter in our original basis

CFSPT(`) = Z

1 + (f y)`

0

= Z

1 + (f y)`

1 + f y + · · ·+ (f y)`−1

(8.44)

which is pictorially shown for the Sierpinski FSPT in Figure 8.6, and approaches a

constant in the FSPT phase, or zero in the trivial paramagnet, as ` = 2l → ∞. By

207

Z

Z

ZZ

Z Z

Z Z Z Z

Z

Z Z

Z

ZZ

ZZ

Z

ZZ

Z

Z

Z

Z

Z

ZZZZ

Figure 8.6: Illustration of the fractal order parameter CFSPT(`) for detecting theFSPT phase of the Sierpinski FSPT, for ` = 23. The operator is a product of Z onthe highlighted sites.

the self-duality of HQuantum, we also know the FSPT to trivial transition happens at

exactly h = 1.

Finally, this duality allows us to determine the full phase diagram even as hx 6=

hz. Keeping hx small and making hz large, one of the HQuantum is driven into its

paramagnetic phase where spins are polarized as Z(b)ij = 1. The Hamiltonian HFSPT

then looks like a single HQuantum, and therefore has spontaneously symmetry broken

ground states. By the duality transformation, we know this transition happens at

exactly hz = 1. The phase diagram is summarized in Fig 8.7(left).

8.5 Three dimensions

Here, we briefly examine the possible physics available in higher dimension. We

consider our symmetry-defining CA in 3D in two ways: via one 2D CA, or two 1D

CA. The first will have similar properties to our earlier models, while the latter in

certain limits also lead to exotic fractal spin liquids introduced by Yoshida [144]

and Haah [40], and may be thought of as (Type-II [10]) symmetry-enriched fracton

topologically ordered (FSET) phases.

208

8.5.1 One 2D cellular automaton

A 2D CA has a two-dimensional state space, combined with one time direction.

The state of such a CA may be straightforwardly represented by a polynomial in two

variables, st(x, z), where the state of the (i, k)th cell is given by the coefficient of xizk.

The update rule is given as a two variable polynomial f(x, z), such that st+1 = fst as

before. Two dimensional CA also result in a rich variety of fractal structures [158].

The classical Hamiltonian takes the form

H1CA = −∑

ijk

Z(xiyjzk[1 + f(x, z)y]) (8.45)

with symmetries on the semi-infinite system (with yj≥0) given by

S(q(x, z)) = X(q(x, z)[1 + fy + (fy)2 + . . . ]) (8.46)

which commutes with H1CA everywhere. On an infinite system, an inverse evolution

f−1 may be defined analogous to Eq 8.21 and the symmetry takes the form

S(q(x, z)) = X(q(x, z)F(x, y, z)) (8.47)

with

F(x, y, z) = · · ·+ (f−1(x, z)y)−2 +f−1(x, z)y+ 1 +f(x, z)y+ (f(x, z)y)2 + . . . (8.48)

The discussion of Sec 8.3 and 8.4 may then be generalized in a straightforward manner.

The phase diagram is exactly the same as in 2D, given by Fig 8.7(left).

209

As an example model, consider the Sierpinski Tetrahedron model, given by the

update rule f(x, z) = 1 + x+ z. The Hamiltonian is given by

HSier-Tet = −∑

ijk

Zi,j,kZi,j−1,kZi−1,j−1,kZi,j−1,k−1 (8.49)

The fractal structure of the symmetries for this model are Sierpinski Tetrahedra,

with Hausdorff dimension d = 2. The quantum model may be constructed which

exhibit the same properties: self-duality about h = 1, spontaneous fractal symmetry

breaking, and instability to non-zero temperatures. A cluster FSPT version may also

be constructed, with the Hamiltonian

HSier-Tet-FSPT = −∑

ijk

Z(a)i,j,kZ

(a)i,j−1,kZ

(a)i−1,j−1,kZ

(a)i,j−1,k−1Z

(b)i,j,k

−∑

ijk

X(b)i,j,kX

(b)i,j+1,kX

(b)i+1,j+1,kX

(b)i,j+1,k+1X

(a)i,j,k (8.50)

This cluster FSPT also has the nice interpretation of being the cluster model (Eq 8.29)

on the diamond lattice. In the presence of an edge, terms in the Hamiltonian must

be excluded leading to degeneracies, and in exactly the same way as in 2D one finds

these degeneracies along a surface cannot be gapped, thus leading to a 2O(L2) overall

symmetry protected degeneracy for an open system.

8.5.2 Two 1D cellular automata

Symmetries defined through two 1D CA allow for a wide variety of possibilities. This

may be thought of as evolving a 1D CA through two time directions, with potentially

different update rules along the two time directions. Let the state of the 1D CA at

time (t1, t2) be represented by a polynomial st1t2(x). The update rules along the two

time directions are given as two polynomials f1(x) and f2(x), with st1+1,t2 = f1(x)st1,t2

and st1,t2+1 = f2(x)st1,t2 . Interpreting the y, z, directions as the t1, t2, directions, the

210

hx

hz1

1

FSPT

Z(b)2 SSB

Z(a)2 SSB

Trivial

hx

hz

FSPT

Z(a)2 FSET

Z(a)2 SSB

Trivial

???

Figure 8.7: (left) Phase diagram of our 2D or 3D FSPT models generated by one CA,under hx/z ≥ 0 perturbations. Possible phases include the FSPT phase symmetric

under all Z(a)2 and Z(b)

2 symmetries, two spontaneous symmetry broken (SSB) phaseswhere either of the two types of symmetries are spontaneously broken, and the trivialparamagnetic phase. (right) Sketch of the phase diagram for the 3D models withsymmetries generated by two 1D CA. There exists the FSPT phase at small hx/z, a

SSB phase at large hz, a fracton topologically ordered phase enriched with with Z(a)2

symmetry (FSET) at large hx, and a trivial phase at both large hx and hz. For thismodel, we do not know what the phase diagram looks like outside of these limits.

classical 3D Hamiltonian takes the form

H2CA = −∑

ijk

Z(xiyjzkα)−∑

ijk

Z(xiyjzkβ)

= −∑

ijk

Z(α)−∑

ijk

Z(β) (8.51)

where α = 1 + f1y and β = 1 + f2z are defined, and in the second line for notational

convenience we have suppressed the xiyjzk factor, when summation over translations

is apparent (and we will continue to do so). The fractal symmetries on a semi-infinite

system (with xiyj≥0zk≥0 are of the form)

S(q(x)) = X(q(x)[1 + f1y + (f1y)2 + . . . ][1 + f2z + (f2z)2 + . . . ]

)(8.52)

211

Z Z

Z

Z Z

Z

Z

Z Z

X XX

X

X

X

X

X

abc

zk

yj

xi

Figure 8.8: The first three terms in the 3D FSPT Hamiltonian HFSPT (Eq 8.54)generated from two CA, using f1 = 1 + x the Sierpinski rule and f2 = x + 1 + x theFibonacci rule. There are three spins on each site of the cubic lattice, labeled by a(blue), b (red), and c (green). Terms are composed of products of X and Z Paulioperators as shown. The Hamiltonian is a sum of translations of these terms.

which can be readily verified to commute with everything in the Hamiltonian. On an

infinite system some inverse may again be defined and the symmetry takes the form

S(q(x)) = X(q(x)F1(x, y)F2(x, z)) (8.53)

with F1/2 each defined as in Eq 8.22 with f1/2.

The decorated defect construction starting from H2CA results in the following

Hamiltonian, with three spins per unit cell, on which we have operators Z(a/b/c)ij and

X(a/b/c)ij ,

HFSPT = −∑

ijk

Z

α

1

0

−∑

ijk

Z

β

0

1

−∑

ijk

X

1

α

β

−∑

ijk

hxX

1

0

0

+ hzZ

0

1

0

+ hzZ

0

0

1

(8.54)

which is illustrated in Fig 8.8, for f1 = 1 + x and f2 = x + 1 + x (the Sierpinski-

Fibonacci model). The first three terms all mutually commute, and hx, hz are small

212

perturbations. The symmetries come in three types: first, we still have the original

symmetries

Z(a)2 : S(a)(q(x)) = X

q(x)F1(x, y)F2(x, z)

0

0

(8.55)

but now the remaining symmetries are more complicated, which arises because there

is a further local operator that commutes with the first three terms in HFSPT, given

by

Bijk = Z

xiyjzk

0

β

α

(8.56)

Due to the existence of Bijk, given any symmetry operation S, BijkS is also a valid

symmetry. Thus, these should be thought of as higher form fractal symmetries [159].

Consider the analogy with, say, a 1-form symmetries in 3D: these are symmetries

which act along a 2 dimensional manifold which may be deformed by local operations.

Here, we have the symmetry operations acting on only b or only c sublattice sites

which may be made to live on a single plane,

Z(b)2 : S(b)(q(x, z)) = Z

0

q(x, z)F1(x, y)

0

Z(c)2 : S(c)(q(x, y)) = Z

0

0

q(x, y)F2(x, z)

(8.57)

213

but we are also free to deform such symmetries using products of Bijk. Such higher

form fractal symmetries are an interesting subject by themselves, and we leave a more

thorough investigation as a topic for future study.

One may confirm that when hx = hz = 0, all these symmetries are products of

terms in the Hamiltonian, and therefore must have expectation value 1 in the ground

state. As every term is independent, and there are three terms that must be satisfied

per unit cell of three sites, the ground state is unique. This model in fact describes

an FSPT protected by the combination of the “global” fractal symmetries Z(a)2 , along

with the set of higher form fractal symmetries Z(b/c)2 . To see this, one may examine

the boundary theory. Let’s consider the simplest case of f1 = f2 = 1 + x the double

Sierpinski. On the top surface, with edge Pauli operators Z,X , one finds that Z(a)2

acts as a 2D Sierpinski fractal symmetry S(a) =∏X , while the Z(b/c)

2 symmetries

may be chosen to act as Z on a single site. Thus, the only Hamiltonian we can write

down on the surface must be composed of Z (to commute with a local Z) and must

commute with the fractal symmetry. The only possibility is therefore the classical

Hamiltonian (as in Eq 8.10), which exhibits spontaneous fractal symmetry breaking

in the ground state. Thus, the surface is non-trivial and must either be gapless or

spontaneous symmetry breaking.

Figure 8.7(right) shows a sketch the phase diagram for this model. Increasing

hx/z drives this model out of the FSPT phase. If we increase only hz while keeping

hx small, we arrive at the spontaneously fractal symmetry broken phase like in the

2D FSPT. Increasing both hx and hz too large will result in the trivial paramagnetic

phase. However, if we only increase hx while keeping hz small, the system enters into

a symmetric fracton topologically ordered phase, which is the subject of the following

discussion.

214

8.5.3 Connection to fracton topological order

The decorated defect approach of the previous sections may be thought of alterna-

tively as the following process:

1. Start with a classical Hamiltonian and some symmetries involving flipping some

spins

2. Introduce additional degrees of freedom at each site and couple them to the

interaction terms via a cluster-like interaction (this is exactly what one would

get following the gauging procedure of Refs [41, 10], and adding the gauge

constraint as a term in the Hamiltonian).

3. The resulting theory still has the original symmetries, along with some addi-

tional symmetry which we may define acting on the new spins, which we take

to be the defining symmetries our model.

4. Perturbations respecting these symmetries may then be added to the Hamil-

tonian (note these may break the gauge constraint from earlier: we are now

interpreting both matter and gauge fields as physical).

Most of our models, except the preceding one, were special under this gauging

procedure as they allowed for no local gauge fluctuations terms and exhibited a self-

duality between the topological and trivial phases. As we will show, in 3D with

symmetries defined by two 1D CA, gauge fluctuations are allowed (these are the

Bijk operators we found in Eq 8.56) and there is a phase in which these models

exhibit fracton topological order. They may be thought of as the simplest fractal

symmetry enriched topological (FSET) phases (this possibility was already hinted at

in Ref [41]). The phenomenology of the resulting topological orders are the same as

those of the Yoshida fractal codes [144]. The Z(a)2 symmetry will serve the purpose

215

of the enriching symmetry, while the other symmetries will have the interpretation of

being logical operators for the underlying Yoshida code.

To avoid complications, let specialize to an L× L× L 3-torus with fL1 = fL2 = 1

(xL = yL = zL = 1). The symmetries in this case are given by Eq 8.55 and 8.57, but

with F1 =∑L

l=0(f1y)l and F2 =∑L

l=0(f2z)l instead of F1, F2, with q still arbitrary.

There are L independent Z(a)2 symmetries, and 2L independent higher-form Z(b/c)

2

symmetries. An independent basis for these symmetries are, for α = 0 . . . L−1, given

by

S(a)α = X

xαF1(x, y)F2(x, z)

0

0

(8.58)

and

S(b)α = Z

0

xαF1(x, y)

0

; S(c)α = Z

0

0

xαF2(x, z)

(8.59)

All symmetries may be written as products of these and Bijk (as S(b/c) are higher

form fractal symmetries).

The fracton topologically ordered phase corresponds to the limit in which we take

hx in Eq 8.54 to be large. Expanding about this limit, the Hamiltonian looks like

HFSET = −hx∑

ijk

X

1

0

0

−G

∑

ijk

X

1

α

β

−K

∑

ijk

Z

0

β

α

+ (perturbations)(8.60)

where we have now specified an energy scale G for the second term, the third term is

the leading order perturbative correction to the Hamiltonian, and we neglect all the

216

other perturbations. Fixing all X(a)ij = 1 results in exactly the Yoshida code

HYoshida = −∑

ijk

X

α

β

−

∑

ijk

Z

β

α

(8.61)

which exhibits a ground state degeneracy (with our geometry and choice of f1/2) of

2k with k = 2L.

