Tensor Network States, Entanglement,

and Anomalies of Topological Phases

of Matters

Yunqin Zheng

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

Adviser: Bogdan Andrei Bernevig

June 2020

c© Copyright by Yunqin Zheng, 2020.

All rights reserved.

Abstract

This dissertation investigates two aspects of topological phases of matter: 1) the

tensor network state (TNS) representations of the ground states as well as their en-

tanglement entropies of gapped Hamiltonians in diverse dimensions; 2) the anomalies

and dynamics of strongly coupled quantum field theories.

For the first aspect, we first show an efficient method of analytically deriving the

translation invariant TNS and matrix product state (MPS) representation for the

ground state of translation invariant stabilizer code Hamiltonians in both 1d and

higher dimensions. These TNS/MPS states have minimal virtual bond dimension.

Using the TNS, we derive the entanglement entropy for a variety of stabilizer codes,

including the fracton models the Haah code. We further go beyond the stabilizer codes

and study the structure of entanglement entropy for generic 3d gapped Hamiltonians.

In particular, an explicit formula for a universal physical observable – topological

entanglement entropy (TEE) – has been derived, which sharpens previous results.

Our formula shows that the TEE across an arbitrary entanglement surface is linearly

proportional to the TEE across a torus.

For the second aspect, we use the global symmetries and their ’t Hooft anomalies

of the SU(2) Yang-Mills theory with a theta term to constrain its dynamics. In

particular, we point out that there are four different such theories, distinguished

by Lorentz symmetry enrichments of the Wilson loops in the SU(2) fundamental

representation. We further derive a new mixed anomaly between time reversal and

one form symmetry which can only be seen on an unorientable manifold. We further

use the anomalies to explore various possible dynamics, such as nontrivial degrees

of freedom localized on the domain wall due to spontaneously broken time reversal

symmetry, as well as a potentially possible but exotic quantum phase transition —

Gauge Enhanced Quantum Critical Point.

iii

Acknowledgements

First and formost, I would like to thank my advisor Prof. B. Andrei Bernevig

for his guidance and support. His sharp and insightful questions are often the major

driving force which pushes our project to a deeper level. I am significantly influenced

and benefited from his persistence in research: every step should be solid, rigorous

and crystal-clearly presented; never give up easily until we find a satisfying answer.

Nevertheless, I am also extremely grateful that Andrei gives me sufficient freedom to

think independently, and encourages me to seek my own collaborations.

I’d also like to thank Prof. Nicolas Regnault and Dr. Huan He, who guided me to

the world of quantum entanglement. Huan is my major collaborator in the projects

related to quantum entanglements. These projects will never be possible without his

uncountable insightful inputs, his patience in answering my naive questions and his

encouragements from time to time. For Nicolas, I am always impressed by his ability

to detect possible loop-holes in my arguments. Whenever our project got stuck, he

always has enough patience to hearing my confusions and pointing out useful ideas.

I own a lot to Dr. Juven Wang, who I regard as my second advisor. Since the

first time I met him at IAS in my first year, he provided me unprecedented support

and guidance. I benefited significantly from our enormous discussions, and his unique

insight in both condensed matter physics and quantum field theories, his kindness in

sharing numerous unpublished notes, which consequently guides me to the fascinating

world of QFT frontiers. I also thank him for sharing his enthusiasm in arts (piano)

and sports (juggling), his hospitality during my visits in Harvard and kindly providing

me accommodation for free.

I’d like to express my gratitude to my numerous colleagues: Jie Wang and Jingyu

Luo have been my classmate for over 9 years, and our friendship will not terminate

as I graduate. I thank Curt von Keyserlingk, Barry Bradlyn, Titus Neupert and

Jennifer Cano for the kind helps during the early years of my PhD. I thank Ho-Tat

iv

Lam, Zheyan Wan, Jie Wang and Yi-Zhuang You for various simulating discussions at

various stages, it is really enjoyable to discuss and collaborate with you guys and I’ve

learned a lot. My gratitude also goes to my collaborators German Sierra, Kantaro

Ohmori, Pavel Putrov, Meng Guo, Xueda Wen, Apoorv Tiwari and Peng Ye. I thank

the members of the “Bernevig group”: Sanjay Moudgalya, Fang Xie, Zhida Song and

Biao Lian for discussions.

I’d also like to thank all the wonderful guys around Princeton. Ilya Belopolski,

Ksenia Bulycheva, Duyu Chen, Xiaowen Chen, Yiming Chen, Zijia Cheng, Clay Cor-

dova, Dui Da, Trithep Devakul, Yale Fan, Tong Gao, Akash Goel, Pranay Gorantla,

Jung Pyo Hong, Po-Shen Hsin, Yuwen Hu, Luca Iliesiu, Ziming Ji, Jiaqi Jiang, Zhaoqi

Leng, Biao Lian, Sihang Liang, Xinran Li, Yaqiong Li, Jingjing Lin, Yingyu Liu,

Zheng Ma, Kelvin Mei, Alexey Milekhin, Fedor Popov, Hao Qian, Justin Ripley, Shu-

Heng Shao, Yu Shen, Xue Song, Zhida Song, Suerfu, Siwei Wang, Wudi Wang, Yantao

Wu, Yang-Le Wu, Xin Xiang, Jun Xiong, Bin Xu, Zhenbin Yang, Yizhi You, Junyi

Zhang, Sonia Zhang, Bo Zhao, Wenli Zhao, Hao Zheng, Xinan Zhou, Liujun Zou,

thank you all for your friendship and support. Special thank also goes to wonderful

administrative staffs Antonia (Toni) Sarchi, Catherine (Kate) Brosowsky, Barbara

Mooring and Jessica Heslin. I appreciate Prof. Robert Austin for leading me the

experimental project.

Finally, I thank Prof. David Huse to be the second reader, and Prof. Waseem

Bakr and Prof. Silviu Pufu for being the FPO committee members.

v

To my parents.

vi

Contents

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

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

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Preliminaries 1

1.1 Representing the Ground State Wavefunctions . . . . . . . . . . . . . 2

1.1.1 Tensor Network States . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Restricted Boltzmann Machine . . . . . . . . . . . . . . . . . 6

1.2 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Anomalies and SPT phases . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Matrix Product State of A Stabilizer Code in 1D 13

2.1 An Example of Stabilizer Codes: ZZXZZ Model . . . . . . . . . . . 14

2.2 MPS for the ZZXZZ Model . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 General Stabilizer Code Convention . . . . . . . . . . . . . . . . . . . 27

2.4 General Algorithm to Construct MPS . . . . . . . . . . . . . . . . . . 29

3 Restricted Boltamann Machine State for Stabilizer Code in 1D 35

3.1 (Restricted) Boltzmann Machine . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 Relation to MPS . . . . . . . . . . . . . . . . . . . . . . . . . 40

vii

3.2 More on ZZXZZ Model . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 An Inequality for Rank of MPS . . . . . . . . . . . . . . . . . . . . . 43

3.4 Restricted Boltzmann Machine State of a Stabilizer Code . . . . . . . 45

3.4.1 MPS Matrix Rank For Cocycle SPT Models . . . . . . . . . . 46

3.4.2 An Example: ZZXZZ Model Revisited . . . . . . . . . . . . 50

3.4.3 RBM States of Cocycle Hamiltonians . . . . . . . . . . . . . . 54

3.4.4 RBM Construction for Zq−1XZq−1 Model . . . . . . . . . . . 60

4 Tensor Network States, Entanglement Entropy of CSS Stabilizer

Codes and Fracton Models in 3D 65

4.1 Stabilizer Code Tensor Network States . . . . . . . . . . . . . . . . . 65

4.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.2 CSS Stabilizer Code and TNS Construction . . . . . . . . . . 68

4.2 Entanglement properties of the stabilizer code TNS . . . . . . . . . . 72

4.2.1 TNS as an exact SVD . . . . . . . . . . . . . . . . . . . . . . 72

4.2.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . 76

4.3 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.3.1 Hamiltonian of 3D Toric Code Model . . . . . . . . . . . . . . 77

4.3.2 TNS for 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . 79

4.3.3 Concatenation Lemma . . . . . . . . . . . . . . . . . . . . . . 84

4.3.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Haah Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4.1 Hamiltonian of Haah code . . . . . . . . . . . . . . . . . . . . 92

4.4.2 TNS for Haah Code . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.3 Entanglement Entropy for SVD Cuts . . . . . . . . . . . . . . 100

4.4.4 Entanglement Entropy for Cubic Cuts . . . . . . . . . . . . . 106

viii

5 Topological Entanglement Entropy of (3+1)D Gapped Phases of

Matter 115

5.1 Reduction formulas for Entanglement Entropy . . . . . . . . . . . . . 115

5.1.1 Strong Sub-Additivity . . . . . . . . . . . . . . . . . . . . . . 116

5.1.2 Topological Entanglement Entropy . . . . . . . . . . . . . . . 120

5.2 Application: Entanglement Entropy of Generalized Walker-Wang The-

ories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.1 Wave Function of GWW Models . . . . . . . . . . . . . . . . 127

5.2.2 Entanglement Entropy of GWW Models . . . . . . . . . . . . 135

6 Anomaly and Dynamics of (3 + 1)d SU(2) Yang-Mills Theory 149

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Standard Lore of SU(N) Yang-Mills . . . . . . . . . . . . . . . 150

6.1.2 New Aspects: Lorentz Symmetry Enrichments . . . . . . . . . 151

6.2 SU(2) Yang-Mills Theory at θ = π . . . . . . . . . . . . . . . . . . . . 154

6.2.1 Time Reversal Symmetry . . . . . . . . . . . . . . . . . . . . . 154

6.2.2 One-form Symmetry . . . . . . . . . . . . . . . . . . . . . . . 155

6.2.3 Formulating on Unorientable Manifold and Lorentz Symmetry

Fractionalization . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.2.4 Anomaly on an Unorientable Manifold . . . . . . . . . . . . . 158

6.2.5 Low Energy Dynamics: Overview and Questions . . . . . . . . 161

6.3 Domain Wall from Time Reversal Spontaneously Broken . . . . . . . 163

6.3.1 Domain Wall for (K1, K2) = (0, 0): Semion with U2 = 1 . . . . 164

6.3.2 Domain Wall for (K1, K2) = (1, 0): Semion with U2 = −1 . . . 168

6.3.3 Domain Wall for (K1, K2) = (0, 1): Anti-Semion with U2 = 1 . 172

6.3.4 Domain Wall for (K1, K2) = (1, 1): Anti-Semion with U2 = −1 174

6.3.5 Remarks On CP⊥ and T , and Summary . . . . . . . . . . . . 175

6.4 Application I: Domain Wall Theory Nf < NCFT . . . . . . . . . . . . 176

ix

6.4.1 Lorentz Symmetry Fractionalization, K2 = 1 . . . . . . . . . . 177

6.4.2 U Unitary Symmetry Fractionalization . . . . . . . . . . . . . 178

6.5 Deconfined Gapless U(1) Gauge Theory . . . . . . . . . . . . . . . . . 180

6.5.1 U(1) Gauge Theory and Spin Liquids at θ = 0 . . . . . . . . . 181

6.5.2 U(1) Gauge theory and Spin Liquids at θ = 2π . . . . . . . . 184

6.6 Application II: Gauge Enhanced Quantum Critical Point Nf ≥ NCFT 187

6.6.1 SU(2) QCD4 and Higher Order Interactions: U(1) Spin Liquid

Phases From Higgsing . . . . . . . . . . . . . . . . . . . . . . 188

6.6.2 Symmetries Realizations and Symmetry Enriched U(1) Spin

Liquids in the Infrared . . . . . . . . . . . . . . . . . . . . . . 191

6.6.3 Gauge Enhanced Quantum Critical Points . . . . . . . . . . . 196

A Appendices for Chapter 2 199

A.1 Conventions for MPS and Canonical MPS . . . . . . . . . . . . . . . 199

A.1.1 Conventions for MPS and Transfer Matrix . . . . . . . . . . . 199

A.1.2 Review of Canonical MPS . . . . . . . . . . . . . . . . . . . . 200

A.2 Correlation Functions and Transfer Matrix Eigenvalues . . . . . . . . 202

A.3 Stabilizer Operator Acts on MPS Locally . . . . . . . . . . . . . . . . 215

A.4 The Action of L and R Operators on the MPS Matrices . . . . . . . 218

A.5 Commutation Relations of U Operators . . . . . . . . . . . . . . . . . 222

A.6 Linear Equations for Local Tensors . . . . . . . . . . . . . . . . . . . 223

A.7 Virtual U Operators as Tensor Products of Pauli Matrices . . . . . . 227

B Appendices for Chapter 3 230

B.1 Projective Representations and 1D Symmetry Protected Topological

Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

B.1.1 Projective Representations and Cocycles . . . . . . . . . . . . 230

B.1.2 Cocycle States . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

x

B.1.3 Cocycle Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . 232

B.2 Some Useful Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 237

B.3 More Examples of RBM for Cocycle Model . . . . . . . . . . . . . . . 239

C Appendices for Chapter 4 242

C.1 Proof for the Concatenation Lemma for the 3D Toric Code Model . . 242

C.2 Numerics for Haah Code . . . . . . . . . . . . . . . . . . . . . . . . . 245

D Appendices for Chapter 5 248

D.1 Review of Entanglement Entropy and Spectrum . . . . . . . . . . . . 248

D.2 Local Contributions to the Entanglement Entropy . . . . . . . . . . . 249

D.3 Derivation of the Reduction Formula . . . . . . . . . . . . . . . . . . 253

D.3.1 Recurrence for Genus . . . . . . . . . . . . . . . . . . . . . . . 254

D.3.2 Recurrence for b0 . . . . . . . . . . . . . . . . . . . . . . . . . 257

D.4 Vanishing of the Mean Curvature Contribution in KPLW Prescription 260

D.5 Review of Lattice TQFT . . . . . . . . . . . . . . . . . . . . . . . . . 267

D.6 Surfaces in the dual lattice . . . . . . . . . . . . . . . . . . . . . . . . 271

D.7 Mutual and Self-Linking Numbers . . . . . . . . . . . . . . . . . . . . 275

D.7.1 Intersection and Linking . . . . . . . . . . . . . . . . . . . . . 277

D.7.2 Self-linking Number . . . . . . . . . . . . . . . . . . . . . . . . 279

D.8 NA(CE)NAc(CE) is Independent of CE . . . . . . . . . . . . . . . . . . 281

D.9 A Case Study of the Conjecture Between GSD and TEE . . . . . . . 286

Bibliography 291

xi

List of Tables

5.1 Constant part and topological part of the entanglement entropy for

generalized Walker-Wang models. STQFTc is the constant part of the

EE for the TQFT, while Stopo is the TEE for a general theory which

belongs to the same phase of the TQFT. b0 is the zeroth Betti number

of entanglement surface b0 =∑g∗

g=0 ng. χ =∑g∗

g=0(2 − 2g)ng is the

Euler characteristic of the entanglement surface. In particular, we

have Stopo(S2) = Stopo(T 2). . . . . . . . . . . . . . . . . . . . . . . . . 146

6.1 Symmetry fractionalization and anomalies on the domain wall theory

for four siblings of Yang-Mills. . . . . . . . . . . . . . . . . . . . . . 176

C.1 Entanglement entropies for various bipartitions of the |TNS〉 of the

Haah code. The second to fourth column list the coordinates of vertices

in region A. The column of ”Left/Right” labels the spin on the left or

right position on the vertex (x, y, z), where 0 and 1 corresponds to the

left and right position respectively. We used the coordinate frame as

shown in Eq. 4.49 and Fig. 4.6. . . . . . . . . . . . . . . . . . . . . . 247

xii

List of Figures

1.1 Examples of TNS lattice wave functions in 1D and 2D. Each node is a

tensor whose indices are the lines connecting to it. The physical indices

- of the quantum Hilbert space - are the lines with arrows, while the

lines without any arrows are the virtual indices. Connected lines means

the corresponding indices are contracted. Panel (a) is an MPS for 1D

systems. Panel (b) is a PEPS on a 2D square lattice. . . . . . . . . . 3

1.2 A tensor T s~rv1,~rv2,~rv3,~rv4,~ron a 2d square lattice with one physical index

s~r and four virtual indices v1,~rv2,~rv3,~rv4,~r. . . . . . . . . . . . . . . . . 4

1.3 Tensor network on a 2d 2× 2 torus lattice. . . . . . . . . . . . . . . 5

2.1 A graphical representation of the matrix T gr1gr2gr3 . We denote each phys-

ical index by an arrow. The shaded region represents a unit cell, and

the virtual left and right indices are represented by the horizontal line.

The virtual indices are not explicitly shown here. . . . . . . . . . . . 16

2.2 A graph representation of Eq. (2.4). . . . . . . . . . . . . . . . . . . 18

2.3 A graphical representation of Eqs. (2.17), (2.18) and (2.19). . . . . . 20

2.4 Graphical representation of Eq. (2.39). The shaded purple region rep-

resents the operator Orα acting on the physical indices. . . . . . . . . 29

2.5 Graphical representation of Eq. (2.40). The virtual operator U ri,τ and

(U ri,τ )−1 act on the right virtual index between the r + τ − 1 and r-th

unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xiii

2.6 An illustration of the operators Lr−(τ−1)α,τ and Rr−(τ−1)

α,τ with fixed r and

α, and all 1τ ≤ Pα−1. The blue blocks represent unit cells. The purple

blocks represent the operators Lr−(τ−1)α,τ , and the operators Rr−(τ−1)

α,τ . 31

3.1 An example of RBM state corresponding to q = 3,M = 2, M = 2. The

red circles represent visible spins. The black rectangles are the hidden

spins connecting visible spin belonging to different unit cells, which are

linked to the purple and orange lines representing the weights Aia and

Bia respectively. The black triangles are the hidden spins connecting

visible spins within the same unit cell, which are linked to the green

lines representing the weights Cib. The blue region represents a unit

cell. Notice that the nonzero weights are only between the hidden spins

and the visible spins. . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Graphical representation of the RBM state of the ZZXZZ model. . 54

3.3 Graphical representation of the RBM state of the ZXZ model. The

red circles represent visible spins, the black rectangles represent the

hidden spins connecting visible spin belonging to different unit cells,

and the black triangles represent the hidden spins connecting visible

spins within the same unit cell. . . . . . . . . . . . . . . . . . . . . . 63

4.1 TNS gauge in MPS. (a) A part of an MPS. A1 and A2 are two local

tensors contracted together. (b) We insert the identity operator I =

UU−1 at the virtual level - it acts on the virtual bonds. The tensor

contraction of A1 and A2 does not change. (c) We further multiply U

with A1 and U−1 with A2, resulting in A1 and A2 respectively in Panel

(d). The tensor contraction of A1 and A2 is the same as the tensor

contraction of A1 and A2. The TNS wave function does not change as

well. Similar TNS gauges also appear in other TNS such as PEPS. . . 67

xiv

4.2 (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The

lines with arrows are the physical indices. The connected lines are

the contracted virtual indices, while the open lines are not contracted.

On each vertex, there lives a T tensor, and on each bond, we have a

projector g tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 The Hamiltonian terms of the 3D toric code model. Panel (a) is Av

which is a product of 6 Z operators, and Panel (b) is Bp which is a

product of 4 X operators. The circled X and Z represent the Pauli

matrices acting on the spin-1/2’s. The toric code Hamiltonian includes

Av terms on all vertices v and Bp terms on all plaquettes p. . . . . . . 78

4.4 Contraction of two local T tensors in the z-direction. . . . . . . . . . 85

4.5 The splitting of tensors near the entanglement cut. . . . . . . . . . . 87

4.6 Tensor contraction for the Haah Code TNS. (a) The lattice size is

2× 3× 3. (b) The lattice size is 3× 3× 3 . . . . . . . . . . . . . . . 99

4.7 Region A contains all the spins connecting with l− 1 T tensors which

are contracted along z direction. The figure shows an example with

l = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.8 Region A contains all the spins connecting with T tensors which are

contracted in a “tripod-like” shape, where three legs extend along

x, y, z directions. There are three legs extending along x, y, z direc-

tions respectively. In general, three legs can have different length, each

with lx−1, ly−1, lz−1 cubes along three directions. This figure shows

an example where lx = ly = lz = 3. . . . . . . . . . . . . . . . . . . . 101

4.9 Transferring the Pauli X operators of the Bc operator from the region

A (a) to the region A (b). . . . . . . . . . . . . . . . . . . . . . . . . 108

xv

5.1 KPLW prescription of entanglement surface T 2. The space inside the

two torus is divided into three regions, A, B and C, each being a solid

torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2 A schematic figure of the topology of spacetime M4 and space S3.

Inside S3, we schematically draw a loop l representing the loop con-

figurations C of the B field in the dual lattice. The dashed surface S

bounding the loop l extends into the spacetime bulkM4, representing

the B field configuration in the dual lattice of spacetime. S ′ repre-

sents the B field configurations that form closed surfaces away from

the boundary of the spacetime ∂M4. The boundary condition in the

path integral Eq. (5.23) is specified by a fixed B configuration C on

S3. The path integral should integrate over all the configurations in

the spacetime bulk M4 with the boundary configuration C on S3 fixed. 128

5.3 A tetrahedron is drawn with solid lines, and its dual is drawn in dash

and gray lines. The 2-simplex (ijk) in the original lattice is dual to

the 1-simplex (ab) in the dual lattice. Similarily, (ikl) is dual to (ad),

(ijl) is dual to (ca) and (jkl) is dual to (ea). The colored dash ar-

rows indicate the orientations of the four 2-simplices, where (ijk) and

(ikl) share the same orientation, and (ijl) and (jkl) share the opposite

orientation. The orientations of the dual-lattice 1-simplices are also

indicated by the arrows on the grey/dashed lines. . . . . . . . . . . . 130

xvi

5.4 An example of the lattice structure of an entanglement cut in (2+1)D.

The green simplices form the entanglement cut Σ, which partitions the

lattice into region A and region Ac. We include Σ as part of region A.

B = π on the red simplices, while B = 0 elsewhere. The dotted loop is

the dual lattice configuration of the red simplices. In this example, the

configuration CE contains two B = π 1-simplices at the entanglement

cut Σ, which are the fourth and eighth 1-simplices of Σ (counting from

the left side) as shown in the figure. . . . . . . . . . . . . . . . . . . . 136

5.5 A particular spatial configuration with one loop γ1 (dashed line)

threading through the hole (the hole itself belongs to region Ac) inside

the region A and one loop γ2 (grey line) threading through the hole

inside the region Ac. γ3 and γ4 are two linked contractible loops, where

γ3 locates inside region A, and γ4 locates both in region A and Ac.

The two blue points are the intersection of l4 with Σ. The simplices

(gray triangles) are living in the real lattice where B = π. The lines

perpendicular to the simplices are living in the dual lattice where

B = π and they form loops in the dual lattice. This configuration

corresponds to α = o, β = o. . . . . . . . . . . . . . . . . . . . . . . . 138

6.1 Lorentz symmetry fractionalization on the Wilson line. The left panel

is the Wilson line with K1 = K2 = 0. When the background field B

for the one-form symmetry is activated, the Wilson line is attached

to a surface Σ bounded by γ. This means that the Wilson line carries

charge 1 under Z2,[1]. K1 = K2 = 0 implies that W1/2 is the worldline of

a boson and a Kramers singlet. The right panel is the Wilson line with

nontrivial (K1, K2). The quantum number of the Lorentz symmetry is

shown in (6.17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

xvii

6.2 When time reversal is spontaneously broken, there are two vacua. We

consider a configuration where each vacuum occupies half of the space,

and there is a domain wall in between. Time reversal exchanges the

two vacua. The anomaly (6.23) in the bulk induces an anomaly (6.24)

on the domain wall, which consequently constrains that there is an

Abelian semion TQFT on the wall. . . . . . . . . . . . . . . . . . . . 165

6.3 Schematic RG flow diagram around the QCD4 fixed point for odd Nf

and Nf > 11. Possible IR fates are listed for completeness, although

some (such as the U(1) SL on the θ = 0 side) may be extremely unlikely.189

A.1 Graphical representation of Eq. (A.7). . . . . . . . . . . . . . . . . . 202

A.2 Graphical representation of (a) Eq. (A.64) and (b) Eq. (A.67). . . . 217

A.3 Graphical representation of (a) Eq. (A.70) and (b) the virtual operator

U r1,1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

A.4 Graphical representation of (a) Eq. (A.86) and (b) Eq. (A.88), and (c)

Eq. (A.91). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

B.1 RBM network for cocycle model with q = 3, P12 = P13 = 1, P23 = 0. . 241

B.2 RBM network for cocycle model with q = 4, P12 = P13 = P14 = 1, P23 =

P24 = P34 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

D.1 Entanglement surfaces used in the application of strong sub-additivity

to derive the recurrence relation Eq. (D.17). In (a), A is a general

3-manifold (as an example, we draw A with 1 genus 3 surface and 2

genus 0 surfaces), B is 3-ball and C is a solid torus. In (b), A′ is a

general 3-manifold (as an example, we draw A′ with 1 genus 3 surface

and 2 genus 0 surfaces), B′ is a solid torus, and C′ is a 3-ball, which is

located exactly at the hole of B′. . . . . . . . . . . . . . . . . . . . . 254

xviii

D.2 Entanglement surfaces used in the application of strong sub-additivity

to derive Eq. (D.22). In (a), A is a 3-manifold with multiple genus

zero surfaces, B is a 3-ball, C is a 3-ball with small 3-ball removed. In

(b), A′ is an open 3-manifold with multiple genus zero surfaces, B′ is

a 3-ball with a small 3-ball removed and C′ is a 3-ball located exactly

in the empty 3-ball inside B′. . . . . . . . . . . . . . . . . . . . . . . 257

D.3 KPLW prescription of regularized entanglement surface T 2. . . . . . . 260

D.4 Left: Regularization of a rectangular hinge with small arcs. Right:

One choice of regularization of each hinge in Fig. D.3. The numbers

label various hinges. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

D.5 Dual lattice of a tetrahedron (ijkl). (ijkp), (ijlq), (iklr), (jkls) are

four adjacent tetrahedra to (ijkl), which are dual to (b), (c), (d), (e), (a)

respectively. The red dots are the intersection between 2-simplices in

the real lattice and the 1-simplices in the dual lattice. For example,

the red dot on (ab) is the intersection point of (ab) and (ijk). . . . . . 271

xix

D.6 We illustrate the geometric meaning of the Hodge dual in a two-

dimensional space example. Suppose A is a 1-cochain, which equals

π on 1-simplices in the dual lattice and 0 elsewhere. A = π ∗2 Σ(l1) +

π ∗2 Σ(l2), where l1 and l2 are loops in the dual lattice drawn in dashed

lines. Σ(l1) and Σ(l2) are 1-cochains living on the 1-simplices in the

dual lattice. ∗2 is a lattice version of Hodge star, which transforms

the 1-cochain living on the dual lattice (dashed lines) to a 1-cochain

living on the lattice (green and purple bold lines). Correspondingly,

A = π ∗2 Σ(l1) + π ∗2 Σ(l2) is a 1-cochain living on the green and pur-

ple bold lines. We use the dual lattice configuration Si, li to label the

B,A-cochains because the dual lattice configurations are easier to vi-

sualize. The interpretation of the 2-cochain B can be straightforwardly

generalized to three spatial dimensions. . . . . . . . . . . . . . . . . . 276

D.7 Regularization of a spatial lattice. The blue arrow represents the con-

stant vector (ax, ay, az). The dashed lattice is obtained from the solid

lattice by the translation (x, y, z)→ (x+ ax, y + ay, z + az). . . . . . 280

D.8 An example of lattice regularization of a trefoil knot. l is a knot (drawn

in the dual lattice), while la is the knot obtained by lattice regulariza-

tion. The underlying lattice is omitted for clarity. . . . . . . . . . . . 280

D.9 A configuration associated with nontrivial CE (on panel (a)) can be

reduced to a configuration associated with trivial CE (on panel (b)). 283

D.10 A configuration associated with trivial CE (on panel (a)) can be reduced

to a configuration associated with a nontrivial CE (on panel (b)). . . 283

xx

D.11 Configurations on a 2 × 2 lattice with periodic boundary conditions.

There are two entanglement cuts, denoted by two green lines. The oc-

cupied bonds in the real lattice are shown in red, and occupied bonds in

the dual lattice are shown as dotted lines. (a), (b), (c), (d) are configu-

rations with no bonds occupied on the entanglement cut. (e), (f), (g),

(h) are configurations with two bonds occupied on the entanglement

cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

xxi

Chapter 1

Preliminaries

Topological phases of matter has recently emerged as one of the central themes of con-

densed matter physics[1, 2, 3, 4, 5]. Prior to the discovery of topological phases, the

consensus in the physics community (dubbed the Landau Paradigm) was that gapped

phases could be classified by symmetry breaking order parameters[6, 7] where the

ground states spontaneously break the global symmetry. The discovery of topological

order[8, 9, 10] revealed that two gapped systems can reside in distinct phases ab-

sent any global symmetries. The discovery of symmetry protected topological (SPT)

order[11, 12, 13, 2, 14, 15] further enriched the family of topological phases of matter:

two systems with the same global symmetry can be in different phases even with

trivial topological order. Both the topological order and the SPT phases (which de-

scribe the phases of gapped Hamiltonians) belong to the family of topological phases

of matter. There are other candidates of topological phases of matters that describe

the phases of gapless Hamiltonians[16, 17, 18, 19], which will not be discussed in this

thesis.

1

1.1 Representing the Ground State Wavefunctions

The ground state encodes the key information of the topological phases of the gapped

Hamiltonian. For instance, in two spatial dimensions, the subleading term of the

entanglement entropy of the ground state (i.e., the topological entanglement entropy

(TEE)) reveals the total quantum dimension D of the topological order[20, 21]. For

example, when D = 2, there are only two possible topological orders: Z2 topological

order (described by the toric code model) [22], and twisted Z2 topological order

(described by the double semion model)[23]. The overlap between different ground

states encodes the information of quasi-particle statistics. [24]

In this thesis, we will study the two representations of the ground state wavefunc-

tions, i.e., the Tensor Network State (TNS) and the Restricted Boltzmann Machine

(RBM). In particular, when a wavefunction describes the ground state of a one di-

mensional Hamiltonian, the TNS is usually dubbed the Matrix Product State (MPS).

1.1.1 Tensor Network States

TNS has been heavily used in condensed matter physics in the past decade, especially

in the study of 1D and 2D topological phases[25].1 Amongst many examples,

1. Numerical simulations of the 1D Haldane chain led to the discovery of symmetry

protected topological phases (SPT)[26].

2. Fractional quantum Hall states can be exactly written as MPS[27, 28, 29, 30,

31, 32, 33, 34, 35, 36, 37, 38] which allows performing numerical calculations

not accessible by exact diagonalization techniques.

3. A large class of spin liquids wave functions can be constructed using TNS with

global spin rotation symmetries and lattice symmetries[39, 40, 41, 42, 43, 44, 45].

1We use nD to denote n spatial dimension, and nd to label n spacetime dimension. We will alsosometimes use (n+ 1)d to label n spatial dimension and 1 time dimension.

2

(a)

(b)

Figure 1.1: Examples of TNS lattice wave functions in 1D and 2D. Each node is a

tensor whose indices are the lines connecting to it. The physical indices - of the quan-

tum Hilbert space - are the lines with arrows, while the lines without any arrows are

the virtual indices. Connected lines means the corresponding indices are contracted.

Panel (a) is an MPS for 1D systems. Panel (b) is a PEPS on a 2D square lattice.

The definition of the TNS is as follows. For convenience, we will focus on the TNS

defined on a 2d square lattice M, see (b) in Figure 1.1. M can be the infinite 2d

plane M = R2, or 2d torus M = T 2. (The definitions of TNS in other dimensions

and in other lattices are similar.) Each vertex supports a physical degree of freedom

(dof) labeled as s~r where ~r denotes the location of the site.2 For instance, if the dof at

each site is a spin-S, then s~r can take 2S+1 values, which can be conveniently labeled

as s~r ∈ {0, 1, ..., 2S}. We denote 2S+ 1 as the physical bond dimension. Graphically,

we use an arrow to represent this physical dof at each site as in Figure 1.1.

2The physical dof can also be chosen to define on links, which will be the case in Chapter 4.

3

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Figure 1.2: A tensor T s~rv1,~rv2,~rv3,~rv4,~ron a 2d square lattice with one physical index s~r

and four virtual indices v1,~rv2,~rv3,~rv4,~r.

To write down the wavefunction, one associates a tensor T s~rv1,~rv2,~rv3,~rv4,~rwith one

physical index s~r and four virtual indices v1,~rv2,~rv3,~rv4,~r on each site, see Figure 1.2.

Suppose each virtual index can be valued in P distinct values, e.g. va,~r ∈ {1, ..., P},

we denote P as the virtual bond dimension. The tensor network state is

|TNS〉 =∑

{s~r}

CM(∏

~r∈M

T s~r

)|{s~r}〉 (1.1)

Let us explain the notations in Eq. (1.1).

1. As explained above, M can be for instance the infinite 2d plane R2 or a 2d

torus T 2 depending on the choice of boundary conditions.

2. |{s~r}〉 is the short hand notation of the direct product state ⊗~r∈M|s~r〉, which

serves as a set of complete, orthogonal and normalized basis of the TNS.

3.∑{s~r} represents the sum over all possible configurations of the spin configura-

tions on the lattice M.

