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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
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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
100
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).
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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
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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.
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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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