From the perspective of the original FSPT, one finds that the charge of all the

S(a/b)α (Eq 8.59) in the ground state of this phase no longer has to be +1, but instead

may be ±1. These are exactly the logical operators of the Yoshida fractal code [144].

This transition may also be thought of as some kind of non-local spontaneous sym-

metry breaking of the higher form fractal symmetries Z(b/c)2 .

The ground state must still be uncharged under the Z(a)2 . We define the fracton

excitation as an excitation of only the first term in the HFSET (these are the relevant

charge excitations when G is large). Such an excitation may be created in multiplets

by (for example) an operator of the form

Z

1 + (f1y)`

1 + (f1y) + (f1y)2 + · · ·+ (f1y)`−1

0

(8.62)

which creates only excitations of the first term at locations given by the non-zero

coefficients in 1 + (f1y)`, and is a few-body creation operator whenever ` = 2l. A

single such excitation clearly carries charge −1 under some Z(a)2 symmetries. This

Hamiltonian therefore describes a fracton topologically ordered phase, enriched by

an additional Z(a)2 symmetry, and is a genuine FSET. In exactly the same way, a

relaxed Ising gauge theory may be interpreted as an SPT protected by a global Z2

217

and 1-form Z2 symmetry, and in a certain limit describe an SET phase enriched by a

global Z2 [29].

A single charge is immobile, as discussed in Sec 8.4.4, provided that f1 and f2

are not algebraically related, the same condition which implies the lack of a string-

like logical operator in the Yoshida code [144]. Finally, we note that Haah’s cubic

code [40] is isomorphic to this type of model, but with a second-order CA along one

time direction [144].

8.6 Conclusion

We have constructed and characterized a family of Ising Hamiltonians that are sym-

metric under symmetry operations which involve acting on a fractal subset of spins.

Fractal structures on a lattice are taken to be those defined by cellular automata with

linear update rules. We discuss some possible phases in systems with such symmetries.

These include the trivial symmetric and spontaneously symmetry broken phases

which are symmetric under a set of fractal symmetries. These fractal symmetries

together form the total symmetry group (Z2)k, where k will depend strongly on

system size and topology. We then construct non-trivial symmetric phases, FSPT

phases, via a decorated defect approach. For symmetries generated by a single CA,

the decorated defect construction leads to a family of cluster type Hamiltonians which

have a non-trivial gapped ground state under the symmetry group (Z2 × Z2)k of

fractal symmetries. We characterize such a phase by means of symmetry-twisting,

ungappable edge modes, and immobile excitations protected by the set of all fractal

symmetries.

In three dimensions, our construction leads to an FSPT protected by a combina-

tion of the usual fractal symmetry along with a higher form fractal symmetry. Aside

from the FSPT phase one also has the possibility of fracton topological order, en-

218

riched by the fractal Z2 symmetries. The topological order in these models are those

of the Yoshida fractal codes [144]. While maintaining our fractal symmetries, these

topologically ordered phases may be thought of as simple fractal symmetry enriched

topological phases (FSET), in which an elementary excitation is charged under the

fractal symmetries.

219

Chapter 9

Classification of 2D Fractal SPTs

9.1 Introduction

In this chapter, we view the results of the previous chapter more generally and ask:

what SPT phases are possible with a given fractal symmetry group? This chapter is

based on the paper

[30] T. Devakul, “Classifying local fractal subsystem symmetry-protected topologi-

cal phases”, Phys. Rev. B 99, 235131 (2019).

A classification of regular subsystem SPT phases (Chapter 6 and 7) relied on the

definition of a modified (weaker) equivalence relation between phases. The reason

this was needed in this case is due to the existence of “subsystem phases”: cases

where two states which differ along only a subsystem may belong to distinct phases

of matter. For instance, consider a D = 2 trivial symmetric state, but along some of

the (d = 1) subsystems, we place a 1D SPT (in such a way that all symmetries are

still respected). This, now, as a whole represents a non-trivial 2D phase of matter

protected by the subsystem symmetries, despite looking trivial in most of the bulk.

Furthermore, the existence of such phases means that in the thermodynamic limit

where system size is taken to infinity, there are an infinite number of subsystems,

220

and so an infinite number of possible phases. The problem with this is that it now

takes a subextensive (growing as O(L) in local systems of size L × L) amount of

information to convey exactly what phase a system is in, without assuming any form

of translation invariance. In Chapter 6, it was shown that there existed some intrinsic

global “data”, called β, which is insensitive to the presence of subsystem phases. All

the infinite phases of such a system could therefore be grouped into equivalence classes

and classified according to β. This classification has the nice interpretation of being

a classification of phases modulo lower-dimensional SPT phases, and is related to the

problem of classifying 3D (type-I) fracton topological orders modulo 2D topological

orders [21, 22, 23, 24, 25]. There is also a connection between this classification and

the appearance of a spurious topological entanglement entropy [123, 19, 120, 73, 74].

The key idea is that a new tool, in this case the modified phase equivalence relation,

was necessary in the classification of these subsystem SPT phases.

Our main finding in this chapter is that systems with fractal subsystem symmetries

are free from subsystem phases and the associated problems that existed for line-

like d = 1 subsystem SPTs. The key factor at play here is locality. Although the

total number of phases is still infinite (a result of the total symmetry group being

infinitely large), the vast majority of these phases are highly non-local and therefore

unphysical. If we fix a degree of locality (what we mean by this will be explained)

then the number of allowed phases remains finite in the thermodynamic limit. This

allows for the classification of phases directly, without needing to define equivalence

classes of phases like before (essentially due to the lack of any “weak” subsystem SPT

phases [11, 19]).

We first begin by reviewing some necessary preliminary topics in Sec 9.2 (the

notation is slightly different from the previous chapter). We then define fractal sym-

metries in Sec 9.3, and discuss the possible local SPT phases in Sec 9.4. In Sec 9.5

we give a explicit constructions for local models realizing an arbitrary local SPT

221

phase. Sec 9.6 deals with irreversible fractal symmetries and introduces the concept

of pseudo-symmetries and pseudo-SPTs. A summary and discussion of the results is

presented in Sec 9.8. The technical proof of the main result is included in Sec 9.9.

9.2 Preliminaries

9.2.1 Linear Cellular Automata

We first describe a class of fractal structures which determine the spatial structure

of all our symmetries in this work (see Ref [144] for a nice introduction to such

fractals and their polynomial representation). These fractal structures, which are

embedded on to a 2D lattice, are generated by the space-time evolution of a 1D cellular

automaton (CA). In particular, the update rule for this 1D cellular automaton will be

linear, translation invariant, local, and reversible. These terms will all be explained

shortly.

Let a(j)i ∈ Fp denote the state of the cell at spatial index i at time index j. Each

a(j)i can take on values 0, . . . , p − 1 for some prime p (p = 2 in the cases with Ising

degrees of freedom). We take periodic boundary conditions in i such that 0 ≤ i < Lx,

and define a(j)i+Lx≡ a

(j)i . The state of the full cellular automaton at a time j is given

by the vector a(j) ∈ FLxp with elements (a(j))i = a(j)i , We will use the notation vi to

denote the ith element of a vector v. Bold lowercase letters will denote vectors, while

bold uppercase letters will denote matrices.

The key ingredient of the cellular automaton is its update rule: given the state a(j)

at time j, how is the state a(j+1) at the next time step calculated? We will consider

only the family of update rules of the form

a(j+1)i =

kb∑

k=ka

cka(j)i−k (9.1)

222

where ck ∈ Fp is a set of coefficients only non-zero for ka ≤ k ≤ kb. Note that

all addition and multiplication is modulo p, following the algebraic structure of Fp.

Linearity refers to the fact that each a(j+1)i is determined by a linear sum of a

(j)i .

Thus, we may represent Eq 9.1 as

a(j+1) = Fa(j) (9.2)

where F ∈ FLx×Lxp is an Lx × Lx matrix with elements given by Fi′i = ci′−i. For a

given initial state a(0), the state at any time j ≥ 0 is simply given by a(j) = Fja(0).

Translation invariance refer to the fact that the update rules do not depend on

the location i, only on the relative location: Fi′i = Fi′+n,i+n. Locality means that Fi′i

is only non-zero for small |i′− i| of order 1. In our case, this means that |ka| and |kb|

should be small O(1) values. Finally, reversibility means that only one a(j) can give

rise to a a(j+1). In other words, the kernel of the linear map induced by F is empty,

and one can define an inverse F−1 (which will generically be highly non-local) such

that F−1F = FF−1 = 1. This is a rather special property which will depend on the

particular update rule as well as choice of Lx.

While we assume reversibility for much of this paper, we note that fractal SPTs

exist even when the underlying CA is irreversible. We call such phases pseudo-SPT

phases, and are discussed in Sec 9.6.

9.2.2 Polynomials over finite fields

Cellular automata with these update rules may also be represented elegantly in terms

of polynomials with coefficients in Fp. By this we mean polynomials q(x) over a

dummy variable x of the form

q(x) =

δq∑

i=0

qixi (9.3)

223

where each qi ∈ Fp, and the degree δq ≡ deg q(x) is finite. The space of all such

polynomials is denoted by the polynomial ring Fp[x]. A state a(j) of the cellular

automaton may be described by such a polynomial, a(j)(x),

a(j)(x) =Lx−1∑

i=0

a(j)i xi. (9.4)

In the case of periodic boundary conditions one should also work with the identity

xLx = 1.

Application of the update rule is expressed most simply in the language of poly-

nomials. Let us define f(x) to be a Laurent polynomial, i.e. f(x) = f(x)xka where

f(x) ∈ Fp[x] is a polynomial (and ka may be negative), given by

f(x) =

kb∑

k=ka

ckxk (9.5)

after which the update rule may be expressed simply as multiplication

a(j+1)(x) = f(x)a(j)(x) (9.6)

Given an initial state a(0)(x) then, the state at any future time is simply given by

a(j)(x) = f(x)ja(0)(x). We will assume cka and ckb are non-zero, and kb 6= ka (so that

f(x) is not a monomial).

The key property of such polynomials that guarantees fractal structures is that

for q(x) ∈ Fp[x], one has that

q(x)pn

= q(xpn

) (9.7)

224

also known as the “freshman’s dream”. Suppose we start off with the initial state

a(0)(x) = 1. After some possibly large time pn, the state has evolved to

a(pn)(x) = f(x)pn

= f(xpn

) =

kb∑

k=ka

ckxkpn (9.8)

which is simply the initial state at positions separated by distances pn. At time pn+1,

this repeats but at an even larger scale. Thus, the space-time trajectory, a(j)i , of this

cellular automaton always gives rise to self-similar fractal structures.

There are various other useful properties, one of which is that any polynomial

q(x) ∈ Fp[x] (without periodic boundary conditions) may be uniquely factorized up

to constant factors as

q(x) = q1(x)q2(x) . . . qn(x) (9.9)

where each qi(x) is an irreducible polynomial of positive degree. A polynomial is

irreducible if it cannot be written as a product of two polynomials of positive degree.

This may be thought of as a “prime factorization” for polynomials.

9.2.3 Projective Representations

The final topic which should be introduced are projective representation of finite

abelian groups. Bosonic SPTs in 1D are classified by the projective representations

of their symmetry group on the edge [110, 117]. Similarly, subsystem SPTs for which

the subsystems terminate locally on the edges (i.e. line-like subsystems) may also

be described by projective representations of a subextensively large group on the

edge [11, 19]. The same is true for fractal subsystem symmetries [29].

LetG by a finite abelian group. A non-projective (also called linear) representation

of G is a set of matrices V (g) for g ∈ G that realize the group structure: V (g1)V (g2) =

V (g1g2) for all g1, g2 ∈ G. A projective representation is one such that this is only

225

satisfied up to a phase factor,

V (g1)V (g2) = ω(g1, g2)V (g1g2) (9.10)

where ω(g1, g2) ∈ U(1) is called the factor system of the projective representation,

and must satisfies the properties

ω(g1, g2)ω(g1g2, g3) = ω(g1, g2g3)ω(g2, g3)

ω(1, g1) = ω(g1, 1) = 1

(9.11)

for all g1, g2, g3 ∈ G. A different choice of U(1) prefactors, V ′(g) = α(g)V (g) leads to

the factor system

ω′(g1, g2) =α(g1g2)

α(g1)α(g2)ω(g1, g2). (9.12)

for V ′(g). Two factor systems related in such a way are said to be equivalent, and

belong to the same equivalence class ω.

Suppose we have a factor system ω1(g1, g2) of equivalence class ω1, and a factor

system ω2(g1, g2) of class ω2. A new factor system can be obtained as ω(g1, g2) =

ω1(g1, g2)ω2(g1, g2), which is of class ω ≡ ω1ω2. This gives them a group structure:

equivalence classes are in one-to-one correspondence with elements of the second

cohomology group H2[G,U(1)], and exhibit the group structure under multiplication.