4. CM(∏

~r∈M T s~r)

is the coefficient of the basis |{s~r}〉, which represents the con-

traction over all the virtual indices shared by the neighorhood tensors. As a

simple example, let us consider a 2d torus with two sites along each direction,

4

see figure 1.3. Then

CT 2

(∏

~r∈M

T s~r

)=

P∑

v1,v2,...,v8=1

Ts~0v1v3v2v4T

s~xv2v4v1v8

Ts~yv5v7v6v3T

s~x+~yv6v8v5v4 (1.2)

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Figure 1.3: Tensor network on a 2d 2× 2 torus lattice.

In this thesis, we will study the TNS and MPS representation of the ground state

of a particular class of Hamiltonians: the translation invariant stabilizer codes. The

stabilizer codes are a class of spin Hamiltonians where every term mutually commute.

These are the simplest class of Hamiltonians with a finite energy gap. We will address

the following question: Given a stabilizer code Hamiltonian, how do we find the

MPS/TNS of its ground state? We will address this question in Chapter 2 for an

arbitrary 1D translation invariant stabilizer code based on various nice properties of

1D MPS, following [46]. In Chapter 4, following [47], we work out the TNS for a large

class of stabilizer codes in higher dimensions — the CSS codes — whose Hamiltonian

terms are either products of purely Pauli X or purely Pauli Z operators (and there

are no mixing terms). We will find simple algorithms to find the tensors/matrices

directly from the Hamiltonian whose virtual bond dimension is minimal.

5

1.1.2 Restricted Boltzmann Machine

Restricted Boltzmann machines (RBM)[48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59,

60, 61, 62, 63, 64] and more generally neural networks, have recently gained lots

of attention as numerical tools for studying quantum many-body physics, boosted

by the fast paced progress in machine learning. An RBM is a restriction from a

Boltzmann machine (BM). The BM is defined on a bipartite graph, whose vertices

are grouped into two classes: the visible vertices and the hidden vertices. Suppose

there are n visible vertices andm hidden vertices, and we associate the visible variables

g ∈ {0, 1}n and the hidden variables h ∈ {0, 1}m on the visible and hidden vertices

respectively. The variables {g, h} obey the Boltzmann distribution,

P (g, h) =1

Zexp (−E(g, h)) , (1.3)

where E(g, h) is a real function mimicking the “energy” in the Boltzmann distribution,

and Z =∑

g,h exp (−E(g, h)) is the partition function. As the name suggests, only the

visible variables will show up in the physical probability distribution, while the hidden

variables are summed over and thus hidden. Given Eq. (1.3), the BM is defined to be

the marginal distribution P (g) over the visible variables g by summing over all the

hidden variables {h}

P (g) =∑

h

P (g, h) =1

Z

∑

h

exp (−E(g, h)) . (1.4)

The RBM further requires that the “energy” function E(g, h) depends linearly on g

and h. The most important property of RBM is its representing power. It has been

proven[65] in the machine learning context that any probability distribution P0(g) of

an n number of Z2 variables, i.e., g ∈ {0, 1}n, can be approximated arbitrarily well

by an RBM P (g) given enough number of hidden spins. See Ref. [65] for details.

6

For the purposes of the quantum physics, it is natural to change the “energy”

function E(g, h) from a real function to a complex one. Then we can interpret the

“complex probability distribution” P (g) as the coefficients of a quantum many-body

wave function:3 4

|Ψ〉 =∑

g

P (g)|g〉, (1.5)

where |g〉 is the basis to expand the quantum states |Ψ〉. The RBM state refers to

the ansatz in Eq. (1.5).

In chapter 3, following [46], we study the RBM representation of the ground state

of the 1D translation invariant stabilizer code, the simplest class of models one can

possibly study. We make progresses toward answering the following questions, when

the stabilizer codes have one ground state with periodic boundary condition (PBC):

1. How to map a translational invariant and finitely connected RBM to an MPS?

2. Given a stabilizer code, can we cast the ground state as an RBM state mini-

mizing the number of hidden spins?

For the first question, we give a necessary condition for a MPS that can be written as

an RBM state. For the second one, we find a sufficient condition where the ground

state of a stabilizer code can be exactly written as an RBM state with minimal hidden

spins.

1.2 Entanglement Entropy

Entanglement entropy is a key ingredient in characterizing the “nontrivialness” of

the ground state wavefunction. If a ground state of a gapped Hamiltonian does not

have entanglement entropy, it means that the state is a product state, hence by

3To obtain a normalized state, we need to rescale P (g) by a common factor irrelevant of g.4The construction of the quantum wave-function from a classical Hamiltonian has been discussed

in Ref. [66] following the work of Rokhsar and Kivelson[67].

7

definition the Hamiltonian belongs to a trivial phase. Moreover, one can also use

the entanglement entropy to probe the topological phase, which will be discussed in

detail in this thesis.

Let us define the entanglement entropy for a generic state |ψ〉. To define the

entanglement unambiguously, one needs to discretize the space M into a lattice. As

an example, we consider a 2D lattice in (b) of Figure 1.1 where each site supports

some dof which expand a local Hilbert space H~r. (When the dof in each site is a

spin 12, then the local Hilbert space is two dimensional, expanded by | ↑〉, | ↓〉. ) The

entire Hilbert space HM is the tensor product of the local Hilbert spaces at each site:

HM = ⊗~rH~r (1.6)

We further divide the lattice into two dis-adjoint regions, denoted as A and B respec-

tively. M = A∪B, where ∪ represents ”union”. Hence a given site either belongs to

region A or region B. Then the total Hilbert space HM splits into the direct product

of the Hilbert space in each region:

HM = ⊗~r∈A ⊗~r∈B H~r = HA ⊗HB (1.7)

where HA = ⊗~r∈AH~r and HB = ⊗~r∈BH~r. The density matrix ρ of the state |ψ〉 is

simply ρ = |ψ〉〈ψ|. The reduced density matrix of region A is given by the trace of ρ

on HB.

ρA = TrHB |ψ〉〈ψ| (1.8)

8

The entanglement entropy of region A is given by the Von-Neumann entropy of the

reduced density matrix ρA

SA = −TrρA log ρA (1.9)

In particular, when |ψ〉 is the ground state of a gapped Hamiltonian, the entan-

glement entropy SA typically obeys the area law [68]: the entanglement entropy of

the ground states with respect to a subregion A grows linearly with the area of the

boundary Area(∂A) of subregion A:

SA ∼ Area(∂A).

Specifically, in 1D, the entanglement entropy of the subsystem A is a constant (which

is proven in [68]):

Sl ∼ Const,

since the boundary of A contains only two points. In 2D, the entanglement entropy

of the subsystem A obeys:

SA ∼ l,

where l is the perimeter length of the boundary of A. For topological ordered phases

in 2D, the area law gets supplemented by a sub-leading constant contribution dubbed

topological entanglement entropy (TEE) [69, 70], which contains the total quantum

dimension of the ground state. An important feature of the TEE is that it is invariant

under the renormalization group flow of the Hamiltonian as well as the deformation

of the entanglement cut. Therefore. the TEE is one of the universal characterizations

of phases of matters.

In higher spatial dimensions than 2D, SA exhibits more exotic properties:

9

1. For a Hamiltonian with conventional topological order (where the ground state

degeneracy on a 3-torus is finite), the subleading terms to the area law term

can host non-topological information of the system. This is in contrast to the

entanglement entropy of the 2D ground state, where the subleading term is

topological. Hence it is more involved to understand the 3D topological entan-

glement entropy. This aspect will be addressed in chapter 5.

2. There are more exotic topological orders in 3D, where the ground state degen-

eracy on a 3-torus depends on the system size. Recently, 3D so-called fracton

models[71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90,

91, 92] represented by Haah code[71] and X-cube model have been proposed,

attracting the attention of both quantum information[93] and condensed matter

community[94, 95, 96, 97, 98]. They can be realized by stabilizer code Hamilto-

nians, whose fundamental property is that they consist solely of sums of terms

that commute with each other. They are hence exactly solvable. The defining

features of fracton models include (but are not restricted to):

(a) Fracton models have an energy gap, since they can be realized by com-

muting Hamiltonian terms.

(b) The ground state degeneracy on the torus changes as the system size

changes. Hence, fracton models seem not to have thermodynamic limits.

(c) The low energy excitations can have fractal shapes, other than only points

and loops available in conventional topological phases.

(d) The excitations of fracton models are not fully mobile: they can only

move either along submanifold of the 3D lattice (Type I fracton model), or

completely immobile without energy dissipation (Type II fracton model).

We will discuss the entanglement entropy of 3D stabilizer codes extensively in chapter

4 using the TNS, following [47].

10

1.3 Anomalies and SPT phases

Another interesting aspect of topological phase of matter is that when the space

has a boundary, there can be interesting phenomena (such as gapless mode, sponta-

neous symmetry breaking, nontrivial topological order, etc) localized on the boundary.

[2, 14, 99, 100] This relation between the topological phases and boundary phenomena

has been established for a particular subclass of topological phases: the SPT phases.

Concretely, the boundary of the (d+ 1) dimensional system in a SPT phase can not

be a trivially gapped insulator (whose ground state can be adiabatically connected

to a product state). The nontrivial degree of freedom on the boundary should be

anomalous in order to componsate the nontrivial anomaly inflow from the SPT in

the bulk [101, 102, 5, 103, 104]. Recently there are extensive progress toward under-

standing the boundary of topological order, which the boundary can support degrees

of freedom with “non-invertible anomaly”[105, 106, 107].

A canonical example in condensed matter physics is the topological insulator in

(3+1)d. [2, 14, 99, 100] A topological insulator in (3+1)d is within a nontrivial SPT

phase protected by (U(1)oZT4 )/Z2, where U(1) is the charge conservation symmetry,

ZT4 is the time reversal symmetry which satisfies T 2 = (−1)F , and the Z2 in the

denominator means that the subgroup Z2 ⊂ U(1) and the subgroup ZT2 ⊂ ZT4 are

identical, which is precisely the fermi parity generated by (−1)F . On the boundary,

it is well known that the bands have to cross and the vicinity of the crossing point

can be described by a Dirac fermion. The symmetry (U(1)oZT4 )/Z2 in such a Dirac

fermion is know to be anomalous: gauging the U(1) symmetry of a Dirac fermion

necessarily breaks time reversal symmetry. If we want to gauge U(1) while preserving

the time reversal, we need to consider the combined system of (2 + 1)d Dirac fermion

and (3+1)d topological insulator in the bulk. For this combined system, gauging U(1)

symmetry does not break the time reversal, hence the combined system is anomaly

free.

11

In this thesis, following [108, 109], we study another example of the correspondence

“anomalous dof on the boundary ←→ SPT in the bulk”. The “anomalous dof” will

be (3 + 1)d Yang-Mills (YM) theory with a theta term at (θ = π), which has a Z2,[1]

one-form symmetry and time reversal symmetry ZT2 . We will study the anomaly of

the global symmetry Z2,[1] × ZT2 , and identify the SPT in the (4 + 1)d bulk. We will

also discuss the dynamics of this (3 + 1)d YM theory, which is constrained by the

anomaly.

12

Chapter 2

Matrix Product State of A

Stabilizer Code in 1D

In this chapter, we present our algorithm to find an MPS for the ground state of a

translational invariant stabilizer code. We illustrate our algorithm using an example,

the ZZXZZ model, and then discuss of the general case. Each step of the algorithm

is proven in App. A.2, A.3, A.4, A.5 and A.6. We derive and prove our results based

on the following assumptions throughout this chapter:

Assumption 2.0.1. We only consider the translational invariant stabilizer codes that

have a unique ground state with periodic boundary condition (PBC).

Assumption 2.0.2. The MPS matrices of the translational invariant stabilizer codes

become independent of the system size for sufficiently large system sizes.

Assumption 2.0.3. The MPS matrices are independent of the boundary condition.

In other words, in the bulk, the MPS matrices for PBC are the same as those for the

open boundary condition (OBC).

We begin by stating the notations of spin chains. We mainly consider spin models

defined on a finite chain with L unit cells and PBC. Each unit cell contains q spin-12’s.

13

For the i-th spin (i = 1, . . . , q) in the r-th unit cell (r = 0, . . . , L− 1), we associate a

two dimensional Hilbert space spanned by |gri 〉, where gri = 0, 1 corresponds to spin

up and spin down respectively. |gri 〉 satisfies

Zri |gri 〉 = (−1)g

ri |gri 〉, Xr

i |gri 〉 = |1− gri 〉, (2.1)

where Zri and Xr

i are Pauli Z and X matrices acting on |gri 〉.

2.1 An Example of Stabilizer Codes: ZZXZZ

Model

To define the ZZXZZ model, we place 3 physical spin-12’s in each unit cell, i.e.,

q = 3. (We choose q = 3 since it fits naturally into the discussion of general cocycle

models. See App. B.1 for details.) We introduce three sets of commuting operators

Orα (α = 1, 2, 3) defined as

Or1 = Zr2Z

r3X

r+11 Zr+1

2 Zr+13

Or2 = Zr3Z

r+11 Xr+1

2 Zr+13 Zr+2

1

Or3 = Zr1Z

r2X

r3Z

r+11 Zr+1

2 ,

(2.2)

where r is defined modulo L due to PBC. Using these operators, the Hamiltonian

reads

HZZXZZ = −L−1∑

r=0

(Or1 +Or2 +Or3). (2.3)

All the terms in the Hamiltonian Eq. (2.3) mutually commute, and have eigenvalues

±1. Thus its ground state is the common positive eigenstate of Orα for any r and α,

14

i.e.,

Orα|GS〉 = |GS〉, α = 1, 2, 3, r = 0, 1, . . . , L− 1. (2.4)

For example, one can construct |GS〉 as

|GS〉 =L−1∏

r=0

3∏

α=1

(1 +Orα

2

)|0〉, (2.5)

where

|0〉 =L−1⊗

r=0

3⊗

i=1

|0ri 〉. (2.6)

It is straightforward to verify that the |GS〉 in Eq. (2.5) satisfies Eq. (2.4).

Our goal in this section is to express the ground state |GS〉 as an MPS

|GS〉 =∑

{gri }

Tr

( L−1∏

r=0

T gr1gr2gr3

)|{gri }〉, (2.7)

where

|{gri }〉 ≡L−1⊗

r=0

3⊗

i=1

|gri 〉. (2.8)

The matrix T gr1gr2gr3 is labeled with three physical indices gr1, g

r2 and gr3 in the r-th

unit cell. The left and right virtual indices of T gr1gr2gr3 and matrix elements will be

solved in Sec. 2.2. The product of two T matrices amounts to contracting the pair of

virtual indices between them. The coefficient of |{gri }〉 is determined by contracting

all virtual indices with the same configuration of physical spins {gri }.

15

Figure 2.1: A graphical representation of the matrix T gr1gr2gr3 . We denote each physical

index by an arrow. The shaded region represents a unit cell, and the virtual left

and right indices are represented by the horizontal line. The virtual indices are not

explicitly shown here.

The matrix T gr1gr2gr3 is graphically represented in Fig. 2.1. Some notations and general

properties of MPS for stabilizer codes are given in App. A.1.

2.2 MPS for the ZZXZZ Model

To derive the MPS for the ground state |GS〉 of the ZZXZZ model Eq. (2.3), we

start with Eq. (2.4). Ori acts on the basis |{gri }〉 in each summand of |GS〉 in Eq. (2.7).

By re-arranging the summation, we derive the action of Ori on the T -matrices. For

16

example, let us consider the action of Or1,

Or1|GS〉 =∑

{gr′i }

Tr

( L−1∏

r′=0

T gr′1 g

r′2 g

r′3

)Or1|{gr

′

i }〉

=∑

{gr′i }

Tr

( L−1∏

r′=0

T gr′1 g

r′2 g

r′3

)(−1)g

r2+gr3+gr+1

2 +gr+13

×∣∣∣∣{gr

′

i }|r′≤r, {(1− gr+11 )gr+1

2 gr+13 }, {gr′′i }|r′′>r+1

⟩

=∑

{gr′i }

Tr

((∏

r′<r

T gr′1 g

r′2 g

r′3

)· T gr1 gr2 gr3 · T (1−gr+1

1 )gr+12 gr+1

3

·( ∏

r′>r+1

T gr′1 g

r′2 g

r′3

))(−1)g

r2+gr3+gr+1

2 +gr+13 |{gr′i }〉

≡∑

{gr′i }

Tr

((∏

r′<r

T gr′1 g

r′2 g

r′3

)· Or1 ◦

(T g

r1 gr2 gr3 · T gr+1

1 gr+12 gr+1

3

)

·( ∏

r′>r+1

T gr′1 g

r′2 g

r′3

))|{gr′i }〉.

(2.9)

In the second equality, we use the definition of Or1 in Eq. (2.2), and in the last equality

we defined the action of Or1 on T gr1gr2gr3 · T gr+1

1 gr+12 gr+1

3 , via

Or1 ◦(T g

r1gr2gr3 ·T gr+1

1 gr+12 gr+1

3

)≡ (−1)g

r2+gr3+gr+1

2 +gr+13

(T g

r1gr2gr3 ·T (1−gr+1

1 )gr+12 gr+1

3

). (2.10)

For Or2 and Or3, we similarly define

Or2 ◦(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3 · T gr+21 gr+2

2 gr+23

)

≡ (−1)gr3+gr+1

1 +gr+13 +gr+2

1

(T g

r1gr2gr3 · T gr+1

1 (1−gr+12 )gr+1

3 · T gr+21 gr+2

2 gr+23

)

Or3 ◦(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3

)≡ (−1)g

r1+gr2+gr+1

1 +gr+12

(T g

r1gr2(1−gr3) · T gr+1

1 gr+12 gr+1

3

).

(2.11)

17

Figure 2.2: A graph representation of Eq. (2.4).

Using the definitions Eqs. (2.10) and (2.11), we find that a sufficient condition for the

stabilizer condition Eq. (2.4) is

Or1 ◦(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3

)=(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3

)

Or2 ◦(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3 · T gr+21 gr+2

2 gr+23

)=(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3 · T gr+21 gr+2

2 gr+23

)

Or3 ◦(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3

)=(T g

r1gr2gr3 · T gr+1

1 gr+12 gr+1

3

).

(2.12)

We prove in App. A.2 and A.3 that Eq. (2.12) is also a necessary condition for

Eq. (2.4). A graphical representation of these equations is given in Fig. 2.2.

We now construct a solution of T matrices from Eq. (2.12). It is difficult to solve

Eq. (2.12) directly, as it is a set of nonlinear equations of the T matrices. In the

following, we will derive a new set of equations equivalent to Eq. (2.12), which are

linear in the T matrices and only contain the matrices in the r-th unit cell.

18

The idea to derive the equations linear in the T matrices is to decompose the

Hamiltonian terms Orα into separate parts, where each part acts only on one unit cell,

and then derive their action on a single T matrix. (Notice that we are only allowed

to cut in between the unit cells, not inside one unit cell.) As a first step, we start by

cutting the operator Orα into two parts: Or1 and Or3 can be cut into Or1 = Lr1,1Rr1,1

and Or3 = Lr3,1Rr3,1, such that the operators

Lr1,1 = Ir1 ⊗ Zr2 ⊗ Zr

3

Rr1,1 = Xr+1

1 ⊗ Zr+12 ⊗ Zr+1

3

(2.13)

and

Lr3,1 = Zr1 ⊗ Zr

2 ⊗Xr3

Rr3,1 = Zr+1

1 ⊗ Zr+12 ⊗ Ir+1

3

(2.14)

only act on a given unit cell. The second subscript τ of Lrα,τ labels the position of

bipartition of Orα. Since Or1 and Or3 are supported on two unit cells, there is only

one way to cut them into two parts and hence τ only takes one value, i.e., τ = 1.

Such a unique bipartition is not possible for Or2 since Or2 is supported on 3 unit cells.

Nevertheless, we can define two bipartitions as follows:

Or2 = Lr2,1Rr2,1, Or2 = Lr2,2Rr

2,2, (2.15)

where Lr2,τ ,Rr2,τ (τ = 1, 2) are

Lr2,1 = Ir1 ⊗ Ir2 ⊗ Zr3

Rr2,1 = Zr+1

1 ⊗Xr+12 ⊗ Zr+1

3 ⊗ Zr+21 ⊗ Ir+2

2 ⊗ Ir+23

Lr2,2 = Ir1 ⊗ Ir2 ⊗ Zr3 ⊗ Zr+1

1 ⊗Xr+12 ⊗ Zr+1

3

Rr2,2 = Zr+2

1 ⊗ Ir+22 ⊗ Ir+2

3 .

(2.16)

19

Figure 2.3: A graphical representation of Eqs. (2.17), (2.18) and (2.19).

Notice that Rr2,1 and Lr2,2 have a support over two unit cells while Rr

2,2 and Lr2,1 have

a support over a single unit cell.

Let us consider the action of Lrα,τ ’s and Rrα,τ ’s on the T matrices. First we focus

on Or1. The product of two matrices T gr1gr2gr3 · T gr+1

1 gr+12 gr+1

3 should be invariant under

the combined action of Lr1,1Rr1,1, where Lr1,1 acts only on T g

r1gr2gr3 while Rr

1,1 only on

T gr+11 gr+1

2 gr+13 . In App. A.4, we prove in a general setting of stabilizer codes that the

following condition is both necessary and sufficient of Eq. (2.12): the action of Lr1,1 on

T gr1gr2gr3 can be encoded by a transformation U r

1,1 on the right virtual index of T gr1gr2gr3 ,

while the action of Rr1,1 on T g

r+11 gr+1

2 gr+13 can be encoded by the inverse of the same

transformation (U r1,1)−1 on the left virtual index of T g

r+11 gr+1

2 gr+13 . Concretely, we have

20

Lr1,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

1,1

Rr1,1 ◦ T g

r+11 gr+1

2 gr+13 = (U r

1,1)−1 · T gr+11 gr+1

2 gr+13 ,

(2.17)

where ◦ represents the action on the physical indices (see Eqs. (2.10) and (2.11)), and

· represents the matrix multiplication (i.e., the contraction over the virtual indices).

See Fig. 2.3. Similarly, for Lr2,1,Rr2,1 and Lr2,2,Rr

2,2,

Lr2,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

2,1 (2.18a)

Rr2,1 ◦ (T g

r+11 gr+1

2 gr+13 · T gr+2

1 gr+22 gr+2

3 )

= (U r2,1)−1 · (T gr+1

1 gr+12 gr+1

3 · T gr+21 gr+2

2 gr+23 ) (2.18b)

Lr2,2 ◦ (T gr1gr2gr3 · T gr+1

1 gr+12 gr+1

3 )

= (T gr1gr2gr3 · T gr+1

1 gr+12 gr+1

3 ) · U r2,2 (2.18c)

Rr2,2 ◦ T g

r+21 gr+2

2 gr+23

= (U r2,2)−1 · T gr+2

1 gr+22 gr+2

3 . (2.18d)

For Lr3,1,Rr3,1, we get

Lr3,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

3,1

Rr3,1 ◦ T g

r+11 gr+1

2 gr+13 = (U r

3,1)−1 · T gr+11 gr+1

2 gr+13 .

(2.19)

Eqs. (2.17), (2.18) and (2.19) are graphically represented in Fig. 2.3.

Let us use the translational invariance and focus on the operators that act on the

virtual indices between the r-th and (r+1)-th unit cells, i.e., U r1,1, U

r2,1, U

r−12,2 and U r

3,1.

For Eqs. (2.17), (2.18) and (2.19) being consistent, the virtual U r′

α′,τ ′ and (U r′

α′,τ ′)−1

operators on the right hand side (RHS) should satisfy the same commutation relations

as the physical Lr′α′,τ ′ and Rr′

α′,τ ′ operators on the left hand side (LHS) respectively.

21

As a result, Lr′α′,τ ′ and Rr′

α′,τ ′ share the same commutation relation.1 Here,

Lr′α′,τ ′ ∈ {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1},

Rr′

α′,τ ′ ∈ {Rr1,1,Rr

2,1,Rr−12,2 ,Rr

3,1}

U r′

α′,τ ′ ∈ {U r1,1, U

r2,1, U

r−12,2 , U

r3,1}

(2.20)

This statement is proved in App. A.5 in a general setting of stabilizer codes. The

commutation relations can be encoded using the compact notations:

Lr′α′,τ ′Lr′′

α′′,τ ′′ = (−1)tr′,r′′

(α′τ ′),(α′′τ ′′)Lr′′α′′,τ ′′Lr′

α′,τ ′ ,

Rr′

α′,τ ′Rr′′

α′′,τ ′′ = (−1)tr′,r′′

(α′τ ′),(α′′τ ′′)Rr′′

α′′,τ ′′Rr′

α′,τ ′ ,

U r′

α′,τ ′Ur′′

α′′,τ ′′ = (−1)tr′,r′′

(α′τ ′),(α′′τ ′′)U r′′

α′′,τ ′′Ur′

α′,τ ′ ,

(2.21)

Since R operators obey the same commutation relations as the L’s, we just focus on

the L operators. The coefficients tr′,r′′

(α′τ ′),(α′′τ ′′) form an anti-symmetric t matrix, which

under the basis (Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1)T is given by2

t =

0 0 1 1

0 0 0 1

−1 0 0 0

−1 −1 0 0

. (2.22)

We first determine the dimension of irreducible representation of the algebra

Eq. (2.21) that {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1} and {U r

1,1, Ur2,1, U

r−12,2 , U

r3,1} obey. It is conve-

1This is because the commutation relations of L’s and those of the R’s are both related to thecommutation relations of U ’s.

2We only focus on the commutation relations between the virtual U operators which act onthe same virtual bond, e.g. {Ur1,1, Ur2,1, Ur−1

2,2 , Ur3,1}. According to Eq. (2.21), they have the same

commutation relations as {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1}. These commutation relations constrain the rep-

resentation of the virtual U operators. The commutation relations between {Lr1,1,Lr2,1,Lr−12,2 ,Lr3,1}

and other L operators, e.g. Lr+11,1 do not constrain the representation of the virtual U operators,

thus we do not consider them.

22

nient to introduce:

L1 = Lr3,1, U1 = U r3,1,

L2 = Lr2,1, U2 = U r2,1

L3 = Lr−12,2 , U3 = U r−1

2,2

L4 = Lr1,1Lr2,1, U4 = U r1,1U

r2,1.

(2.23)

The new operators {Lα, Uα, α = 1, 2, 3, 4} satisfy a simpler algebra,

LαLβ = (−1)tαβ LβLα

UαUβ = (−1)tαβ UβUα,

(2.24)

with

t =

0 1 0 0

−1 0 0 0

0 0 0 1

0 0 −1 0

. (2.25)

The algebra of {Lα} (or {Uα}) is decoupled into two subalgebras, generated by

{L1, L2} (or {U1, U2}) and {L3, L4} (or {U3, U4}) respectively. Each subalgebra has

a two dimensional irreducible representation. Hence the total dimension of the ir-

reducible representation of {Lα} (or {Uα}) is 2 × 2 = 4. Finally, noticing that

the transformation between L’s and L’s is invertible. The inverse transformation of

Eq. (2.23) is

Lr3,1 = L1, Ur3,1 = U1,

Lr2,1 = L2, Ur2,1 = U2

Lr−12,2 = L3, U

r−12,2 = U3

Lr1,1 = L4(L2)−1, U r1,1 = U4(U2)−1

(2.26)

23

Since L’s form a irreducible representation, the physical operators Lr+11,1 ,Lr+1

2,1 ,Lr2,2and Lr3,1 (and thus by Eq. (2.21), U r+1

1,1 , Ur+12,1 , U

r2,2 and U r

3,1 as well) also provide a 4

dimensional irreducible representation.

For the rest of the task, we need to first find a matrix representation of

U r1,1, U

r2,1, U

r−12,2 and U r

3,1 satisfying the algebra Eq. (2.21), and then solve for T gr1gr2gr3

in Eqs. (2.17), (2.18) and (2.19). Since the irreducible representation of the algebra

is 4-dimensional, the virtual U operators should be 4× 4 matrices.

The matrix representations for the U operators are not unique. If U r′

α′,τ ′ satisfies

the algebra Eq. (2.21) and T gr′1 g

r′2 g

r′3 is the solution of Eqs. (2.17), (2.18) and (2.19),

then S · U r′

α′,τ ′ · S−1, with S independent of r′, α′, τ ′, also satisfies Eq. (2.21) and the

corresponding solution for the T matrices is given by S ·T gr′

1 gr′2 g

r′3 ·S−1. Hence without

loss of generality, let us choose the virtual U operators {U r′

α′,τ ′} to be:3

U r1,1 = (X ⊗X)r,

U r2,1 = (I ⊗X)r,

U r−12,2 = ((−Y )⊗ I)r.

U r3,1 = (I ⊗ (−Y ))r.

(2.27)

where the superscript r on the RHS indicates that the operators acts on the virtual

bonds connecting the the r-th and r + 1-th unit cell. We denote the corresponding

MPS matrix elements as Tgr1g

r2gr3

h1h2,h3h4where h1, h2 ∈ {0, 1} represent the left virtual

indices and h3, h4 ∈ {0, 1} represent the right virtual indices. In Eq. (2.27), the first

Pauli matrices act on the virtual indices h1 and h3, while the second Pauli matrices

act on the virtual indices h2 and h4.

So far, we have only considered bipartition of the Hamiltonian terms Orα. For

the operators that have support over three or more unit cells, we can take the com-

3We choose these U operators in order to compare with the MPS matrices derived from the RBMstates in Sec. 3.4.

24

binations of L and R operators so that Orα can be decomposed into a product of

operators which only act within a single unit cell. We enumerate the decompositions

for all three types of operators:

Or1,1 = Lr1,1 · Rr1,1

Or2 = Lr2,1 ·(Rr

2,1(Rr2,2)−1

)· Rr

2,2

= Lr2,1 ·((Lr2,1)−1Lr2,2

)· Rr

2,2

Or3,1 = Lr3,1 · Rr3,1

(2.28)

where all the terms on the RHS of Eq. (2.28), in particular(Rr

2,1(Rr2,2)−1

)and

((Lr2,1)−1Lr2,2

), are supported within one unit cell. In App. A.6, we show in a general

setting of stabilizer codes, that Eq. (2.18) is equivalent to the following equations

linear to the T matrix in the r-th unit cell:

Lr1,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

1,1

Rr−11,1 ◦ T g

r1gr2gr3 = (U r−1

1,1 )−1 · T gr1gr2gr3

Lr2,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

2,1

((Lr−12,1 )−1Lr−1

2,2 ) ◦ T gr1gr2gr3 = (U r−12,1 )−1 · T gr1gr2gr3 · U r−1

2,2

(Rr−12,1 (Rr−1

2,2 )−1) ◦ T gr1gr2gr3 = (U r−12,1 )−1 · T gr1gr2gr3 · U r−1

2,2

Rr−22,2 ◦ T g

r1gr2gr3 = (U r−2

2,2 )−1 · T gr1gr2gr3

Lr3,1 ◦ T gr1gr2gr3 = T g

r1gr2gr3 · U r

3,1

Rr−13,1 ◦ T g

r1gr2gr3 = (U r−1

3,1 )−1 · T gr1gr2gr3 ,

(2.29)

25

where by translational invariance

U r1,1 = (X ⊗X)r, U r−1

1,1 = (X ⊗X)r−1,

U r2,1 = (I ⊗X)r, U r−1

2,1 = (I ⊗X)r−1,

U r−12,2 = ((−Y )⊗ I)r, U r−2

2,2 = ((−Y )⊗ I)r−1,

U r3,1 = (I ⊗ (−Y ))r, U r−1

3,1 = (I ⊗ (−Y ))r−1.

(2.30)

More concretely, the equations in (2.29) are

Tgr1g

r2gr3

h1h2,h3h4(−1)g

r2+gr3 = T

gr1gr2gr3

h1h2,(1−h3)(1−h4)

T(1−gr1)gr2g

r3

h1h2,h3h4(−1)g

r2+gr3 = T

gr1gr2gr3

(1−h1)(1−h2),h3h4

Tgr1g

r2gr3

h1h2,h3h4(−1)g

r3 = T

gr1gr2gr3

h1h2,h3(1−h4)

Tgr1(1−gr2)gr3h1h2,h3h4

(−1)gr1+gr3 = −iT g

r1gr2gr3

h1(1−h2),(1−h3)h4(−1)1−h3

Tgr1(1−gr2)gr3h1h2,h3h4

(−1)gr1+gr3 = −iT g

r1gr2gr3

h1(1−h2),(1−h3)h4(−1)1−h3

Tgr1g

r2gr3

h1h2,h3h4(−1)g

r1 = −iT g

r1gr2gr3

(1−h1)h2,h3h4(−1)h1

Tgr1g

r2(1−gr3)

h1h2,h3h4(−1)g

r1+gr2 = −iT g

r1gr2gr3

h1h2,h3(1−h4)(−1)1−h4

Tgr1g

r2gr3

h1h2,h3h4(−1)g

r1+gr2 = −iT g

r1gr2gr3

h1(1−h2),h3h4(−1)h2 .

(2.31)

Thus we have derived a set of linear equations Eq. (2.31) of the T -matrices from the

non-linear ones Eq. (2.12). Solving Eq. (2.31), we obtain a solution up to a total scale

26

factor:

T 000 =

1 1 1 1

i i i i

i i i i

−1 −1 −1 −1

, T 001 =

i −i i −i

−1 1 −1 1

−1 1 −1 1

−i i −i i

,

T 010 =

−1 −1 1 1

i i −i −i

−i −i i i

−1 −1 1 1

, T 011 =

i −i −i i

1 −1 −1 1

−1 1 1 −1

i −i −i i

,

T 100 =

−1 −1 −1 −1

i i i i

i i i i

1 1 1 1

, T 101 =

i −i i −i

1 −1 1 −1

1 −1 1 −1

−i i −i i

,

T 110 =

1 1 −1 −1

i i −i −i

−i −i i i

1 1 −1 −1

, T 111 =

i −i −i i

−1 1 1 −1

1 −1 −1 1

i −i −i i

.