In the case of finite abelian groups, a much simpler picture may be obtained in

terms of the quantities

Ω(g1, g2) ≡ ω(g1, g2)

ω(g2, g1)(9.13)

which is explicitly invariant under the transformations of Eq 9.12. They have a nice

interpretation of being the commutative phases of the projective representation

V (g1)V (g2) = Ω(g1, g2)V (g2)V (g1). (9.14)

226

Ω(g1, g2) has the properties of bilinearity and skew-symmetry in the sense that

Ω(g1g2, g3) = Ω(g1, g3)Ω(g2, g3) (9.15)

Ω(g1, g2g3) = Ω(g1, g2)Ω(g1, g3) (9.16)

Ω(g1, g2) = Ω(g2, g1)−1 (9.17)

These properties mean that Ω(g1, g2) is completely determined by its value on all

pairs of generators of G. Suppose a1, a2 ∈ G are two independent generators with

orders n1, n2, respectively. Then, one can show that Ω(a1, a2)n1 = Ω(a1, a2)n2 = 1,

and so Ω(a1, a2) = e2πiw/ gcd(a1,a2) for integer w. The value of w for every pair of

generators provides a complete description of the projective representation, and each

of them may be chosen independently.

By the fundamental theorem of finite abelian groups, G may be written as a direct

product

G = Zn1 ⊗ Zn2 ⊗ · · · ⊗ ZnN (9.18)

where each ni are prime powers. Let ai be the generator of the ith direct product of

G with order ni, and define mij through Ω(ai, aj) = e2πimij/ gcd(ni,nj). Each choice of

0 ≤ mij < gcd(ni, nj) for i < j corresponds to a distinct projective representation.

Indeed, applying the Kunneth formula, one can compute the second cohomology

group

H2[G,U(1)] =∏

i<j

Zgcd(ni,nj) (9.19)

There is therefore a one-to-one correspondence between choices of mij and elements

of H2[G,U(1)].

Hence, we may simply refer to the commutative phases Ω(g1, g2) of the generators,

mij, as a proxy for the whole projective representation.

227

9.2.4 1D SPTs and twist phases

Let us now connect our discussion of projective representations to the classification

of 1D SPT phases. There are various ways this connection can be made, for instance,

by looking at edges or matrix product state representations [117, 112]. Here, we will

be using symmetry twists [110, 117, 119, 36, 154, 155, 156, 157], which turn out to

be a natural probe in the case of 2D fractal symmetries [29].

Suppose we have a 1D SPT described by the unique ground state of the local

Hamiltonian H and global on-site symmetry group G. Let us take the chain to be of

length Lx (taken to be large) with periodic boundary conditions. The symmetry acts

on the system as

S(g) =Lx−1∏

i=0

ui(g) (9.20)

for g ∈ G, where ui(g) is the on-site unitary linear representation of the symmetry

element g on site i, and [H,S(g)] = 0. A local Hamiltonian may always be written as

H =Lx−1∑

i=0

Hi (9.21)

where the sum is over local terms Hi with support only within some O(1) distance

of i.

The twisting procedure begins by constructing a new Hamiltonian, Htwist(g), for

a given g ∈ G. We pick a cut across which to apply the twist, xcut, which can be

arbitrary. Then, define the truncated symmetry operator

S≥(g) =xcut+R∏

i=xcut

ui(g) (9.22)

228

for some 1 R Lx. The twisted Hamiltonian is given by

Htwist(g) =Lx−1∑

i=0

S≥(g)HiS≥(g)† if Hi crosses xcut

Hi else

(9.23)

thus, the Hamiltonian is modified for Hi near xcut, but remains the same elsewhere.

We can now define the twist phase

T (g1, g2) =〈S(g1)〉Htwist(g2)

〈S(g1)〉H(9.24)

which is a pure phase representing the charge of the symmetry g1 in the ground state

of the g2 twisted Hamiltonian, relative to in the untwisted Hamiltonian. Here, 〈O〉Hmeans that expectation value of the operator O in the ground state of the Hamiltonian

H. It is straightforward to show that T (g1, g2) does not depend on where we place the

cut, xcut (this fact will be used to our advantage when twisting fractal symmetries).

The set of twist phases T (g1, g2) is a complete characterization of the state. Indeed,

the correspondence of the twist phases to the projective representation characterizing

a phase can be made by simply

Ω(g1, g2) = T (g1, g2). (9.25)

as such, we refer to Ω(g1, g2) itself as the twist phases.

An alternate, but equivalent, view is to examine the action of S≥(g2) on the ground

state |ψ〉. The action of S≥(g2) on |ψ〉 must act as identity on the majority of the

system, except near xcut and xcut +R, where it may act as some unitary operation,

S≥(g2) |ψ〉 = Ug2Ug2 |ψ〉 . (9.26)

229

where Ug2 acts near xcut, and Ug2 acts near xcut +M . Then, the twisted Hamiltonian

acting on the ground state can be thought of as

Htwist(g2) |ψ〉 = Ug2HU†g2|ψ〉 (9.27)

such that the ground state of Htwist(g2) is given by Ug2 |ψ〉. The twist phase is then

given by

Ω(g1, g2) =〈ψ|U †g2S(g1)Ug2 |ψ〉〈ψ|S(g1) |ψ〉

= 〈ψ|S(g1)†U †g2S(g1)Ug2 |ψ〉(9.28)

which measures the charge of the excitation created by Ug2 under the symmetry S(g1).

Thus, all information regarding the phase is contained within this local unitary matrix

Ug2 that appears due to a truncated symmetry operator.

9.3 Fractal Symmetries

We can now discuss fractal symmetries. The fractal symmetries we consider may be

thought of as a combination of an on-site symmetry group imbued with some spatial

structure.

Let us first consider a system with one fractal symmetry, described by the cellular

automaton polynomial f(x) over Fp, which we will denote by

G = Z(f,y)p (9.29)

which means that the on-site symmetry group is Zp, while the superscript, (f, y),

denotes the associated spatial structure: f denotes a cellular automaton described

230

by the polynomial f(x), and y denotes the positive “time” direction of this cellular

automaton (in this case, the positive y direction).

Our systems have degrees of freedom placed on the sites of an Lx × Ly square

lattice with periodic boundary conditions. Each site is labeled by its index along

the x and y direction, (i, j), and transforms as an on-site linear representation uij(g)

under g ∈ G. For simplicity, we will only consider the cases where Lx = pN is a power

of p, and Ly chosen such that f(x)Ly = 1. The latter is not difficult to accomplish, as

f(x)Lx = f(xLx) = f(1), so we may simply choose Ly = kLx > 0 such that f(1)k = 1.

Note that reversibility of f(x) implies f(1) 6= 0.

The symmetries of the system are in one-to-one correspondence with valid space-

time histories of the cellular automaton. The choices of Lx and Ly made earlier

means that any state a(0) (on a ring of circumference Lx) is cyclic in time with period

dividing Ly: a(Ly) = a(0). Given a valid trajectory a(j), the operator∏

ij uij(ga(j)i ) for

g ∈ G represents a valid symmetry operator. The entire space-time trajectory a(j) is

determined solely by its state at a particular time j0, a(j0), which can be in any of

pLx states. The total symmetry group will therefore be given by Gtot = (Zp)Lx .

Let us identify a particular element g as a generator for Zp. Then, let a set of Lx

generators for Gtot = ZLxp , defined with respect to j0, be g (j0)i 0≤i<Lx . We may then

define a vectorial representation of group elements via the one-to-one mapping from

vectors v ∈ FLxp to group elements,

g (j0)[v] =Lx−1∏

i=0

(g(j0)i )vi ∈ Gtot (9.30)

The action of each of these symmetry elements on the system is defined as

S(g (j0)[v]) =Lx−1∏

j=0

uj[g ;F j−j0v] (9.31)

231

j0

j0

i

S(g(a, j0)i )

S(g(b, j0)i )

Figure 9.1: Example of a symmetry generator (top) S(g(a,j0)i ) or (bottom) S(g

(b,j0)i )

for the fractal generated by f(x) = x + 1 + x with p = 2. Sites with blue or redsquares are acted on by uij(g (a)) or uij(g (b)), respectively, and form a valid space-timetrajectory of a cellular automaton.

where we have introduced the vectorial representation for uij(g) on a row j,

uj[g ; v] ≡Lx−1∏

i=0

uij(gvi) (9.32)

Thus, S(g (j0)[v]) is the unique symmetry operator that acts as uj(g)[v] on the row

j0. It can be viewed as the symmetry operation corresponding to the space-time

trajectory of a CA which is in the state v at time j0. Because f(x)Ly = 1 due to our

choice of Lx and Ly, any initial state is guaranteed to come back to itself after time

Ly, representing a valid cyclic space-time trajectory.

We may choose as a generating set the operators defined with respect to row j0,

S(g (j0)[ei]) = S(g(j0)i ) (9.33)

232

where ei is the unit vector (ei)i′ = δii′ . These act on only a single site on the row

j0, and an example of which is shown in Figure 9.1 (top). However, notice that this

choice of basis is only “most natural” when viewed on the row j0. Suppose we wanted

to change the row which we have defined our generators with respect to from j0 to j1.

How are the new operators related to our old ones? Well, one can readily show that

S(g (j1)[v]) =∏

j

uj[g ;F j−j1v] (9.34)

=∏

j

uj[g ;F j−j0F j0−j1v] (9.35)

= S(g (j0)[F j0−j1v]) (9.36)

is simply related via multiplication of v by powers of F . Thus,

g (j1)[v] = g (j0)[F j0−j1v] (9.37)

In general, we can have systems with multiple sets of fractal symmetries. The

other main situation we consider is the case of two fractal symmetries of the form

G = Z(f,y)p × Z(f ,y)

p (9.38)

where x ≡ x−1 and f(x) ≡ f(x). This is the form of fractal symmetry known to

protect non-trivial fractal SPTs [29, 153]. The first fractal represents a CA evolving in

the positive y direction with the rule f(x), and the second represents a CA evolving in

the opposite y direction with the rule f(x) (they are spatial inversions of one another).

In this case, we have one generator from each Zp, g (a) and g (b), and we can define two

sets of fractal symmetry generators as above with respect to a row j0. Let us call the

two sets of generators g (a,j0)i i and g (b,j0)

i i, and define their corresponding vectorial

233

representation. A general a or b type symmetry acts as

S(g (a,j0)[v]) =Lx−1∏

j=0

uj[g(a);F j−j0v]

S(g (b,j0)[v]) =Lx−1∏

j=0

uj[g(b); (F T )j0−jv]

(9.39)

where we have used the fact that the matrix form of f(x) is given by F T . A generator

for an a and a b type symmetry are shown in Figure 9.1. The generalization of Eq 9.37

for moving to a new choice of basis j1 for an a or b type symmetry is

g (a,j1)[v] = g (a,j0)[F j0−j1v]

g (b,j1)[v] = g (b,j0)[(F T )j1−j0v]

(9.40)

9.4 Local phases

Consider performing the symmetry twisting experiment on a system with fractal

symmetries. We can view the system as a cylinder with circumference Lx and consider

twisting the symmetry as discussed in Sec 9.2.4. We separately discuss the cases of

one or two fractal symmetries of a specific form first, and then go on to more general

combinations. Our main findings in this section are summarized as:

1. For the case of one fractal symmetry, G = Z(f,y)p , no non-trivial SPT phases

may exist

2. For the case of two fractal symmetries, G = Z(f,y)p × Z(f ,y)

p , if we only allow

for locality up to some lengthscale `, then there are a only a finite number of

possible SPT phases (scaling exponentially in `2)

3. For the case of more fractal symmetries, it is sufficient to identify pairs of

symmetries of the form Z(f,y)p × Z(f ,y)

p , and apply the same results from above.

234

9.4.1 One fractal symmetry

Let us take G = Z(f,y)p and consider twisting by a particular element g

(j0)i ∈ Gtot.

Since the twist phase doesn’t depend on the position of the cut, we can choose to

make the cut on the row jcut = j0. The twisted Hamiltonian Htwist(g(j0)i ) is then

obtained by conjugating terms in the Hamiltonian which cross jcut by the truncated

symmetry operator S≥(g(j0)i ).

Let the Hamiltonian be written as a sum

H =∑

i,j

Hij (9.41)

where each Hij is a local term with support near site (i, j). Now, consider twisting

the Hamiltonian by g(j0)i across the cut which also goes along the row j0. As can be

seen in Figure 9.2 (left), S≥(g(j0)i ) acts on a single site on row j0, and extends into

the fractal structure on the rows above. The important point is that S≥(g(j0)i ) only

acts differently from an actual symmetry operator at the point (i, j0) (and on some

row j0 +R far away). Thus, the twisted Hamiltonian may be written as

Htwist(g(j0)i ) |ψ〉 = U

g(j0)i

HU †g(j0)i

|ψ〉 (9.42)

when acting on the ground state |ψ〉, for some unitary Ug(j0)i

with support near the

site (i, j0). Note that there is always some freedom in choosing this unitary.

Then, consider measuring the charge of a symmetry g(j0−ly)i′ in response to this

twist, as in Eq 9.28. Clearly, only those symmetry operators whose support overlaps

with the support of Ug(j0)i

may have picked up a charge. Suppose the support of every

Ug(j0)i

is bounded within some (2lx + 1) × (2ly + 1) box centered about (i, j0), such

that only sites (i′, j′) with |i′− i| ≤ lx and |j′− j0| ≤ ly lie in the support. As can be

235

seen in Figure 9.2 (left), S(g(j0−ly)i′ ) only overlaps with this box for i′ in the range

− lx − 2lykb ≤ i′ − i ≤ lx − 2lyka (9.43)

and therefore, Ω(g(j0−ly)i′ , g

(j0)i ) may only be non-trivial if i′ − i is within some small

range. This places a constraint on the allowed twist phases. In addition, this must be

true for all choices of j0. It turns out this is a very strong constraint, and eliminates

all but the trivial phase in the case of G = Z(f,y)p , and only allows a finite number of

specific solutions for the case G = Z(f,y)p × Z(f ,y)

p , as we will show.