(2.32)

2.3 General Stabilizer Code Convention

We will generalize the method in Sec. 2.2 to a generic 1d translation invariant stabilizer

code. A generic translational invariant stabilizer code is described by the Hamiltonian

H = −L−1∑

r=0

t∑

α=1

Orα (2.33)

where t is the total number of types of interactions and α ∈ 1, . . . , t labels the type.

Each unit cell contains q spin-12’s. Orα is a product of Pauli X and Z operators such

27

that (1) Orα is Hermitian and (Orα)2 = 1 for any r, α; and (2) Orα and Or′α′ commute

for any r, r′, α, α′, i.e.,

[Orα,Or′

α′ ] = 0, ∀ α, α′, r, r′. (2.34)

Each interaction term Orα is supported over the unit cells r, r + 1, ..., r + Pα − 1, and

can be written as an ordered product of Pα number of local operators orα,τ

Orα =Pα∏

τ=1

orα,τ , (2.35)

where orα,τ is a product of Pauli matrices only supported on the (r + τ − 1)-th unit

cell. For convenience, we define Lrα,τ and Rrα,τ as follows

Lrα,τ =τ∏

µ=1

orα,µ,

Rrα,τ =

Pα∏

µ=τ+1

orα,µ, τ = 1, . . . , Pα − 1.

(2.36)

By Assumption 2.0.1, we consider only cases where Eq. (2.33) has a unique ground

state when using PBC. The ground state |GS〉 is the common eigenstate of all Orα’s

with eigenvalue 1 for all r and α,

Orα|GS〉 = |GS〉, ∀ r, α. (2.37)

Our goal is to express the ground state |GS〉 as an MPS

|GS〉 =∑

{gri }

Tr

( L−1∏

r=0

T gr1 ...g

rq

)|{gri }〉. (2.38)

where grα, (α = 1, ..., q), labels the value of the α-th physical spin in the r-th unit cell.

28

2.4 General Algorithm to Construct MPS

The calculation algorithm to construct an MPS representation is divided into in 4

steps. These 4 steps will follow those of Sec. 2.2 for the ZZXZZ model.

1. We start with the stabilizer condition Eq. (2.37). A sufficient condition for the

MPS of Eq. (2.38) to satisfy Eq. (2.37) is

Orα ◦( r+Pα−1∏

r′=r

T gr′1 ...g

r′q

)=( r+Pα−1∏

r′=r

T gr′1 ...g

r′q

). (2.39)

Eq. (2.39) is graphically represented as Fig. 2.4. In fact, Eq. (2.39) is not only

sufficient, but also necessary for Eq. (2.37), as derived in App. A.3.

2. To find a solution of Eq. (2.39), we consider a bipartition of the Hamiltonian

term Orα into the product of the left and right part, i.e., Lrα,τ and Rrα,τ . The two

Figure 2.4: Graphical representation of Eq. (2.39). The shaded purple region repre-sents the operator Orα acting on the physical indices.

29

parts act solely on two disjoint and contiguous sets of unit cells. For Pα > 1, since

Orα is supported on the unit cells between r and r + Pα − 1, Lrα,τ is chosen to be

supported from r to r + τ − 1-th unit cell, and Rrα,τ is supported from (r + τ) to

(r + Pα − 1)-th unit cell. The definitions of Lrα,τ and Rrα,τ are in given Eq. (2.36).

Either Lrα,τ or Rrα,τ can act nontrivially on the MPS, although their product leaves

the MPS invariant. This nontrivial action can be captured by a transformation on

the virtual index exactly across the cut (between the (r+ τ −1)-th and the (r+ τ)-th

unit cell). From Eq. (2.39), we find

Lrα,τ ◦( r+τ−1∏

r′=r

T gr′1 ...g

r′q

)=( r+τ−1∏

r′=r

T gr′1 ...g

r′q

)· U r

α,τ (2.40)

Rrα,τ ◦

( r+Pα−1∏

r′=r+τ

T gr′1 ...g

r′q

)= (U r

α,τ )−1 ·

( r+Pα−1∏

r′=r+τ

T gr′1 ...g

r′q

).

Eq. (2.40) is graphically represented as Fig. 2.5. We prove in App. A.4 that Eq. (2.40)

is both necessary and sufficient for Eq. (2.39).

Figure 2.5: Graphical representation of Eq. (2.40). The virtual operator U ri,τ and

(U ri,τ )−1 act on the right virtual index between the r + τ − 1 and r-th unit cell.

30

For convenience, for each choice of (α, τ) we can shift r to r− τ + 1 in Eq. (2.40)

by translational invariance such that the U r−τ+1α,τ acts on the virtual bond between

the r-th and (r+1)-th unit cell. See Fig. 2.6 for the operators that are obtained from

shifting Orα. Under the shift r → r − τ + 1, Eq. (2.40) becomes

Lr−τ+1α,τ ◦

( r∏

r′=r−τ+1

T gr′1 ...g

r′q

)=( r∏

r′=r−τ+1

T gr′1 ...g

r′q

)· U r−τ+1

α,τ

Rr−τ+1α,τ ◦

( r+Pα−τ∏

r′=r+1

T gr′1 ...g

r′q

)= (U r−τ+1

α,τ )−1 ·( r+Pα−τ∏

r′=r+1

T gr′1 ...g

r′q

),

1 ≤ α ≤ t, 1 ≤ τ ≤ Pα − 1

(2.41)

Figure 2.6: An illustration of the operators Lr−(τ−1)α,τ and Rr−(τ−1)

α,τ with fixed r and α,

and all 1τ ≤ Pα−1. The blue blocks represent unit cells. The purple blocks represent

the operators Lr−(τ−1)α,τ , and the operators Rr−(τ−1)

α,τ .

When the operator Orα is supported only over 1 unit cell, i.e., Pα = 1, Eq. (2.39)

is already linear. Without loss of generality, we take Lrα,1 = Orα, Rrα,1 = Ir and

U rα,1 = Ir where Ir is an identity operator acting on the r-th unit cell.

31

3. We further determine the minimal bond dimension of T gr1 ...g

rq . We prove in

App. A.5 that the commutation and anti-commutation relations of the virtual U op-

erators on RHS of Eq. (2.41) should match those of the physical L and R operators

on the LHS,

Lr−(τ ′−1)α′,τ ′ Lr−(τ ′′−1)

α′′,τ ′′ = (−1)tr−(τ ′−1),r−(τ ′′−1)

(α′τ ′),(α′′τ ′′) Lr−(τ ′′−1)α′′,τ ′′ Lr−(τ ′−1)

α′,τ ′ ,

Rr−(τ ′−1)α′,τ ′ Rr−(τ ′′−1)

α′′,τ ′′ = (−1)tr−(τ ′−1),r−(τ ′′−1)

(α′τ ′),(α′′τ ′′) Rr−(τ ′′−1)α′′,τ ′′ Rr−(τ ′−1)

α′,τ ′ ,

Ur−(τ ′−1)α′,τ ′ U

r−(τ ′′−1)α′′,τ ′′ = (−1)

tr−(τ ′−1),r−(τ ′′−1)

(α′τ ′),(α′′τ ′′) Ur−(τ ′′−1)α′′,τ ′′ U

r−(τ ′−1)α′,τ ′ ,

1 ≤ α′, α′′ ≤ t, 1 ≤ τ ′ ≤ Pα′ − 1, 1 ≤ τ ′′ ≤ Pα′′ − 1

(2.42)

The parameter

tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) = 0, 1 mod 2 (2.43)

encodes whether Ur−(τ ′−1)α′,τ ′ and U

r−(τ ′′−1)α′′,τ ′′ commute or anti-commute. We ensemble

the parameters tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) into an anti-symmetric matrix t. 4

The algebra Eq. (2.42) is a generalization of the Clifford algebra, where the

standard Clifford algebra is generated by mutually anti-commuting operators. In

Ref. [110], it was shown that any integer-valued antisymmetric matrix t can be block

diagonalized by a unimodular integer matrix V , such that each nontrivial block is a

2 × 2 anti-symmetric matrix with integer off-diagonal elements. Due to Eq. (2.43),

only the modulo 2 values of the off-diagonal elements of the nontrivial 2 × 2 blocks

4In general, if the set of operators {Uλ} satisfy Uλ′Uλ′′ = eitλ′,λ′′Uλ′Uλ′′ , where tλ′,λ′′ is a realnumber characterizing the commutation relations between {Uλ}, tλ′,λ′′ is antisymmetric: tλ′,λ′′ =−tλ′′,λ′ . This is because from the above commutation relation, one move the phase to the left handside as e−itλ′,λ′′Uλ′Uλ′′ = Uλ′Uλ′′ . Combining with the definition Uλ′′Uλ′ = eitλ′′,λ′Uλ′′Uλ′ , wederive that e−itλ′,λ′′ = eitλ′′,λ′ . This yields that tλ′′,λ′ = −tλ′,λ′′ mod 2π. Applying to our case,

tr−(τ ′−1),r−(τ ′′−1)(α′τ ′),(α′′τ ′′) = −tr−(τ ′′−1),r−(τ ′−1)

(α′′τ ′′),(α′τ ′) mod 2.

32

matter. The nontrivial blocks can therefore be written as follows:5

V tV T =

0 1

−1 0

⊕

0 1

−1 0

· · · ⊕

0 1

−1 0

⊕ 0 · · · . (2.44)

Here we explicitly keep the minus signs to make the antisymmetry manifest. In the

new basis, the operators Lrα,τ become decoupled pairs of anti-commuting operators

(such as Eq. (2.23)); there are rank(t)2

such pairs. Since each pair provides a two

dimensional irreducible representation, the dimension of the irreducible representation

of the generalized Clifford algebra Eq. (2.42) is given by

D = 2rank(t)

2 . (2.45)

Since the dimension of an irreducible representation of the algebra Eq. (2.42) is D, the

matrices of the U rα,τ operators, as well as the MPS matrix T g

r1 ...g

rq under the irreducible

representation should be D×D matrices.6 Since the representation is irreducible, D

is also the minimal bond dimension. For the ZZXZZ model with 3 spins per unit

cell discussed in Sec. 2.2, the t matrix is given by Eq. (2.22), which is of rank 4. By

Eq. (2.45), the minimal bond dimension of the MPS is D = 24/2 = 4, which matches

the MPS explicitly derived in Eq. (2.32).

4. We solve Eq. (2.41) for the MPS matrices T with the minimal bond dimension D.

Let us first determine the form of U . The matrix elements of U can be obtained by

finding the representation of the algebra Eq. (2.42). Here we focus only on irreducible

5When there is an operator Lr−τ+1α,τ commuting with all other Lr−τ ′+1

α′,τ ′ for any (τ ′, α′), the t-matrix is not full rank. For instance, when there is an operator Orα which is only supported overone unit cell, i.e., Pα = 1, then by our convention in the previous paragraph, Lrα,1 = Orα,Rrα,1 = 1.

Then Lrα,1 commutes with Lr−τ ′+1α′,τ ′ for any (τ ′, α′). Hence there are 0 blocks in the decomposition

Eq. (2.44), i.e., t matrix is not full rank.6One may consider Urα,τ = 0 (for all α, τ and r) to be a solution of Eq. (2.42). However, due to

Eq. (2.41), the T gr1 ...g

rq would be zero, hence the MPS is a null state. So we do not consider this

solution.

33

representations such that the bond dimension is minimal. Notice that there exist

multiple choices of U operators satisfying the same algebra Eq. (2.42). However,

since we only consider models with a single ground state, different solutions of T

from different choices of U should correspond to the same ground state. Hence it

is sufficient to work with one choice of U . As shown in App. A.7, U can always

be constructed as a tensor product of rank(t)2

Pauli matrices. After specifying the

virtual U operators, we manipulate the equations in Eq. (2.41) such that all the

physical operators on the LHS only act on the r-th unit cell, and all the equations are

linear in T gr1 ···grq . For instance, using the definition Eq. (2.36), Lr−τ+1

α,τ =∏τ

µ=1 or−τ+1α,µ

and Lr−τ+1α,τ−1 =

∏τ−1µ=1 o

r−τ+1α,µ , the combination

((Lr−τ+1

α,τ−1 )−1Lr−τ+1α,τ

)= or−τ+1

α,τ is only

supported on the r-th unit cell. In App. A.6, we show that Eq. (2.41) are equivalent

to

Lrα,1 ◦ T gr1 ...g

rq = T g

r1 ...g

rq · U r

α,1

((Lr−τ+1

α,τ−1 )−1Lr−τ+1α,τ

)◦ T gr1 ···grq = (U r−τ+1

α,τ−1 )−1 · T gr1 ···grq · U r−τ+1α,τ

(Rr−τ+1α,τ−1 (Rr−τ+1

α,τ )−1)◦ T gr1 ...grq = (U r−τ+1

α,τ−1 )−1 · T gr1 ...grq · U r−τ+1α,τ

Rr−(Pα−1)α,Pα−1 ◦ T gr1 ...grq = (U

r−(Pα−1)α,Pα−1 )−1 · T gr1 ...grq ,

1 ≤ α ≤ t, 2 ≤ τ ≤ Pα − 1

(2.46)

Since Eq. (2.46) is a set of linear equations in T , they can be numerically solved

efficiently. For all the models we have explicitly checked (e.g. Zq−1XZq−1 with

2 ≤ q ≤ 6), Eq. (2.46) has one non-zero solution up to an overall scaling.

34

Chapter 3

Restricted Boltamann Machine

State for Stabilizer Code in 1D

In this chapter, we develop a criteria judging when the ground state of a 1D stabilizer

code can be exactly written as a RBM state. Once the criteria is satisfied, we find

the RBM state with the minimal number of hidden spins.

3.1 (Restricted) Boltzmann Machine

In this section, we introduce the notion of Boltzmann machine (BM) states, restricted

Boltzmann machine (RBM) states and their connection to MPS.

3.1.1 Definitions

A BM state is a state defined by a classical Ising model on a graph. Each vertex of

the graph carries a classical Ising spin sr = 0, 1 where r is the index of the vertex.

Each edge of the graph carries a weight Wrr′ ∈ C that mimics the Ising “interaction”

between sr and sr′, and each vertex also carries a bias αr ∈ C that mimics “an

35

external magnetic field”. The “energy” for such an Ising model is:

EBM({sr}) =∑

r,r′

Wrr′srsr′+∑

r

αrsr, (3.1)

where the summation runs over all spins. In turn, a BM can be efficiently represented

by a graph: (1) the vertices of the graph represent the spins {sr}; (2) the nonzero

weight of sr and sr′

is represented by the link connecting sr and sr′. The set of spins

is divided into two disjoint subsets: the visible spins whose set is denoted by V and

the hidden spins denoted by H. We denote gr the visible spins and hs the hidden

spins. Using these notations, the BM state is defined as:

|BM〉 = C∑

{gr}r∈V

∑

{hs}s∈H

exp

(− EBM({hs}, {gr})

)|{gr}〉, (3.2)

where C is a normalization constant that we will drop for simplicity. The states

|{gr}〉 are the basis states over the visible spins, i.e., a given |{gr}〉 is the direct

product of Pauli Z eigenstates with eigenvalues {(−1)gr}. The “energy” terms in

EBM({hs}, {gr}) can be split into

EBM({hs}, {gr}) =∑

r,r′∈V

Rrr′grgr

′+∑

s,s′∈H

Sss′hshs

′+∑

r∈Vs∈H

Wrsgrhs +

∑

r∈V

βrgr +

∑

s∈H

αshs,

(3.3)

where Wrs, Rrr′ , Sss′ ∈ C are the weights between visible and hidden, visible and

visible, hidden and hidden spins respectively. βr ∈ C is the bias of the visible spin

gr, and αs ∈ C is the bias of the hidden spin hs.

A restricted Boltzmann machine (RBM) state is a special BM state satisfying

Rrr′ = 0, ∀ r, r′ ∈ V ; Sss′ = 0, ∀ s, s′ ∈ H. (3.4)

36

Thus an RBM state reads

|RBM〉 =∑

{gr}r∈V

∑

{hs}s∈H

exp

(− ERBM({hs}, {gr})

)|{gr}〉

(3.5)

with

ERBM({hs}, {gr}) =∑

r∈Vs∈H

Wrsgrhs +

∑

r∈V

βrgr +

∑

s∈H

αshs. (3.6)

In this article, we will consider RBM states for 1D translational invariant systems.

For this reason, we use r, s to label the unit cells, and i, a to label the visible spin

and hidden spins within a unit cell (which are dubbed “orbitals”) respectively. We

further require the RBM to be finitely connected, and by properly enlarging the unit

cell, we can always choose the RBM to be nearest unit cell connected. Due to the

requirement of translational invariance and nearest neighbor connectivity, we label

the visible spins, the hidden spins, the weights and the biases as follows:

1. The visible spins within the unit cell at r are labeled by gri where i = 1, . . . , q

labels the orbitals within the unit cell. q is the number of visible spins within

each unit cell.

2. The hidden spins are divided into two categories:

(a) hra, a ∈ {1, . . . ,M}, labels the hidden spins connecting to the visible spins

from the unit cell at r−1 and those from the unit cell at r, i.e., hra connects

to both {gr−1i } and {gri }. M is the total number of such hidden spins within

the unit cell. Since we assume that the RBM is nearest unit cell connected,

hra does not connect to the visible spins of another unit cell. We will dub

such hidden spins as type-h hidden spins.

(b) hrb, b ∈ {1, . . . , M}, labels the hidden spins connecting to the visible spins

within the unit cell at r, i.e., hrb only connects to {gri }. M is the total

37

number of such hidden spins within the unit cell. We will dub such hidden

spins as type-h hidden spins.

3. The weight connecting hra and gri is labeled by Aia, i ∈ {1, . . . , q}, a ∈

{1, . . . ,M}.

4. The weight connecting hra and gr−1i is labeled by Bia, i ∈ {1, . . . , q}, a ∈

{1, . . . ,M}.

5. The weight connecting hrb and gri is labeled by Cib, i ∈ {1, . . . , q}, b ∈

{1, . . . , M}.

6. The bias of the visible spin gri is βi, i ∈ {1, . . . , q}.

7. The bias of the hidden spin hra is αa, a ∈ {1, . . . ,M}.

8. The bias of the hidden spin hrb is αb, b ∈ {1, . . . , M}.

Due to translational invariance, the weights Aia, Bia, Cib and the biases βi, αa and

αb are all independent of the position of the unit cell r. We have distinguished the

hidden spins into type-h and type-h because, as will be explained in Sec. 3.1.2, the

hidden spins of type-h contribute to the entanglement, while those of type-h do not.

Correspondingly, we distinguish the weights Aia which connect the visible spin gri to

the hidden spins of type-h, i.e., hra, and Cib which connect the visible spin gri to the

hidden spins of type-h, i.e., hrb. In Fig. 3.1, we show an example of such an RBM

state with q = 3,M = 2 and M = 2. The visible spins (i.e., gri ) are represented by

red circles. The hidden spins connecting to the visible spins from the neighboring

unit cells (i.e., hra) are represented by the rectangles and the hidden spins connecting

to the visible spins from a single unit cell (i.e., hrb) are represented by triangles.

38

Figure 3.1: An example of RBM state corresponding to q = 3,M = 2, M = 2. The redcircles represent visible spins. The black rectangles are the hidden spins connectingvisible spin belonging to different unit cells, which are linked to the purple and orangelines representing the weights Aia and Bia respectively. The black triangles are thehidden spins connecting visible spins within the same unit cell, which are linked tothe green lines representing the weights Cib. The blue region represents a unit cell.Notice that the nonzero weights are only between the hidden spins and the visiblespins.

With the notations introduced above, a translational invariant and nearest neigh-

bor connected RBM state is

|RBM〉 =∑

{gri }

∑

{hra,hrb}

exp

(− ERBM({hra, hrb}, {gri })

)|{gri }〉, (3.7)

with

ERBM =∑

r

q∑

i=1

M∑

a=1

(Aiagri h

ra +Biag

ri h

r+1a ) +

M∑

b=1

Cibgri h

rb

+∑

r

q∑

i=1

βigri +

M∑

a=1

αahra +

M∑

b=1

αbhrb

.

39

3.1.2 Relation to MPS

The RBM state defined by Eq. (3.7) can be cast into an MPS by mapping the hidden

spins of the RBM to the virtual indices of the MPS. We name such MPS an RBM-

MPS. Specifically, Eq. (3.7) can be rewritten as

|RBM〉 =∑

{gri }

Tr

(∏

r

T gr1 ...g

rq

)|{gri }〉, (3.8)

where

Tgr1 ...g

rq

hr1...hrM ,h

r+11 ...hr+1

M

= e−∑q,Mi,a=1(Aiagri hra+Biag

ri hr+1a )−

∑qi=1 βig

ri−

∑Ma=1 αah

ra

∑

{hrb}

e−∑q,Mi,b=1 Cibg

ri hrb−

∑Mb=1 αbh

rb .

(3.9)

The bond dimension of the RBM-MPS is determined by the number of type-h hidden

spins, i.e., M . Hence only the hidden spins of type-h contribute to the entanglement,

while those of type-h do not. The optimal M will be determined in Sec. 2. For

instance, if each hra ∈ {0, 1} is Z2 valued, the bond dimension is 2M . The tensor T

satisfies two useful properties:

Theorem 3.1.1. (a) T gr1 ...g

rq in Eq. (3.9) is either strictly zero or all its matrix el-

ements are non-vanishing. (b) If T gr1 ...g

rq is non-vanishing, it is of rank 1. If T g

r1 ...g

rq

vanishes, it is of rank 0.

Proof. To prove (a), we notice that each matrix element of T gr1 ...g

rq is a com-

mon multiplicative factor∑{hrb}

e−∑q,Mi,b=1 Cibg

ri hrb−

∑Mb=1 αbh

rb independent of the hid-

den spins {hra, hr+1a } for all a, times a strictly nonzero expression of {hra, hr+1

a }:

e−∑q,Mi,a=1(Aiagri hra+Biag

ri hr+1a )−

∑qi=1 βig

ri−

∑Ma=1 αah

ra . If the common multiplicative factor is

zero then T gr1 ...g

rq vanishes. If the common multiplicative factor is nonzero, all matrix

elements are non-vanishing.

40

To prove (b), we observe that, when the matrix elements of T gr1 ...g

rq are non-

vanishing, the ratio

Tgr1 ...g

rq

hr1...hrM ,h

r+11 ...hr+1

M

Tgr1 ...g

rq

hr1...hrM ,h

′r+11 ...h′r+1

M

(3.10)

is independent of hr1 . . . hrM , for any hr+1

1 . . . hr+1M and h′r+1

1 . . . h′r+1M . Hence any two

rows of the matrix T gr1 ...g

rN are proportional to each other, and thus the matrix is of

rank 1. When T gr1 ...g

rq vanishes, by definition, it is of rank 0.

Since the non-vanishing matrices of the RBM-MPS are of rank 1, it is natural to

ask if the reverse statement also holds true, i.e., whether an MPS can be expressed

as an RBM-MPS if the non-vanishing MPS matrices are of rank 1. In the rest of this

article, we study this problem in the context of stabilizer codes. We conjecture that

if the non-vanishing MPS matrices of the ground state of a translational invariant

stabilizer code are of rank 1, such a ground state can also be found as an RBM state.

In Sec. 2, we first determine the condition for the non-vanishing MPS matrices of a

stabilizer code to be of rank 1. In Sec. 3.4, we give an algorithm to generate the RBM

state for a large class of models (the cocycle models) whose MPS matrices are of rank

1.

3.2 More on ZZXZZ Model

In section 2.2, we derived the MPS for the ZZXZZ model, as shown in Eq. (2.32).

These matrices are of rank 1 and all the tensor elements are nonzero. We emphasize

that they match the two properties (a) and (b) in Theorem 3.1.1, and this match

depends on the proper choice of the matrices for U operators. Indeed, if there is a U

operator containing Pauli Z matrix, for instance U r1,1 = X ⊗X,U r

2,1 = X ⊗ I, U r−12,2 =

I⊗Z,U r3,1 = Z⊗I, then the MPS matrix elements can have both zeros and nonzeros.

The appearance of zero matrix elements makes it difficult to match the MPS to the

41

RBM element-wise, because the matrix elements of RBM-MPS are all non-vanishing

as shown in Theorem 3.1.1.

In fact, we do not have to solve the matrices and then find their ranks. We

can immediately find the rank of the matrices from the Hamiltonian terms. From

Eq. (2.31) used only for T 000,

T 000h1h2,h3h4

= T 000h1h2,(1−h3)(1−h4)

T 000h1h2,h3h4

= T 000h1h2,h3(1−h4)

T 000h1h2,h3h4

= −iT 000(1−h1)h2,h3h4

(−1)h1

T 000h1h2,h3h4

= −iT 000h1(1−h2),h3h4

(−1)h2 .

(3.11)

The physical indices are unchanged on both sides of Eq. (3.11) simply because the

four equations are coming from acting with the physical operators on the LHS of

Eq. (3.11),

Lr1,1 = Ir1 ⊗ Zr2 ⊗ Zr

3 ,

Lr2,1 = Ir1 ⊗ Ir2 ⊗ Zr3 ,

Rr2,2 = Zr

1 ⊗ Ir2 ⊗ Ir3 ,

Rr3,1 = Zr

1 ⊗ Zr2 ⊗ Ir3 ,

(3.12)

which contain only Pauli Z operators and identities. Hence, the left indices h1 and h2

of the matrix T 000 obey two independent constraints, and the right indices h3 and h4

obey two independent constraints. Therefore, the rank of the matrix T 000 can be at

most 1, since each constraint for the left (or right) indices eliminates half of the total

rank. Acting with Eq. (3.12) on the T -matrices with other physical indices, we also

get two independent constraints on the left and right indices respectively. Hence the

T matrices with any physical indices are of rank 1. For other general models obeying

the assumptions (2.0.1), (2.0.2) and (2.0.3), we can similarly find the constraints on

42

the rank of the matrices by counting the independent L or R operators with only

Pauli Z matrices without solving explicitly the matrices by brute-force. We elaborate

this idea in Sec. 3.3.

We finally comment that for the ZZXZZ model with one spin per unit cell, i.e.,

H1−siteZZXZZ =

r−1∑

i=0

Zr−2Zr−1XrZr+1Zr+2 (3.13)

Using the same calculation in this section, the ground state of H1−siteZZXZZ can be ex-

pressed as an MPS,

|GS〉1−site =∑

{gr}

Tr

( L−1∏

r=0

T gr

)|{gr}〉 (3.14)

where the MPS matrices are

T 0 =

1 0 1 0

−i 0 −i 0

0 1 0 1

0 −i 0 −i

, T 1 =

0 −1 0 1

0 −i 0 i

−1 0 1 0

−i 0 i 0

(3.15)

Notice that both T 0 and T 1 are of rank 2. By theorem 3.1.1, it is impossible to

express the MPS Eq. (3.14) as an RBM state.

3.3 An Inequality for Rank of MPS

As discussed in Sec. 3.1, a necessary condition for the existence of a finitely connected

RBM of a stabilizer code ground state is Theorem 3.1.1. In this section, we propose an

inequality which allows us to directly constrain the rank of the MPS without solving

for the MPS matrices.

43

Before we state and prove our theorem, it is convenient to introduce two notations.

Denote a set of operators:

L =

{Lr1,1,Lr2,1, . . . ,Lrt,1

}. (3.16)

In particular, L contains a special subset dubbed as LZ such that the operators in

LZ are only the tensors products of Pauli Z and the identity I matrices. Denote

NLZ as the number of independent operators in LZ . Notice that due to translational

invariance, NLZ is independent of r.

Theorem 3.3.1. For the matrices T gr1 ...g

rq satisfying Eq. (2.46) where the U matrices

are tensor product of Pauli matrices, the rank of T gr1 ...g

rq is upper bounded:

rank(T gr1 ...g

rq ) ≤ D

2NLZ

= 2rank(t)

2−NLZ , ∀{gri }. (3.17)

where NLZ is the number of independent operators in LZ.

Proof. To constrain rank(T gr1 ...g

rq ), we only focus on a subset of Eq. (2.46) satisfying:

(1) the physical operator on LHS of Eq. (2.46) only involves the operators in LZ ;

(2) the virtual operator on RHS of Eq. (2.46) only acts on the right virtual index.

Explicitly, this subset of equations are all included in the following equations:

Lrα,1 ◦ T gr1 ...g

rq = T g

r1 ...g

rq · U r

α,1, ∀ Lrα,1 ∈ LZ . (3.18)

This subset is useful to constrain rank(T gr1 ...g

rq ) because

(1) both LHS and RHS of this subset of equations only involve the same matrix

T gr1 ...g

rq . Indeed, since Lrα,1 belongs to LZ , the LHS is proportional to the matrix

T gr1 ...g

rq ;

44

(2) only the columns of T gr1 ...g

rq are constrained.

Using Theorem A.6.2 of App. A.6, the number of independent equations among

Eq. (3.18)(i.e., the number of independent constraints for the columns ) is given

by the number of independent operators in LZ , i.e., NLZ . We know that U operators

form a generalized Clifford algebra, and as proven in App. A.7, their matrices are

tensor products of the Pauli matrices. More precisely, each virtual U operator either

swaps and/or multiplies by some factors (±i or ±1) on half of the columns. Hence,

each independent constraint eliminates half of the rank. Therefore, the rank of the

MPS T matrix is upper bounded:

rank(T gr1 ...g

rq ) ≤ D

2NLZ

. (3.19)

This completes proving Theorem 3.3.1.

In the 1D stabilizer codes we have studied, the upper bound in Eq. (3.17) always

saturates.

3.4 Restricted Boltzmann Machine State of a Sta-

bilizer Code

In this section, we discuss how to express the ground states of a class of stabilizer

codes, which we dub as cocycle models, as RBM states. They are a special class

of Hamiltonians describing 1D symmetry protected topological phases. We first use

Theorem 3.3.1 to prove that the rank of the ground state MPS is 1. Then we use the

ZZXZZ model as an example to illustrate the construction of the RBM state with

the RBM-MPS bond dimension 4. We further present a general and explicit algorithm

to construct the RBM states for an arbitrary cocycle model, with the minimal RBM-

MPS bond dimension. We finally conjecture that for any stabilizer code which satisfies

45

Assumptions 2.0.1, 2.0.2 and 2.0.3 and also the necessary condition 3.1.1, it is possible

to express its ground state as an RBM state with the minimal RBM-MPS bond

dimension.

3.4.1 MPS Matrix Rank For Cocycle SPT Models

In this section, we apply Theorem 3.3.1 to a particular family of stabilizer codes —

the cocycle Hamiltonians for symmetry protected topological phases — and show that

their MPS matrices are of rank 1. In App. B.1, we provide some backgrounds about

the cocycle Hamiltonians, including the projective representations of the global sym-

metry G, cocycles ω2, cohomology group H2(G,U(1)) and 1D SPT phases. The cocy-

cle ω2 ∈ H2(G,U(1)) classifies the 1D SPT phases with the discrete onsite symmetry

G. In this paper, we restrict G to be (Z2)q. The group elements are parametrized

by g = (g1, g2, . . . , gq) with gi ∈ Z2 = {0, 1}, and the generic form of the cocycle is

[111, 112]:

ω2(g, g′) = exp

(−iπ

∑

1≤i<j≤q

Pijgjg′i

), g, g′ ∈ G, (3.20)

where Pij can be either 0 or 1. The cocycles can also be used to construct representa-

tive SPT wave functions and representative parent Hamiltonians which are stabilizer

codes. For simplicity, we dub the representative states and representative Hamiltoni-

ans as cocycle states and cocycle Hamiltonians respectively. See App. B.1 for a brief

overview.

The cocycle Hamiltonian for a (Z2)q SPT phase (with q spin-12’s per unit cell)

with a given generic cocycle ω2 Eq. (3.20) is

H(Z2)q ,ω2 = −L−1∑

r=0

q∑

α=1

Orα, (3.21)

46

with

Orα =

∏

1<l≤q

(Zr+1l Zr

l )P1lXr+1

1 α = 1

∏α<l≤q(Z

r+1l Zr

l )PαlXr+1

α

∏1≤k<α(Zr+2

k Zr+1k )Pkα 1 < α < q

Xrq

∏

1≤k<q

(Zr+1k Zr

k)Pkq α = q.

(3.22)

For 1 < α < q, Orα are supported on 3 unit cells; while for α = 1, q, Or1 and Orqare supported on 2 unit cells. The Hamiltonian H(Z2)q ,ω2 has the ground state (see

App. B.1 for details)

|GS〉(Z2)q ,ω2 =∑

{gri }

exp

(iπ

L−1∑

r=0

∑

1≤i<j≤q

Pij(grj − gr−1

j )gri

)|{gri }〉. (3.23)

When

P =

0 1 1

0 0 1

0 0 0

, (3.24)

the Hamiltonian Eq. (3.21) reduces to the Hamiltonian of the ZZXZZ model, i.e.,

Eq. (2.3).

Theorem 3.4.1. For the stabilizer codes of Eq. (3.21), if T gr1 ...g

rq is not null, then

rank(T gr1 ...g

rq ) = 1. (3.25)

Proof. To calculate rank(T gr1 ...g

rq ), we apply Theorem 3.3.1, where the upper bound

of the rank of T gr1 ···grq is given by 2

rank(t)2−NLZ . We will first compute rank(t) and NLZ

respectively, and show that the upper bound is 1. We further show that the upper

bound is saturated, which completes the proof of the theorem.

47

We first compute rank(t). To calculate the t-matrix, we enumerate all possible

Lr−τ+1α,τ with all possible (α, τ) and fixed r. For 1 < α < q, τ = 1, 2; for α = 1 or q,

τ = 1. Hence there are 2(q − 1) L operators:

Lr1,1 ≡ (Zr2)P12 ⊗ (Zr

3)P13 ⊗ · · · ⊗ (Zrq−1)P1(q−1) ⊗ (Zr

q )P1q

...

Lrq−2,1 ≡ (Zrq−1)P(q−2)(q−1) ⊗ (Zr

q )P(q−2)q

Lrq−1,1 ≡ (Zrq )P(q−1)q

Lr−12,2 ≡ (Zr−1

3 )P23 ⊗ · · · ⊗ (Zr−1q )P2q ⊗ (Zr

1)P12 ⊗Xr2 ⊗ (Zr

3)P23 ⊗ (Zr4)P24 ⊗ · · · ⊗ (Zr

q )P2q

Lr−13,2 ≡ (Zr−1

4 )P34 ⊗ · · · ⊗ (Zr−1q )P3q ⊗ (Zr

1)P13 ⊗ (Zr2)P23 ⊗Xr

3 ⊗ (Zr4)P34 ⊗ (Zr

5)P35 ⊗ · · · ⊗ (Zrq )P3q

...

Lr−1q−1,2 ≡ (Zr−1

q )P(q−1)q ⊗ (Zr1)P1(q−1) ⊗ (Zr

2)P2(q−1) ⊗ · · · ⊗ (Zrq−2)P(q−2)(q−1) ⊗Xr

q−1 ⊗ (Zrq )P(q−1)q

Lrq,1 ≡ (Zr1)P1q ⊗ (Zr

2)P2q ⊗ · · · ⊗ (Zrq−1)P(q−1)q ⊗Xr

q .

(3.26)

We have suppressed the identity operators for simplicity. Among all the op-

erators in Eq. (3.26), the first q − 1 and the last one act only on the r-th

unit cell, while the remaining act both on the r − 1-th and r-th unit cells.

It is straightforward to compute the commutation relation and determine the

t matrix. In the basis where the operators are listed as in Eq. (3.26), i.e.,

{Lr1,1, · · · ,Lrq−2,1,Lrq−1,1,Lr−12,2 ,Lr−1

3,2 , · · · ,Lr−1q−1,2,Lrq,1}, the t matrix reads

t =

0 Λ

−ΛT 0

, (3.27)

48

where 0 is a (q − 1) × (q − 1) dimensional zero matrix, and Λ is a (q − 1) × (q − 1)

upper triangular matrix:

Λ =

P12 · · · P1(q−1) P1q

. . ....

...

P(q−2)(q−1) P(q−2)q

P(q−1)q

. (3.28)

Therefore, by Eq. (3.27), we have:

rank(t) = 2rank(Λ). (3.29)

Counting rank(Λ) is simply counting the number of independent rows in Λ.

We proceed to evaluate NLZ . Recall that NLZ is defined to be the number of

independent operators among LZ . In this case, we have:

LZ = {Lr1,1,Lr2,1, . . . ,Lrq−1,1}. (3.30)

A crucial observation is that the powers of the Z’s among the operators in Eq. (3.30)

are in one-to-one correspondence with the rows of the Λ matrix in Eq. (3.28). Hence,

the number of independent operators among Eq. (3.30) coincides with the number of

independent rows of the Λ matrix Eq. (3.28), i.e.,

NLZ = rank(Λ). (3.31)

Using Theorem 3.3.1 and Eqs. (3.29) and (3.31), we obtain

rank(T g1...gq) ≤ 2rank(t)

2−NLZ = 2

2rank(Λ)2

−rank(Λ) = 1. (3.32)

49

We have assumed that T g1...gq is not null. rank(T g1...gq) is thus assumed to be positive.

Constrained by Eq. (3.32), we conclude that

rank(T g1...gq) = 1. (3.33)

Since in the ground state Eq. (3.23) for any spin configuration {gri } the coefficient

of the basis |{gri }〉 is a non-vanishing number, the MPS matrices are non-vanishing

for any physical indices gr1 . . . grq . This shows that the matrices T g

r1 ...g

rq are indeed not

null. Hence the MPS matrix rank is 1 for the ground state MPS of an arbitrary

cocycle Hamiltonian in Eq. (3.21) with the global symmetry (Z2)q.

3.4.2 An Example: ZZXZZ Model Revisited

In this section, we derive the RBM for the ZZXZZ model with the RBM-MPS bond

dimension 4.

We start with the ground state |GS〉 of the ZZXZZ model Eq. (2.5). Concretely,

by restricting Eq. (3.23) to q = 3, and using P12 = P23 = P13 = 1, we obtain the

ground state

|GS〉ZZXZZ =∑

{gri }

exp

(iπ

L−1∑

r=0

∑

1≤i<j≤3

(grj − gr−1j )gri

)|{gri }〉. (3.34)

The coefficient of the configuration |{gri }〉 is an exponent of a quadratic function of

the physical spins. The idea to write Eq. (3.34) in the form of an RBM state is to

introduce hidden spins and to transform the quadratic terms in g to linear terms.

This is achieved by applying a series of identities proved in App. B.2. The identities

50

can be summarized as

exp

(iπSym(g1, · · · , gn)

)

=1√2

1∑

h=0

exp

(iπ

2(1− 2h)

n∑

i=1

gi − iπ

4(1− 2h)

),

(3.35)

where gi ∈ {0, 1}, and Sym(g1, · · · , gn) is a symmetric summation of quadratic ex-

pressions in gi, i.e.,

Sym(g1, · · · , gn) ≡∑

1≤i<j≤n

gjgi. (3.36)

We introduce the following definitions to simplify the discussion below:

1. The on-site terms : the quadratic terms involving only the visible spins from a

single unit cell. For example: grjgri , g

r−1j gr−1

i , etc.

2. The inter-site terms : the quadratic terms involving the visible spins from dif-

ferent unit cells. For example: gr−1j gri , g

rjgr−1i , etc.

3. The on-site symmetric expressions : the symmetric expressions involving only

visible spins from a single unit cell. For example: Sym(gri , grj , g

rk), etc.

4. The inter-site symmetric expressions : the symmetric expressions involving vis-

ible spins from different unit cells. For example: Sym(gr−1i , grj , g

rk), etc.

To convert Eq. (3.34) into an RBM state, our strategy is as follows. We group all

the quadratic terms in the exponent of Eq. (3.34) into a sum of symmetric expressions,

and apply the identity Eq. (3.35) to each symmetric expression. For the inter-site

symmetric expression, applying Eq. (3.35) introduces a hidden spin of type-h; for the

on-site symmetric expression, applying Eq. (3.35) introduces a hidden spin of type-

h. As discussed in Sec. 3.1, each hidden spin of type-h doubles the bond dimension

once we write the RBM state as an MPS (i.e., RBM-MPS), while the hidden spin of

type-h does not contribute to the bond dimension. Hence, to obtain the RBM state

51

whose RBM-MPS bond dimension is as small as possible, we are aiming to group

the quadratic expressions in Eq. (3.34) to as few inter-site symmetric expressions as

possible, together with some additional on-site symmetric expressions.

We first discuss the inter-site terms in Eq. (3.34), i.e.,∑

1≤i<j≤3 gr−1j gri , because

on-site terms do not contribute to the inter-site symmetric expressions. There are

different ways to decompose the inter-site terms in the exponent of Eq. (3.34) as

a summation of symmetric expressions. Superficially, there are 3 inter-site terms,∑

1≤i<j≤3 gr−1j gri = gr−1

2 gr1 + gr−13 gr1 + gr−1

3 gr2, and it seems that one has to introduce 3

hidden variables by applying Eq. (3.35) to the three terms separately. However, it is

possible to organize the three inter-site terms into the sum of two inter-site symmetric

expressions and one on-site symmetric expression. Concretely,

∑

1≤i<j≤3

gr−1j gri =

Sym(gr−12 , gr1) + Sym(gr−1

3 , gr1, gr2)− Sym(gr1, g

r2).

(3.37)

Under the decomposition Eq. (3.37) and applying Eq. (3.35), we need to introduce

2 hidden spins of type-h, which we denote as hr1 and hr2. From the discussion in

the last paragraph, the bond dimension of the RBM-MPS is 22 = 4, which precisely

matches the minimal bond dimension of the ZZXZZ model derived in Sec. 2.2. This

shows that there is no way to decompose the quadratic expression∑

1≤i<j≤3 gr−1j gri

in Eq. (3.34) as a sum of at most one inter-site symmetric expression, together

with some additional on-site symmetric expressions. Different decompositions of∑

1≤i<j≤3 gr−1j gri should include at least two inter-site symmetric expressions. We

will provide a general recipe of grouping the inter-site terms in Sec. 3.4.3 for all the

1D cocycle models and show that the grouping is optimal.

52

We further consider the on-site terms∑

1≤i<j≤3 grjgri . We use the same decompo-

sition as Eq. (3.37) by replacing gr−1j with grj , and obtain

∑

1≤i<j≤3

grjgri = Sym(gr2, g

r1) + Sym(gr3, g

r1, g

r2)− Sym(gr1, g

r2). (3.38)

Applying Eq. (3.35) for all the symmetric expressions, the ground state |GS〉ZZXZZcan be rewritten as an RBM state

|GS〉ZZXZZ

=∑

{gri }

exp

(iπ

L−1∑

r=0

−Sym(gr−12 , gr1)− Sym(gr−1

3 , gr1, gr2) + Sym(gr2, g

r1) + Sym(gr3, g

r1, g

r2)

)|{gri }〉

=∑

{gri }

∑

{hr1,hr2}{hr1,hr2}

L−1∏

r=0

exp

(− iπ

2(1− 2hr1)(gr−1

2 + gr1) + iπ

4(1− 2hr1)

− iπ2

(1− 2hr2)(gr−13 + gr1 + gr2) + i

π

4(1− 2hr2) + i

π

2(1− 2hr1)(gr1 + gr2)

− iπ4

(1− 2hr1) + iπ

2(1− 2hr2)(gr1 + gr2 + gr3)− iπ

4(1− 2hr2)

)|{gri }〉

(3.39)

We have suppressed the overall normalization constant. From the discussion in

Sec. 3.1, the RBM state Eq. (3.39) can further be written as an MPS with the MPS

matrix:

Tgr1g

r2gr3

hr1hr2,h

r+11 hr+1

2

=

∑

{hr1,hr2}

exp

(− iπ

2(1− 2hr1)gr1 − i

π

2(1− 2hr+1

1 )gr2 + iπ

4(1− 2hr1)− iπ

2(1− 2hr2)(gr1 + gr2)

− iπ2

(1− 2hr+12 )gr3 + i

π

4(1− 2hr2) + i

π

2(1− 2hr1)(gr1 + gr2)− iπ

4(1− 2hr1)

+ iπ

2(1− 2hr2)(gr1 + gr2 + gr3)− iπ

4(1− 2hr2)

).

(3.40)

53

The bond dimension of the RBM-MPS Eq. (3.40) is indeed 4, which matches the

bond dimension derived from the RBM state Eq. (3.39). Since we have shown in

Sec. 2.2 that the minimal bond dimension of the ZZXZZ MPS is 4, there can not

be an RBM state with the number of hidden spin of type-h per unit cell less than 2.

This implies that our RBM state is the most optimal, in the sense that the number

of hidden spins of type-h is minimal.

Fig. 3.2 is a graphical representation of the RBM state Eq. (3.39). In fact, the

RBM-MPS matrices Eq. (3.40) are the same as the MPS matrices Eq. (2.32) in derived

in Sec. 2.2. As we will see in the next subsection, for more general models Zq−1XZq−1,

each unit cell contains q visible spins. Our construction yields the RBM-MPS bond

dimension 2q−1, and we need to introduce 2(q − 1) hidden spins on average for each

unit cell. Among them, (q − 1) are of the type-h while the remaining (q − 1) are of

the type-h.

3.4.3 RBM States of Cocycle Hamiltonians

In Sec. 3.4.1, we have shown that the MPS matrices of the (Z2)q cocycle Hamiltonians

(with q spin-12’s per unit cell) are all of rank 1. Then it is natural to ask if the ground

state of the cocycle Hamiltonians can always be expressed as an RBM state, whose

Figure 3.2: Graphical representation of the RBM state of the ZZXZZ model.

54

RBM-MPS bond dimension being D defined in Eq. (2.45). In this subsection, we

describe a procedure to obtain the RBM states with minimal number of hidden spins.

In particular, we generalize and apply the procedures of Sec. 3.4.2, and we present

explicit RBM states for Zq−1XZq−1 cocycle Hamiltonians with arbitrary q.

The cocycle Hamiltonian in Eq. (3.21) has the ground state |GS〉(Z2)q ,ω2 in

Eq. (3.23). To convert it to an RBM state, we follow the same procedures in Sec. 3.4.2.

The core idea is that we need to group the inter-site terms∑

1≤i<j≤q Pijgr−1j gri as a

sum of the rank(Λ) inter-site symmetric expressions together with some on-site terms.

Since each inter-site symmetric expression contributes a hidden spin of type-h which

doubles the bond dimension of the RBM-MPS, the bond dimension of the RBM-MPS

is thus 2rank(Λ) ≡ 2rank(t)

2 . This is precisely the minimal bond dimension derived in

Sec. (2.4), which in turn implies that the decomposition of the inter-site terms is

optimal, i.e, the number of type-h hidden spins is minimal in our construction.

Lemma 3.4.2. For an inter-site quadratic term,

(gr−1

)T · Γ · gr =

q∑

i,j=1

Γijgr−1i grj , Γij ∈ {0, 1}, (3.41)

there exists a unimodular transformation G such that Γ transforms to

Γ→ Γ = (G)T · Γ ·G =

γ

0

mod 2, (3.42)

where the integer matrix γ of size rank(Γ)× q has full row rank:

rank(γ) = rank(Γ). (3.43)

55

The vectors gr−1 and gr transform as

gr−1i → gr−1

i =

q∑

j=1

G−1ij g

r−1j , gri → gri =

q∑

j=1

G−1ij g

rj . (3.44)

and

(gr−1

)T · Γ · gr =(gr−1

)T · Γ · gr (3.45)

Proof. Our proof is based on the Gaussian elimination algorithm. For simplicity, we

first introduce the matrix notations: I represents the identity q×q matrix, and E(i, j)

represents a q × q matrix whose elements are

(E(i, j))m,n = δm,iδn,j, ∀ m,n = 1, 2, . . . , q. (3.46)

In other words, the only nonzero value of E(i, j) is 1 located at the i-th row and j-th

column. Moreover, we use the following two types of matrix row transformations:

G1(i, j) = I + E(i, j) + E(j, i)− E(i, i)− E(j, j)

G2(i, j) = I + E(j, i), i 6= j.

(3.47)

It is obvious that both G1 and G2 are unimodular, i.e.,

| det(G1(i, j))| = 1, | det(G2(i, j))| = 1. (3.48)

The products of G1’s and G2’s are also unimodular.

The first transformation G1(i, j) interchanges the i-th row and the j-th row of

Γ, and the second transformation G2(i, j) adds the i-th row to the j-th row.1 There

1Notice that the matrix determinant det(G2(i, i)) = det(I +E(i, i)) = 0. Since we only consideruni-modular transformation, we do not allow i = j in G2(i, j).

56

exists a sequence of G1(i, j) and G2(i, j) such that:

∏

m

Gkm(im, jm) · Γ =

γ′

0

mod 2, (3.49)

where the matrix γ′ of size rank(Γ)× q has full row rank, and its elements are either

0 and 1. Denote:

G =

(∏

m

Gkm(im, jm)

)T

. (3.50)

Using Eq. (3.49), we have:

Γ = GT · Γ ·G =

γ′

0

·G =

γ

0

mod 2, (3.51)

where

γ = γ′ ·G, (3.52)

and γ of size rank(Γ)× q has full row rank.

Lemma 3.4.3. The inter-site term in the ground state |GS〉(Z2)q ,ω2 Eq. (3.23)∑

1≤i<j≤q Pijgr−1j gri can be grouped into rank(Λ) number of inter-site symmetric

expressions and rank(Λ) on-site symmetric expressions, where Λ is defined in

Eq. (3.28).

Proof. We first define the Γ matrix:

Γ ≡

0 0 · · · 0 0

P12 0 · · · 0 0

P13 P23 · · · 0 0

......

. . ....

...

P1q P2q · · · P(q−1)q 0

=

0 0 · · · 0 0

0

0

ΛT ...

0

. (3.53)

57

The matrix Γ is a q × q matrix, whose each element is defined modulo 2. There are

0s in the first row and last column because gr−11 and grq do not appear in the sum

∑1≤i<j≤q Pijg

r−1j gri . The bottom-left (q − 1) × (q − 1) block of Γ is ΛT where Λ is

defined in Eq. (3.28). In particular,

rank(Γ) = rank(Λ). (3.54)

Using this notation, we have:

∑

1≤i<j≤q

Pijgr−1j gri = (gr−1)T · Γ · gr. (3.55)

Using Lemma 3.4.2, Eq. (3.55) can be simplified:

∑

1≤i<j≤q

Pijgr−1j gri =

rank(Λ)∑

i=1

gr−1i

q∑

j=1

Γij grj . (3.56)

It can be decomposed by the symmetric expressions:

∑

1≤i<j≤q

Pijgr−1j gri =

rank(Λ)∑

i=1

Sym(gr−1i , Γi1g

r1, . . . , Γiqg

rq)−

rank(Λ)∑

i=1

Sym(Γi1gr1, . . . , Γiqg

rq).

(3.57)

The first rank(Λ) terms are inter-site symmetric expressions, and the remaining

rank(Λ) terms are the on-site terms. This completes the proof.

Theorem 3.4.4. There exists an RBM for the state Eq. (3.23) whose RBM-MPS has

the minimal bond dimension 2rank(Λ) where Λ is defined in Eq. (3.28).

58

Proof. Using Lemma 3.4.2 and 3.4.3, we obtain

exp

(− iπ

∑

1≤i<j≤q

Pijgr−1j gri

)= exp

(− iπ(gr−1)T · Γ · gr

)

= exp

(− iπ

rank(Λ)∑

i=1

Sym(gr−1i , Γi1g

r1, . . . , Γiqg

rq) + iπ

rank(Λ)∑

i=1

Sym(Γi1gr1, . . . , Γiqg

rq)

).

(3.58)

Applying Eq. (3.35) to the inter-site symmetric expressions leads to:

exp

(− iπ

∑

1≤i<j≤q

Pijgr−1j gri

)

=

rank(Λ)∏

i=1

[1√2

1∑

hri=0

exp

(− iπ

2(1− 2hri )(g

r−1i +

q∑

j=1

Γij grj ) + i

π

4(1− 2hri )

)

× exp

(− iπSym(Γi1g

r1, · · · , Γiqgrq)

)].

(3.59)

Notice that further introducing the hidden spins by linearizing the on-site terms on

RHS of Eq. (3.58) does not increase the bond dimension of the RBM-MPS. Hence we

have shown that the RBM-MPS derived via the above algorithm has rank(Λ) hidden

spins of type h, which corresponds to the RBM-MPS bond dimension D = 2rank(Γ) =

2rank(Λ). This matches the bond dimension Eq. (2.45) associated with the irreducible

representation in Sec. 2.

We use the rest of this section to express the state Eq. (3.23) as an RBM explicitly.

exp

(iπ

∑

1≤i<j≤q

Pij(grj − gr−1

j )gri

)

= exp

(− iπ

rank(Λ)∑

i=1

Sym(gr−1i , Γi1g

r1, . . . , Γiqg

rq) + iπ

rank(Λ)∑

i=1

Sym(gri , Γi1gr1, . . . , Γiqg

rq)

).

(3.60)

59

Applying Eq. (3.35) to Eq. (3.60), we can write the ground state |GS〉(Z2)q ,ω2 as an

RBM state

|GS〉(Z2)q ,ω2

=∑

{gri },{hri },{hri }

rank(Λ)∏

i=1

exp

(− iπ

2(1− 2hri )(g

r−1i +

q∑

j=1

Γij grj )

+ iπ

4(1− 2hri ) + i

π

2(1− 2hri )(g

ri +

q∑

j=1

Γij grj )− i

π

4(1− 2hri )

)|{gri }〉.

(3.61)

We find that in the particular construction Eq. (3.61), the number of inter-site hid-

den spin is the same as the number of on-site hidden spin, for an arbitrary cocycle

Hamiltonian. The relation between {gri } and {gri } depends on the cocycle parameters

Pij, as per Eq. (3.44).

3.4.4 RBM Construction for Zq−1XZq−1 Model

To exemplify our RBM construction, we apply the above algorithm to the stabilizer

code Zq−1XZq−1 for an arbitrary cocycle. Another example is discussed in App. B.3.

The Zq−1XZq−1 model corresponds to the cocycle Hamiltonian with Pij = 1 for any

1 ≤ i < j ≤ q.

The Hamiltonian of the Zq−1XZq−1 model is

HZq−1XZq−1 =−L−1∑

r=0

(q−1∏

i=1

ZriX

rq

q−1∏

i=1

Zr+1i +

q∏

i=2

ZriX

r+11

q∏

i=2

Zr+1i

+

q∑

s=3

( q∏

i=s

Zri

s−2∏

j=1

Zr+1j Xr+1

s−1

q∏

k=s

Zr+1k

s−2∏

l=1

Zr+2l

) ).

(3.62)

Its ground state is

|GS〉Zq−1XZq−1 =∑

{gri }

L−1∏

r=0

exp

(iπ

∑

1≤j<i≤q

(gri − gr−1i )grj

)|{gri }〉. (3.63)

60

The the q × q Γ matrix and the (q − 1)× (q − 1) Λ matrix are

Γ =

0 0 · · · 0

1 0

1 1 0

......

. . ....

1 1 · · · 1 0

, Λ =

1 1 · · · 1

1 · · · 1

. . ....

1

. (3.64)

To transform the Γ matrix to the form as in Eq. (3.42), we switch the rows using

GT = G1(q − 1, q) · · ·G1(1, 2). (3.65)

The visible spins transform as

gr1

gr2...

grq−1

grq

→

gr1

gr2...

grq−1

grq

= G−1 ·

gr1

gr2...

grq−1

grq

=

gr2

gr3...

grq

gr1

. (3.66)

The Γ matrix transforms as

Γ→ Γ = GT · Γ ·G =

1 0

1 1 0

......

. . ....

1 1 · · · 1 0

0 0 · · · 0 0

. (3.67)

61

All the q − 1 rows in the top (q − 1)× q block of Γ are independent,

rank(Γ) = rank(Γ) = rank(Λ) = q − 1. (3.68)

As a result, the exponents in Eq. (3.63) can be written as

∑

1≤j<i≤q

(gri − gr−1i )grj =−

q−1∑

i=1

Sym(gr−1i+1 , g

ri , g

ri−1, . . . , g

r1)

+

q−1∑

i=1

Sym(gri+1, gri , g

ri−1, . . . , g

r1).

(3.69)

On RHS of the equality, the first q−1 symmetric functions are inter-site terms. Using

Eq. (3.35) we introduce q − 1 hidden spins of type-h contributing to 2rank(Γ) = 2q−1

bond dimension of the RBM-MPS. The remaining q − 1 symmetric functions only

contain on-site quadratic terms. Using Eq. (3.35), we introduce q− 1 hidden spins of

type-h. Combining these two operations, we have:

|GS〉Zq−1XZq−1

=∑

{gri }

∑

{hr1}...{hrq−1}

L−1∏

r=0

exp

(− iπ

2

q−1∑

i=1

(1− 2hri )(gr−1i+1 +

i∑

j=1

grj ) + iπ

4

q−1∑

i=1

(1− 2hri )

)

×∑

{hri }

exp

(iπ

2

q−1∑

i=1

(1− 2hri )i+1∑

j=1

grj − iπ

4

q−1∑

i=1

(1− 2hri )

)|{gri }〉.

(3.70)

62

Figure 3.3: Graphical representation of the RBM state of the ZXZ model. The redcircles represent visible spins, the black rectangles represent the hidden spins con-necting visible spin belonging to different unit cells, and the black triangles representthe hidden spins connecting visible spins within the same unit cell.

This RBM can be casted into an rank-1 MPS, and the matrix elements of the RBM-

MPS are:

Tgr1 ,...,g

rq

hr1...hrq−1,h

r+11 ...hr+1

q−1

= exp

(− iπ

2

q−1∑

i=1

(1− 2hri )(i∑

j=1

grj )− iπ

2

q−1∑

i=1

(1− 2hr+1i )gri+1 + i

π

4

q−1∑

i=1

(1− 2hri )

)

×∑

{hri }

exp

(iπ

2

q−1∑

i=1

(1− 2hri )i+1∑

j=1

grj − iπ

4

q−1∑

i=1

(1− 2hri )

).

(3.71)

We discuss two particular cases. When q = 2, the model corresponds to the ZXZ

model. A graphical representation of the ZXZ model is shown in Fig. 3.3. We notice

that the corresponding RBM-MPS has bond dimension 2. In the RBM derived in

Ref. [113], the corresponding bond dimension is 4, which is not minimal. When q = 3

which corresponds to the ZZXZZ model, we find that the RBM-MPS matrices in

Eq. (3.71) precisely agrees with the MPS matrices in Eq. (2.32).

In summary, we have shown that for cocycle Hamiltonians, the ground state can

be expressed as an RBM state with the minimal RBM-MPS bond dimension. We

63

further conjecture, that for an arbitrary translational invariant stabilizer code with

non-degenerate ground state with PBC, if its ground state MPS matrix is of rank

1, then it is possible to express its ground state as an RBM state with the minimal

RBM-MPS bond dimension matching Eq. (2.45). We leave the proof of this conjecture

for future work.

64

Chapter 4

Tensor Network States,

Entanglement Entropy of CSS

Stabilizer Codes and Fracton

Models in 3D

4.1 Stabilizer Code Tensor Network States

In this section, we provide an overview of the stabilizer codes and the tensor network

state description of their ground states. In this article, we focus on a few ”main”

stabilizer codes in three dimensions : the toric code[114] and the Haah code[71].

(In [47], we also discussed another stabilizer code: the X-cube model[115]. ) The

TNS for these models have similarities in their derivation and they share several (but

importantly not all!) common features. Both aspects are presented in this section.

4.1.1 Notations

We first fix the notations, to which we will refer throughout the chapter:

65

1. We introduce a g tensor, which denotes the projector from a physical index to

virtual indices. g tensors are essentially the same (up to the number of indices)

for all stabilizer codes. g tensors have two virtual indices and one physical index

for the 3D toric code model and the X-cube model, while g tensors for Haah

code have 4 virtual indices and 1 physical index. They are depicted in Eq. (4.4),

(4.52) and (4.53).

2. We introduce the T tensor, which denotes the local tensor for each model. It

has only virtual indices and no physical index. The specific tensor elements

differ for different models.

3. Since we consider mostly models on cubic lattices, the indices of T tensors will

be denoted as x, x, y, y, z and z in the 3 directions (forward and backward)

respectively. The indices will be collectively denoted using curly brackets. For

instance, the physical indices are collectively denoted as {s}, while the virtual

indices are denoted as {t}. The virtual indices which are not contracted over

are called “open indices”. Both the physical indices and the virtual indices are

non-negative integer values.

4. Graphically, the physical indices are denoted by arrows, while the virtual indices

are not associated with any arrows.

5. The contraction of a network of tensors over the virtual indices is denoted as

CM ( ) whereM is the spatial manifold that the TNS lives on. The correspond-

ing wave function that arises from the contraction is denoted as |TNS〉M. When

evaluating the TNS wave function norms or any other physical quantities, we

contract over the virtual indices from both the bra and the ket layer. This

contraction is still denoted by CM ( ).

66

U-1U

A1 A2

A1 A2

(a)

(b)

U-1UA1 A2

A1 A2

(c)

(d)

Figure 4.1: TNS gauge in MPS. (a) A part of an MPS. A1 and A2 are two localtensors contracted together. (b) We insert the identity operator I = UU−1 at thevirtual level - it acts on the virtual bonds. The tensor contraction of A1 and A2 doesnot change. (c) We further multiply U with A1 and U−1 with A2, resulting in A1 andA2 respectively in Panel (d). The tensor contraction of A1 and A2 is the same as thetensor contraction of A1 and A2. The TNS wave function does not change as well.Similar TNS gauges also appear in other TNS such as PEPS.

6. Lx, Ly and Lz refer to the system sizes in the three directions (the bound-

ary conditions will be specified), while lx, ly and lz refer to the sizes of the

entanglement cut. Both are measured in units of vertices.

7. TNS gauge is defined as the gauge degrees of freedom of TNS such that the

wave function stays invariant while the local tensors change. One can insert

identity operators I = UU−1 on the virtual bonds, where U is any invertible

matrix acting on the virtual index, multiplying U and U−1 to nearby tensors

respectively. The local tensors then change but the wave function stays invari-

ant. We refer to this gauge degree of freedom as TNS gauge. TNS gauge exists

in MPS, PEPS etc. See Fig. 4.1 for an illustration. In our calculations, we only

fix the tensor elements up to a choice of TNS gauge.

67

4.1.2 CSS Stabilizer Code and TNS Construction

We now summarize the general idea of constructing TNS for stabilizer codes. In the

following, we assume that the physical spins are defined on the bonds of the cubic

lattice (such as the 3D toric code). The cases where the physical spins are defined on

vertices can be analyzed similarly. The generic philosophy of any CSS stabilizer code

model (named after Robert Calderbank, Peter Shor and Andrew Steane) is captured

by the following exactly solvable Hamiltonian:

H = −∑

v

Av −∑

p

Bp (4.1)

where the Hamiltonian is the sum of the Av terms composed of only Pauli Z operators

and the Bp terms composed of only Pauli X operators, and v and p denotes the

positions of the lattice. (There are more general non-CSS stabilizer codes that contain

the terms with both X and Z operators. We considered non-CSS models in 1D (see

chapter 2 and 3), but we will not discuss them in higher dimensions.) In the 3D toric

code, v is the vertex of the cubic lattice, while p is the plaquette. In the Haah code,

both v and p are cubes. See Sec. 4.3.1 and 4.4.1 for the definition of Hamiltonians of

these three models. All these local operators commute with each other:

[Av, Av′ ] = 0, ∀ v, v′

[Bp, Bp′ ] = 0, ∀ p, p′

[Av, Bp] = 0, ∀ v, p

(4.2)

The Hamiltonian eigenstates are the common eigenstates of these local terms indi-

vidually. In particular, any ground state |GS〉 should satisfy:

Av|GS〉 = |GS〉, ∀ v

Bp|GS〉 = |GS〉, ∀ p(4.3)

68

for all positions labeled by v and p.

The ground states for the stabilizer codes with Hamiltonian as in Eq. (4.1) can be

written exactly in terms of TNS. Our construction, when restricted to the 2D toric

code model, is the same as in the literature[116, 117]. In the following, we provide one

possible general construction for such TNS wave functions. We introduce a projector

g tensor with one physical index s and two virtual indices i, j:

gsij =

i j

s=

1 s = i = j

0 otherwise

(4.4)

where the line with an arrow represents the physical index, and the lines without

arrows correspond to the virtual indices. The physical index s = 0, 1 represents

the Z-eigenstates of |↑〉, |↓〉 respectively where Z|↑〉 = |↑〉, and Z|↓〉 = −|↓〉. The

projector g tensor maps the physical spin into the virtual spins exactly. As a result,

the virtual index has a bond dimension 2. 1 When a Pauli operator acts on the

physical index of a projector g tensor, its action transfers to the virtual indices of g.

For instance, a Pauli operator X acting on the physical index of a g tensor amounts

to two Pauli operators X acting on both virtual indices of the same g tensor, and

a Pauli operator Z acting on the physical index of a g tensor amounts to a Pauli

operator Z acting on either virtual index of the same g tensor.

1This construction is limited to the models where the entanglement entropy of the ground statescales as S / Area ln 2. If the entanglement area law is higher than Area ln 2, for example 2Area ln 2,than this construction does not work. However, for the stabilizer codes to be discussed in this chapter,the entanglement entropy of the ground states indeed satisfy the above scaling property.

69

Tg

T

g

(a) (b)

xzy

xzy

Figure 4.2: (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The lines

with arrows are the physical indices. The connected lines are the contracted virtual

indices, while the open lines are not contracted. On each vertex, there lives a T

tensor, and on each bond, we have a projector g tensor.

To each vertex, we associate a local tensor T which only has virtual indices. To

each bond, we associate a projector g tensor. The TNS is obtained by contracting

the g and T tensors as depicted in Fig. 4.2 (a) and (b). We define the TNS wave

function as:

|TNS〉 =∑

{s}

CR3

(gs1gs2gs3 . . . TTT . . .) |{s}〉 (4.5)

where CR3denotes the contraction over all virtual indices on R3 as illustrated in

Fig. 4.2 (b); |{s}〉 is a wave function basis for spin configurations on the cubic lattice

in Pauli Z basis. The TNS can be put on other spatial manifolds such as T 3 and

T 2 ×R. In our notation, they are denoted by changing CR3to CT 3

and CT 2×R. The

TNS for the ground states satisfies:

Av|TNS〉 = |TNS〉, ∀ v

Bp|TNS〉 = |TNS〉, ∀ p(4.6)

for all positions labeled by v and p.