We also do not strictly require that the support of Ug(j0)i

be bounded in a box. This

will generally not be the case, as the operator may have an exponentially decaying

tail. Consider a unitary U which has some nontrivial charge eiφ 6= 1 under S, meaning

SUS† = eiφU (9.44)

when acting on the ground state. Clearly, if the support of U and S are disjoint, this

cannot be true. Next, consider any decomposition of U into a sum of matrices Uk,

U =∑

k Uk, and suppose that some of the Uk had disjoint support with S. Then, we

may write

U =∑

k∈D

Uk +∑

k∈DUk (9.45)

where k ∈ D are all the k for which Uk and S have disjoint support, and k ∈ D are

all the k for which they do not. But then

SUS† =∑

k∈D

SUkS† +∑

k∈DUk (9.46)

6= eiφU (9.47)

236

as the disjoint component has not picked up a phase eiφ, and SUkS† for k ∈ D cannot

have disjoint support with S (since only identity maps to identity under unitary

transformations) and so can’t affect the disjoint component of U . Thus, let us define

a subset of sites, A(U), defined as

A(U) =⋂

decompsU=

∑k Uk

⋂

k

Supp(Uk) (9.48)

where the first intersection is over all possible decompositions U =∑

k Uk, and

Supp(Uk) is the support of Uk (the subset of sites for which it acts as non-identity).

U can only have nontrivial charge under S if A(U) overlaps with the support of S.

In our case, lx and ly should actually be chosen such that A(Ug(j0)i

) may always be

contained within the (2lx + 1, 2ly + 1) box. An exponentially decaying tail of U is

therefore completely irrelevant, as A(U) only cares about the smallest part, before the

decay begins. The exact value of lx or ly is not too important — what is important

is that it is finite and small.

We also note that the twist phases obtained when twisting along a cut in the y

direction will be different, but are not independent of our twist phases for a cut along

the x direction. To see why this is, consider a truncated symmetry operator which

has been truncated by a cut in the y direction. This may alternatively be viewed as

an untruncated symmetry operator, multiplied by S≥(g(j)i at various (i, j)s located

near the cut. The action of twisting this symmetry for a cut along the y direction is

then also fully determined by the same set of Ug(j)i

from before, and is therefore not

independent of the twist phases for a cut along the x direction. Thus, it is sufficient to

examine only the set of twist phases for a cut parallel to x, as we have been discussing.

As we chose y to be the “time” direction of our CA, twisting along the x direction is

far more natural.

237

Let us make some definitions which will simplify this discussion. Notice that

Ω(g(j0)[v], g(k0)[w]) may be described by the bilinear form FLxp ×FLxp → Fp represented

by the skew-symmetric matrix W (j0,k0) ∈ FLx×Lxp defined according to

Ω(g(j0)[v], g(k0)[w]) = e2πip

vTW (j0,k0)w (9.49)

and that W (j0,k0) for any (j0, k0) contains full information of the twist phases. Fur-

thermore, since g(j1)[v] = g(j0)[F j0−j1v], we can deduce that W transforms under this

change of basis as

W (j1,k1) = (F j0−j1)TW (j0,k0)F k0−k1 (9.50)

We say that a matrix W (j0−ly ,j0), for a particular choice of j0, is local if its only

non-zero elements W(j0−ly ,j0)i′i 6= 0 are within a small diagonal band given by Eq 9.43.

A stronger statement, which we will call consistent locality, is that this is true for all

j0. The matrix W (j0−ly ,j0) for a physical state must be consistently local.

Let us adopt a polynomial notation which will be useful to perform computations.

We may represent the matrix W (j0,k0) by a polynomial W (j0,k0)(u, v) over Fp as

W (j0,k0)(u, v) =∑

ii′

W(j0,k0)i′i ui

′vi′−i (9.51)

with periodic boundary conditions uLx = vLx = 1. Locality is simply the statement

that the powers of v in this polynomial must be bounded by Eq 9.43 (modulo Lx).

Now, consider what happens to this polynomial as we transform our basis choice from

j0 → j0 − n,

W (j0−n−ly ,j0−n)(u, v) = f(v)nf(uv)nW (j0−ly ,j0)(u, v) (9.52)

238

which must be local for all n if W (j0−ly ,j0)(u, v) is to be consistently local.

Let us start with j0 = 0, and suppose that we have some W (−ly ,0)(u, v) that is non-

zero and local. By locality, W (−ly ,0)(u, v) may always be brought to a form where the

powers of v are all within the range given by Eq 9.43. Let va and vb be the smallest

and largest powers of v in W (ly ,0)(u, v) once brought to this form, which must satisfy

− lx − 2lykb ≤ a ≤ b ≤ lx − 2lyka (9.53)

Now, consider W (−ly−n,−n)(u, v) for small n,

W (−ly−n,−n)(u, v) = f(v)nf(uv)nW (ly ,0)(u, v) (9.54)

which (by adding degrees) will have va−nδf and vb+nδf as the smallest and largest

powers of v, where δf = deg(x−kaf(x)) > 0. The smallest and largest powers will

therefore keep getting smaller and larger, respectively, as we increase n. Thus, there

will always be some finite n beyond which locality is violated, and so W (−ly ,0)(u, v) can

never be consistently local. The only consistently local solution is therefore given by

W (−ly ,0)(u, v) = 0, which corresponds to the trivial phase. We have therefore shown

that no non-trivial local SPT phase can exist protected by only G = Z(f,y)p symmetry.

9.4.2 Two fractal symmetries

Let us now consider the more interesting case, G = Z(f,y)p ×Z(f ,y)

p , for which we know

non-trivial SPT phases can exist. In this case, we have the symmetry generators

g(α,j0)i for α ∈ a, b, and 0 ≤ i < Lx. As we showed in the previous section, the twist

phase between two a or two b symmetries must be trivial. The new ingredient comes

in the form of non-trivial twist phases between a and b symmetries.

239

j0j0 − ly

i i′

j0

j0 + ly

i i′

S≥ (g(a, j0)i )

S (g (a, j0−ly)i′ )

S≥ (g(a, j0)i )

S (g (b, j0+ ly)i′ )

Figure 9.2: Measurement of the twist phases for (left) Ω(g(a,j0−ly)i′ , g

(a,j0)i ) and (right)

Ω(g(b,j0+ly)i′ , g

(a,j0)i ). Due to locality, the twist phase may only be non-trivial if the

support of (left) S(g(a,j0−ly)i′ or (right) S(g

(b,j0+ly)i′ ) has some overlap with the yellow

box of size (2lx + 1)× (2ly + 1) about (i, j0). This implies that the twist phase mustbe trivial for i′ outside of a small region around i, a property which we call locality.However, this must be true for all choices of j0, which greatly constrains the allowedtwist phases. In the case of twist phases between the same type of symmetry (left),

only the trivial set of twist phases, all Ω(g(a,j0−ly)i′ , g

(a,j0)i ) = 1 is allowed. Between an

a and a b type symmetry (right), we show that only a finite number of solutions exist.

As can be seen in Fig 9.2, by the same arguments as before, the twist phase

Ω(g(b,j0+ly)i′ , g

(a,j0)i ) (9.55)

may only be non-trivial if i′ − i lies within some finite range,

− lx + 2lyka ≤ i′ − i ≤ lx + 2lykb. (9.56)

Let us again define the matrix W (k0,j0), but this time only between the a and b

symmetries via

T (g(b,k0)[w], g(a,j0)[v]) = e2πip

wTW (k0,j0)v (9.57)

240

note that W (k0,j0) need not be skew-symmetric like before. From Eq 9.40, the chang-

ing of basis is given by

W (k1,j1) = F k1−k0W (j0,j1)F j0−j1 (9.58)

We are looking for matrices W(j0+ly ,j0)i′i which are local (only non-zero within the

diagonal band Eq 9.56), and also consistently local, meaning that this is true for all

j0. Starting with j0 = 0, then, we are searching for a local matrix W (ly ,0), for which

W (ly+n,n) = F nW (ly ,0)F−n (9.59)

is also itself local for all n.

Let us again go to a polynomial representation

W (ly ,0)(u, v) =∑

i′i

W(ly ,0)i′i ui

′vi′−i (9.60)

which leads to the relation

W (ly+n,n)(u, v) = f(v)−nf(uv)nW (ly ,0)(u, v) (9.61)

which must have only small (in magnitude) powers of v for all n. However, f(v)−1 ≡

f(v)Ly−1 contains arbitrarily high powers of v, and therefore simply adding degrees

as before does not work and we may expect that a generic W (ly ,0)(u, v) will become

highly non-local immediately. Instead, what must be happening is that, at each step,

f(uv)W (j0+ly ,j0)(u, v) must contain some factor of f(v) (when viewed as a polynomial

without periodic boundary conditions) such that the f(v)−1 can divide out this factor

cleanly, producing a local W (j0+1+ly ,j0+1)(u, v).

241

How does this work in the case of the known fractal SPT [29]? In that case, W (0,0)

is already local and is given by the identity matrix. Then, clearly W (n,n) = W (0,0)

as it is invariant under Eq 9.59, and remains local for all n. In the polynomial

language, the identity matrix corresponds to the polynomial W (0,0)(u, v) =∑

i ui,

which has the property of translation invariance: W (0,0)(u, v) = uW (0,0)(u, v). In this

case, f(uv)W (0,0)(u, v) = f(v)W (0,0)(u, v), and so can be safely multiplied by f(v)−1.

In fact, any translation invariant solution, W (0,0)(u, v) = g(v)∑

i ui for any g(v), is

invariant under multiplying by f(v)−1f(uv).

We now state the main result of this paper: a special choice of basis functions

vkKm(u, v) with the property that W (u, v) is consistently local if and only if in the

unique decomposition

W (u, v) =Lx−1∑

k=0

Lx−1∑

m=0

Ck,mvkKm(u, v) (9.62)

where Ck,m ∈ Fp are constants, each Ck,mvkKm(u, v) is itself individually local.

Km(u, v) is given by

Km(u, v) = (u− 1)Lx−1−mVm(v) (9.63)

Vm(v) =

Nf−1∏

i=0

fi(v)mi (9.64)

where fi(x) are the Nf unique irreducible factors of the polynomial f(x) ≡ x−kaf(x)

appearing ri times, f(x) =∏

i fi(x)ri , and mi = bm/pαicpαi where αi is the power of p

in the prime decomposition of ri. The proof of this is rather technical and is delegated

to Sec 9.9. Thus, any phase can simply be constructed by finding all vkKm(u, v) that

are local, and choosing their coefficients Ck,m freely.

Let us go back to the matrix representation, and define the corresponding matrices

K(k,m) ↔ vkKm(u, v), following the same mapping as Eq 9.60. The elements of the

242

K(0,0) K(0,1)

K(0,2)

K(0,3)

Figure 9.3: Visualization of the matrices K(k,m) for the example of f(x) = 1 + x+ x2

and p = 2, for k = 0 and m = 0, 1, 2, 3 (other k can be obtain by shifting every elementk steps to the left). Each blue cell (i′, i), counting from the top left, represents a non-

zero matrix element K(0,m)i′i = 1. The arrows indicate evolution by K → FKF−1,

under which they exhibit cycles of period 2dlog2(m+1)e, as can be seen. Each of themare only non-zero about a small diagonal band (non-gray squares) of width given byDm = 1 + 2m. A K(k,m) is local if this white band fits inside some diagonal band(Eq 9.56). If a K(k,m) is local, then it can be seen that under evolution it retainslocality (the white band never gets larger), a property which we call consistent locality.The main result of this paper is that any consistently local matrix can be written asa linear sum of local K(k,m). Since there are only a finite number of local K(k,m),there are only a finite number of consistently local matrices that can be written, andtherefore a finite number of distinct phases in the thermodynamic limit. The numberwill depend on the choice of (lx, ly), i.e. how local the model is. Notice that consistentlocality is non-generic: if we just pick a local matrix by filling in elements along thediagonal band at random, it will generically quickly become highly non-local after afew steps of evolution.

243

matrix K(k,m)i′i are non-zero if and only if k ≤ i′− i ≤ k+Dm, where Dm is the degree

of Vm(v). Dm increases monotonically with m, and is bounded by Dm ≤ mδf . This

bound is saturated when v−kaf(x) is a product of irreducible polynomials, each of

which appear only once. Our main result (Theorem 9.9.1) states that any consistently

local W (ly ,0) can be written as a linear sum of local K(k,m). Thus, it is straightforward

to enumerate all possible W (ly ,0), which is simply all matrices in the subspace of

FLx×Lxp spanned by the set of local K(k,m) (note that the full set of K(k,m)km for

all 0 ≤ k,m < Lx forms a complete basis for this space). Figure 9.3 shows K(0,m) for

m = 0, 1, 2, 3 for a specific example, and how they evolve from one row to the next

while maintaining locality.

A property of the matrices K(k,m) is that they are periodic with period pNm ,

meaning K(k,m)

i+pNm ,i′+pNm = K(k,m)ii′ , where Nm ≡ dlogp(m+1)e. They also have cycles of

period pNm , meaning K(k,m) = F pNmK(k,m)F−pNm

. Since Dm increases monotonically

with m, only m up to some maximum value, M , are local and may be included in

W (ly ,0). We therefore see that W (ly ,0) must be periodic with period pNM . Thus,

locality enforces that the projective representation characterizing the phase, W (ly ,0),

be pNM -translation invariant! This is a novel phenomenon that does not appear in,

say, subsystem SPTs with line-like symmetries where the projective representation

does not have to be periodic (and as a result there are an infinite number of distinct

phases in the thermodynamic limit, even with a local model).