Since we have projector g tensors contracted with all virtual indices of a T tensor,

the actions of Av and Bp operators on the TNS can be transferred to the virtual

70

indices, using the definition of the g tensor. Then the actions of Av and Bp on the

physical indices will be transferred to actions on the local tensors T . By enforcing

the local tensors T to be invariant under Av and Bp actions, we obtain Eq. (4.6), and

|TNS〉 belongs to the ground state manifold. For the three models analyzed in this

paper, we have found that up to TNS gauge, the elements of the local tensor T can

be reduced to two values, either 1 or 0. The first equation of Eq. (4.6) restricts the

local T tensor to be:

Txxy...

6= 0 if the indices xxy . . . satisfy

some constraints

= 0 otherwise

(4.7)

Applying the second equation of Eq. (4.6) will further restrict the local T tensor to

be:

Txxy... =

1 if the indices xxy . . . satisfy

some constraints

0 otherwise

(4.8)

For simplicity, we calculate the entanglement entropies of the wave function on R3.

We emphasize that in this chapter, we are only concerned about the bulk wave

functions and their entanglement entropies. In principle, the TNS of Eq. (4.5) re-

quires boundary conditions, i.e. the virtual indices at infinity on R3. The boundary

conditions are assumed not to make a difference to the reduced density matrices in

the bulk. (Note that this is true as long as the region considered for the reduced

density matrices does not contain any boundary virtual index.) Hence, we do not

need to specify the boundary conditions for the TNS in the following calculations of

entanglement entropies.

71

4.2 Entanglement properties of the stabilizer code

TNS

The specific structure of the TNS discussed in the previous section allows to derive

its entanglement properties. In this section, we show that for a large class of en-

tanglement cuts the TNS is already in Schmidt form i.e. is exactly a singular value

decomposition.

4.2.1 TNS as an exact SVD

We propose a general sufficient condition that the TNS is SVD with respect to partic-

ular entanglement cuts. Suppose we denote the TNS wave function with open virtual

indices {t} as:

|{t}〉 =∑

{s}

CM (TTT . . . gs1gs2gs3 . . .) |{s}〉 (4.9)

whereM is an open manifold which the TNS lives on, CM stands for the contraction

over the virtual indices inside M, but not over the open ones {t} that straddle the

boundary of M. In Eq. (4.9), the T tensors and g tensors are the tensors inside M

such that the nodes of the local T tensors and the projector g tensors are inside M.

For example, when M is a cube, we have a TNS figure:

|{t}〉 =

{t}

(4.10)

where inside the cube is a network of contracted tensors which are not explicitly

drawn, and the red lines denote the open virtual indices {t}. With this notation of

72

|{t}〉, the TNS wave function can be written as:

|TNS〉 =∑

{t}

|{t}〉A ⊗ |{t}〉A (4.11)

with respect to a region A and its complement A. |{t}〉A is the TNS wave function

in region A with open indices {t}, while |{t}〉A is the TNS wave function in region A

with the same open indices {t} due to tensor contraction. In other words, the TNS

naturally induces a bipartition of the wave functions. However, the two partitions do

not need to each form orthonormal sets.

We now propose a simple sufficient (but not generally necessary) condition to

determine when Eq. (4.11) is an exact SVD for the TNS constructed in this paper.

We first have to make an assumption, satisfied by all our TNS:

Local T tensor assumption: We assume that the indices of the nonzero elements

of the local T tensor are constrained: if all the indices of the element T...t... except for

t are fixed, then there is only one choice of t such that T...t... is nonzero.

We are now ready to express our SVD condition:

SVD condition: If there are no two open virtual indices in {t} (see Eq. (4.10))

of the region A that connect to the same T tensor in the region A, i.e. if every

open virtual index in {t} belongs to different T tensors, then the non-vanishing states

|{t}〉A span an orthogonal basis. Similarly, if there are no two open virtual indices in

{t} of the region A that connect to the same T tensor in the region A, i.e. if every

open virtual index in {t} belong to different T tensors, then the non-vanishing states

|{t}〉A form an orthogonal basis. Therefore, Eq. (4.11) is an exact SVD.

Proof :

We first prove the statement for region A. Suppose that |{t}〉A and |{t′}〉A are

two non-vanishing TNS wave functions in the region A. Any open index in {t}

of the region A must connect to either a projector g tensor or a local tensor T . We

73

discuss the two situations respectively, and examine the overlap of two different states

A〈{t′}|{t}〉A as a function of the two indices configurations {t′} and {t}.

(1) If the open virtual index m in the ket layer (i.e. |{t}〉A) connects to a projector

g tensor, then the open virtual index m′ in the bra layer (i.e. A〈{t′}|), at the same

place as the index m, also connects to a projector g tensor. If we “zoom in” on the

local area of A〈{t′}|{t}〉A near the index m and m′, we have the following diagram:

A

m'

m

A

Entanglement Cut

ket layer

bra layer

(4.12)

By using the projection property of the g-tensor Eq. (4.4), we can conclude that

m = m′, otherwise A〈{t′}|{t}〉A = 0.

(2) If the open virtual index m0 in the ket layer connects to a local T tensor, we

require by the SVD condition that there are no other open virtual indices connecting

to this T tensor. Then the other indices of this T tensor are all inside the region A.

Similarly for the index m′0 in the bra layer. In terms of a diagram, A〈{t′}|{t}〉A near

the area of the index m0 and m′0 can be represented as:

m'

T

T

A

ket layer

bra layer

A

mi

m'i

m0

0

Entanglement Cut

(4.13)

where mi and m′i with i = 1, 2, 3 . . . denote the other virtual indices of the T tensor in

the bra and ket layer respectively, except m0 and m′0. Note that in the ket layer, the

74

virtual indices mi (i = 1, 2, . . .) of the T tensor (all indices except the index m0) are

all connected with contracted projector g tensors inside region A. Correspondingly,

in the bra layer, the virtual indices m′i (i = 1, 2, . . .) are also all connected with the

same contracted projector g tensors. Hence, due to these projector g tensors, all

the indices except m0 of the T tensor in the ket layer are equal to their respective

analogues in the bra layer:

mi = m′i, i = 1, 2, . . . (4.14)

otherwise the overlap would be A〈{t′}|{t}〉A = 0. The only remaining question is

whether the open indices m0 and m′0 should be identified in order to have a non-

vanishing overlap A〈{t′}|{t}〉A.

Using the local T tensor assumption:, mi (i = 1, 2, . . .) will uniquely determine

m0 in order to have nonzero element of the T tensor in the ket layer. Similarly,

m′i (i = 1, 2, . . .) will uniquely determine m′0 in order for the T tensor in the bra layer

to give a nonzero element. Therefore, Eq. (4.14) implies that:

m0 = m′0 (4.15)

such that the overlap A〈{t′}|{t}〉A is nonzero.

Therefore, both situations (1) and (2) lead to the conclusion that the open indices

{t} and {t′} should be identical in order to have a nonzero overlap A〈{t′}|{t}〉A. The

non-vanishing states |{t}〉A are orthogonal basis. A similar proof can be derived for

the region A. The orthogonality of each set |{t}〉A and |{t}〉A implies that Eq. (4.11)

is indeed an SVD. However, the singular values are not clear at this stage since the

basis may not be orthonormal (i.e., the states might not be normalized). 2

In the following specific discussions of the 3D toric code model and the Haah code,

we will show that we can select a region A and a cut on the TNS such that |{t}〉A

75

and |{t}〉A are not only orthogonal, but also normalized. In particular for the 3D

toric code model, we can just select the region A to be a cube which satisfies the SVD

condition directly. See respectively Sec. 4.3.4 for detailed discussions. However, the

Haah code is different: a cubic region A does not fulfill the SVD condition However,

in Sec. 4.4.3 we can generalize the SVD condition to the Generalized SVD Condition

and apply Bc operators to make the TNS wave function an SVD.

4.2.2 Summary of the results

We now summarize the major results derived in this paper for the three stabilizer

codes. Fundamentally, our calculations come down to the fact that the indices of the

nonzero elements of the local tensor T and g are constrained. More specifically, when

we calculate the entanglement entropies with a TNS which is an exact SVD, the only

task is to count the number of independent Schmidt states |{t}〉A. The number of

independent Schmidt states |{t}〉A is determined by the Concatenation lemma,

i.e., when a network of T tensors and g tensors are concatenated, the open indices of

the nonzero elements of the resulting tensors are constrained as well.

1. The TNS is the exact SVD for the ground states with respect to particular

entanglement cuts. The entanglement spectra are flat for models studied in

this paper.

2. The entanglement of TNS is bounded by the area law:

S ≤ Area× log(D),

where D is the virtual index dimension and Area is measured in the units of

vertices. For the models studied in this paper, the entanglement entropies are

strictly smaller than the area law when one is computing in terms of vertices. For

the toric code, the correction is a negative constant, − log(2). For Haah code,

76

the correction includes a negative term linear with the system size, presented

in Sec. 4.4.4.

4.3 3D Toric Code

In this section, we construct the TNS for 3D toric code model and then calculate

the entanglement entropy and GSD of the toric code model, both deriving from the

Concatenation lemma. The results are the immediate generalizations of those in

2D toric code model. We find a topological entanglement entropy in accordance to

that obtained by Ref. [118] using field theoretic approach.

This section is organized as follows: In Sec. 4.3.1, we briefly review the toric code

model in a cubic lattice. In Sec. 4.3.2, we construct the TNS for the toric code model.

In Sec. 4.3.3, we prove a Concatenation lemma for toric code TNS, which is useful

in the following calculations. In Sec. 4.3.4, we calculate the entanglement entropies

on R3.

4.3.1 Hamiltonian of 3D Toric Code Model

The 3D toric code model can be defined on any random lattice. However, for sim-

plicity, we only work on the cubic lattice. On a cubic lattice, the physical spins are

defined on the bonds of the lattice, and the Hamiltonian is built from two types of

terms:

H = −∑

v

Av −∑

p

Bp (4.16)

Av is defined around a vertex v, Bp is defined on a plaquette p.

Av =∏

i∈v

Zi, Bp =∏

i∈p

Xi (4.17)

77

Z ZZ

Z

x

x

xx

x

z

y

Z

Z

(a) (b)

Figure 4.3: The Hamiltonian terms of the 3D toric code model. Panel (a) is Av whichis a product of 6 Z operators, and Panel (b) is Bp which is a product of 4 X operators.The circled X and Z represent the Pauli matrices acting on the spin-1/2’s. The toriccode Hamiltonian includes Av terms on all vertices v and Bp terms on all plaquettesp.

where Zi and Xi are Pauli matrices for the i-th spin. On a cubic lattice, Av is

composed of 6 Pauli Z operators while Bp is composed of 4 Pauli X operators. These

two terms are depicted in Fig. 4.3. In the 2D toric code, Av is composed of 4 Pauli

Z operators on a square lattice. The Hamiltonian is the sum of Av operators on all

vertices v and Bp operators on all plaquettes p.

It is easy to verify that all the Hamiltonian terms commute:

[Av, Av′ ] = 0, ∀ v, v′

[Bp, Bp′ ] = 0, ∀ p, p′

[Av, Bp] = 0, ∀ v, p

(4.18)

and their eigenvalues are ±1:

A2v = 1, B2

p = 1. (4.19)

78

The ground states |GS〉 should satisfy:

Av|GS〉 = |GS〉, ∀ v

Bp|GS〉 = |GS〉. ∀ p(4.20)

These two sets of equations are enough to derive the local T tensor and to construct

TNS for the toric code model. In particular, one of the ground states on the torus

that we will find is

|ψ〉 =∏

v

(1v + Av)|0x〉 (4.21)

where |0x〉 is the tensor product of all X = 1 eigenstates defined on each link.

4.3.2 TNS for 3D Toric Code

We first introduce a projector g tensor Eq. (4.4) on each bond of the lattice. The

range of the virtual index is only from 0 to 1. The local physical indices are |0〉 ≡ |↑〉

and |1〉 ≡ |↓〉. The projector g tensor satisfies:

Z = =

x x=x

(4.22)

79

In terms of algebraic equations, these diagrams correspond to:

gsi,j(−1)s = gsi,j(−1)i = gsi,j(−1)j

g1−si,j = gs1−i,1−j

(4.23)

These two sets of equations are true, because (1) the indices s, i and j are identified for

nonzero gsi,j, (2) the nonzero gsi,j are always 1 Eq. (4.4). We can use these conditions

to transfer the action of the physical operators to the virtual operators. Now we

introduce additional T tensors on each vertex of the cubic lattice, these T tensors

have six virtual indices. Graphically, we represent this T tensor as:

Txy

z

z

y

x (4.24)

Next we need to fix the elements of the T tensor, up to the TNS gauge freedom. The

method to fix the T tensor is to make it invariant under the actions of Av and Bp

operators, in order to implement the local conditions for ground states Eq. (4.20).

80

The actions of Av and Bp operators on local tensors are:

TZ

ZZ

Z

Z

Z

Z

Z Z

ZZ

Z

g

gg

g

g

g

=

Tg

gg g

x

x

x x

x xx

xx x

x

x=

(4.25)

where we have used Eq. (4.22) to transfer the physical operators to the virtual ones.

We require a strong version of the solution to of the above equations. We want the

tensors in the dashed red rectangles to be invariant under the actions of any of the

Av and Bp (this is a sufficient constraint that guarantees that the tensors form the

81

ground state) , which leads to the following equations:

T TZ Z

Z

Z=

T T

x x

T

x

xT

x

x

T

x

xT

xx

Tx

x

Tx

Tx

xx

= = =

=

=

==

Z

Z

=

Tx

x T

xx=== =

Tx

x

Tx

T

x

x

x

===T

xxTx

x

(4.26)

In the second set of equations, the first 12 equalities are obvious from the red dashed

squares, and the last 3 equalities can be derived from the first 12 ones. Expanding

the first set of conditions by using Zij = δij(−1)i, we have:

Txx,yy,zz = (−1)x+x+y+y+z+zTxx,yy,zz

⇔

Txx,yy,zz

= 0, if x+ x+ y + y + z + z = 1 mod 2

6= 0, if x+ x+ y + y + z + z = 0 mod 2

(4.27)

82

where x, x, y, y, z, z are the six indices of T in the three directions respectively. We

emphasize for notation clarity that x is not−x, they are notations for different indices.

The second set of conditions in Eq. (4.26) further enforces that an even number of

index flipping of the virtual indices of a tensor does not change the value of the tensor

elements. For instance, in terms of components, we have:

Txx,yy,zz =T(1−x)(1−x),yy,zz

=T(1−x)x,(1−y)y,zz

=Txx,yy,(1−z)(1−z)

= . . .

(4.28)

Hence, the elements of the T tensor are all equal. Up to an overall normalization, we

have the unique solution:

Txx,yy,zz =

0, if x+ x+ y + y + z + z = 1 mod 2

1, if x+ x+ y + y + z + z = 0 mod 2

(4.29)

The ground state TNS wave function is then Eq. (4.5) with the local T to be Eq. (4.29).

The local T tensors are the same on other spatial manifolds, such as T 3.

A similar set of conditions as the first equality in Eq. (4.26) have been intro-

duced by several other names in tensor network literature: Z2-injectivity[119], MPO-

injectivity[120], Z2 gauge symmetry[24] etc. The previous studies were in 2D, and

our condition is the 3D generalization. Note that the first equation in Eq. (4.26)

alone will not necessarily lead to topological order. It only implies that the ground

state is Z2 symmetric. The state which only satisfies the first condition in Eq. (4.26)

could also be a topological trivial state by tuning the relative strength of the nonzero

elements of T tensor. This can be interpreted as a condensation transition from topo-

83

logical phases to trivial phases. See Refs. [24, 121, 122, 123, 124] for explanations and

examples in the case of 2D TNS.

4.3.3 Concatenation Lemma

In this section, we consider some contraction of a network of local T tensors with

open virtual indices. Since the elements of a local T tensor are 0 for the odd sector

and 1 for the even sector (see Eq. (4.29)), we will show that, in general, a network of

contracted T tensors obeys a similar rule: some elements are zeros while the others

are identical and nonzero. A Concatenation lemma is proposed to define the rule

for the contraction of several tensors in general and will be frequently used in the

following discussions.

Concatenation Lemma: For a network of contracted T tensors Eq. (4.29)

with open indices, the open indices need to sum to 0 mod 2, otherwise the element

of the network tensor is zero. Moreover, if nonzero, the elements of the network

tensor are constants, independent of open indices.

This lemma can be easily proved by using Z2 symmetry Eq. (4.29) and induction.

The proof is in App. C.1. We explain this lemma by a simple example. Suppose we

have two T tensors contracted over a pair of indices:

Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2

=∑

z1,z2

Tx1x1,y1y1,z1z1Tx2x2,y2y2,z2z2δz1z2(4.30)

84

T

T

x

z

y

Figure 4.4: Contraction of two local T tensors in the z-direction.

Graphically, the tensor T is represented by Fig. 4.4. The open indices of the

tensor T need to sum to an even number in order for the elements of the T tensor to

be nonzero. This comes out of writing the constraints of each of the T tensors:

x1 + x1 + y1 + y1 + z1 + z1 = 0, mod 2

x2 + x2 + y2 + y2 + z2 + z2 = 0, mod 2

z1 = z2

⇒ x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2

=0, mod 2

(4.31)

Otherwise, the tensor element of T is zero. Moreover, the elements of the contracted

tensor are 1, if nonzero:

Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2 =

0 if x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2 = 1, mod 2

1 if x1 + x1 + y1 + y1 + z1 + x2 + x2 + y2 + y2 + z2 = 0, mod 2

(4.32)

85

For a more complicated contraction of T tensors, we have:

T{t} =

0 if∑

i ti = 1, mod 2

Const if∑

i ti = 0, mod 2

(4.33)

where {t} denotes all the indices of the tensor T. Note that the nonzero constant

does not depend on {t} .

4.3.4 Entanglement

We now show that Eq. (4.5) is exactly the SVD for the wave function with respect

to the entanglement cut illustrated in Fig. 4.5. For simplicity, suppose that the TNS

is on infinite R3. As we have emphasized at the end of Sec. 4.1.2, we do not specify

the boundary conditions of TNS, since we are only concerned with the bulk wave

functions whose reduced density matrices are assumed not to be influenced by the

boundary conditions. If we put the wave function on a large but finite R3, we have

to specify the boundary conditions of the TNS by fixing the indices on the boundary.

Suppose the open indices on the boundary are denoted as {tb}. The norm of the

TNS on open R3, which can be expressed as a network of contracted T tensors with

open virtual indices {tb}, is zero when∑

i tbi = 1 mod 2 and nonzero when

∑i tbi = 0

mod 2, according to the Concatenation lemma of the 3D toric code model. Hence,

we can only fix the boundary indices {tb} to be∑

i tbi = 0 mod 2. Calculating

the entanglement on a nontrivial manifold is ambiguous since multiple degenerate

ground states, which cannot be distinguished locally, appear. Their superpositions

have different entanglement entropies.

We rewrite Eq. (4.5) by separating the tensor contractions to a spatial region A

and its complement region A. Region A contains the g tensors near the entanglement

86

cut as illustrated in Fig. 4.5:

|TNS〉R3 =∑

{t}

|{t}〉A ⊗ |{t}〉A (4.34)

where

|{t}〉A =∑

{s}∈A

∑

{i}∈A

CA(gs1t1i1gs2t2i2

. . . gs3i3i4gs4i5i6Ti7...Ti8... . . .)|{s}〉 (4.35)

Indices denoted by s are the physical indices; indices denoted by t are the open virtual

indices going out of the entanglement cut from the region A; indices denoted by i are

the contracted virtual indices inside the region A. The tensors gs1t1i1 and gs2t1i2 etc are

the projector g tensors near the entanglement cut on the region A side as illustrated

in Fig. 4.5; gs3i3i4 and gs4i5i6 are the projector g tensors inside the region A; for this cut,

all the T tensors are inside the region A. The summation is over all physical indices

{s} inside the region A.

Tg

A Acut

Figure 4.5: The splitting of tensors near the entanglement cut.

Thereby, |{t}〉 is the TNS wave function for region A with open virtual indices {t}.

We choose a convention of splitting tensors whereby g tensors near the entanglement

cut belong to the region A, as illustrated in Fig. 4.5. For instance, when the region A

87

is a cube, we can graphically denote the basis |{t}〉 as Eq. (4.10), where in the bulk

of this cube is a TNS, and the red lines are the outgoing virtual indices {t}. The g

tensors connecting with these red lines are inside the cube. Similarly for the region

A:

|{t}〉A =∑

{s}∈A

∑

{i}∈A

CA(gs1i1i2gs2i3i4Tt1i5...Tt2i6... . . .)|{s}〉 (4.36)

Since the TNS for region A and A share the same boundary virtual indices {t}, then

in Eq. (4.34) the two basis for region A and A have the same label {t}. For the TNS

wave function of Eq. (4.5), the boundary virtual indices {t} of the regions A and A

are contracted over, and thus in Eq. (4.34) {t} are summed over.

We now show that |{t}〉A and |{t}〉A are an orthonormal basis (normalized up to

constant) for the region A and the region A respectively. Therefore, Eq. (4.34) is

exactly the SVD for the ground state wave function, i.e.,

A〈{t′}|{t}〉A ∝ δ{t′},{t}δ(∑

i

ti = 0 mod 2). (4.37)

Proof:

Applying the SVD condition to the toric code TNS, we can immediately conclude

that the |{t}〉A span an orthogonal basis, and the TNS is exactly an SVD. However,

the SVD condition does not tell us whether the basis is orthonormal. In the follow-

ing, we show that |{t}〉A is not only orthogonal, but also orthonormal with a norm

independent on t, which leads to the flat singular values. Following the definition of

88

our basis:

A〈{t′}|{t}〉A =

∑

{s′}∈A

∑

{j}∈A

CA(gs′1?

t′1j1gs′2?

t′2j2. . . g

s′3?j3j4

gs′4?j5j6

T ?j7...T?j8...

. . .)〈{s′}|

∑

{s}∈A

∑

{i}∈A

CA(gs1t1i1gs2t2i2

. . . gs3i3i4gs4i5i6Ti7...Ti8... . . .)|{s}〉

(4.38)

When the open virtual indices {t′} 6= {t}, the overlap is clearly zero, as the spin

configurations on the boundary are different due to the projector g tensors. Hence,

the basis |{t}〉A are orthogonal.

Next we show that A〈{t}|{t}〉A is zero when(∑

ti∈{t} ti

)is odd. Using the defi-

nition of the g-tensor, we have:

A〈{t}|{t}〉A = CA (. . . TTT . . .) (4.39)

with the open virtual indices {t}. The contraction CA is over the T tensors in the

region A. Applying the Concatenation lemma, A〈{t}|{t}〉A is zero if the open

indices {t} are summed to be 1 mod 2:

∑

i

ti = 1 mod 2 ⇒ A〈{t}|{t}〉A = 0 (4.40)

Moreover,

A〈{t}|{t}〉A = Const, when∑

i

ti = 0 mod 2 (4.41)

Hence |{t}〉 is orthonormal basis up to an overall normalization factor that can be

obtained by the normalization of |TNS〉. 2

The same proof works for the region A and |{t}〉A. Therefore, we can conclude

that Eq. (4.34) is indeed an SVD, and the singular values are all identical. Hence,

89

for a entanglement cut, we only need to count the number of singular vectors in

Eq. (4.34). For a connected entanglement surface with N open virtual indices, the

number of singular vectors in Eq. (4.34) is 2N−1, because the open virtual indices

need to sum to be 0 mod 2. Hence, the entanglement entropy for a region whose

entanglement surface is singly connected is:

S = N log(2)− log(2) (4.42)

If the entanglement surface still has N open virtual indices but is separated into n

disconnected surfaces, then the entanglement entropy is:

S = N log(2)− n log(2)

= Area× log(2)− n log(2)

(4.43)

The above is true because the condition that the open indices need to have an even

summation holds true for each component of the entanglement cut. Furthermore, if

we place our TNS ground state on a 3D cylinder T 2xy ×Rz, and the entanglement cut

splits the cylinder into two halves z > 0 and z < 0, then the entanglement entropy of

the either side is also S = Area× log(2)− log(2). The results can be easily generalized

to ZK lattice gauge models on R3:

S = Area× log(K)− n log(K) (4.44)

with the same equation holding on a cylinder T 2xy ×Rz. The entanglement spectrum

is also flat. The area is measured by the number of open virtual indices going out of

the entanglement cut.

Following the same logic, for the toric code in (d + 1) dimensions, all the open

virtual indices of region A, {ti}, have to satisfy a single constraint∑

i ti = 0 mod 2,

90

because they have to obey the Concatenation lemma. If there are N open virtual

indices on the surface of region A, there are N − 1 independent open virtual indices.

Hence the rank of the reduced density matrix is still 2N−1, because each independent

open index can take 2 values. The entanglement entropy is

S = N log(2)− log(2) (4.45)

The topological entanglement entropy Stopo[T d−1] is independent of the dimensional-

ity, and it obeys the conjecture presented in Ref. [118]:

exp(−dStopo[T d−1]) = GSD[T d] (4.46)

where GSD[T d] = 2d.

4.4 Haah Code

In this section, we derive the TNS for Haah code following a similar prescription as

that in Sec. 4.3.2. We then compute the entanglement entropies using the TNS for

several types of entanglement cuts. In Sec. 4.4.1, we review the Haah code and the

Hamiltonian terms. In Sec. 4.4.2, we present the construction of TNS for the Haah

code. In Sec. 4.4.3, we discuss the entanglement cuts for which the tensor network

wave function is an exact SVD. In Sec. 4.4.4, we discuss the cubic entanglement cut,

where the tensor network wave function is not an exact SVD. The calculation for

entanglement entropies proceeds in the same way as that for the Toric code: one

counts the number of constraints for open indices.

91

4.4.1 Hamiltonian of Haah code

The Haah code is defined on a cubic lattice. As opposed to the two previous models,

there are two spin-1/2’s defined on each vertex of a cubic lattice. Each spin-1/2

is associated with a two-dimensional local Hilbert space. Similar to the toric code

and the X-cube model discussed in Sec. 4.3, the Hamiltonian is a sum of commuting

operators where each term is the product of Pauli X or Z operators. Specifically,

there are two types of the Hamiltonian terms:

H = −∑

a,b,c

Aabc −∑

a,b,c

Babc (4.47)

The A and B operators are defined on each cube in the cubic lattice, and the indices

a, b, c represent the vertex coordinates. If we choose the space to be the infinite

without periodic boundary condition, i.e., R3, then a, b, c ∈ Z. If we choose the space

to be a 3D torus of the size Lx × Ly × Lz with periodic boundary condition on each

side, then a ∈ ZLx , b ∈ ZLy and c ∈ ZLz . The operators defined on a = 0, b = 0, c = 0

are

A000 = ZL110Z

L101Z

L011Z

L111Z

R100Z

R010Z

R001Z

R111

B000 = XL000X

L110X

L101X

L011X

R000X

R100X

R010X

R001

(4.48)

The up-indices L/R represent the left or the right spin on a vertex where the Pauli

operators act on. The bottom-indices (ijk) ∈ Z2×Z2×Z2 represent the coordinate of

vertices (on a cube). All other operators Aabc and Babc can be obtained by translation.

92

Pictorially the two types of terms are:

x

z

y

(4.49)

It is straightforward to check that all the Hamiltonian terms commute.

4.4.2 TNS for Haah Code

The ground state |GS〉 is obtained by requiring

Aabc|GS〉 = |GS〉 (4.50)

Babc|GS〉 = |GS〉 (4.51)

for every a, b, c. We can solve these two equations similarly to the toric code model

Sec. 4.3.2 to obtain a TNS representation, although, since the model geometry is

different (spins on sites rather than on bonds), the form of the TNS is also changed.

We now specify the projector g tensor and the local T tensor.

There are 2 types of g tensors gL and gR associated with the left and right physical

spins on each vertex. Each g tensor has 1 physical index s and 4 virtual indices i, j, k, l.

93

The reason for these 4 virtual indices (rather than 2 virtual indices as in the toric

code and the X-cube examples) is that, for each vertex, the virtual indices from T

tensors (to be defined below) in the neighboring 8 octants need to be fully contracted;

this requires the g tensor to have 4 virtual indices. The index assignment of the Left

and Right- spin g tensor gLsijkl and gRsijkl are:

gLsijkl =

s

i

j

k l

I

IIIII

IV

V

V IV II

V III

(4.52)

and

gRsijkl =

l

j

ki

s

I

IIIII

IV

V I

V

V II

V III

(4.53)

94

where s is the physical index in {|0〉 = |↑〉, |1〉 = |↓〉}, and ijkl are virtual indices.

We use a blue dot for the right spin and a red dot for the left spin. The green dots

at the center of each cube represent T tensors (which we define below). Similar to

the toric code model and the X-cube model, we require that the g tensor acts as a

projector from the physics index to the four virtual indices:

gLsijkl =

1 i = j = k = l = s

0 otherwise

, gRsijkl =

1 i = j = k = l = s

0 otherwise

(4.54)

The four virtual indices of gLsijkl extend along the III, VIII, VII, VI octants (as shown

in Eq. (4.52)), and the four virtual indices of gRsijkl extend along the II, VII, IV, V

octants (as shown in Eq. (4.53)).

The tensor T{i} is defined at the center of each cube, and every T tensor has 8

virtual indices. Graphically, the T tensor is:

Ti1i2i3i4i5i6i7i8 = T (4.55)

The T tensor is contracted to 8 of the total 16 (8 vertices times 2 degrees of freedom

per vertex) g tensors located at the cube corners via the virtual indices. The reason

for only 8 virtual indices (instead of 16 virtual indices) in the T tensor is that among

16 spins around the cube (a, b, c) only eight of them are addressed by the Pauli Z

operators in the Aabc term of the Hamiltonian. The elements of the T tensor for

a given set of virtual indices i1i2i3i4i5i6i7i8 are determined by solving Eq. (4.50)

95

and Eq. (4.51). Imposing the condition Eq. (4.50) and transferring the physical Z

operators to the virtual level, we find that:

TZ

Z Z

Z Z

Z

ZZ

= T (4.56)

which amounts to

Ti1i2i3i4i5i6i7i8 = (−1)∑8n=1 inTi1i2i3i4i5i6i7i8 (4.57)

where i1, · · · , i8 are the eight virtual indices of the T tensor defined in Eq. (4.55).

Hence,

Ti1i2i3i4i5i6i7i8 = 0, if8∑

n=1

in = 1 mod 2 (4.58)

96

Imposing the condition Eq. (4.51) and transferring the physical X operators to the

virtual level, we find that

T = Tx x

= T

x

x

= T

x

x

= T

x

x = Tx x

= T

x x

= T

x x

= T

x

x

= T

x

x = T

x

x

= Tx

x

= Tx

x

= Tx

x

= T

x x

x

x

xx

(4.59)

97

In terms of components, Eq. (4.59) means that Ti1i2i3i4i5i6i7i8 = Ti′1i′2i′3i′4i′5i′6i′7i′8 where

i′1i′2i′3i′4i′5i′6i′7i′8 are obtained by flipping arbitrary pairs of indices from i1i2i3i4i5i6i7i8.

For example,

Ti1i2i3i4i5i6i7i8 = T(1−i1)(1−i2)i3i4i5i6i7i8

= Ti1(1−i2)(1−i3)i4i5i6i7i8

= Ti1i2(1−i3)(1−i4)i5i6i7i8

= ...

(4.60)

Combining Eq. (4.57) and (4.60), we find that any configuration of Ti1i2i3i4i5i6i7i8

satisfying the condition∑8

k=1 ik = 0 mod 2 are equal. We can rescale the T tensor

such that Ti1i2i3i4i5i6i7i8 = 1 for∑8

k=1 ik = 0 mod 2, i.e.,

Ti1i2i3i4i5i6i7i8 =

1∑8

n=1 in = 0 mod 2

0∑8

n=1 in = 1 mod 2

(4.61)

For simplicity, we consider the space to be R3. Since there is no non-contractible

spatial cycle, there is only one ground state:

|TNS〉 =∑

{s}

CR3 (gL,s1gR,s2gL,s3gR,s4 . . . TTT . . .

)|{s}〉 (4.62)

98

T T

T T

(a)

x

z

y

T T

T T

T T

T T

(b)

Figure 4.6: Tensor contraction for the Haah Code TNS. (a) The lattice size is 2×3×3.

(b) The lattice size is 3× 3× 3

Note that the contraction of the Haah code TNS is quite different from that of the

3D toric code model and the X-cube model. The main difference is that the g tensor

has 4 virtual indices for the Haah code, while it has only 2 virtual indices for the toric

code and the X-cube code. As an example of contraction, we take two blocks of size

2× 2× 1 and 2× 2× 2 in Fig. 4.6. The T tensors with their virtual indices are drawn

99

explicitly. Each red or blue node in the two figures is a projector g tensor, whose

physical index is not drawn; we only draw the virtual legs that are connected to the

T tensors inside the blocks. In the block 2 × 2 × 2, all the 8 virtual indices of the

two g tensors (4 per each g tensor) in the middle of all the cubes are contracted with

T tensors, while other g tensors have open virtual indices (which are not explicitly

drawn).

4.4.3 Entanglement Entropy for SVD Cuts

In this section, we compute the entanglement entropy for two types of cuts for which

the TNS wave function is an SVD.

Two types of SVD Cuts

To compute the entanglement entropy, we use the same convention which was adopted

in the discussion of the toric code (in Sec. 4.3 ): the open virtual indices of the region

A connect directly to the g tensors while the open virtual indices of the region A

connect with T tensors. We further choose a region A such that the TNS is an SVD,

and compute the entanglement entropy. We found two types of entanglement cuts for

which the Haah code TNS is an exact SVD. For more general regions, there exists an

extra step required to make other cuts SVD. This step will be presented in Sec. 4.4.4.

1. Region A only consists of the spins connecting to a set of (l − 1) T tensors

which are contracted along a certain direction. Figure 4.7 shows an example

with l− 1 = 3 contracted along the z direction. (Since in Sec. 4.3, we used l as

the number of vertices along each side of region A, so there are l− 1 bonds (or

cubes) along each side.)

100

T

T

T

Figure 4.7: Region A contains all the spins connecting with l− 1 T tensors which are

contracted along z direction. The figure shows an example with l = 4.

2. Region A contains all the spins connecting with T tensors which are contracted

in a “tripod-like” shape, where three legs extend along x, y, z directions. If there

are lx− 1 cubes in the x leg, ly − 1 cubes in the y leg, and lz − 1 cubes in the z

leg, then there are 1 + (lx − 2) + (ly − 2) + (lz − 2) = lx + ly + lz − 5 cubes (or

T tensors) region A. Figure 4.8 shows an example with lx = ly = lz = 3.

T T

T

T

Figure 4.8: Region A contains all the spins connecting with T tensors which are

contracted in a “tripod-like” shape, where three legs extend along x, y, z directions.

There are three legs extending along x, y, z directions respectively. In general, three

legs can have different length, each with lx−1, ly−1, lz−1 cubes along three directions.

This figure shows an example where lx = ly = lz = 3.

101

In the first case and for l = 2, 3, we used brute-force numerics to find that the

reduced density matrix is diagonal (see App. C.2 for details), which shows that the

wave function is an exact SVD.

In order to show that the above cuts correspond to an SVD, we follow the argu-

ments developed in Sec. 4.2.1. In Sec. 4.2.1, we proposed a SVD Condition. However,

we find that the region A of both types, shown in Fig. 4.7 and 4.8, do not satisfy

the SVD Condition: Two open virtual indices in region A connects with the same T

tensor, which violates the SVD Condition. For instance, the g1 and g2 in Fig. 4.8 con-

nects to the same T tensor in their upper-left cube which is in the region A. Here, we

propose a Generalized SVD Condition which suffice to prove that the entanglement

cut corresponding to Figure 4.7 and 4.8 are SVD.

Generalized SVD condition: Let {t} be the set of open virtual indices. Given a

set of physical indices {s} inside region A, if {t} can be uniquely determined by the

{s} inside region A via the g tensor projection condition Eq. (4.54) and T tensor

constraints Eq. (4.61), then |{t}〉A is orthogonal. Since |{t}〉A is orthogonal because

all the open virtual indices are connected with g tensors, the TNS wave function

|TNS〉 =∑{t} |{t}〉A ⊗ |{t}〉A is SVD.

To prove the Generalized SVD Condition, we notice that if we have two different

sets of open virtual indices {t}A and {t′}A, the physical indices {s}A and {s′}A which

connect (via g tensors) to the T tensors on the boundary of region A cannot be the

same. Otherwise, if {s}A = {s′}A, since the physical indices {s}A and {s′}A in the

region A uniquely determine the open virtual indices {t}A and {t′}A, {t}A = {t′}A,

hence it is in contradiction with our assumption {t}A 6= {t′}A. Therefore, {t}A 6=

{t′}A implies {s}A 6= {s′}A, and hence A〈{t}|{t′}〉A = 0. This is in the same spirit of

the proof in Sec. 4.2.1. The proof of normalization of the wave function is independent

of {t} is also the same as in Sec. 4.2.1. Furthermore, A〈{t}|{t′}〉A = 0 for {t} 6= {t′}

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is the straightforward because {t}A are connected with g tensors. In summary, if the

entanglement cut satisfies the Generalized SVD Condition, we have

1. A〈{t}|{t′}〉A ∝ δ{t},{t′} when |{t}〉A and |{t′}〉A are not null vectors;

2. A〈{t}|{t′}〉A ∝ δ{t},{t′} when |{t}〉A and |{t′}〉A are not null vectors.

This shows that the TNS wave function is an SVD.

We explain the Generalized SVD Condition in the simplest example, i.e., l = 2 in

case 1. There is only one T tensor, and region A contains 8 physical spins.

T

All other spins apart from the eight connecting with the T tensor belong to region

A. Because the virtual indices and physical indices are related by the g tensor which

is a projector, we use i1 to denote the values of both virtual indices and physical

indices connecting with left g tensor located at (x, y, z) = (0, 0, 1). Here, we use

the coordinate convention where the (x, y, z) = (0, 0, 0) is located at the left down

frontmost corner as in Fig. 4.6. Similarly we use i2, i3, i4, i5, i6, i7, i8 to label the values

of the virtual/physical indices on the remaining seven nodes connecting with the same

T tensor. Hence the set of open indices is effectively {i1, i2, i3, i4, i5, i6, i7, i8} (after

identified by the g tensors). We further consider how the physical indices from the

region A constraint the open indices. Consider the T tensor in the region A (which

we denote by T ′) which shares two spins i7, i8 with region A (The T ′ tensor lives in

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the lower right corner):

T

T'

(4.63)

Since six among the eight virtual indices of T ′ are contracted with g tensors inside

region A, the remaining two open virtual indices, i.e., i7 and i8 are subject to one

constraint from the T ′ tensor:

i7 + i8 = fixed (4.64)

where “fixed” means that the sum is fixed by the physical indices inside the region

A. We can similarly consider the constraints coming from other T tensors in region

A. The whole set of constraints are listed as follows:

i7 + i8 = fixed, i1 + i2 = fixed, i5 = fixed, i6 = fixed, i6 + i7 = fixed

i2 + i3 = fixed, i8 = fixed, i1 = fixed, i4 = fixed, i3 = fixed, i7 = fixed

(4.65)

The “fixed” on the right hand side of the equations means that the virtual indices

or the sum of the virtual indices are fixed by the physical indices in the region A.

All variables and equations are defined module 2. The above equations uniquely

determine all the open virtual indices i1...i8. Therefore, such a choice of region A of

the entanglement cut satisfies the Generalized SVD Condition.

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For the first type of region A with general l, and the second type of region A

with general lx, ly, lz, we can similarly check that the TNS wave function satisfies

the Generalized SVD Condition. Numerically, we checked TNS wave function indeed

satisfies the Generalized SVD Condition for 2 ≤ l ≤ 9 for the first type, and 3 ≤ lx ≤

8, 3 ≤ ly ≤ 8, 3 ≤ lz ≤ 8 for the second type.

Entanglement entropy

We now compute the entanglement entropy for the exact SVD TNS wave funtions. We

first consider the case 1 with general l, such as in Fig. 4.7. All the spins connecting

with l − 1 contracted T tensors along the z directions are in region A, and the

remaining belong to region A. The number of open virtual indices, after identified by

the local g tensors, is 8 + 7(l− 2) = 7l− 6. The number of constraints from the local

T tensors is simply the number of T tensors l − 1, because they are all independent.

Hence the number of independent open virtual indices is 7l − 6 − (l − 1) = 6l − 5.

Therefore, the entanglement entropy is

S(A)

log 2= 6l − 5 (4.66)

In appendix. C.2, we numerically brute-force compute the reduced density matrix for

l = 2 and l = 3, and find that the results match the general formula Eq. (4.66).

We further consider the case 2 — region A of tripod shape. The legs in the x, y, z

direction contains lx− 1, ly − 1, lz − 1 T tensors respectively. We first count the total

number of open virtual indices. When lx = ly = lz = 3 as shown in Fig. 4.8, there are

26 physical spins (or g tensors) in total. However, there is one g tensor (at the left spin

of (x, y, z) = (1, 1, 1)) whose four virtual indices are all contracted by the T tensors

within region A. Hence the number of open virtual indices, after identified by the local

g tensor, is 25. Moreover, we notice that adding one T tensor in one of the three legs of

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region A brings 7 extra spins. Therefore the total number of open virtual indices (after

identified by the g tensor) is (26−1)+7(lx−3)+7(ly−3)+7(lz−3) = 7lx+7ly+7lz−38.

We further numerically count the number of constraints that these open virtual indices

satisfy. We find the number of constraints is the number of cubes minus 1, i.e.,

(lx + ly + lz − 5) − 1 = lx + ly + lz − 6. Therefore the number of independent open

virtual indices is (7lx + 7ly + 7lz − 38)− (lx + ly + lz − 6) = 6lx + 6ly + 6lz − 32. The

entanglement entropy is

S(A)

log 2= 6lx + 6ly + 6lz − 32. (4.67)

4.4.4 Entanglement Entropy for Cubic Cuts

In this section, we consider the case where the region A is a cube of size l × l × l,

where l is the number of vertices in each direction of the cube. The cut is chosen

such that all the open virtual indices coming out of the region A are connected to g

tensors in the region A (i.e., all the physical spins near the boundary belong to the

region A). For example, for l = 2 as shown in (4.55), all 16 physical spins belong to

the region A. For l = 3 as shown in Fig. 4.6 (b), all 54 physical spins belong to the

region A. For the simplicity of notations, in this section, we denote the Hamiltonian

terms as Ac and Bc where the subindex refers to a cube c.

SVD for TNS

For the cubic region A, we find that the TNS for the Haah code is different from that

for the toric code and X-cube models: the TNS for the Haah code is not an exact

SVD. The TNS basis in the region A, |{t}〉A, are orthonormal, since the open virtual

indices are connected with g tensors. However, the TNS basis |{t}〉A in the region A

are not orthogonal. In other words, the basis |{t}〉A is over complete.

106

The subtlety that the TNS bipartition is not an exact SVD manifests as follows:

the singular vectors in the region A for the ground states of the Haah code have to be

the eigenvectors of all Ac and Bc operators that actually lie in the region A, and the

corresponding eigenvalues should all be 1. Note that our TNS basis state |{t}〉A, if

not null, are the eigenvectors of all Ac operators inside the region A with eigenvalues

1, and are also the eigenvectors of Bc operators with eigenvalues 1 when Bc operators

are deep inside the region A, i.e., when they do not act on any spin at the boundary

of A. However, |{t}〉A are not the eigenvectors of Bc operators, when Bc operators

are inside the region A but also adjacent to the region A’s boundary. The reason

is that the Bc operators adjacent to the region A’s boundary, when acting on the

TNS basis |{t}〉A, will flip the physical spins on the boundary, and thus flip the open

virtual indices {t} due to the projector g tensors. Therefore, the basis |{t}〉A is no

longer the singular vectors for the Haah code. This is not an a priori problem, but

a result of the geometry of the Haah code, whose spins cannot be written on bonds

but have to be written on sites. A similar situation would occur if the 2D toric code

model would be re-written to have its spins on sites.

The method to find the correct SVD for the TNS wave function is to use the |{t}〉Ato construct the eigenvectors of Bc operators by projection. We prove the following

statement:

If |{t′}〉A = Bc|{t}〉A when Bc is inside the region A and also adjacent to the

region A’s boundary, then A〈{t′}|{t}〉A = 0 and |{t′}〉A = |{t}〉A.

The proof is as follows. The first part of the statement is a consequence of the

|{t}〉A basis state orthogonality. Indeed, Bc flips physical spins located at the region

A’s boundary. Thus the two sets {t} and {t′} are distinct. The second part of the

statement is more involved. Suppose for simplicity that we consider two nearest

neighbor T tensors for the region A and A in Fig. 4.9. The Bc operator acts on the

right cube Fig. 4.9 (a). The physical spins on the boundary of the region A which

107

T T

Cut

x

x

x

xx

T T

Cut

x

x

(a)

(b)

Figure 4.9: Transferring the Pauli X operators of the Bc operator from the region A(a) to the region A (b).

are flipped by Bc are those covered by circled X in Fig. 4.9 (a). Then these Pauli X

operators can be transferred to the virtual indices due to projector g tensors, and the

virtual indices of the T tensor in the region A obtain two X operators as in Fig. 4.9

(b). Note the T tensor for the Haah code is invariant under this action (see the 12th

cube in Eq. (4.59)). This is also true for other T tensors in the region A that are

affected by Bc. The transfer of X operators from the region A to the region A gives

exactly the same equations in Eq. (4.59) when we solve for the T tensors. Hence, the

X operators transferred to the open virtual indices in the region A do not change the

state at all, i.e., |{t′}〉A = |{t}〉A. As a consequence, we can perform the following

factorization

|{t}〉A ⊗ |{t}〉A + |{t′}〉A ⊗ |{t′}〉A

=[(1 +Bc)|{t}〉A

]⊗ |{t}〉A

(4.68)

108

The left part of the tensor product is an eigenstate of Bc with eigenvalue 1.

Therefore, in the TNS decomposition Eq. (4.11), we can group the basis state |{t}〉Awhich are connected by this Bc operator. This factorization can be extended to

any product of Bc operators inside the region A and also adjacent to the region A’s

boundary. Note that any such product has at least one X operator belonging to

only one Bc and so is different from the identity. When acting with all the possible

products of these Bc operator (including the identity) on a given |{t}〉A will generate

as many unique states as there are Bc’s. The TNS can be brought to the following

form

|TNS〉 =∑

{t}′

[∏

c

(1 +Bc

2

)|{t}〉A

]⊗ |{t}〉A (4.69)

where the product over c only involves the Bc operators inside the region A and also

adjacent to the region A’s boundary and the sum over {t}′ is over the open virtual

index configurations that are not related by the action of these Bc operators.

Counting the number of TNS basis in region A: Notations

To find the upper bound to the entanglement entropy, we need to find the number of

basis states in the region A that are also eigenstates of any Bc operators fully lying

in the region A. This number that we denote as basis(TNS(A)) is

basis(TNS(A)) = 2N−NB (4.70)

where N is the number of independent open virtual indices and NB is the number

of Bc operators inside the region A and also adjacent to the region A’s boundary.

Every open virtual index connected to a g tensor located in A and at the boundary

of this region. Since each g tensor has a unique independent virtual index, we have

N = Ng −Nc where Ng is the number of g tensors in A and at the boundary of this

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region and Nc are the number of constraints on the open indices coming from the T

tensors within the region A. We this get

log2(basis(TNS(A))) = Ng −Nc −NB (4.71)

and the upper bound on the entanglement entropy reads

S(A) = (Ng −Nc −NB) log 2 (4.72)

Counting Ng and NB

We first count Ng. The number of g tensors can be computed by looking at Fig. 4.6

(b). We consider the region A with size lx × ly × lz (Notice that lx, ly, lz are the

number of vertices in each direction). In eight corners, there are 8× 2 = 16 vertices.

On the four hinges along x direction, there are 2 × 4 × (lx − 2) vertices, where 2

means there are two spins on each vertex, and 4 means four hinges. And similar for

2 × 4 × (ly − 2) and 2 × 4 × (lz − 2) in the y and z directions respectively. For the

xy-plane, there are 2× 2× (lx− 2)(ly− 2), where the first 2 comes from two spins per

vertex, and the second 2 comes from two xy-planes. Similarly 2× 2× (lx− 2)(lz − 2)

and 2 × 2 × (ly − 2)(lz − 2) for xz and yz plane respectively. Therefore, the total

number of g tensors is

Ng =16 + 8(lx − 2) + 8(ly − 2) + 8(lz − 2)

+ 4(lx − 2)(ly − 2) + 4(lx − 2)(lz − 2)

+ 4(ly − 2)(lz − 2)

=4lxly + 4lxlz + 4lylz − 8lx − 8ly − 8lz + 16

(4.73)

We further count NB. As explained in Sec. 4.4.4, NB is the number of Bc operators

inside the region A and adjacent to the boundary of the region A. For a cubic region

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A with size l × l × l (which is the case we consider below), the number of such Bc

operators are

NB = (l − 1)3 − (l − 3)3 = 6l2 − 24l + 26,∀l ≥ 3 (4.74)

For l = 2, we just have 1 Bc operator. Hence we have

NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2 (4.75)

Counting Nc: Contribution from the T tensors

The open indices may be constrained by the T tensor fully inside the region A. In the

following, we will discuss the specific entanglement cuts where lx = ly = lz = l. We

rely on numerical calculations to evaluate Nc. We first consider the examples l = 2

and l = 3 in details, and then we describe our algorithm to search the number of the

linear independent constraints.

For l = 2, no g tensor has all virtual indices contracted. The reader can refer to

Fig. 4.6 (a) as an example. There is only one T tensor. The element of the T tensor

is

Ti1i2i3i4i5i6i7i8 (4.76)

where i1, i2, i3, i4, i5, i6, i7, i8 are all contracted virtual indices. Because they are con-

tracted with g tensors where at least one virtual index is open, all the contracted

virtual indices i1, i2, i3, i4, i5, i6, i7, i8 are equal to some open indices, and we denote

them as

i1 = t1, i2 = t2, i3 = t3, i4 = t4,

i5 = t5, i6 = t6, i7 = t7, i8 = t8

(4.77)

111

The constraints on {i}’s are hence equivalent to the constraints on {t}’s, i.e.,

t1 + t2 + t3 + t4 + t5 + t6 + t7 + t8 = 0 mod 2 (4.78)

There is only one constraint from the T tensor. Hence Nc = 1 for l = 2.

For l = 3, as shown in the Fig. 4.6 (b), we have eight constraints from eight T

tensors which involve the open indices via the g tensors. The eight equations are

8∑

n=1

i(x,y,z)n = 0 mod 2, x, y, z ∈ {0, 1} (4.79)

where the up-index (x, y, z) represents the position of the T tensor, and n counts the

eight indices of each cube in the 2 × 2 × 2 cut. All the i’s are contracted virtual

indices. However, except the virtual indices that are connected with the central two

g tensors (which are defined on the two spins at the vertex (x, y, z) = (1, 1, 1)), all

other indices (which are defined on two spins at vertices (x, y, z), x, y, z ∈ {0, 1, 2}

except (x, y, z) = (1, 1, 1)) are equal to some open indices via g tensors. Specifically,

the virtual indices that are connected with the two center g tensors are

i0004 = i100

3 = i0101 = i001

7 mod 2

i0005 = i110

2 = i1018 = i011

6 mod 2

(4.80)

Since we only count the number of constraints for the open indices, we need to Gauss-

eliminate all these eight virtual indices i0004 , i100

3 , i0101 , i001

7 , i0005 , i110

2 , i1018 , i011

6 from the

above 8 equations. Therefore, we obtain 8− 2 = 6 independent equations in terms of

open indices only. Hence there are 6 constraints for the open indices.

For the general l, we apply the same principle. We first enumerate all possible

constraints from the T tensors, and then we Gauss-eliminate all the virtual indices

that are contracted within region A. Hence we obtain a set of equations purely in

112

terms of the open indices. The number of constraints is the rank of these set of

equations, and we list the number of linear independent constraints for the open

indices as follows:

l(≥ 3) 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nc 6 12 18 24 30 36 42 48 54 60 66 72 78 84(4.81)

Hence, for l ≥ 3, there are

6l − 12 (4.82)

linearly independent constraints for the open indices. Taking into account the fact

that when l = 2 the number of constraints is 1, we infer that the number of constraints

for a generic l is:

6l − 12 + δl,2 (4.83)

Entanglement entropy

We are ready to collect all the data we have obtained and compute the entanglement

entropy. For the entanglement cut of size l × l × l, the total number of g tensors is

Ng = 12l2 − 24l + 16 (4.84)

The number of of T tensor constraints is

Nc = 6l − 12 + δl,2,∀l ≥ 2 (4.85)

The number of Bc operators is

NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2 (4.86)

113

Therefore entanglement entropy reads

S

log 2=Ng −Nc −NB

=6l2 − 6l + 2,∀l ≥ 2

(4.87)

The entanglement entropies also have negative linear corrections.

If the region A is much larger than the region A, we conjecture that the region A

will not impose any additional constraint. In that case, the upper bound would be

saturated. The numerical calculations in App. C.2 also support this conjecture.

114

Chapter 5

Topological Entanglement Entropy

of (3+1)D Gapped Phases of

Matter

5.1 Reduction formulas for Entanglement Entropy

In this section, we study the general structure of the EE for gapped phases of matter in

(3+1)D. The definitions of the entanglement entropy and the entanglement spectrum

are reviewed in Appendix D.1. We are inspired by the fact that for a (2+1)D system,

the EE of the ground state of a local, gapped Hamiltonian obeys the area law. In

particular, if we partition our system into two subregions, A and Ac, the EE of

subregion A with the rest of the system Ac takes the form

S(A) = αl + γ +O(1/l), (5.1)

where αl is the area term, and l is the length of the boundary of region A. Importantly

the constant term −γ− is topological and thus dubbed “topological entanglement

entropy” [20, 21]. We would like to understand whether an analogous formula holds

115

for gapped phases of matter in (3+1)D. In particular, we ask how the constant part

of the EE depends on the topological properties of both the Hamiltonian and the

entanglement surface.

Our approach to this question relies on the SSA inequality for the entanglement

entropy. We also make certain locality assumptions about the form of the entropy,

detailed in Appendix D.2. This allows us to derive an expression for the constant part

of the EE of a subregion A for a TQFT, STQFTc (A), which depends on the topological

properties (e.g. Betti numbers) of the entanglement surface ∂A ≡ Σ.1

We start by reviewing some general facts about the EE and then use SSA inequal-

ities to determine the formula for the EE across a general surface in Sec. 5.1.1. In

Sec. 5.1.2, we discuss the implications of our EE formula, especially regarding models

away from a renormalization group (RG) fixed point. Our approach is inspired by

Ref. [125].

5.1.1 Strong Sub-Additivity

Structure of the EE of Fixed Point TQFTs

As reviewed in Appendix D.2, for a generic theory with an energy gap, the EE for a

subregion A can be decomposed as

S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ)

+4F ′2

∫

Σ

d2x√hH2 +O(1/|Σ|), (5.2)

where the coefficients F0, F2 and F ′2 are constants that depend on the system under

study. The first term is the area law term, where |Σ| is the area of the entangle-

ment surface, Σ. The second term is the topological entanglement entropy, which

is independent of the details of the entanglement surface and of the details of the

1In this paper, we will denote a generic entanglement surface as Σ.

116

Hamiltonian. The third term is proportional to the Euler characteristic χ(Σ) of the

entanglement surface. Although it only depends on the topology of Σ, it is not uni-

versal, and we expect that the coefficient, F2, will flow under the RG. The fourth

term is proportional to the integral of the mean curvature, H = (k1 +k2)/2, of Σ (see

Appendix D.2 for a derivation of the local contributions). It depends on the geometry

(in contrast to the topology) of Σ, and its coefficient F ′2 also flows under the RG in

general. The remaining terms are subleading in powers of the area |Σ|, and vanish

when we take the size of the entanglement surface to infinity. One of the main goals

of this paper is to understand the structure of the topological entanglement entropy,

Stopo(A), and how it can be isolated from the Euler characteristic term and the mean

curvature term.

In this section, unless otherwise stated, we consider (3+1)D TQFTs describing

the low energy physics of a gapped topologically ordered phase. In this case the

constant part of the EE depends only on the topology of the entanglement surface.

The reason is the following: since a TQFT does not depend on the spacetime metric, it

is invariant under all diffeomorphisms, including dilatations as well as area-preserving

diffeomorphisms. Hence, the term related to the mean curvature (which depends on

the shape of Σ) should not appear. This implies that the coefficient F ′2 flows to

zero at the fixed point. When we regularize the theory on the lattice, we explicitly

break the scaling symmetry while maintaining the invariance under area preserving

diffeomorphisms. Hence the area law term can survive, i.e. F0 can flow to a non-

vanishing value at the fixed point. (We relegate the explanation of this subtlety in

Sec. 5.2.2.) Since the Euler characteristic is topological, F2 can also flow to a non-

vanishing value. In summary, the possible form of the EE for a low energy TQFT

(when regularized on the lattice) is

S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ) +O(1/|Σ|). (5.3)

117

For the sake of clarity, we denote the constant part of the EE for a generic theory

as Sc(A) = Stopo(A) − 4πF2χ(Σ) + 4F ′2∫

Σd2x√hH2, and the constant part of the

EE for a TQFT as STQFTc (A) = Stopo(A) − 4πF2χ(Σ). We point out that the value

of F2 for a general theory and for a TQFT are not the same, since its value flows

under renormalization to the one in the TQFT, which will be specified in Sec. 5.1.2.

Furthermore, the area law part of the EE, F0|Σ|, is denoted as Sarea(A).

For any quantum state, there are several information inequalities relating EEs

between different subsystems that are universally valid[126], such as sub-additivity,

strong sub-additivity, the Araki-Lieb inequality[127] and weak monotonicity[128].

Special quantum states, such as quantum error correcting codes[129] and holographic

codes[130, 126, 131], obey further independent information inequalities. The major

constraint on the EE utilized in this paper is the strong sub-additivity inequality,

which is typically used in quantum information theory. Explicitly, the SSA inequality

is

S(AB) + S(BC) ≥ S(ABC) + S(B), (5.4)

where the space is divided into four regions A,B,C, and (ABC)c. Here, (ABC)c is

the complement of ABC ≡ A ∪ B ∪ C. SSA strongly constrains the structure of the

constant part of S(A), i.e., Sc(A), as we will see below.

Reduction to the Constant Part of the EE

The SSA is universal, and hence it is valid for any choice of the regions A, B and

C. Here we will only need to consider the special cases with A ∩ C = ∅. This

configuration is chosen precisely to cancel the area law part of the EE on both sides

of the SSA inequality, thus giving us information about the constant part Sc(A).

Explicitly, when A ∩ C = ∅, we have

Sarea(AB) + Sarea(BC) = Sarea(ABC) + Sarea(B). (5.5)

118

Equation (5.4) then implies

Sc(AB) + Sc(BC) ≥ Sc(ABC) + Sc(B) . (5.6)

When restricted to a TQFT, we have

STQFTc (AB) + STQFT

c (BC) ≥ STQFTc (ABC) + STQFT

c (B) . (5.7)

Structure of Sc(A)

We need to parametrize STQFTc (A) in order to proceed. For a TQFT (where F ′2 = 0),

we see that STQFTc (A) = Stopo(A) − 4πF2χ(Σ) only depends on the topology of the

entanglement surface Σ through its Euler characteristic. Two-dimensional orientable

surfaces are classified by a set of numbers {n0, n1, n2, . . .}, where ng is the number of

disconnected components (parts) with genus g.2 We will show that this is an over-

complete labeling for STQFTc (A), and that STQFT

c (A) only depends on the zeroth and

first Betti number[132] of Σ defined below in terms of {n0, n1, n2, · · · }.

For the time being, we use the (over-)complete labeling scheme for STQFTc (A)

STQFTc [(0, n0), (1, n1), · · · , (g, ng), · · · ], (5.8)

where in each bracket, the first number denotes the genus, and the second number

denotes the number of disconnected boundary components ∂A with the corresponding

genus. The list ends precisely when ng∗ 6= 0 and ng = 0 for any g > g∗. In other words,

STQFTc [(0, n0), (1, n1), . . . , (g∗, ng∗)] is the constant part of the EE of the region with

n0 genus 0 boundaries, n1 genus 1 boundaries, · · · and ng∗ genus g∗ boundaries. We

emphasize that the region A can have multiple disconnected boundary components.

The set {ng} is related to the Betti numbers bi and the Euler characteristic χ through

2In this paper, the entanglement surfaces do not wrap around non-contractible cycles of the space.

119

g∗∑

g=0

ng = b0,

g∗∑

g=0

ng(2− 2g) = 2b0 − b1 = χ. (5.9)

These numbers will be useful in the following calculations.

By applying the SSA inequality to a series of entanglement surfaces, we derive an

expression for STQFTc in terms of the Betti numbers b0 and b1, as well as the entropies

STQFTc [T 2] and STQFT

c [S2] across the torus and sphere, respectively. Relegating the

details of the derivation to Appendix D.3, we find:

STQFTc [(0, n0), (1, n1), · · · , (g, ng)]

= b0STQFTc [T 2] +

χ

2

(STQFT

c [S2]− STQFTc [T 2]

). (5.10)

Notice that Eq. (5.10) is consistent with the expectation that disconnected parts of

the entanglement surface result in additive contributions due to the local nature of

the mutual information.

5.1.2 Topological Entanglement Entropy

Our first main result is Eq. (5.10), which clarifies two points. First, as we mentioned in

the introduction (and as was also discussed in Ref. [125]), given a general entanglement

surface [(0, n0), (1, n1), ..., (g∗, ng∗)], we can reduce the computation of the constant

part of the EE of a TQFT, STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)], to that of STQFT

c [S2]

and STQFTc [T 2]. Second, using Eq. (5.10), we can identify the topological and universal

part of Sc(A) for a generic theory beyond the TQFT fixed point. We now elaborate

on these points.

STQFTc [S2] and STQFT

c [T 2]

For a TQFT, Eq. (5.10) proves that the constant part of the EE across a general

surface can be reduced to a linear combination of the constant part of the EE across

120

S2 and T 2. Whether STQFTc [S2] and STQFT

c [T 2] are independent of each other depends

on the type of TQFT. As we show in Sec. 5.2, for a BF theory [see Eq. (5.22)] in

(3+1)D, STQFTc [S2] = STQFT

c [T 2]. For the GWW models [see Eq. (5.19)] in (3+1)D,

we show in Sec. 5.2 that STQFTc [S2] and STQFT

c [T 2] are different in general. Thus,

Eq. (5.10) is the simplest expression that is universally valid for any TQFT.

Away from the Fixed Point

In Sec. 5.1.1 and Appendix D.2, we revisited the arguments presented in Ref. [125]

that the constant part of the EE for a theory away from the fixed point is generically

not topological. The structure of the EE of a generic theory was shown in Eq. (5.2).

Combining Eq. (5.2) and Eq. (5.10), we now extract more information about the

structure of the EE.

First, we argued in Sec. 5.1.1 that

F ′2 → 0, (5.11)

when the theory is renormalized to a TQFT fixed point.

Second, by setting F ′2 = 0 in Eq. (5.2) and comparing the TEE and the coefficient

of the Euler characteristic χ in Eq. (5.2) and Eq. (5.10), we find that

Stopo[(0, n0), · · · , (g∗, ng∗)]

= b0STQFTc [T 2] =

( g∗∑

i=0

ni

)STQFT

c [T 2],(5.12)

and

F2 → −1

8π

(STQFT

c [S2]− STQFTc [T 2]

). (5.13)

121

Equation (5.12) suggests that the TEE across an arbitrary entanglement surface (for

a generic theory) is proportional to STQFTc [T 2]; in particular, the TEE across T 2 (for

a generic theory) equals STQFTc [T 2], i.e., Stopo[T 2] = STQFT

c [T 2]. Equation (5.13)

shows that while F2 can flow when the theory is renormalized, it converges to a

nontrivial value − 18π

(STQFT

c [S2] − STQFTc [T 2]

)at the RG fixed point. Our iden-

tification of the TEE Eq. (5.12) further elaborates on the result from Ref. [125],

which showed that the TEE across a genus g entanglement surface Σg is Stopo[Σg] =

gStopo[T 2]− (g−1)Stopo[S2]. Our result Eq. (5.12) suggests that Stopo[S2] = Stopo[T 2]

and therefore further simplifies the result of Ref. [125] to Stopo[Σg] = Stopo[T 2] for any

g. Our identification of the TEE also works for entanglement surfaces with multiple

disconnected components.

Extracting the TEE

Equation (5.12) suggests an “algorithm” to compute the TEE for a generic theory:

1) take a ground state wavefunction |ψ〉 for a generic system; 2) renormalize |ψ〉 to

the fixed point; 3) compute the entanglement entropy for an entanglement surface

T 2, STQFT[T 2]. The constant part STQFTc [T 2] is the TEE across T 2. Notice that this

is consistent with our definition STQFTc [T 2] = Stopo[T 2]− 4πF2χ(T 2) since χ(T 2) = 0.

The TEE across an arbitrary surface immediately follows from Eq. (5.12).

In this section, we will explain a more practical algorithm for extracting the TEE

(across T 2) which is applicable to the groundstate wavefunction of any generic the-

ory, and does not require renormalization to the TQFT fixed point. Our algorithm

(which is termed the KPLW prescription) builds upon the study of the topological

entanglement entropy in (2+1)D systems initiated by Kitaev, Preskill, Levin and

Wen[20, 21](KPLW) and the proposal in Ref. [125] in (3+1)D. We compute a par-

ticular combination of the EE of different regions, which we call SKPLW[T 2], and

demonstrate that this combination equals Stopo[T 2]. The same KPLW prescription

122

was studied in Ref. [125], but here we provide a rigorous proof of the equivalence

between the entanglement entropy from the KPLW prescription Eq. (5.14) and the

TEE Stopo[T 2], as we derive in Eq. (5.17). Via Eq. (5.12), we can then obtain the

TEE across a general surface.

A CB

Figure 5.1: KPLW prescription of entanglement surface T 2. The space inside the two

torus is divided into three regions, A, B and C, each being a solid torus.

We generalize the KPLW prescription to (3+1)D by considering the configuration

of the entanglement regions shown in Fig. 5.1 and computing the combination of EEs

SKPLW[T 2] ≡ S(A) + S(B) + S(C)− S(AB)

−S(AC)− S(BC) + S(ABC). (5.14)

Following similar arguments in Ref. [20], it can be shown that SKPLW[T 2] satisfies two

properties:

1. SKPLW[T 2] is insensitive to local deformations of the entanglement surface.

2. SKPLW[T 2] is insensitive to local perturbations of the Hamiltonian.

We first argue that the property 1 holds. If we locally deform the common bound-

ary of region A and B (but away from the common boundary of region A, B and C,

123

which is a line), the deformation of SKPLW[T 2] is

∆SKPLW[T 2] = [∆S(A)−∆S(AC)]

+ [∆S(B)−∆S(BC)].