How many possible phases may exist for a given (lx, ly)? This is given by the

number of K(k,m) that can fit within a diagonal band of width ` ≡ 1+2lx+4lyδf . For

each m, K(k,m) is local if 0 ≤ k < `−Dm. Thus, there are Cm ≡ max`−mDm, 0

possible k values for each m. The total number of local K(k,m) is then∑

mCm.

Consider the case where f(x) = xkaf1(x)f2(x) . . . where each unique irreducible

factor fi(x) only appears once. Then, Dm = mδf . The total number of local K(k,m)

244

is then

Nloc =∞∑

m=0

max `−mδf , 0 (9.65)

=δf2dCe (2C − dCe+ 1) (9.66)

where we have just summed m to infinity since ` Lx/δf , and C ≡ `/δf . Notice

that Nloc only depends only on the combination `, and not specifically what lx and ly

are. The W (ly ,0) describing each phase is therefore a linear sum of these Nloc matrices

K(k,m), and so the total number of possible phases is pNloc . These phases are in

one-to-one correspondence with elements of the group ZNlocp , and exhibit the group

structure under stacking. Note that this number is an upper bound on the number of

possible phases with a given (lx, ly).

Consider the example in Figure 9.3, which has f(x) = 1 + x + x2 and p = 2.

Suppose we were interested in phases that have locality (lx, ly) = (1, 0), then the

matrix W(ly ,0)i′i may only be non-zero if −1 ≤ i′ − i ≤ 1. The only local K(k,m)

matrices are then K(−1,0), K(0,0), K(1,0), and K(−1,1). Then, our result states that

all consistently local W (ly ,0) are a linear sum (with binary coefficients) of these four

K(k,m) matrices. There are therefore only 24 possible phases, and they all have twist

phases that are periodic with a period of 2 sites (or 1 if the coefficient of K(−1,1) is

zero).

9.4.3 More fractal symmetries

Beyond these two cases, we may imagine more general combinations of fractal sym-

metries of the form

G =N−1∏

i=0

Z(fi,yηi )

p (9.67)

245

where we have N different fractals fi(x), which each have positive time direction yηi

given by ηi = ±1. We again assume none of fi(x) are monomials. In this language, the

previous case of two fractal symmetries is given by N = 2 with f0(x) = f1(x) = f(x),

and η0 = −η1 = 1. Note that we could have allowed p to vary among the fractals —

the reason we do not consider this case is that the twist phases between generators of

Zp and Zq are gcd(p, q)th roots of unity, but since p and q are both prime, this phase

must be trivial.

By an argument similar to that given in Sec 9.4.1, we may show that any twist

phase between the two generators of Z(fi,yηi )

p and Z(fj ,yηj )

p for which ηi = ηj must be

trivial. What about when ηi 6= ηj? In that case, we can show that there may only

exist non-trivial twist phases between them if fi(x) = fj(x).

Suppose we have some matrix W (ly ,0) describing twist phases between symmetry

generators of Z(fi,yηi )

p and Z(fj ,yηj )

p . Going to a polynomial representation W (ly ,0)(u, v)

(as in Eq 9.60), the change of basis to a different row is

W (ly+n,n)(u, v) = fi(uv)nfj(v)−nW (ly ,0)(u, v) (9.68)

which must be local for all n. Suppose that W (ly ,0)(u, v) is pk-periodic, such that

upkW (ly ,0)(u, v) = W (ly ,0)(u, v). Then,

W (ly+npk,npk)(u, v) = (fi(v)pk

fj(v)−pk

)nW (ly ,0)(u, v) (9.69)

should also be local for all n (recall that locality in the polynomial language is a

statement about the powers of v present). This implies that fi(v)pkfj(v)−p

k= 1,

or fi(vpk) = fj(v

pk). If pk Lx, then this means that we must have fi(x) =

fj(x). In the case where pk 6 Lx, we may simply consider larger system sizes

Lx, Ly → pmLx, pmLy, but with the same periodicity pk, and come to the same

246

conclusion. Hence, there can only exist non-trivial twist phases between symmetries

with fi(x) = fj(x) and ηi = −ηj.

For the more general group G in Eq 9.67, to find all the possible phases with

a fixed locality (lx, ly), we should simply find all pairs (i, j) where ηi = −ηj and

fi(x) = fj(x), and construct a local W (ly ,0) matrix for each such (i, j) pair. Thus, the

case with two fractal symmetries G = Z(f,y)p ×Z(f ,y)

p already contains all the essential

physics.

9.5 Constructing commuting models for arbitrary

phases

In this section, we show that it is indeed possible to realize all the phases derived in

the previous section for systems with two fractal symmetries, G = Z(f,y)p × Z(f ,y)

p , in

local Hamiltonians. We show how to construct a Hamiltonian, composed of mutually

commuting local terms, for an arbitrary phase characterized by the matrix W (ly ,0).

These Hamiltonians are certainly not the most local models that realize each phase,

but they are quite conceptually simple and the construction works for any given

W (ly ,0).

Let us define Zp generalizations of the Pauli matrices X and Z satisfying the

following algebra,

XN = ZN = 1 (9.70)

XZ = e2πip ZX (9.71)

and may be represented by a p× p diagonal matrix Z whose diagonals are p-th roots

of unity, and X as a cyclic raising operator in this basis.

247

The local Hilbert space on each site (i, j) of the square lattice is taken to be

two such p-state degrees of freedom, labeled a and b, which are operated on by the

operators Z(α)ij and X

(α)ij , for α ∈ a, b. Each Z

(α)ij only has non-trivial commutations

with X(α)ij on the same site and α.

Let us also define a vectorial representation of such operators: functions of vectors

v ∈ FLxp to operators on the row j as

Z(α)j [v] =

Lx−1∏

i=0

(Z(α)ij )vi (9.72)

X(α)j [v] =

Lx−1∏

i=0

(X(α)ij )vi (9.73)

One can verify that the commutation relations in this representation are

X(α)j [v]Z

(α)j [w] = e

2πip

vTwZ(α)j [w]X

(α)j [v] (9.74)

for two operators on the same row j with the same α ∈ a, b, and trivial otherwise.

The onsite symmetry group is G = Z(f,y)p × Z(f ,y)

p . Let us label the first Zp factor

as a, and the second as b, and let g (a) and g (b) be generators for the two. Then, we

take the onsite representation

uij(g (α)) = X(α)ij (9.75)

As always, we take Lx to be a power of p, and Ly such that f(x)Ly = 1. The total

symmetry group is Gtot = ZLxp × ZLxp . An arbitrary element of the first ZLxp factor,

with basis defined with respect to row j0, is given by

S(g (a,j0)[v]) =

Ly−1∏

j=0

X(a)j [F j−j0v] (9.76)

248

and of the second by

S(g (b,j0)[v]) =

Ly−1∏

j=0

X(b)j [(F T )j0−jv] (9.77)

Suppose we are given a consistently local matrix W (ly ,0) representing the twist

phase. For convenience, let us denote Wj ≡W (ly+j,j). Recall that consistent locality

implies (Wj)i′i is only non-zero if i′ − i is within some small range, for all j. Then,

let us define the operators

Aij = X(a)j [ei]Z

(b)j+ly

[−Wjei]Z(b)j+ly+1[FWjei]

Bij = X(b)j+ly

[ei]Z(a)j [−W T

j ei]Z(a)j−1[F TW T

j ei]

(9.78)

Notice that these are local operators, as Wj is consistently local. Consider the Hamil-

tonian

H = −∑

ij

Aij −∑

ij

Bij. (9.79)

which we will now show is symmetric, composed of mutually commuting terms, and

has a unique ground state which realizes the desired twist phase Wj.

First, let us show that Aij and Bij commute with all S(g(a,j0)[v]) and S(g(b,j0)[v]).

Note that Aij commutes with all a type symmetries, and Bij commutes with all b

type symmetries trivially. To see that Aij commutes with S(g(b,j0)[v]), note that the

phase factor obtained by commuting the symmetry through the Z(b)j+ly

term exactly

cancelled out by the phase from the Z(b)j+ly+1 term. In the same way, it can be shown

the Bij commutes with all S(g(a,j0)[v]). Thus, H is symmetric.

Next, we verify that all terms are mutually commuting. One can verify that Aij

and Bi′j with the same j commutes, as the phases from commuting each component

individually cancels. For Aij and Bi′,j+1, one finds that AijBi′,j+1 = αBi′,j+1Aij, where

α = e2πip

(eTi FTW T

j+1ei′−eTi W Tj F T ei′ ) = 1 using the fact that W T

j+1 = (F−1)TW Tj F

T . All

249

other terms commute trivially. Therefore, this Hamiltonian is composed of mutually

commuting terms. The set Aij ∪ Bij may therefore be thought of as generators

of a stabilizer group, and the ground state is given by the unique state |ψ〉 that is a

simultaneous eigenstate of all operators, Aij |ψ〉 = Bij |ψ〉 = |ψ〉. Uniqueness of the

ground state follows from the fact that all Aij and Bij are all independent operators,

which can be seen simply from the fact that Aij is the only operator which contains

X(a)ij , and Bi,j−ly is the only operator which contains X

(b)ij (all other operators act as

Z(α)ij or identity on the site ij).

Let show that the ground state is uncharged under all symmetries: S(g(α,j0)[v]) |ψ〉 =

|ψ〉. We do this by showing that every symmetry operation can be written as a prod-

uct of terms Aij and Bij in the Hamiltonian. Let us define a vectorial representation

for Aij and Bij,

Aj[v] =∏

i

Aviij = X(a)j [v]Z

(b)j+ly

[−Wjv]Z(b)j+ly+1[FWjv]

Bj[v] =∏

i

Bviij = X

(b)j+ly

[v]Z(a)j [−W T

j v]Z(a)j−1[F TW T

j v]

(9.80)

and note that

∏

j

Aj[Fj−j0v]

=S(g(a,j0)[v])∏

j

Z(b)j+ly

[−WjFj−j0v]Z

(b)j+ly+1[FWjF

j−j0v]

=S(g(a,j0)[v])

×[∏

j

Z(b)j+ly

[−WjFj−j0v]

][∏

j

Z(b)j+ly

[WjFj−j0v]

]

=S(g(a,j0)[v])

(9.81)

250

where we have again used the evolution equation Wj = FWj−1F−1. Similarly, we

may show that∏

j

Bj[(FT )j−ly−j0v] = S(g(b,j0)[v]) (9.82)

Thus, all symmetries may be written as a product of Aij and Bij, so therefore the

ground state |ψ〉 has eigenvalue +1 under all symmetries.

Next, let us measure the twist phases to verify that this model indeed describes

the desired phase. Consider twisting by the symmetry g(a,0)[ei]. Let us conjugate

every term in the Hamiltonian crossing the j = 0 cut by the truncated symmetry

operator S≥(g(a,0)[ei]). The only terms which are affected by this conjugation are

Bi′,0 for which eTi WT0 ei′ = W

(ly ,0)i′i 6= 0, which are transformed as

Bi′,0 → B′i′,0 = e−2πipW

(ly,0)

i′i Bi′,0 (9.83)

on the zeroeth row, and B′i′,j = Bi′,j on all other j 6= 0, in the twisted Hamiltonian.

Now, we are curious about the charge of a symmetry g(b,ly)[ei′ ] in the ground state of

this twisted Hamiltonian, which acts as

S(g(b,ly)[ei′ ]) =∏

j

Bj[(FT )jei′ ] (9.84)

= e2πipW

(ly,0)

i′i∏

j

B′j[(FT )jei′ ] (9.85)

(9.86)

since the symmetry only includes a single Bi′,0 on the zeroeth row. Thus, the ground

state of the twisted Hamiltonian (which has eigenvalue 1 under B′ij), has picked up

a nontrivial charge under the symmetry S(g(b,ly)[ei′ ]), relative to in the untwisted

251

Hamiltonian. Indeed, this phase is

Ω(g(b,ly)i′ , g

(a,0)i ) = e

2πipW

(ly,0)

i′i (9.87)

which is exactly as desired. Thus, this model indeed realizes the correct projective

representation and describes the desired phase of matter.

Note that these models bear resemblance to the cluster state, and can be under-

stood as a Zp version of the cluster state on a particular bipartite graph. Suppose we

have a graph defined by the symmetric Zp-valued adjacency matrix Adj(r, r′) ∈ Zp,

where r, r′ label two particular sites. Then, the Hamiltonian of a generalized cluster

state on this graph is given by

Hclus =∑

r

UXrU† (9.88)

where U =∏

rr′(CZrr′)Adj(r,r′), and CZrr′ is a generalized controlled-Z (CZ) operator

on the bond connecting sites r and r′. We define the Zp generalization of the CZ

operator by CZrr′ |zrzr′〉 = e2πipzrzr′ |zrzr′〉, where |zrzr′〉 is the eigenstate of Zr and

Zr′ with eigenvalues e2πizrp and e

2πizr′p respectively, such that CZrr′XrCZ†rr′ = XrZr′ .

Let us label a site by r = (i, j, α), its xy-coordinate and its sublattice index α ∈ a, b.

Then, the graph relevant to this model is given by the adjacency matrix

Adj((i, j, a), (i′, j′, b)) =

(−Wj)i′i if j′ = j + ly

(FWj)i′i if j′ = j + ly + 1

0 else

= Adj((i′, j′, b), (i, j, a))

(9.89)

252

Hence, one can think of each site (i, j, a) as being connected to sites (i′, j+ly, b) by the

adjacency matrix given by −Wj, and also sites (i′, j+ ly+1, b) via FWj. Generically,

this graph will be complicated and non-planar.

9.6 Irreversibility and Pseudosymmetries

In this section, we discuss fractal symmetries described by a non-reversible linear

cellular automaton (for which fractal SPTs do still exist [153, 29]), or even originally

reversible cellular automata that become irreversible when put on different system

sizes (e.g. Lx or Ly that are not powers of p).

Fractal symmetries are drastically affected by the total system size. For example,

consider the Sierpinski fractal SPT [29], which is generated by a non-reversible f(x) =

1+x with p = 2, on a lattice of size Lx = Ly = 2N . In this scenario, there are no non-

trivial symmetries at all! The total symmetry group Gtot = Z1 is simply the trivial

group. Yet, we can still define large operators that in the bulk look like symmetries

(i.e. they obey the local cellular automaton rules), but violate the rules only within

some boundary region. The total symmetry group being trivial may be thought of

as simply an incommensurability effect, whereby the space-time trajectory of the CA

cannot form any closed cycles. Thus, there is still a sense in which this model obeys

a symmetry locally. This effect is exemplified when one notices that, upon opening

boundary conditions, there are no obstacles in defining fractal symmetries and non-

trivial SPTs. In this way, it should still be possible to extract what the SPT phase

“would have been” if the CA rules had been reversible or if the total system sizes had

been chosen more appropriately such the total symmetry group had been non-trivial.

To generalize away from the fixed point and to an actual phase, we must formulate

what it means for a perturbation to be “symmetric” in a system with a potentially

trivial total symmetry group. We will say that such a model is symmetric under a

253

pseudosymmetry, as a symmetry of the full system may not even exist. Thus, a system

may respect a pseudosymmetry, and be in a non-trivial pseudosymmetry protected

topological phase (pseudo-SPT), despite not having any actual symmetries!

Let us define what we mean when we say that a system is symmetric under a

fractal pseudo-symmetry. Let us work in the case of two fractal symmetries, so

G = Z(f,y)p × Z(f ,y)

p . As always, we may decompose the Hamiltonian into a sum of

local terms

H =∑

ij

Hij (9.90)

where each Hij has support within some bounded box. Suppose Hij has support only

on sites with (x, y) coordinates i0 ≤ x ≤ i1 and j0 ≤ y ≤ j1, where `x ≡ i1 − i0 and

`y ≡ j1 − j0 are of order 1. Then, we say that Hij is symmetric under the fractal

pseudo-symmetry if it commutes with every

S(a,j0,j1)i =

j1∏

j=j0

uj[g(a);F j−j0ei] ; i0 − kb`y ≤ i ≤ i1 − ka`y

S(b,j1,j0)i =

j1∏

j=j0

uj[g(b); (F T )j1−jei] ; i0 + ka`y ≤ i ≤ i1 + kb`y

(9.91)

which is enough to replicate how any fractal symmetry would act on this lx×ly square,

if they existed for the total system. Notice that these only involve positive powers of

F , as we do not assume an inverse exists. Thus, even in the extreme case where Gtot

is trivial, a Hamiltonian may still be symmetric under the fractal pseudo-symmetry G

in this way. In the opposite extreme case where f(x) is reversible and Gtot = GLx (as

was the topic of the rest of this paper), Hij commuting with all pseudo-symmetries is

equivalent to it commuting with all the fractal symmetries in Gtot. Thus, it is natural

to expect that pseudo-symmetries may also protect non-trivial SPT phases.

Indeed, notice that one can perform a twist of a pseudo-symmetry. Given a cut,

j0, we may use the operator S(a,j0,j0+M)i , for some 1 M Lx, in place of the

254

truncated symmetry operator S≥(g(a,j0)i ) from Sec 9.4. This can be used to obtain a

twisted Hamiltonian as before by conjugating each term

Hij → S(a,j0,j0+R)i Hij(S

(a,j0,j0+R)i )† (9.92)

for some 1 R Lx if Hij crosses j0. Each Hij commuting with all their respective

pseudo-symmetries (Eq 9.91) means that the only terms which may no commute with

S(a,j0,j0+R)i are those near (i, j0) and those at the far-away row j0 + R which are not

affected by the twisting process.

Measuring the charge of a pseudo-symmetry is a trickier process, since there is

no “symmetry operator” which we can measure the charge of in the ground state.

Hence, the charge of a pseudo-symmetry is not so well defined. However, we may still

measure the charge relative to what it would be in the ground state on the untwisted

Hamiltonian, |ψ〉, which turns out to be well defined. One approach is to again

express the twisted process as the action of some local unitary near (i, j0), Htwist |ψ〉 =

UHU † |ψ〉, where as before A(U) is contained within some (2lx + 1) × (2ly + 1) box

(A(U) is defined in Eq 9.48). If the support of U were entirely in this box, then we

could measure the change in the charge of a b type symmetry by

Ω(j0+ly ,j0)i′i = 〈ψ|S†U †SU |ψ〉 (9.93)

where S = S(b,j0+ly ,j0)i′ , and |ψ〉 is the ground state of H (for convenience we have

suppressed the dependence of U and S on i, i′, etc). However, if the support of U

is not confined to this box, this expectation value may not yield a pure phase. One

solution is to use a family of larger pseudo-symmetry operators which act the same

way within A(U), and take the limit of the sizes going to infinity. For example, using

255

S(n) instead of S in the above, defined as

S(n) ≡ S(b,j0+ly+pn,j0−pn)i′+kapn (9.94)

which is shown in Figure 9.4. For large n and i′ within

− lx + kaly ≤ i′ − i ≤ lx + kbly, (9.95)

this operator acts in the same way as S within A(U), but is also a valid pseudo-

symmetry operator elsewhere as well, except on rows j0 + ly + pn and j0 − pn which

are far away, as shown in Figure 9.4. Then, we may define

Ω(j0+ly ,j0)i′i ≡ lim

n→∞〈ψ|S(n)†U †S(n)U |ψ〉 (9.96)

which, in the large n limit (while keeping pn Lx), is a pure phase. In the case

where Gtot = GLx , this will coincide with the twist phases

Ω(j0+ly ,j0)i′i = Ω(g

(b,j0+ly)i′ , g

(a,j0)i ) (9.97)

discussed earlier.

However, the key ingredient to showing that this pseudo-SPT phase is truly stable

to local pseudo-symmetric perturbations is to show that Ω(j1+ly ,j1)i′i for all j1 is com-

pletely determined by its value at j0. Define (like before) the matrix W (j1,j0) ∈ FLx×Lxp

by

Ω(j1,j0)i′i = e

2πipW

(j1,j0)

i′i (9.98)

Starting with j0 = 0, the matrix W (ly ,0) would normally be evolved to W (ly+1,1) using

Eq 9.59. However, in this case there is no inverse F−1 which we can use. Instead, we

256

j0

j0 + ly + pn

S(a,j0,j0+ R)i

i′j0 − pn

i i′ + kapn

S(b,j0+ ly+ pn ,j0− pn )i′+ kapn

U j0 + ly

Figure 9.4: A visualization of the process to defining a twist phase for pseudosymme-tries S. Twisting with respect to S

(a,j0,j0+R)i can be thought of as acting via a unitary

U , which has some region A(U) shown as the yellow square. To measure the changein charge of another symmetry, we take the expectation value of the commutator

(Eq 9.96) of U with S(b,j0+ly+pn,j0−pn)i′+kapn , as shown, for large n. This may be non-trivial

for small |i′ − i| when they overlap, and is the sign of a non-trivial pseudo-SPT.

have the relation

W (ly+1,0) = W (ly+1,1)F (9.99)

which does not uniquely determine W (ly+1,1), as we may add vT to any row of

W (ly+1,1), for v ∈ ker(F T ). However, it is easy to show that any v ∈ ker(F T )\0 is

highly non-local, by which we mean that there are no integers a and b for which vi is

only non-zero for a ≤ i ≤ b, and 0 ≤ b − a Lx (essentially, any non-zero vector v

for which F Tv = 0 needs to be exploiting the periodic boundary conditions). Thus,

adding any non-zero vector v ∈ ker(F T ) to a row of a local W (ly+1,1) will necessarily

make it non-local. Thus, if there exist a local matrix W (ly+1,1) satisfying Eq 9.99,

then it is the only local one. The matrices K(k,m) can be defined even for irreversible

f(x). Therefore, for a matrix W (ly ,0) composed of a sum of local K(k,m), a local

W (ly+1,1) does exist and is unique. This can be reiterated to uniquely determine the

set of local W (j0+ly ,j0) for all j0, assuming it is commensurate with the system size.

257

Thus, we have shown that Ω(j0+ly ,j0)i′i is indeed a global invariant (knowing it for one

j0 determines it for all j0). It therefore cannot be changed via a local pseudo-symmetry

respecting perturbation (or equivalently a pseudo-symmetry respecting local unitary

circuit), and such a phase can indeed be thought of as a non-trivial pseudo-SPT.

To define K(k,m) for cases where f(x) may not be reversible, we may simply note

that each solution is pNm-periodic in both directions. Thus, it is straightforward to

generalize K(k,m) for m where pNm divides Lx and Ly. In the proof, we are careful

to show that f(x) is only ever divided out of polynomials of finite degree in x which

contained f(x) as a factor anyway, so polynomial division by f(x) is remainder-less

and results in another polynomial. Thus, the results apply equally well for non-

reversible f(x), as long as Lx and Ly are commensurate with the periodicity. This

commensurability requirement may greatly reduce the number of possible pseudo-SPT

phases, for example, if Lx or Ly are coprime to p, then only m = 0 is allowed. Note

that on such system sizes is also possible to have periodicity that is not a power of p

in non-generic cases, for example, the special case where f(x) = g(xν) is a function

of only xν and ν is not a power of p.

9.7 Identifying the phase

Suppose we are given an unknown system with G = Z(f,y)p × Z(f ,y)

p , how do we deter-

mine what phase it belongs to and how do we convey compactly what phase it is in?

Recall that for the case with line-like subsystem symmetries (the topic of Ref [29]),

to describe a specific phase required information growing with system size, and so a

modified phase equivalence relation was introduced to deal with this. Such a modi-

fied phase equivalence was not needed in this case, and we will show that indeed a

specific local phase may be described with a finite amount of information. Suppose

we are given an unknown Hamiltonian H. It is possible to compute the full set of

258

twist phases and construct the Lx × Lx matrix W (0,0). If the only non-zero matrix

elements of W (0,0) are within some diagonal band, then we are set. Otherwise, find

the smallest integer ly ≥ 0 for which W (ly ,0) = F lyW (0,0) is only non-zero within a

diagonal band of width ` ∼ O(1). This is guaranteed to be the case for some ly (also

of O(1)) due to locality. Note that ` and ly are independent of which row we call the

zeroeth row. From the fact that W (j0+ly ,j0) must also be non-zero only within this

diagonal band for all j0, our main result (Theorem 9.9.1) states that W (ly ,0) must be

a sum

W (ly ,0) =∑

(k,m)∈locCkmK

(k,m) (9.100)

where Ckm ∈ Fp, and loc is the finite set of all pairs (k,m) where K(k,m) is also only

non-zero within the same band of width `. Thus, this phase is specified fully by our

choice of origin, ly, and the finite set of non-zero Ckm. Furthermore, this description

does not depend on Lx and Ly, and so it makes sense to say whether two systems

of different sizes belong to the same phase. However, note that unlike with ordinary

phases, the choice of origin is important here. This procedure may also be done in

cases where the symmetry is irreversible, the matrix W (ly ,0) will instead be defined

from the pseudo-symmetry twist phases Ω(ly ,0)i′,i .

9.8 Discussion

We have therefore asked and answered the question of what SPT phases can exist

protected by fractal symmetries for the type G = Z(f,y)p , G = Z(f,y)

p × Z(f ,y)p , or

combinations thereof. If we completely ignore locality along the x direction, effectively

compactifying our system into a quasi-1D cylinder with global symmetry group Gtot =

GLx , we would have found that the possible phases are classified by H2[GLx , U(1)]

which is infinitely large as Lx →∞. What we have shown, however, is that the vast

majority of these phases require highly non-local correlations that cannot arise from

259

a local Hamiltonian. In the case of G = Z(f,y)p , locality disqualifies all but the trivial

phase. In the G = Z(f,y)p ×Z(f ,y)

p case, there exists multiple non-trivial phases that are

allowed. If we restrict the twist phases to be local up to some degree, (lx, ly), then

there are only a finite number of possible phases, independent of total system size Lx

and Ly. The number of phases depends only on the combination ` ≡ 1 + 2lx + 4lyδf ,

which is linear in both lx and ly (thus demonstrating a kind of holographic principle).

For more general combinations of such fractal symmetries, we have shown that the

classification of phases simply amounts to finding pairs of fractal symmetries of the

form (Z(f,y)p ,Z(f ,y)

p ) and repeating the analysis above.

Where do other previously discovered 2D systems with fractal symmetries fall

into our picture? The quantum Newman-Moore paramagnet [148] is described by the

Hamiltonian

HNM = −∑

ij

ZijZi,j+1Zi−1,j−1 − h∑

ij

Xij (9.101)

Xij, Zij, are Pauli matrices acting on the qubit degree of freedom on site (i, j).