(5.15)

Because the deformation is far away from region C (farther than the correlation

length ξ ' 1/m, where m is the energy gap), ∆S(A) − ∆S(AC) = 0, and similarly

∆S(B) − ∆S(BC) = 0. Hence SKPLW[T 2] is unchanged under the deformation of

common boundary of A and B, away from the line which represents the common

boundary of A, B and C. If we now locally deform the common boundary of regions

A, B and C 3 (the line A ∩ B ∩ C),

∆SKPLW[T 2] ≡∆S(A) + ∆S(B) + ∆S(C)−∆S(AB)

−∆S(AC)−∆S(BC)

= [∆S(DBC)−∆S(BC)] + [∆S(DAC)

−∆S(AC)] + [∆S(DAB)−∆S(AB)],

(5.16)

where region D is the complement of the region ABC, i.e., D = (ABC)c, and we

have used Ac = DBC and S(A) = S(Ac). Since the deformation is far from region D

(farther than the correlation length ξ) as it is acting only on the line A∩B∩C, each of

three square brackets vanishes separately. Hence SKPLW[T 2] is unchanged under the

deformation of the common boundary line of A, B and C. In summary ∆SKPLW[T 2] =

0 under an arbitrary deformation of the entanglement surface. Therefore property 1

holds.

We now argue that property 2 holds. As suggested in Refs. [20, 21], when we

locally perturb the Hamiltonian far inside one region4, for instance region A, the

3We should distinguish between the common boundary of A, B and C, which is a line A∩B∩C,and the boundary of region ABC, which is a surface

4Quantitatively, the shortest distance d between the position of the local deformation and theentanglement surface should be much longer than the correlation length ξ ' 1/m, i.e., d� ξ.

124

finiteness of the correlation length ξ guarantees that the perturbation does not affect

the reduced density matrix for the region Ac. Therefore the entanglement entropy

S(A) = S(Ac) is unchanged. If a perturbation of the Hamiltonian occurs on the

common boundary of multiple regions, for example region A and B, one can deform

the entanglement surface using property 1 such that the perturbation is non-vanishing

in one region only. This shows that SKPLW[T 2] is invariant under local deformations

of the Hamiltonian which does not close the gap (i.e., those which leave ξ <∞), and

property 2 holds. In summary SKPLW[T 2] is a topological and universal quantity.

Lastly we show that the combination SKPLW[T 2] equals the TEE, Stopo[T 2], i.e.,

SKPLW[T 2] = Stopo[T 2], (5.17)

where Stopo[T 2] is defined in Eq. (5.12). We insert the expansion of the EE (5.2) in the

definition of SKPLW[T 2]. First, it is straightforward to check that the KPLW combina-

tion of the area law terms cancel. Second, the KPLW combination of the Euler char-

acteristic terms vanish since each region in the KPLW combination is topologically a

T 2, and χ(T 2) = 0. Third, as we prove in Appendix D.4, the KPLW combination of

the mean curvature terms vanishes as well, i.e,

4F ′2

∫∂A+∂B+∂C−∂AB−∂AC−∂BC+∂ABC

d2x√hH2 = 0. (5.18)

This was assumed implicitly in Ref. [125], but we demonstrate it explicitly here so

as to close the loop in the argument.

Finally, the KPLW combination simplifies to Stopo[T 2]: it is given by the sum

of the TEE across the four tori ∂A, ∂B, ∂C and ∂ABC, minus the TEE across the

three tori ∂AB, ∂AC and ∂BC. Therefore, Eq. (5.17) holds. In summary, we have

demonstrated that the KPLW prescription, Eq. (5.14), gives a concrete method to

extract the TEE for a generic (non-fixed-point) theory.

125

5.2 Application: Entanglement Entropy of Gener-

alized Walker-Wang Theories

In this section, we construct lattice ground state wave functions for a class of TQFTs

known as the generalized Walker-Wang (GWW) models, whose actions are given by

Eq. (5.19) below. We then compute the EE across various two dimensional entangle-

ment surfaces. The calculations in this section are independent of the SSA inequality

used in Sec. 5.1. The calculations in this section provide support for our assumptions

about the entanglement entropy for fixed-point models, and suggest a conjecture

about higher dimensional topological phases.

The GWW models are described by a TQFT with the action[133, 134, 135]

SGWW =

∫n

2πB ∧ dA+

np

4πB ∧B, n, p ∈ Z. (5.19)

The Walker-Wang models correspond to the special cases p = 0 and p = 1. In

Eq. (5.19) B is a 2-form U(1) gauge field and A is a 1-form U(1) gauge field. (When

we formulate the theory on a lattice, they will be Zn valued. See Appendix D.5 for

details.) The gauge transformations of the gauge fields are

A→ A+ dg − pλ,

B → B + dλ,

(5.20)

where λ is a u(1) valued 1-form gauge field (where u(1) is the Lie algebra of U(1))

with gauge transformation λ → λ + df (where f is a scalar satisfying f ' f + 2π),

and g is a compact scalar (i.e., g ' g + 2π). The gauge invariant surface and line

126

operators are respectively

exp(ik

∮

Σ1

B), k ∈ {0, 1, ..., n− 1},

exp(il

∮

γ

A+ ilp

∫

Σ2

B), l ∈ {0, 1, ..., n− 1},

(5.21)

where Σ1 is a closed two dimensional surface, γ is a closed one dimensional loop and Σ2

is an open two dimensional surface whose boundary is γ. The gauge invariance follows

from the compactification of the scalar g and the standard Dirac flux quantization

condition of U(1) gauge field λ:∮γdg ∈ 2πZ and

∮Σ1dλ ∈ 2πZ.5 We will use

canonical quantization to explain that exp(in∮

Σ1B) and exp(in

∮γA+ inp

∫Σ2B) are

trivial operators in App. D.5.

5.2.1 Wave Function of GWW Models

BF Theory: (n, 0)

For simplicity, we first discuss the special case when p = 0, which is referred to as a

BF theory. The action is

SBF =

∫

M4

n

2πB ∧ dA, (5.22)

where A is a 1-form gauge field and B is a 2-form gauge field. The theory is defined

on a spacetime which is topologically a four ball, M4 ' B4, whose boundary S3 is

a spatial slice, as shown in Fig. 5.2. In the following, we formulate the theory on a

triangulated spacetime lattice. The 1-form gauge field A corresponds to 1-cochains

A(ij) ∈ 2πnZn living on 1-simplices (ij). The 2-form gauge field B corresponds to

5The Dirac flux quantization of the U(1) gauge field λ can be derived as follows:∮

Σ1dλ =∫

Σ+1dλ+ −

∫Σ−1

dλ− =∫∂Σ+

1λ+ −

∫∂Σ−1

λ−, where Σ+1 ∪ Σ−1 = Σ1 and the minus sign of the Σ−1

term is due to orientation. We use λ+ and λ− to emphasis that the gauge field are evaluated in Σ+1

and Σ−1 respectively. The U(1) gauge symmetry implies that λ+ − λ− on the common boundary∂Σ+

1 = ∂Σ−1 = Σ+1 ∩ Σ−1 does not have to vanish, but it can be a pure gauge df . Therefore,∮

Σ1dλ =

∮Σ+

1 ∩Σ−1df ∈ 2πZ. This proves the Dirac flux quantization.

127

M4

S3

l

S

S0

Figure 5.2: A schematic figure of the topology of spacetimeM4 and space S3. InsideS3, we schematically draw a loop l representing the loop configurations C of the Bfield in the dual lattice. The dashed surface S bounding the loop l extends into thespacetime bulkM4, representing the B field configuration in the dual lattice of space-time. S ′ represents the B field configurations that form closed surfaces away fromthe boundary of the spacetime ∂M4. The boundary condition in the path integralEq. (5.23) is specified by a fixed B configuration C on S3. The path integral shouldintegrate over all the configurations in the spacetime bulk M4 with the boundaryconfiguration C on S3 fixed.

2-cochains B(ijk) ∈ 2πnZn living on 2-simplices (ijk)6. We define the Hilbert space

to be H = ⊗(ijk)H(ijk), where H(ijk) is a local Hilbert space on the 2-simplex (ijk)

spanned by the basis |B(ijk)〉 = |2πq/n〉, q ∈ Zn.7 More details about the lattice

formulation of the TQFT are given in Appendix D.5.

We now discuss the ground state wave function for this theory. The ground state

wave function is defined on the boundary of the open spacetime manifold S3 = ∂M4

as[136, 137]

|ψ〉 = C∑

C,C′

∫

C′|∂M4

DA∫

C|∂M4

DB exp(in

2π

∫

M4

B ∧ dA)|C〉, (5.23)

6We use i, j, k to label vertices, and (ij), (ijk) to label 1-simplices and 2-simplices with thespecified vertices.

7Note that the Hilbert space on each 1-simplex is defined independently, and does not have tosatisfy the closed loop (Gauss law) constraint Eq. (5.25).

128

where C ′ and C indicate the boundary configurations for the A and B fields respec-

tively, i.e., the value of A and B fields on ∂M4. We integrate over all A and B

subject to the boundary conditions C ′ and C. C is a normalization factor. Because

A and B are canonically conjugate, the states are specified by the configuration of B

only; |C〉 is a specific state corresponding to the particular B field configuration C on

∂M4. The summation over C ranges over all possible configurations of B-cochain with

weights determined by the path integral. C|∂M4 means the path integral is subject

to the fixed boundary conditions C on ∂M4, and similarly for C ′|∂M4 . If we take the

spacetime M4 to be a closed manifold, Eq. (5.23) reduces to the partition function

overM4. Because the spacetime is topologically a 4-ball B4, there is only one ground

state associated with the boundary S3.8

We first work out the wavefunction for the BF theory with n = 2 explicitly as a

generalizable example. We use B field values as a basis to express |C〉. Integrating

out A (notice that we both integrate over the configurations of the A-field with

fixed boundary configurations and also sum over the boundary configurations, i.e.,∑C′∫C′|∂M4

DA, which is tantamount to integrating over all configurations of A), we

get the constraint δ(dB),

|ψ〉 = C∑

C

∫

C|∂M4

DBδ(dB)|C〉. (5.24)

where the delta function δ(dB) constrains dB(ijkl) = 0 mod 2π on each tetrahedron

(ijkl) in M4. Concretely,

dB(ijkl) = B(jkl)−B(ikl) +B(ijl)−B(ijk)

= 0 mod 2π.

(5.25)

8Topologically degenerate ground states are the representation of line and surface operators whichwrap around the nontrivial spatial cycles. Since there are no nontrivial 1-cycles and 2-cycles in thespatial manifold S3 that line and surface operators can wrap around, the ground state is topologicallynon-degenerate.

129

Any B configuration satisfying this constraint is said to be flat (see Appendix D.5

for details). Since B(ijk) ∈ {0, π},∀i, j, k for the n = 2 theory, Eq. (5.25) means

that for each tetrahedron, there are an even number of 2-simplices where B(ijk) = π

mod 2π, and an even number of 2-simplices with B(ijk) = 0 mod 2π. We refer to

the π 2-simplices as occupied and to the 0 2-simplices as unoccupied.

i

j l

k

d

a

e

b

c

Figure 5.3: A tetrahedron is drawn with solid lines, and its dual is drawn in dash

and gray lines. The 2-simplex (ijk) in the original lattice is dual to the 1-simplex

(ab) in the dual lattice. Similarily, (ikl) is dual to (ad), (ijl) is dual to (ca) and

(jkl) is dual to (ea). The colored dash arrows indicate the orientations of the four

2-simplices, where (ijk) and (ikl) share the same orientation, and (ijl) and (jkl)

share the opposite orientation. The orientations of the dual-lattice 1-simplices are

also indicated by the arrows on the grey/dashed lines.

It is more transparent to consider the configurations in the dual lattice of the

spatial slice S3. (In the next paragraph, we will discuss the dual lattice configurations

in the spacetime M4.) As an example, the dual lattice of a tetrahedron is shown in

Fig. 5.3. The 2-simplices in the original lattice are mapped to 1-simplices in the dual

lattice.9 A 2-cochain B(ijk) defined on a 2-simplex in the original lattice is mapped to

9The dual lattice of a triangulation is not necessarily a triangulation. For example, the dual latticeof a triangular lattice in two dimensions is a honeycomb lattice. Therefore, it is inappropriate totalk about cochains and simplices in the dual lattice of a triangulation. However, we will still use

130

a 1-cochain B(ab) defined on an 1-simplex in the dual lattice. If B(ijk) = π, then we

define the corresponding B(ab) = π in the dual lattice. In the dual lattice, Eq. (5.25)

means that there are an even number of occupied bonds (1-simplices) associated with

each vertex, as well as an even number of unoccupied bonds. If we glue different

tetrahedra together, we find that the occupied bonds in the dual lattice form loops.

Pictorially, this is reminiscent of the wave function of the toric code model in one

lower dimension[138, 22, 139].

In the (3 + 1)D spacetime M4 [rather than the 3D space S3], 2-simplices are

dual to the (4 − 2) = 2-simplices [rather than the 1-simplices] in the dual lattice.

Equation (5.25) means the occupied 2-simplices form continuous surfaces in the dual

spacetime lattice. (Continuous means that the simplices in the dual lattice connect via

edges, rather than via vertices. We discuss the continuity of the dual lattice surfaces

in Appendix D.6.) If these surfaces are inside the bulk of the spacetime and do not

touch ∂M4 (such as S ′ in Fig. 5.2), they are continuous and closed surfaces; if the

surfaces intersect with the spatial slice ∂M4 (such as S in Fig. 5.2), the intersections

are closed loops in ∂M4.

For the BF theory with a general coefficient n, the wavefunction is also a su-

perposition of loop configurations. The only difference is that the loops are formed

by 1-simplices in the dual lattice with B = 2πn

. When there is a loop formed by

1-simplices with B = 2πln

in the dual lattice, we regard the loop as composed of l

overlapping loops formed by the same 1-simplices with B = 2πn

. We emphasize that

the loop configuration is enforced by the flatness condition Eq. (5.25). For n > 2,

we need to specify the orientations of the simplices and keep tract of the signs in

Eq. (5.25). The orientation of each simplex is specified in Fig. 5.3, where the ori-

entations of (jkl) and (ijl) are pointing into the tetrahedron, while the orientation

of (ikl) and (ijk) are pointing out of the tetrahedron. For example, if the values

such notions for simplicity as long as the context is clear. In the dual lattice, we use “1-simplex” todenote a link, and “1-cochain” to denote a discretized 1-form on the link.

131

of the B-cochains are B = 2πq1/n, 2πq2/n, 2πq3/n, 0 with q1 − q2 + q3 = 0 on the

2-simplices (jkl), (ikl), (ijl), (ijk) respectively, the dual of (jkl) and (ikl) (i.e., (ea)

and (ad)) belong to one loop in the dual lattice, while the dual of (ijl) and (ikl)

(i.e., (ca) and (ad)) belong to another loop in the dual lattice. Note that the two

loops share the same dual lattice bond (ad) where the value of the B-cochain is the

sum of the B values from the two loops B(ad) = 2π(q1 + q3)/n = 2πq2/n. The gauge

transformation, B(ijk)→ B(ijk)+λ(jk)−λ(ik)+λ(ij), preserves Eq. (5.25). Hence,

although it deforms the position of loops, it never turns closed loops into open lines.

Open lines in the dual lattice violate the flatness condition Eq. (5.25), and so do

not contribute to the wave function Eq. (5.24). Summing over the configurations C

ensures gauge invariance of the wave function. Notice that Eq. (5.24) implies that

the weights associated with different loop configurations C are equal, similar to the

toric code. Thus we see that Eq. (5.24) reduces to

|ψ〉 = C∑

C∈L

|C〉, (5.26)

where the sum is taken over the set L of all possible loop configurations C at the

spatial slice S3 = ∂M4. This is termed “loop condensation”, since the wave function

is the equal weight superposition of all loop configurations in the dual lattice.

General Case: (n, p)

In this section, we consider GWW models with nontrivial p described by the action

in Eq. (5.19), where A is still a 1-form and B a 2-form. Canonical quantization of

the GWW theories implies that B ∈ 2πnZn on the lattice (see Appendix D.5 for more

details).

In order to find the ground state wave function, we still use B as the basis to label

the configurations C and the corresponding states |C〉 on the spatial slice. The wave

132

function is formally given by

|ψ〉 =C∑

C,C′

∫

C′|∂M4

DA∫

C|∂M4

DB exp(in

2π

∫

M4

B ∧ dA+ inp

4π

∫

M4

B ∧B)|C〉. (5.27)

For simplicity, we consider the case n = 2, p = 1 in the following. As in the BF

theory, we first integrate out the A fields, yielding

|ψ〉 = C∑

C

∫

C|∂M4

DB δ(dB)

exp(i

2

4π

∫

M4

B ∧B)|C〉. (5.28)

The difference between this wave function and that of the BF theory, Eq. (5.24), is that

when the flatness condition δ(dB) is satisfied, the states with different configurations

C are associated with different weights. The weights are determined by the integral

exp(i

2

4π

∫

M4

B ∧B), (5.29)

where B must satisfy the flatness condition dB = 0 with the boundary condition

labeled by C.

We proceed to evaluate the integral in Eq. (5.29). Notice that the flatness condi-

tion, Eq. (5.25), implies that the 2-simplices at which B = π form two-dimensional

spacetime surfaces in the dual lattice of M4 whose boundaries on the spatial slice

S3 are closed loops belonging to C. Relegating the details of the derivation to Ap-

pendix D.7, we show that when B = π only at two dual lattice surfaces S1, S2, whose

boundaries are dual lattice loops l1 = ∂S1, l2 = ∂S2 in C, it follows that

exp(i

2

4π

∫

M4

B ∧B)

= exp(iπlink(l1, l2) + i

π

2link(l1, l1) + i

π

2link(l2, l2)

). (5.30)

The first term is associated with the mutual linking number, link(l1, l2), between dif-

ferent loops, while the second and the third terms are associated with the self-linking

133

number, link(li, li), of one loop, li, with itself, defined in Appendix D.7. Equation

(5.30) can be generalized to configurations with many loops, and the weights of differ-

ent configurations are determined by the linking numbers of the loops. In summary,

the ground state wave function for the (n, p) = (2, 1) theory is:

|ψ〉 = C∑

C∈L

(−1)#(Mutual links)i#(Self links)|C〉. (5.31)

For general (n, p), a similar argument can be made. B can now take n different

values 2πkn, k = 0, 1, · · · , n−1 on each 2-simplex in the lattice, or on each 1-simplex in

the dual lattice. Due to the constraint of Eq. (5.25), the 1-simplices where B = 2π/n

form loops in the dual lattice. Similar to the discussion of the case p = 0 and

general n, two dual-lattice loops can touch in one tetrahedron. We also regard a

loop with B = 2πq/n to be q overlapping loops with B = 2π/n. If there are q1

loops with B = 2π/n that are overlapping on l1 (which is equivalent to one loop with

B = 2πq1/n on l1) and q2 loops with B = 2π/n that are overlapping on l2 (which is

equivalent to one loop with B = 2πq2/n on l2), then

exp(inp

4π

∫

M4

B ∧B)

= exp[2inp(2π)2q1q2

4πn2link(l1, l2) + i

np(2π)2q21

4πn2link(l1, l1) + i

np(2π)2q22

4πn2link(l2, l2)

]

= exp[i2πpq1q2

nlink(l1, l2) + i

πpq21

nlink(l1, l1) + i

πpq22

nlink(l2, l2)

].

(5.32)

Therefore after evaluating these weights, the wave function Eq. (5.27) reduces to

|ψ〉 = C∑

C∈L

ei2πpn

#(Mutual links)eiπpn

#(Self links)|C〉, (5.33)

134

where the mutual-linking and self-linking numbers are counted with multiplicities q1

and q2 as given in Eq. (5.32). The sum over C ∈ L contains configurations with all

possible q1 and q2.

5.2.2 Entanglement Entropy of GWW Models

In this section, we show that the constant part of the EE of GWW theories depends on

the topology of the entanglement surface in a nontrivial way. In particular, Sc[S2] 6=

Sc[T2] in general. Hence, Sc[S

2] and Sc[T2] are truly independent quantities.

This section is divided into two parts: In Sec. 5.2.2, we calculate the EE for GWW

models with arbitrary (n, p) across the entanglement surface T 2. In Sec. 5.2.2, we

compute the EE for GWW models across closed surfaces with arbitrary genus and

an arbitrary number of disconnected components. These independent calculations

confirm Eq. (5.10).

EE for the Torus, n = 2, p = 1

In this subsection, we compute the EE of GWW models across Σ = T 2. For simplicity,

we first consider the case n = 2, p = 1, and then generalize to models with arbitrary

n and p.

We start with the wave function obtained in the last section, Eq. (5.31):

|ψ〉 = C∑

C

(−1)#(Mutual links)i#(Self links)|C〉. (5.34)

We choose the subregion A to be a solid torus whose surface is T 2, and Ac to be the

complement of A. We illustrate the microscopic structure of the spatial partitioning

in Fig. 5.4 via a lower-dimensional example. The entanglement surface Σ is chosen to

be a smooth surface in the real spatial lattice (green simplices in Fig. 5.4). The real

space simplices that form the entanglement surface Σ are counted as part of region

135

A

⌃

B=π simplices on the entanglement surface ∑

Ac

Figure 5.4: An example of the lattice structure of an entanglement cut in (2 + 1)D.The green simplices form the entanglement cut Σ, which partitions the lattice intoregion A and region Ac. We include Σ as part of region A. B = π on the red simplices,while B = 0 elsewhere. The dotted loop is the dual lattice configuration of the redsimplices. In this example, the configuration CE contains two B = π 1-simplices atthe entanglement cut Σ, which are the fourth and eighth 1-simplices of Σ (countingfrom the left side) as shown in the figure.

A. 10 We will find the Schmidt decomposition of the wavefunction corresponding to

this spatial partitioning in order to calculate the EE. To do so, we first parametrize

the configurations C appearing in Eq. (5.34) as:

C 7→ {CE, (a, α), (b, β)}, (5.35)

which we now explain. CE labels the real space B-cochain configuration at the entan-

glement surface Σ. (In Fig. 5.4, the fourth and the eighth green 1-simplices (counting

from the left side) are occupied on the entanglement surface Σ, which also belong

to region A according to our partition.) We denote by NA(CE) the number of con-

figurations in the region A (but not including Σ) consistent with the choice of CE.

We label such configurations by (a, α), where α is the parity (even e or odd o) of

the number of occupied loops winding around the nontrivial spatial cycle inside the

10There are other choices of spatial partitioning. For example, we can count the real simplices thatform the entanglement surface as part of region Ac. We will consider only consider the partitioningmentioned in the main text for definiteness.

136

region A in the dual lattice, and the configurations of either parity are enumerated by

a = 1, . . . , NA(CE)/2.11 Similarly, (b, β) labels the NAc(CE) configurations in region

Ac. Figure 5.5 presents a particular configuration where, besides two contractible

dual-lattice loops, there is one dual lattice loop wrapping the non-contractible cycle

in the dual lattice of region A and one dual lattice loop wrapping the non-contractible

cycle in the dual lattice of region Ac, which corresponds to α = o and β = o. Note

that two non-contractible cycles are in different regions A and Ac. To be illustrative,

we also draw 2-simplices in the real lattice where B = π whose dual configurations

form loops in the space. Hence the summation over C splits as:

∑

C

=∑

CE

NA(CE)/2∑

a=1

NAc (CE)/2∑

b=1

∑

α=e,o

∑

β=e,o

. (5.36)

11We can establish a one-to-one correspondence between the configurations of loops in the evenparity sector and the odd parity sector. If we start with a configuration in the even parity sectorin which k dual lattice loops wrap around the non-contractible cycle in region A, we can obtain aconfiguration in the odd parity sector by adding a single loop wrapping the non-contractible cycle sothat there are (k+1) non-contractible dual lattice loops in total. Similarly, we can start with the oddparity sector and obtain the even parity sector. This demonstrates that the number of configurationsin the even parity sector is equal to that of the odd parity sector. Therefore, we denote the numberof configurations in both sectors by NA(CE)/2. This argument can be generalized to the case ofgeneral n.

137

A

Ac

�4

�3 �2

�1

Figure 5.5: A particular spatial configuration with one loop γ1 (dashed line) threading

through the hole (the hole itself belongs to region Ac) inside the region A and one

loop γ2 (grey line) threading through the hole inside the region Ac. γ3 and γ4 are

two linked contractible loops, where γ3 locates inside region A, and γ4 locates both in

region A and Ac. The two blue points are the intersection of l4 with Σ. The simplices

(gray triangles) are living in the real lattice where B = π. The lines perpendicular to

the simplices are living in the dual lattice where B = π and they form loops in the

dual lattice. This configuration corresponds to α = o, β = o.

For convenience we also introduce the notation

lCEa,e =(−1)#(Mutual links with fixed CE configuration of region A in even sector),

lCEa,o =(−1)#(Mutual links with fixed CE configuration of region A in odd sector),

sCEa,e =i#(Self links with fixed CE configuration of region A in even sector),

sCEa,o =i#(Self links with fixed CE configuration of region A in odd sector),

(5.37)

where even/odd sector refers to the set of states with an even/odd number of loops in

the dual lattice threading the non-contractible cycle in region A. Similar definitions

apply to region Ac. See Fig. 5.5 for an illustration. We further define |ACEa 〉α to

be a state associated with one particular configuration in region A, which is labeled

138

by {CE, a, α}, and define |AcCEb 〉β likewise in region Ac. There is a subtlety: we

also need to specify the mutual-linking/self-linking number of loops which cross the

entanglement surface. We specify that when two loops (among which at least one of

them crosses the entanglement surface) are linked, such as γ3 and γ4 in Fig. 5.5, the

mutual-linking number is counted as part of the A side, i.e., lCEa,e and lCEa,o. Additionally,

when a loop crosses the entanglement surface, the self-linking number of the loop

is counted as part of the A side, i.e., sCEa,e and sCEa,o. We are able to make such a

choice because there is a phase ambiguity in the Schmidt decomposition, and phases

can be shuffled between A and Ac by redefining the basis |ACEa 〉e/o and |AcCEb 〉e/o.

(For example, we can define another set of states via |ACEa 〉e/o = sCE−1a,e/o |ACEa 〉e/o, and

|AcCEb 〉e/o = sCEa,e/o|Ac

CEb 〉e/o.) As we will see, the reduced density matrix Eq. (5.39)

does not depend on the choice of phase assignment. Combining the above, we get

|ψ〉 =C∑

CE

NA(CE)/2∑

a=1

NAc (CE)/2∑

b=1

∑

α=e/o

∑

β=e/o

(−1)αβlCEa,αlCEb,βs

CEa,αs

CEb,β|ACEa 〉α|Ac

CEb 〉β. (5.38)

The factor (−1)αβ, which equals −1 when α = β = o and 1 otherwise, reflects the

mutual-linking between the non-contractible loops in region A (such as γ1 in Fig. 5.5)

and the non-contractible loops in region Ac (such as γ2 in Fig. 5.5). Figure 5.5 shows

a special configuration where there is one non-contractible loop in region A and one

non-contractible loop in region Ac.

From this we easily obtain the reduced density matrix for region A by tracing over

the Hilbert space in region Ac,

ρA =|C|2∑

CE

NAc(CE)

2

NA(CE)/2∑

a,a=1

∑

α,α,γ=e,o

(−1)(α−α)γ|ACEa 〉α〈ACEa |α

=|C|2∑

CE

NAc(CE)

NA(CE)/2∑

a,a=1

(|ACEa 〉e〈ACEa |e + |ACEa 〉o〈ACEa |o

),

(5.39)

139

where we have performed unitary transformations on the bases |ACEa 〉e/o and |AcCEb 〉e/o

to absorb the mutual-linking and self-linking factors within region A and region

Ac respectively. The transformed bases are denoted |ACEa 〉α = lCEa,αsCEa,α|ACEa 〉α and

|AcCEb 〉β = lCEb,βs

CEb,β|Ac

CEb 〉β.

Furthermore, the constraint

TrHA(ρA) = |C|2

∑

CE

NAc(CE)NA(CE) = 1 (5.40)

fixes the normalization constant C. For each fixed configuration CE on the entangle-

ment surface, the product of the number of configurations in the region A and the

number of configurations in region Ac, i.e., NAc(CE)NA(CE), is independent of CE (see

Appendix D.8 for details). Thus, to compute C we need only to count the number

of different choices of CE. There are in total 2|Σ|−1 different boundary configurations,

where the 1 comes from the constraint that closed dual lattice loops always inter-

sect the entanglement surface twice (hence the number of occupied 1-simplices on Σ

is even), and |Σ| is the number of 2-simplices on the entanglement surface. Since

|C|2NAc(CE)NA(CE) is independent of CE, and there are 2|Σ|−1 choices of CE,

|C|2NAc(CE)NA(CE) =1

2|Σ|−1. (5.41)

We give a more detailed derivation of this formula in Appendix D.8.

From the reduced density matrix ρA, we can calculate the entanglement entropy

of the ground state |ψ〉 associated with the torus entanglement surface by the replica

trick,

S(A) = −TrHAρA log ρA = − d

dN

(TrHA

ρNA(TrHA

ρA)N

)∣∣∣∣N=1

(5.42)

140

Using Eq. (5.39),

TrHAρNA

= |C|2N∑

CE0

NA0/2∑

a0=1

∑

α0=e,o

〈ACE0a0 |α0

N∏

I=1

(∑

CEI

NAI/2∑

aI ,aI=1

∑

αI=e,o

NAc(CEI )|ACEIaI 〉αI 〈A

CEIaI|αI)|ACE0

a0 〉α0

= |C|2N∑

CE0,a0,α0

N∏

I=1

( ∑

CEI ,aI ,aI ,αI

NAc(CEI )

)δCE0

CE1δCE1

CE2· · · δCEN CE0

× δa0a1δa1a2δa2a3 · · · δaN−1aN δaNa0

× δα0α1δα1α2 · · · δαNα0

= |C|2N∑

CE0

NAc(CE0)N∑

α0=o,e

NA(CE0/2)∑

a1=1

· · ·NA(CE0

/2)∑

aN=1

1

= |C|2N∑

CE0

2NAc(CE0)N(NA(CE0)

2

)N= 2−|Σ|(N−1).

(5.43)

In the first equation, we expand the trace over the Hilbert space in region A. In the

second equation, we use the orthogonal condition 〈ACEa |α|AC′Ea′ 〉α′ = δCEC′Eδaa′δαα′ . In

the third equation, we simplify the formula using the delta functions CE0 = CE1 =

· · · = CEN , α0 = α1 = · · · = αN , and eliminate {a0, aI} by {aI}. In the last equation,

we used Eq. (5.41). Moreover, notice that TrHAρA = 1, we obtain the entanglement

entropy

S(A) = − d

dN2−|Σ|(N−1)|N=1 = |Σ| log 2. (5.44)

Since |Σ| is the number of 2-simplices on Σ, which is proportional to the area of

Σ, hence it is the area law term. Since there is no constant term, the topological

entanglement entropy is trivial, reflecting the absence of topological order in this

model.

141

EE for the Torus: general (n, p)

We carry out the analogous calculations for a general GWW theory with arbitrary

coefficients n and p. We start by writing down the ground state wave function,

|ψ〉 =C∑

CE

NA(CE)/n∑

a=1

NAc (CE)/n∑

b=1

n−1∑

α,β=0

ei2πpαβn lCEa,αl

CEb,βs

CEa,αs

CEb,β|ACEa 〉α|Ac

CEb 〉β, (5.45)

where lCEa,α, lCEb,β, s

CEa,α, s

CEb,β are straightforward generalizations of Eq. (5.37) to the cases

with arbitrary coefficients p and n, c.f. Eq. (5.33). The reduced density matrix is

ρA = |C|2∑

CE

NAc(CE)

n

NA(CE)/n∑

a,a=1

n−1∑

α,α,γ=0

ei2πp(α−α)γ

n |ACEa 〉α〈ACEa |α, (5.46)

where we again performed the unitary transformations to absorb the self-linking and

mutual-linking factors, and denote the resulting new basis as |ACEa 〉α and |AcCEb 〉β.

For the same reason as in Eq. (5.41),

|C|2NAc(CE)NA(CE) =1

n|Σ|−1, (5.47)

where |Σ| is the number of 2-simplices on the entanglement surface.