The symmetry in our notation is given by G = Z(f,y)2 with f(x) = 1 + x (which is

irreversible). HNM has a phase transition from a spontaneously symmetry-broken

phase at |h| < 1 to a trivial symmetric paramagnet at |h| > 1. Our results would

imply that there can be no non-trivial SPT phase in this system. Thus, all the

possibilities in this model are different patterns of broken symmetry. Next, we have

the explicit example of the 2D Sierpinski fractal SPT [29, 133] (which appeared at

a gapped boundary in Ref [153]). This model is isomorphic to the cluster model

on the honeycomb lattice, and is described by symmetries G = Z(f,y)2 × Z(f ,y)

2 with

f(x) = 1 + x. With proper choice of unit cell, the Hamiltonian is given by

Hclus = −∑

ij

X(b)ij Z

(a)ij Z

(a)i,j−1Z

(a)i−1,j−1

−∑

ij

X(a)ij Z

(b)ij Z

(b)i,j+1Z

(b)i+1,j+1

(9.102)

260

Notice that f(x) = 1 + x with p = 2 is irreversible for all system sizes, thus these

phases should be viewed as pseudo-SPT phases (and indeed every term commutes with

all the pseudo-symmetries). Computing the pseudo-SPT twist phases for Hclus gives

Ω(j,j)i′i = (−1)δi′i . Thus, we have simply W (0,0) = 1 = K(0,0). A translation invariant

model must simply be a sum of K(k,0) and this is indeed the case here. The family

of 2D fractal SPT models described in Ref [29] all realize W (0,0) = 1. Our results

here imply the existence of a number of new local phases for which the Hamiltonian

and twist phases are not strictly translation invariant with period 1. Sec 9.5 gave

a construction of such models, which works even when f(x) is not reversible. The

twist phases for these models may be translation invariant with a minimal period of

2n sites along either x or y, but in exchange will also require interactions of range

O(2n).

We show explicitly in Fig 9.5 a few of these additional phases that were previously

undiscovered, which are represented as cluster models on various graphs. Recall

that the usual Z2 cluster model for on an arbitrary graph is simply given by the

Hamiltonian

Hclus = −∑

v

Xv

∏

v′∈adj(v)

Zv′ , (9.103)

where the sum is over vertices v, and adj(v) is the set of vertices connected to v by

an edge.

A signature of subsystem SPT phases is an extensive protected ground state de-

generacy along the edge. That is, for an edge of length Ledge, there is a ground state

degeneracy scaling as log GSD ∼ Ledge which cannot be lifted without breaking the

subsystem symmetries. The dimension of the protected subspace may be thought of

as the minimum dimension needed to realize the projective representation character-

izing the phase on the boundary. For the case of the honeycomb lattice cluster model

(Fig 9.5a), we have exactly GSD = 2Ledge . For the more complicated models, some

261

W0

W0

W1

W2

W3

W0

W1

W2

W3

W0

W1x

y

ab

(a) K(0,0) (b) K(−1,1)

(c) K(−1,2) (d) K(−2,3)

Figure 9.5: Explicit examples of some possible phases for the case ofG = Z(f,y)2 ×Z(f ,y)

2

with f(x) = 1 + x: the Sierpinski triangle symmetry. The models are constructedfollowing the procedure of Sec 9.5, and are simply cluster models defined on someunderlying graph. The models all have ly = 0 and W0 = W (0,0) given by (a) K(0,0),(b) K(−1,1), (c) K(−1,2), and (d) K(−2,3). Each site (gray circle in (a)) consists of ana and a b qubit, which are represented by blue and red vertices. Example of fractalsubsystems on which the symmetries act are also shown in (a): green highlightedvertices for the a type subsystems, and orange for the b. The reader may verify that∏X on these subsystems is indeed a symmetry of the cluster model defined on all

these graphs. The case (a) is simply the previously studied honeycomb lattice clustermodel, which is translation invariant. The other three are previously undiscoveredphases, and are only translation invariant with a period of (b) 2 or (c, d) 4 along the xand y directions. The graphs for phases with K(k,m) for k other than the ones chosenhere may be obtained simply by shifting each blue a vertex left/right by a number ofsites, while maintaining connectivity of the graph. For each case, we also show onecycle of the matrices Wj = F jW0F

−j, the matrix characterizing the twist phasesfor symmetries defined w.r.t. to row j, presented in the same manner as in Fig 9.3.The lower-leftmost a and b vertices of each graph are defined to be at coordinate(x, y) = (0, 0). Although we have only shown examples on an 8× 8 torus, these maybe tiled onto any commensurate system size.

262

of this degeneracy may be lifted, leaving only a fraction GSD = 2αLedge remaining.

Moreover, the degeneracy along the left or right edges will also generally be different.

9.9 Proof of main result

In this section, we will focus on proving the claim in Sec 9.4.2 that any consistently

local matrix W must be a linear sum of K(k,m), each of which are local. We will say

that the set of matrices satisfying this property, K(k,m), serve as an optimal basis

(this term will be precisely defined soon). Recall that we are dealing with the case

where Lx = pN is a power of p and Ly is chosen such that f(x)Ly = 1. We will simply

use L to refer to Lx in this section for convenience.

9.9.1 Definition and statement

We will be using the polynomial representation exclusively. Let W (u, v) be a Lau-

rent polynomial over Fp in u and v representing the twist phases, defined according

to Eq 9.60. Formally, periodic boundary conditions uL = vL = 1 means that we

only care about the equivalence class of W (u, v) in Fp[u, v]/〈uL − 1, vL − 1〉, where

〈uL − 1, vL − 1〉 is the ideal generated by these two polynomials. Rather than deal-

ing with equivalence classes, we will instead deal with canonical form polynomials:

a polynomial q(u, v) is in canonical form if degu q(u, v) < L and degv q(u, v) < L.

Obviously, canonical form polynomials are in one-to-one correspondence with equiv-

alence classes from Fp[u, v]/〈uL − 1, vL − 1〉. Any polynomial with u or v-degree

larger than L can be brought into canonical form via simply taking ua = ua mod L

and va = va mod L. From now on, we will implicitly assume all polynomials have been

brought to their canonical form.

Let us now define what it means for a polynomial to be local.

263

Definition 9.9.1. A Laurent polynomial g(u, v) is (a, b)-local, for integers a ≤ b, if

degv v−ag(u, v) ≤ b− a (9.104)

A Laurent polynomial g(u, v) being (a, b)-local roughly means that the only non-

zero powers of v are va, va+1, . . . , vb (powers mod L). As a shorthand, we will more

often say that a polynomial is `-local to mean (0, `)-local, which can be thought of

as simply an upper bound on its v degree. Whenever something is said to be `-local,

we are usually talking about ` L being some finite value of order 1. Some nice

properties are that if g(u, v) is (a, b)-local, then

1. vkg(u, v) is (a+ k, b+ k)-local;

2. g(u, v)g′(u, v) is (a+ a′, b+ b′)-local, if g′(u, v) is (a′, b′)-local.

3. g(u, v)N is (Na,Nb)-local;

Next, let us define the “evolution operator” th with respect to an (invertible)

Laurent polynomial h(x) which operates on a polynomial W (u, v) as

th : W (u, v)→ th(u, v)W (u, v); (9.105)

th(u, v) = h(v)−1h(uv) (9.106)

By invertible, we mean that there exists an inverse h(v)−1 with periodic boundary

conditions, such that h(v)h(v)−1 = 1. In the case of h(x) = f(x), tf evolves the poly-

nomial W (ly ,0)(u, v) → W (ly+1,1)(u, v). Notice that an overall shift, h(x) → xah(x),

results in th(u, v) → uath(u, v), which does not affect the locality properties (which

only depends on powers of v). For the purposes of this proof we will therefore simply

work with (non-Laurent) polynomials h(x). We can now define consistent locality.

Definition 9.9.2. A Laurent polynomial g(u, v) is consistently (a, b)-local under th

if tnhg(u, v) is (a, b)-local for all n.

264

Physically, the twist phases W (ly ,0)(u, v) must be consistently (−lx + 2lyka, lx +

2lykb)-local (from Eq 9.56) under tf in order to correspond to a physical phase ob-

tained from a local Hamiltonian.

Let us define

Um(u) = (u− 1)L−1−m (9.107)

for m = 0 . . . L− 1, which serves as a complete basis for all polynomials g(u) ∈ Fp[u]

with degree degu g(u) ≤ L. Any polynomial W (u, v) may therefore be uniquely

expanded as

W (u, v) =L−1∑

m=0

Um(u)Wm(v) (9.108)

which we take to be our definition of Wm(v). Since Um(u) are all independent, if

W (u, v) is `-local, each Wm(v) must also be `-local.

Definition 9.9.3. A set of polynomials Vm(v) indexed by m = 0, . . . , L− 1 is said

to generate an optimal basis for th if for every `-local W (u, v), W (u, v) is consistently

`-local if and only if Vm(v) | Wm(v) for all m. The basis set vkUm(u)Vm(v) is then

called an optimal basis.

When we say Vm(v) | Wm(v), we mean that Vm(v) divides Wm(v) as polynomials

in Fp[v] without periodic boundary conditions, i.e. there exists a polynomial q(v)

such that Wm(v) = q(v)Vm(v) and

degv q(v) = degvWm(v)− degv Vm(v) ≤ `− degv Vm(v) (9.109)

which follows by addition of degrees, and since Wm(v) is `-local.

Suppose Vm(v) generates an optimal basis for th. Assuming Vm(v) is invertible,

vkUm(u)Vm(v)k,m for 0 ≤ k,m < L generates a complete basis for canonical form

polynomials. This basis is optimal with respect to th in the sense that all consistently

`-local polynomials under th may be written as a linear sum of `-local basis elements.

265

If there are only a finite number Nloc of `-local basis elements (as will be the case),

then there are also only a finite number pNloc of consistently `-local W (u, v).

We can now restate our main theorem, the proof of which will be the remainder

of this section.

Theorem 9.9.1. The polynomials Vm(u, v), defined in Eq 9.64, generate an optimal

basis for tf .

9.9.2 Proof

Let us first list some relevant properties of Um(u).

1. (u− 1)nUm(u) = Um−n(u) for n ≤ m, or 0 for n > m

2. Um(u) is pNm-periodic, meaning

upNmUm(u) = Um(u) (9.110)

where Nm ≡ dlogp(m+ 1)e. This follows from the fact that

(upNm − 1)Um(u) = (u− 1)p

NmUm(u) = 0 (9.111)

since pNm > m, due to property 1.

3. Um(u) is also cyclic under evolution by th with period dividing pNm , tpNm

h U0(u) =

U0(u). This follows from the fact that upNmU0(u) = U0(u), and so

tpNm

h U0(u) = h(v)−pNmh(uv)p

NmU0(u)

= h(v)−pNmh(v)p

NmU0(u) = U0(u)

(9.112)

266

4. thUm(u) = Um(u) +∑

m′<m qm′(v)Um′(u). Under evolution by th, thUm(u) is

given by simply Um(u), plus a linear combination of Um′(u) for m′ < m.

Using property 1, It is therefore easy to extract each component Wm(v) in the expan-

sion of Eq 9.108 directly from W (u, v) in a straightforward way. Suppose the largest

m for which Wm(v) 6= 0 is m = M . Then, (u − 1)MW (u, v) = WM(v)U0(u) gives

only the m = Mth component multiplying U0(u). Then, we may take W ′(u, v) ≡

W (u, v)− UM(u)VM(v), which has largest m given by M ′ < M . This process can be

repeated on W ′(u, v) to fully obtain Wm(v) for all m. From property 2, we also find

that W (u, v) is actually pNm-periodic.

Property 4 is the most important property (and what makes Um(u) a special basis

for this problem). It follows from Property 3 for m = 0, thU0(u) = U0(u), and the

fact that the mth component of thUm(u) is obtained by

(u− 1)mth(u, v)Um(u) = th(u, v)U0(u) = U0(u) (9.113)

which remains unchanged under evolution by th. Thus, supposing the expansion of

W (u, v) has some largest m value m = M , then defining ∆hW (u, v) according to

thW (u, v) = W (u, v) +∆hW (u, v) (9.114)

we must have that ∆hW (u, v) has a largest m value m < M . Alternatively, (u −

1)M∆hW (u, v) = 0. This fact will be used numerous times as it allows for a proof by

recursion in M in many cases.

Let us first prove two minor Lemmas.

Lemma 9.9.2. Suppose Vm(v) generates an optimal basis for some th. Then,

Vm(v) | Vm′(v) for all m′ ≥ m and V0(v) = 1.

267

Proof. First, any `-local W (u, v) that contains only an m = 0 component, W (u, v) =

U0(u)W0(v), is trivially also consistently `-local under any th, since thU0(u) = U0(u).

Thus, it must be that V0(v) = 1. Next, if W (u, v) is consistently `-local, then

(u− 1)nW (u, v) =L−1∑

m=n

Um−n(u)Wm(v) (9.115)

must also be consistently `-local for any n ≥ 0. However, this implies that Vm(v) |

Wm+n(v). But all we know is that Vm+n | Wm+n(v). For this to always be satisfied,

we must therefore have that Vm(v) | Vm′(v) for all m′ ≥ m.

Lemma 9.9.3. Let W (u, v) be `-local. Then, W (u, v) is consistently `-local under th

if and only if ∆hW (u, v) is also consistently `-local.

Proof. Consider evolving W (u, v),

thW (u, v) = W +∆hW (9.116)

t2hW (u, v) = W +∆hW + th∆W (9.117)

t3hW (u, v) = W +∆hW + th∆hW + t2h∆hW (9.118)

and so on. By definition, if W (u, v) is consistently `-local, then tnhW (u, v) must all be

`-local. But then, this means that each term added in increasing n, tn−1h ∆hW (u, v),

must also be `-local, meaning that ∆hW (u, v) is therefore consistently `-local. If

W (u, v) is not consistently `-local, then that means that there must be some n such

that tnh∆W (u, v) is not `-local, which therefore implies that ∆hW (u, v) is also not

consistently `-local.

The next Lemma gives a family of a consistently `-local polynomials.

Lemma 9.9.4. Let Khm(u, v) = Um(u)h(v)m for some 0 ≤ m < L. Then,

degv tnhW (u, v) = mδh for all n. It is therefore consistently mδh-local under th.

268

Proof. Let us prove by recursion in m. The base case, m = 0, is true since U0(u) is

indeed consistently 0-local. Now, assume m > 0 and we have proved this Lemma for

all m′ < m.

Let us compute ∆hKhm(u, v),

∆hKhm(u, v) = Um(u)h(v)m−1(h(uv)− h(v)) (9.119)

= Um(u)h(v)m−1

δh∑

k=0

hkvk(uk − 1) (9.120)

where h(x) =∑δh

x=0 hkxk, and we have used Property 2 of Um(u) to replace uk → uk,

where k ≡ (k mod pNm) is positive. Then, we may use the identity

uk − 1 =k∑

n=0

(k

n

)(u− 1)n (9.121)

to expand

∆hKhm(u, v) = Um(u)h(v)m−1

δh∑

k=0

k∑

n=0

(k

n

)hkv

k(u− 1)n (9.122)

=

δh∑

k=0

k∑

n=0

hkvkh(v)n−1Kh

m−n(v) (9.123)

and note that by our recursion assumption, vkh(v)n−1Khm−n is consistently (k, (m −

1)δh+k)-local. Since 0 ≤ k ≤ δh, each term is therefore consistently mδh-local. Thus,

∆hKhm(u, v) is consistently mδh-local. By Lemma 9.9.3, Kh

m(u, v) is therefore also

mδh-local. Finally, the v-degree of Khm(u, v) saturates mδh since the mth component

of tnhKhm(u, v) has v-degree mδh for all n. The proof follows for all m by recursion.

Thus, a family of consistently `-local W (u, v) may be created by a linear sum

over of `-local vkKhm(u, v), over k and m. However, this may not be exhaustive:

there may be some consistently `-local W (u, v) that are not in this family. To show

269

exhaustiveness, we need to show that Vm(v) = h(v)m generates an optimal basis

for th. This is not true generally, but is true in the case that h(x) is irreducible,

which our next lemma addresses. Notice that Vm(v) = h(v)m are consistent with the

properties of being generators of an optimal basis from Lemma 9.9.2, V0(v) = 1 and

Vm(v) | Vm′(v) for all m′ ≥ m.

Lemma 9.9.5. Suppose h(x) is an irreducible polynomial. Then, Vm(v) = h(v)m

generates an optimal basis for th.

Proof. To prove that h(v)m generates an optimal basis for th, we need to show that

for any `-local W (u, v), it is consistently `-local if and only if h(v)m | Wm(v) must

hold for all m.

First, the reverse implication follows from Lemma 9.9.4: if W (u, v) is `-local and

each h(v)m | Wm(v), then W (u, v) is also consistently `-local. We must now prove

the forward implication.

Let W (u, v) by consistently `-local under th, with the expansion

W (u, v) =M∑

m=0

Um(u)Wm(v) (9.124)

where M is the largest m value in the expansion, and WM(v) 6= 0. We need to

prove that this implies that h(v)m | Wm(v) for all m. We now prove by recursion,

and assume that this has been proven for all M ′ < M . Note that for the base case

M = 0, h(v)m indeed generates an optimal basis for all M = 0 polynomials W (u, v).

If h(v) = cvk is a monomial, then this proof is also trivial, so we will assume this is

not the case in the following.

Consider ∆hW (u, v), which by Lemma 9.9.3, also has maximum m < M and is

consistently `-local. Take the m = M − 1th component of ∆hW (u, v), obtained by

270

(u− 1)M−1∆hW (u, v), which by a straightforward calculation is given by

(u− 1)M−1∆hW (u, v) = g(v)h(v)−1WM(v)U0(u) (9.125)

where

g(v) =

δh∑

k=0

khkvk (9.126)

δh ≡ degv h(v), hk is defined through h(v) =∑

k hkvk, and k ≡ (k mod pNM ).

Note that since W (u, v) is `-local, despite Eq 9.125 containing h(v)−1, is of v-degree

bounded by `. By our recursion assumption, h(v)M−1 must divide Eq 9.125.

Let us prove that h(v) - g(v) and g(v) 6= 0. First, since degv h(v) = degv g(v) and

h(v) is irreducible, if h(v) is to possible divide g(v), it must be that g(v) = const·h(v).

This can only be the case if (k mod pNM ) ≡ k0 is the same for all k. But then,

h(v) = k0

imax−1∑

i=0

hk0+ipNM vk0+ipNM (9.127)

= k0vk0

(imax−1∑

i=0

hk0+ipNM vi

)pNM

(9.128)

which contradicts with the fact that h(v) is irreducible, as imax > 1 and pNM > 1

(which is the case here). The g(v) = 0 is the k0 = 0 case of this. Thus, g(v) 6= 0 and

h(v) - g(v).

Going back, we have that

h(v)M−1 | g(v)h(v)−1WM(v) (9.129)

=⇒ h(v)M | g(v)WM(v) (9.130)

but since h(v) - g(v), it must be the case that h(v)M | WM(v).

271

Now, consider W ′(u, v) = W (u, v) − UM(u)WM(v), which is a sum of two con-

sistently `-local polynomials (using Lemma 9.9.4), and so is also consistently `-local.

By our recursion assumption, it then follows that h(v)m | Wm(v) for m < M . Thus,

h(v)m | Wm(v) holds for all m in W (u, v).

By recursion in M , we have therefore proved that for all W (u, v), h(v)m | Wm(v)

must be true for all m. Thus, h(v)m generates an optimal basis for th.

If f(x) = x−kaf(x) is irreducible, then Lemma 9.9.5 is sufficient to obtain all

consistently (a, b)-local W (u, v). To do so, we simply have to find all (k,m) where

the basis element vkK fm(u, v) is (a, b)-local, and take a linear combination of them. If

there are Nloc(a, b) such basis elements, then the pNloc(a,b) possible linear combinations

are exhaustive.

In the case that f(x) is not irreducible, there may be consistently (a, b)-local

polynomials that do not fall within this family. However, note that it is always

possible to expand f(x) in terms of its unique irreducible factors

f(x) = f0(x)r0f1(x)r1 . . . (9.131)

The next two Lemmas allows us to use this result construct an optimal basis for f(x),

based on this factorization.

Lemma 9.9.6. Let h(x) be an irreducible polynomial, and r > 0 an integer. Then,

Vm(v) = h(v)m generates an optimal basis for thr , where m = bm/pαcpα and α is

the power of p in the prime factorization of r.

Proof. First, note that if p - r, p is coprime to r, then being consistently `-local under

th is equivalent to being consistently `-local under thr . This follows from the fact

that, if W (u, v) has maximum m value m = M , then tpNM

h W (u, v) = W (u, v). If

W (u, v) is consistently `-local under th, then tnhW (u, v) = tn mod pNm

h W (u, v) is, by

definition, `-local for all n. If W (u, v) is instead consistently `-local under thr = trh,

272

then trnh W (u, v) = trn mod pNm

h W (u, v) is `-local for all n. But, rn takes on all value

mod pNm , and so these two conditions are equivalent. Thus, Lemma 9.9.5 states that

h(v)m generates an optimal basis for th, which therefore also generates an optimal

basis for thr . Indeed, if p - r, h(v)m = h(v)m and the proof is complete.

Next, consider the case where r = pα is a power of p. Notice that trh(u, v) =

th(ur, vr) in this case is a function of only ur and vr. Let W (u, v) be `-local and

decompose it as

W (u, v) =r−1∑

i=0

r−1∑

j=0

(u− 1)r−1−ivjWij(ur, vr) (9.132)

such that each of the ij “block” does not mix under evolution by trh. Thus, each ij

may be treated as an independent system in terms of variables u ≡ ur and v ≡ vr,

with L ≡ L/r. Thus, by Lemma 9.9.5, each ij component (and therefore W (u, v)) is

only consistently `-local if and only if in the decomposition

Wij(ur, vr) =

L/r−1∑

m=0

(ur − 1)L/r−1−mWij,m(vr) (9.133)

h(vr)m | Wij,m(vr) for all i, j, m. Defining m ≡ i+ mr, W (u, v) may be written as

W (u, v) =L−1∑

m=0

Um(u)r−1∑

j=0

vjWij,m(ur, vr) (9.134)

=L−1∑

m=0

Um(u)Wm(v) (9.135)

where Wm(v) =∑r−1

j=0 vjWij,m(ur, vr), so W (u, v) is consistently `-local if and only if

h(vr)m | Wm(v). To eliminate reference to m, we may use the fact that m = bm/rc,

such that m = rm. Therefore, W (u, v) is consistently `-local if and only if h(v)m |

Wm(v) for all m, and h(v)m generates an optimal basis for thr when r = pα as well.

273

Finally, consider the general case r = rpα, where p - r. We have just shown that

h(v)m generates an optimal basis for thpα . Since r is coprime to p, by our first

argument, this also generates an optimal basis for thr .

Lemma 9.9.7. Suppose V1,m(v) and V2,m(v) generate optimal bases for th1 and th2

respectively, and V1,m(v) and V2,m′(v) share no common factors for all m, m′. Then,

Vm(v) = V1,m(v)V2,m(v) generates an optimal basis for th1h2.

Proof. Let W (u, v) be `-local which we expand as

W (u, v) =M∑

m=0

Um(u)Wm(v) (9.136)

where M is the largest m for which Wm(v) 6= 0. If V1,m(v)V2,m(v) | Wm(v), then

W (u, v) is consistently `-local under th1 and th2 , and therefore also under th1h2 . We

then need to prove the reverse implication, that W (u, v) being consistently `-local

under th1h2 implies V1,m(v)V2,m(v) | Wm(v) for all m. We will prove this by recursion in

M . The base case, M = 0, is trivial since V0(v) = V1,0(v)V2,0(v) = 1 is a requirement

from Lemma 9.9.2. Now, suppose this has been proven for all M ′ < M .

First, assume that W (u, v) is consistently `-local under th1 but not th2 . Then,

consider ∆h1h2W (u, v), which has largest m < M and is consistently `-local under

th1h2 by Lemma 9.9.3. Our recursion assumption, then, implies that ∆h1h2W (u, v) is

also consistently `-local under th1 and th2 individually. Then,

tmh1h2W (u, v) = W (u, v) +m−1∑

i=0

tih1h2∆12W (u, v) (9.137)

and so

tnh1tmh1h2

W (u, v) = tnh1W (u, v) +m−1∑

i=0

tnh1tih1h2

∆12W (u, v) (9.138)

which is `-local. But, if we choose n = (k−m mod pNM ), then we get that tkh2W (u, v)

is always `-local. Thus, W (u, v) is consistently `-local under th2 as well, which contra-

274

dicts our initial assumption. Therefore, W (u, v) cannot be consistently `-local under

th1 but not th2 . The same is also true with h1 ↔ h2.

Next, assume W (u, v) is neither consistently `-local under th1 nor th2 . Then,

consider

W ′(u, v) ≡ V1,M(v)W (u, v) (9.139)

which is consistently ` + degv V1,M(v) ≡ `′-local under th1h2 (notice that if ` L,

then `′ L as well). W ′(u, v) is also `′-local under th1 , since V1,m(v) | V1,M(v) for

all m ≤ M by Lemma 9.9.2. However, since V1,M(v) shares no common factors with

any V2,m(v), W (u, v) is still not consistently `′-local under th2 . But, we just showed

previously that we cannot have a situation in which W (u, v) is `′-local under th1h2 and

th1 but not th2 , thus leading to a contradiction. W (u, v) must therefore be consistently

`-local under both th1 and th2 .

This means that V1,m(v) | Wm(u, v) and V2,m(v) | Wm(u, v) for all m. Since V1,m(v)

and V2,m(v) share no common factors, this means that V1,m(v)V2,m(v) | Wm(u, v).

Thus, V1,m(v)V2,m(v) generates an optimal basis for th1h2 .

We may now prove Theorem 9.9.1. Let us factorize f(x) into its Nf unique

irreducible polynomials,

f(x) =

Nf∏

i=0

fi(x)ri (9.140)

Using Lemma 9.9.6, an optimal basis for tfrii , is generated by fi(v)mi, where mi =

bm/pαicpαi , and αi is the power of p in the prime factorization of ri. Since fi(v)mi for

different i share no common factors (as fi(v) are irreducible), Lemma 9.9.7 then says

that f0(v)m0f1(v)m1 generates an optimal basis for tfr00 fr11

. This may be iterated to

construct an optimal basis for tfr00 fr11 f

r22

and so on. Finally, one gets that ∏i fi(v)mi

generates an optimal basis for tf , which is therefore also an optimal basis for tf . This

is exactly Vm(v), and the proof is complete.

275

Chapter 10

Conclusion

Hopefully, this Dissertation has provided some clarity into the space of “subdimen-

sional topological quantum phases of matter”. The main technical contributions of

these works, to my mind, are the results on classification. Although these works

have been primarily focused on a narrow space of possibilities, they paint a clear and

complete picture within this space. However, due to the recency of the field, there

remains much beyond yet to be understood.

276

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