In order to compute the entanglement entropy

SA = −TrHAρA log ρA, (5.48)

we first calculate the entanglement spectrum, i.e., we diagonalize ρA. As a first step,

we carry out the sum over γ in Eq. (5.46). We note that the sum is nonvanishing

only if p(α− α)/n is an integer, in which case the sum takes the value n. Thus,

n−1∑

γ=0

ei2πp(α−α)γ

n = n δ

(α− α = 0 mod

n

gcd(n, p)

). (5.49)

142

We find

ρA = |C|2∑

CE

NAc(CE)

NA(CE)/n∑

a,a=1

n−1∑

α,α

δ

(α− α = 0 mod

n

gcd(n, p)

)|ACEa 〉α〈ACEa |α(5.50a)

=∑

CE,a,α,a,α

[ρCEA

]a,α;aα

|ACEa 〉α〈ACEa |α, (5.50b)

where[ρCEA

]a,α;aα

are matrix elements given by

[ρCEA

]a,α;aα

= |C|2NAc(CE)[1 n

gcd(n,p)⊗ Jgcd(n,p)

]αα⊗[JNA(CE)

n

]aa. (5.50c)

Here, 1m is the m ×m identity matrix, and Jl is an l × l matrix of ones (which has

one nonzero eigenvalue equal to l). The first term in this expression originates from

the periodic delta function in Eq. (5.50a), and the second term comes from the sum

over a, a in the outer product. Noting that each Jm is a rank one matrix with nonzero

eigenvalue m, we see immediately that ρCEA can be put in diagonal form

ρCEA = |C|2NAc(CE)NA(CE)

ngcd(n, p)(1 n

gcd(n,p)⊕ 0NA(CE)−n/gcd(n,p)). (5.51)

The matrix in Eq. (5.51) is

1

1

. . .

1

0

0

. . .

0

0

0

ngcd(n,p) 1’s

NA (C

E )−ngcd(n,p) 0’s

(5.52)

143

Finally, using Eq. (5.47), we find that the nonzero entanglement eigenvalues are

given by

e−ξCE,r =gcd(n, p)

n|Σ|, (5.53)

where r = 1, · · · , n|Σ|/gcd(n, p). With this spectrum, it is straightforward to evaluate

Eq. (5.48) to obtain the entanglement entropy as

S(A) = |Σ| log n− log gcd(n, p). (5.54)

The first term is proportional to the area of the entanglement surface. The second

constant term is the TEE 12:

STQFTc (A) = Stopo(A) = − log gcd(n, p). (5.55)

We see that the TEE depends nontrivially on the parameters n and p. If n and p

are coprime, i.e., gcd(n, p) = 1, the TEE vanishes. If p = 0, using the definition

gcd(n, 0) = n, the constant part of the EE reduces to − log n. Alternatively, we can

also compute the EE of the BF theory using the wave function Eq. (5.26), and we

find the constant part to be − log n.

Note that this result is consistent with Refs. [134] and [140] where the ground

state degeneracy (GSD) on T 3 was computed to be gcd(n, p)3. The ground state

degeneracy suggests that the GWW models can be topologically ordered, which, in

our context, is reflected by the nonzero TEE, − log gcd(n, p). When gcd(n, p) = 1,

the ground state on T 3 is non-degenerate, and the TEE vanishes. In particular, for

the case of the Walker-Wang model n = 2, p = 1, we obtain

S(A) = |Σ| log 2, (5.56)

12The constant part of the EE is STQFTc (A) = − log gcd(n, p). According to the discussion in

Sec. 5.1, because the entanglement surface is T 2, whose Euler characteristic vanishes, Stopo(A) ≡Stopo[T 2] = STQFT

c (A) = − log gcd(n, p).

144

and there is no topological order. We notice the relation between the GSD on T 3 and

the TEE across the torus T 2,

exp(−3Stopo[T 2]) = GSD[T 3], (5.57)

which should be compared to the similar relation, exp(−2Stopo[T 1]) = GSD[T 2], for

the (2+1)D Abelian theories.

For an Abelian theory in (d+ 1)D, our computation leads us to conjecture that

exp(−dStopo[T d−1]) = GSD[T d]. (5.58)

For (d + 1)D BF theory with level n, we have computed both the TEE and the

GSD[T d], and we found Stopo[T d−1] = − log n and GSD = nd. This is consistent

with our conjecture. (See Appendix D.9 for details.) We conjecture that this re-

lationship is true for more general theories such as Dijkgraaf-Witten models, and

higher dimensional Chern-Simons theories as well. For a generic (2 + 1) dimensional

nonabelian Chern-Simons theory, Eq. (5.58) may not hold. For example, the TEE

of the SU(2)3 Chern-Simons theory is Stopo[T 1] = − log(√

5/(2 sin(π/5)))[141], and

exp(−2Stopo[T 1]) is not an integer. Hence Eq. (5.58) can not hold because the GSD

should be an integer. However, we note that for some nonabelian theories, the con-

jecture still holds. For example, for the bosonic Moore-Read quantum Hall state in

(2 + 1)D, GSD[T 2] = 4 (which consists of 3 states from the even parity sector and 1

state from the odd parity sector), and Stopo[T 1] = − log 2, hence Eq. (5.58) holds in

this case.

EE for Arbitrary Genus

Following the same procedure used for the torus, we calculate the EE across a general

entanglement surface with genus g. (The results are summarized in Table 5.1.) For

145

S2 T 2 [(0, n0), · · · , (g∗, ng∗)]n2πBF

STQFTc − log n − log n −b0 log nStopo − log n − log n −b0 log n

n2πBF + np

4πBB

STQFTc − log n − log gcd(n, p) (−b0 + χ

2) log gcd(n, p)− χ

2log n

Stopo − log gcd(n, p) − log gcd(n, p) −b0 log gcd(n, p)

Table 5.1: Constant part and topological part of the entanglement entropy for gen-eralized Walker-Wang models. STQFT

c is the constant part of the EE for the TQFT,while Stopo is the TEE for a general theory which belongs to the same phase of

the TQFT. b0 is the zeroth Betti number of entanglement surface b0 =∑g∗

g=0 ng.

χ =∑g∗

g=0(2 − 2g)ng is the Euler characteristic of the entanglement surface. In

particular, we have Stopo(S2) = Stopo(T 2).

each hole i (i = 1, · · · , g) of the entanglement surface, we introduce a pair of additional

indices αi and βi that count the number of loops (modulo n) winding around the non-

contractible cycles around the hole in region A and region Ac, respectively. Then the

wavefunction is

|ψ〉 =C∑

CE

NA(CE)

ng∑

a=1

NAc(CE)

ng∑

b=1

n−1∑

α1···αg=0

n−1∑

β1···βg=0

g∏

i=1

ei2πpαiβi

n |ACEa 〉α|AcCEb 〉β.

(5.59)

We collect the set of indices α1, · · · , αg into a index vector α. We first consider the

configurations in region A. Since each hole is associated with an index αi, which can

take n different values, the complete set of indices α can take ng different values.

Hence, the NA(CE) configurations are partitioned into ng classes, where each class

contains NA(CE)/ng configurations. For this reason the summation in Eq. (5.59)

reaches only up to NA(CE)/ng. For region Ac, similar arguments hold. Then the

146

reduced density matrix on a genus g surface takes the form

ρA =|C|2∑

CE

NAc(CE)

ng

n−1∑

α1,··· ,αg=0

n−1∑

α1,··· ,αg=0

n−1∑

γ1,··· ,γg=0

NA(CE)/ng∑

a,a=1

g∏

i=1

ei2πp(αi−αi)γi

n |ACEa 〉α〈ACEa |α

=|C|2∑

CE

NAc(CE)n−1∑

α1,··· ,αg=0

n−1∑

α1,··· ,αg=0

NA(CE)/ng∑

a,a=1

g∏

i=1

δ

(αi − αi = 0 mod

n

gcd(n, p)

)|ACEa 〉α〈ACEa |α

=∑

CE

∑

α,α

NA(CE)/ng∑

a,a=1

[ρCEA

]

aα,aα

|ACEa 〉α〈ACEa |α,

(5.60)

where

[ρCEA

]

aα,aα

=|C|2NAc(CE)

g⊗

i=1

[1 n

gcd(n,p)⊗ Jgcd(n,p)

]

αiαi

⊗[JNA(CE)

ng

]

aa

=|C|2NAc(CE) gcd(n, p)gNA(CE)

ng

[1 ng

gcd(n,p)g⊕ 0NA(CE)− ng

gcd(n,p)g

]

aα,aα

=gcd(n, p)g

n|Σ|+g−1

[1 ng

gcd(n,p)g⊕ 0NA(CE)− ng

gcd(n,p)g

]

aα,aα

.

(5.61)

In the second line of Eq. (5.60), we summed over γ1, · · · , γg using Eq. (5.49). In the

last line of Eq. (5.60) and the first line of Eq. (5.61), we reorganized the coefficients

|ACEa 〉α〈ACEa |α into a matrix form, where 1 ngcd(n,p)

is the identity matrix due to the

delta function, and Jgcd(n,p) is because all elements of α = ngcd(n,p)

k, α = ngcd(n,p)

k

with k, k = 0, 1, · · · , gcd(n, p) − 1 are enumerated, and similar for JNA(CE)

ng. In the

second line of Eq. (5.61), we expand the tensor product. In the last line, we use the

normalization condition |C|2NAc(CE)NA(CE) = 1n|Σ|−1 . We see that all of the non-zero

eigenvalues of the entanglement spectrum are given by 1/Nn,p,g;|Σ|, where

Nn,p,g;|Σ| ≡n|Σ|−χ/2

gcd(n, p)g, χ = 2− 2g. (5.62)

147

χ is the Euler characteristic of Σ. Thus, the EE across a general surface of genus g

is:

S[(0, 0),(1, 0), . . . , (g − 1, 0), (g, 1)]

=|Σ| log n− g log gcd(n, p)− (1− g) log n

=|Σ| log n− χ

2log

n

gcd(n, p)− log gcd(n, p).

(5.63)

Equation (5.63) is consistent with Eq. (5.10). We summarize Stopo(A) and STQFTc (A)

for various systems and various entanglement surfaces in Table 5.1.

We note that although Eq. (5.63) is the EE for a low energy TQFT, there is still

an area law term. Since the TQFT is independent of the metric of the entanglement

surface, one may naively expect that the area law term should vanish. The reason that

the area law term appears in Eq. (5.63) is that we formulated our theory on a lattice,

which explicitly broke the scaling symmetry (i.e., changing the area of the cut changes

the number of links passing through Σ). However symmetry under area-preserving

diffeomorphisms was unaffected by the lattice regularization (changing the shape of

the cut does not change the number of links passing through Σ). Because of this, we

get terms that scale like the area of the cut (area law term), but no further shape-

dependent terms. Therefore, we expect, and indeed find, that the mean curvature

term vanishes for the TQFT (F ′2 → 0).

148

Chapter 6

Anomaly and Dynamics of (3 + 1)d

SU(2) Yang-Mills Theory

6.1 Introduction

The SU(N) Yang-Mills theory is a non-Abelian gauge theory with a gauge group

SU(N) described by the action

S = − 1

4g2

∫

M4

Tr(F ∧ ?F ) +θ

8π2

∫

M4

Tr(F ∧ F ), (6.1)

which admits a topological term parameterized by a variable θ. Since the second

Chern number

c2(VSU(N)) =1

8π2Tr(F ∧ F ) (6.2)

of the SU(N) vector bundle integrates to be an integer, θ is 2π periodic[142, 143].

The theory has a Z2,[1] one form center symmetry[144, 145, 146, 147]. When θ = 0, π

mod 2π, it is also time reversal symmetric.

149

SU(N) Yang-Mills is the simplest non-Abelian gauge theory in 3 + 1d that ex-

hibits rich dynamics. In contrast to the Abelian U(1) Maxwell gauge theory which

is free, the SU(N) Yang-Mills is strongly coupled due to negative beta function,

and the low energy dynamics is prohibitive via merely perturbative approaches[148].

However, various evidences including ’t Hooft anomalies[149, 143], deformation of

supersymmetric Yang-Mills[150, 151, 152], and holographic calculation in the large

N limit[153, 154] provide various constraints on the low energy dynamics, which we

summarize as the Standard Lore of Yang-Mills.

6.1.1 Standard Lore of SU(N) Yang-Mills

We review the dynamics of SU(N) Yang-Mills as a function of θ ∈ [0, 2π).

• θ = 0: When θ = 0, the only term is the kinetic energy of the gauge field, which

is time reversal symmetric. Various evidences including lattice simulations,

softly broken supersymmetry and large N holographic models suggest that the

ground state is confining with an unbroken center symmetry, and there is a

mass gap[148, 143, 150, 151, 152].

• θ = π: Another instance which is time reversal symmetric is when θ = π. In

this case, there is a mixed anomaly between the time reversal symmetry and the

ZN center symmetry for even N , and a more subtle global inconsistency for odd

N [143, 108, 155]. Both cases are unified from the point of view of anomaly in the

space of coupling constants[156, 157]. For even N , this anomaly immediately

constrains that SU(N) Yang-Mills with θ = π can not flow to a trivial phase.

It is widely believed that at the low energy, the theory confines and the center

symmetry is unbroken. However time reversal is spontaneously broken, leading

to two degenerate ground states[142, 158, 143]. Such spontaneous broken of

150

time reversal has been shown for large N Yang-Mills, where as one tunes from

θ < π to θ > π a first order phase transition has been observed[153, 154].

• 0 < θ < π, π < θ < 2π: The dynamics in this regime is less clear, due to the lack

of time reversal symmetry and consequently the anomaly. It is believed that the

theory confines for all θ. This is also supported by the large N calculation[153,

154]. The phase at θ = 2π, although is believed to be dynamically trivial

as θ = 0, differs from the phase at θ = 0 by a subtle symmetry protected

topological (SPT) phase [144, 145, 152, 147, 159].

Although the Standard Lore is believed to hold for large N , there are less evidences

supporting the standard lore for small N . In particular, for N = 2, the SU(2) Yang-

Mills at θ = π can flow to one of the several possible scenarios at low energy. The

low energy theory should either spontaneously break time reversal, or be deconfined,

or preserve time reversal symmetry and being confined while being gapless. 1 As far

as we know, none of the above scenarios has been excluded for N = 2. Therefore, it

is desirable to study all possible scenarios of SU(2) Yang-Mills in detail.

6.1.2 New Aspects: Lorentz Symmetry Enrichments

For any gauge theory with Z2,[1] one form symmetry, and in particular the SU(2) Yang-

Mills with any theta parameter[108, 155], can be enriched by the SO(3, 1) Lorentz

symmetry,2 via fractionalizing the Lorentz symmetry on the Wilson line operators.

This phenomena has been previously explored in [163, 164, 165] and others, and

has been recently termed in [166] poetically as Lorentz symmetry fractionalization.

1Gapped and confined TQFT that preserve time reversal symmetry has been ruled out in arecent work[160]. In [108, 155], the authors constructed a H-symmetry extended TQFT via theexact sequence 1 → K → H → Z2,[1] → 1, generalizing [161, 162] to higher form symmetries. Bydynamically gauging K, it has been realized that Z2,[1] is spontaneously broken, which is consistentwith [160].

2There are two branches of SO(3, 1), differed by chirality. These are denoted as SO±(3, 1) in theliterature. For our purposes, the choice of chirality will not play a role. In the rest of the paper, wefocus on the positive chirality +, and will suppress the superscript for simplicity.

151

Fractionalization of the Lorentz symmetry on a Wilson line requires that the Wilson

line transforms projectively under SO(3, 1), i.e., the self statistics is shifted by h =

1/2. This is done by shifting the background field B for the center Z2,[1] one-form

symmetry by the second Stiefel-Whitney class of the tangent bundle of the spacetime

manifold, i.e.,

B → B +K2w2, (6.3)

where K2 = 0, 1 represents trivial/nontrivial fractionalization.

For θ = 0, π, the SU(2) Yang-Mills is time reversal symmetric. Thus one can

further enrich the SU(2) Yang-Mills by the time reversal symmetry (or O(3, 1) if

combined with the SO(3, 1) Lorentz symmetry)[108, 155]. In this case, time reversal

symmetry can be fractionalized on the Wilson line. Nontrivial fractionalization of

time reversal means that the Wilson line is a Kramers doublet. Formally, this is done

by shifting the background field B by the square of the first Stiefel-Whitney class of

the tangent bundle of the spacetime manifold, i.e.

B → B +K1w21, (6.4)

where K1 = 0, 1 represents trivial/nontrivial fractionalization of time reversal symme-

try. Of course, one can consider enriching the SU(2) Yang-Mills at θ = 0, π by both

time reversal and SO(3, 1) Lorentz symmetry. In [108, 155], we denote the four differ-

ent O(3, 1) Lorentz symmetry enrichments, labeled by (K1, K2), of SU(2) Yang-Mills

at θ = 0, π as the Four Siblings, which we use throughout the present work.

Keeping the symmetry enrichments in mind, it is natural to revisit the standard

lore and ask a more refined question: How the dynamics of SU(N) Yang-Mills

depends on the O(3, 1) symmetry enrichment, i.e. the four siblings (K1, K2)?

152

In this work, we study the dynamics of SU(2) Yang-Mills at θ = π, and focus on two

low energy scenarios.

In the first scenario (to be discussed in section 6.3), time reversal is spontaneously

broken, and we study the domain wall theory that is constrained by the ’t Hooft

anomalies. We highlight several features of our results:

1. The domain wall theory is not time reversal symmetric, in contrast to the bulk.

Instead, there is a discrete unitary symmetry U .

2. The four siblings in the bulk corresponds to four different enrichments of the

Lorentz symmetry as well as the unitary symmetry U on the wall.

3. Even though the ’t Hooft anomaly of SU(2) Yang-Mills does not depend on the

SO(3, 1) Lorentz symmetry enrichment, the ’t Hooft anomaly on the wall does.

In section 6.4, we also discuss the consequences of symmetry enrichments on the

domain wall theories for SU(2) QCD within the regime of chiral symmetry breaking

Nf < NCFT .

The second scenario will be discussed in section 6.5, where we assume that the

low energy of SU(2) Yang-Mills with θ = π is deconfined. In particular, we only

discuss the case where the low energy theory is described by a U(1) Maxwell theory,

with certain Lorentz symmetry enrichment. The Lorentz symmetry enriched U(1)

Maxwell theories have been studied in [167, 168, 169, 170] where they classify the

phases of time reversal U(1) quantum spin liquids. We will use the ’t Hooft anomaly

to constrain the correspondence between the symmetry enrichments of SU(2) Yang-

Mills at θ = π and the symmetry enrichments of the Maxwell theory. In section 6.6,

we further apply this correspondence to study the phase transitions between different

U(1) spin liquids, as well as the phase transitions between U(1) spin liquids and

trivial paramagnets. Amusingly, we find that SU(2) QCD with Nf fermions (Nf >

NCFT ) in the fundamental representation can be interpreted as the second order phase

153

transition between the the above phases, where the gauge group is enhanced at and

only at the transition point. We denote such transition as gauge enhanced quantum

critical points.

6.2 SU(2) Yang-Mills Theory at θ = π

The 4d SU(2) Yang-Mills gauge theory with an SU(2) gauge group and a theta term

in the Minkowski spacetime M4 is described by an action 3

S = − 1

4g2

∫

M4

Tr(F ∧ ?F ) +θ

8π2

∫

M4

Tr(F ∧ F ), (6.5)

where we denote a as the SU(2) gauge field and F = da − ia2 is the field strength.

Since the second Chern number c2(VSU(2)) =∫M4

Tr(F ∧F )/8π2 is quantized to be an

integer, the θ parameter has periodicity 2π.

6.2.1 Time Reversal Symmetry

We first focus on the discrete time-reversal symmetry ZT2 and its symmetry transfor-

mation T acting on the gauge field aµ ≡ aαµTα, where Tα is the generator of SU(2).

T acts on aµ as:

T : aα0 → −aα0 , aαi → aαi , (t, xi)→ (−t, xi). (6.6)

Tα → Tα, a0 → −a0, ai → ai.

The components of the field strength Fα0i, F

αij transforms under T as

T : Fαij = ∂ia

αj − ∂jaαi + fαβγaβi a

γj → ∂ia

αj − ∂jaαi + fαβγaβi a

γj = Fα

ij(−t, xi),

Fα0i = ∂ta

αi − ∂iaα0 + fαβγaβ0a

γi → −∂−taαi + ∂ia

α0 − fαβγaβ0aγi = −Fα

0i(−t, xi).(6.7)

3For definiteness, the spacetime signature is taken to be (−1, 1, 1, 1).

154

where fαβγ is the structure constant of the SU(2) Lie algebra. Under T , the kinetic

term∫M4 Tr(F ∧ ?F ) is invariant, while the θ term flips the sign:

T :θ

8π2

∫

M4

Tr(F ∧ F )→ − θ

8π2

∫

M4

Tr(F ∧ F ). (6.8)

Because θ is 2π periodic, (6.5) is time reversal invariant only when θ = 0, π mod 2π.

6.2.2 One-form Symmetry

(6.5) also has a Z2,[1] one-form center symmetry which acts on the gauge-invariant

Wilson line

We = TrRP exp

(i

∮a

). (6.9)

where P stands for the path ordering. R can be any possible representation of SU(2).

If R is an irreducible representation and let l be the number of boxes in the Young

diagram of R, then We transforms under Ze2,[1] as

Z2,[1] : We → (−1)lWe. (6.10)

In particular, for the fundamental representation, there is only one box in the Young

diagram, hence the Wilson line flips sign under one form symmetry.

The generator of the symmetry Z2,[1] is a co-dimension two surface operator Ue.

We will find below that

Ue = exp(iπ

∮Λ), (6.11)

where Λ ∈ H2(M4,Z2).

It is useful to couple (6.5) to the Z2,[1] background gauge field B. Following [146?

? ], we first promote the SU(2) gauge field a to a U(2) gauge field a,

a = a+1

2AI2. (6.12)

155

where I2 is a two dimensional identity matrix. The first Chern class of the U(2)

bundle is c1 ≡ c1(VU(2)) ≡ TrF2π≡ dA

2πwhere F = da − iaa is the U(2) field strength.

Then we couple to B by requiring c1 = B mod 2, which can be done via introducing

a Lagrangian multiplier Λ:

∫DΛ . . . exp

(iπ

∫

M4

Λ ∪ (c1 −B)

). (6.13)

The minimal coupling exp(iπ∫

Λ∪B) implies that the generator of ZeN,[1] is precisely

exp(iπ∫

Λ), which explains (6.11). Notice that integrating out the Lagrangian multi-

plier Λ removes the U(1) degree of freedom, hence the gauge group is SO(3)=PSU(2)

(rather than SU(2)),

U(2)

U(1)=

(SU(2)× U(1))/Z2

U(1)=

SU(2)

Z2

= PSU(2) = SO(3). (6.14)

with the gauge bundle constraint

c1(VU(2)) = w2(VSO(3)) = B mod 2. (6.15)

6.2.3 Formulating on Unorientable Manifold and Lorentz

Symmetry Fractionalization

As we are focusing on the time reversal symmetric theory, one should be tempted

to formulate the theory (6.5) on an unorientable manifold. The Lorentz symmetry

associated with an unorientable manifold is O(3,1). In particular, on a generic un-

orientable manifold, both the first and second Stiefel-Whitney classes, w1 and w2, of

the tangent bundle of the spacetime manifold M4 are allowed to be nontrivial. One

can twist the gauge bundle constraint (6.15) as

c1(VU(2)) = B +K1w21 +K2w2 mod 2, K1, K2 = 0, 1. (6.16)

156

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Figure 6.1: Lorentz symmetry fractionalization on the Wilson line. The left panel isthe Wilson line with K1 = K2 = 0. When the background field B for the one-formsymmetry is activated, the Wilson line is attached to a surface Σ bounded by γ.This means that the Wilson line carries charge 1 under Z2,[1]. K1 = K2 = 0 impliesthat W1/2 is the worldline of a boson and a Kramers singlet. The right panel is theWilson line with nontrivial (K1, K2). The quantum number of the Lorentz symmetryis shown in (6.17).

As explained in section 6.1.2, (K1, K2) labels four distinct O(3, 1) Lorentz symmetry

enrichments of SU(2) Yang-Mills theories. In [108, 155], the authors also referred

(K1, K2) as the Four Siblings of O(3, 1) enriched SU(2) Yang-Mills with θ = 0, π.

One can understand (6.16) as follows. When B is nontrivial, the Wilson line

with SU(2) isospin j = 1/2, W1/2(γ), is attached to a surface operator exp(iπ∫

ΣB)

with ∂Σ = γ. See figure 6.1. The twisted gauge bundle constraint modifies the

above surface operator by decorating an additional 2d invertible TQFT of the Lorentz

symmetry: π(K1w21 + K2w2). The physical meanings of these invertible TQFTs are

well known. πw21 is the worldsheet theory of a time reversal symmetric SPT (a.k.a. the

Haldane chain) whose boundary supports a Kramers doublet. πw2 is the worldsheet

theory whose boundary transforms projectively under the Lorentz symmetry SO(3,1),

i.e., the boundary supports a fermion. We further realize that without the twists from

the Lorentz symmetry O(3,1) (i.e. K1 = K2 = 0), the original SU(2) Wilson line

W1/2(γ) transforms under O(3,1) as Kramers singlet and is a boson. Combining the

157

above physical understandings, under the twists using the O(3,1) Lorentz symmetry,

the statistics h(W ) and the Kramers parity T 2W of W1/2(γ) are

h(W ) =K2

2mod 1, T 2

W = (−1)K1+K2 . (6.17)

It is also illuminating to refer twisting the gauge bundle constraint from (6.15) to

(6.16) as Lorentz symmetry fractionalization. See [166] for the related discussions on

3d Chern-Simons (matter) theories and [169, 170] on 4d U(1) gauge theories.

6.2.4 Anomaly on an Unorientable Manifold

The SU(2) Yang-Mills theory with θ = π, coupled to the two-form background field

B, is

S = − 1

4g2

∫

M4

Tr(F − πBI2) ∧ ?(F − πBI2) +π

8π2

∫

M4

Tr(F − πBI2) ∧ (F − πBI2),

(6.18)

subjected to the gauge bundle constraint (6.16). Here F = da− ia2 is the U(2) field

strength. One further attempts to formulate (6.18) on an unorientable and non-spin

manifold M4, which enables one to prove the full quantum anomalies.

On an unorientable manifold, the top differential form is not well-defined, due

to the lack of the volume form whose definition needs an orientation. To make

sense of (6.18) on an unorientable manifold, we reformulate it in terms of the Chern

characteristic classes. We denote the jth Chern class of the U(2) gauge bundle as

cj(VU(2)). Denote the jth Chern class of the U(N) gauge bundle as cj. For j = 1, 2,

we have

c1 =TrF

2π,

c2 = − 1

8π2Tr(F ∧ F ) +

1

8π2(TrF ) ∧ (TrF ).

(6.19)

158

Replacing 18π2 Tr(F ∧ F ) by c1∪c1

2− c2, we rewrite the topological term in (6.18) in

terms of the cocycles and characteristic classes as

π

∫

M4

(− c2 +

c1 ∪ c1

2− 1

2c1 ∪B +

P(B)

4

)(6.20)

where P(B) is the Pontryagin square. On an unorientable manifold, only Z2 char-

acteristic classes can be integrated. Hence, except the first term in (6.20), the other

terms are all ill-defined. To make sense of the last three terms in (6.20), we can

promote these ill-defined terms to a 5d integral using the Stocks rule:

Sanom ≡π∫

M5

δ

(P(B)

4− 1

2c1 ∪B +

c1 ∪ c1

2

)

=π

∫

M5

δP(B)

4− 1

2c1 ∪ δB −

1

2δc1 ∪B +

δc1 ∪ c1

2+c1 ∪ δc1

2

=π

∫

M5

BSq1B + Sq2Sq1B − c1 ∪ Sq1B − Sq1c1 ∪B + Sq1c1 ∪ c1 + c1 ∪ Sq1c1

=π

∫

M5

BSq1B + Sq2Sq1B − (B +K1w21 +K2w2) ∪ Sq1B − (Sq1B +K2Sq1w2) ∪B

+ (Sq1B +K2Sq1w2) ∪ (B +K1w21 +K2w2)

+ (B +K1w21 +K2w2) ∪ (Sq1B +K2Sq1w2)

=π

∫

M5

BSq1B + Sq2Sq1B +K1Sq1B ∪ w21 +K2Sq1(B ∪ w2)

+K2

((K1w

21 +K2w2) ∪ Sq1w2 + Sq1w2 ∪ (K1w

21 +K2w2)

)

=π

∫

M5

BSq1B + Sq2Sq1B +K1Sq1B ∪ w21 +K2Sq1(B ∪ w2).

(6.21)

In the third line, we have used the gauge bundle constraint (6.16), which implies that

c1, when promoted to 5d, can still be valued in Z2 cohomology H2(M5,Z2) (although

c1 is no-longer valued in a Z cohomology class in M5). This implies that on M5, it

makes sense to define Sq1c1. (6.21) has several properties:

159

1. Every term in the last expression of (6.21) is Z2 valued cohomology class, hence

the integration on unorientable M5 is perfectly well-defined.

2. It is straightforward to check that when M5 is closed (without boundary), (6.21)

is invariant under the background gauge transformation

B → B + δλ. (6.22)

Combining with the first point, (6.21) respects all the symmetries of the SU(2)

Yang-Mills.

3. All terms in (6.21) only depend on the background gauge fields, and independent

of the dynamical gauge fields.

4. Because (6.21) is non-vanishing for closed M5, the extension from M4 to M5

depends on the choice of M5, hence (6.21) is really an 5d SPT of the global

symmetry Z2,[1] × ZT2 .

All these properties leads to the conclusion that the SU(2) Yang-Mills theory with

θ = π is anomalous, with the anomaly polynomial

Sanom = π

∫

M5

BSq1B + Sq2Sq1B +K1w21Sq1B +K2Sq1(w2B). (6.23)

To gauge Z2,[1] while preserving time reversal symmetry, we really have to regard the

SU(2) Yang-Mills as a 4d-5d coupled system, where the structures (w1, w2, B) on M4

are extended to M5.

We emphasize that the anomaly polynomial (6.23) depends on K2 only when M5

has a nontrivial boundary M4. This implies that the term K2Sq1(w2B) does not lead

to a distinguished anomaly. Instead, it is a WZW-like counter term. However, we

will show in section 6.3 that, if time reversal is spontaneously broken at θ = π, the

160

WZW-like counter term leads to a nontrivial ’t Hooft anomaly on the time reversal

domain wall.

6.2.5 Low Energy Dynamics: Overview and Questions

The SU(2) Yang-Mills theory is strongly coupled in the infrared, due to negative beta

function. Thus the low energy fate of the SU(2) dynamics is hardly known. It is

famously conjectured [148] that for any N ≥ 2 the SU(N) Yang-Mills with θ = 0 has

a mass gap.4 Moreover, [143] found that the SU(N) Yang-Mills (for even N) has a

nontrivial ’t Hooft anomaly only at θ = π. Since nontrivial ’t Hooft anomaly implies

that the low energy theory can not be trivially gapped, there should be nontrivial

dynamics at θ = π. In particular, the above analysis also apply to SU(2) Yang-Mills.

For the regime within θ ∈ (0, π)∪ (π, 2π), the dynamics is less clear. In fact, [143]

proposed two scenarios for the SU(2) Yang-Mills dynamics at zero temperature. In

one scenario, SU(2) Yang-Mills is confined for every θ. In the other scenario, SU(2)

Yang-Mills is deconfined within a regime θ ∈ [π − x, π + x] for x ∈ [0, π). In the

following discussion, we will not discuss the generic θ and will exclusively focus on

θ = π, where one can infer more on the dynamics based on the ’t Hooft anomaly.

As mentioned above, an immediate consequence of the ’t Hooft anomaly (6.23) for

Yang-Mills theory at θ = π is that the low energy theory can not be trivially gapped.

What should the low energy theory be at the fixed point? [143, 108] discussed several

scenarios, which we enumerate below.

1. The theory confines, and correspondingly the one-form symmetry Z2,[1] is un-

broken. Time reversal symmetry is spontaneously broken. There are two vacua

which are related by the spontaneously broken time reversal transformation.

This scenario is believed to take place for SU(N) Yang-Mills with large N .

4Though the mass gap is supported by numerous evidences, it still remains a conjecture. Insection 6.6, we contemplate another exotic possibility where the low energy of θ = 0 Yang-Mills isgapless, described by a deconfined U(1) Maxwell theory.

161

2. The theory is gapless and deconfined, and correspondingly the one-form sym-

metry Z2,[1] is spontaneously broken. Time reversal is unbroken. The decon-

finement can be realized by a gapless conformal field theory (CFT) (e.g. U(1)

Maxwell theory). See [167, 164] for discussions of different time reversal enriched

gapless CFTs.

3. The theory is gapped and deconfined, and correspondingly the one-form sym-

metry Z2,[1] is spontaneously broken. Time reversal is unbroken. The decon-

finement can be realized by a gapped TQFT (e.g. Z2 gauge theory). In [108],

the authors have proposed the action of Z2 gauge theory in 4d saturating the

anomaly (6.23).

4. Both Z2,[1] and time reversal are preserved by a gapped TQFT. In [155, 108], the

authors constructed a H-symmetry extended TQFT via the exact sequence 1→

K → H → Z2,[1] → 1, generalizing [161, 162] to higher form symmetries. By

dynamically gauging K, it was realized that Z2,[1] is spontaneously broken. This

suggests a possible no go to construct a symmetric TQFT. More systematically,

this scenario is ruled out by a no-go theorem from Cordova and Ohmori[160],

by making use of the quantum surgery constraints on cutting and gluing the

spacetime manifolds[171, 172] and other criteria.

5. Both Z2,[1] and time reversal are preserved by a gapless CFT.

Though the candidate phases have been proposed, it is worthy to discuss in further

detail the following aspects.

1. [143] only discussed one sibling, i.e. K1 = K2 = 0 among the Four Siblings in

[108]. Thus it is worthwhile to explore further the dynamical consequences for

different siblings.

162

2. In the first scenario, time reversal is spontaneously broken, and there are two

vacua related by time reversal symmetry. Hence there can be a domain wall

interpolating between the two vacua. The anomaly of 4d Yang-Mills (6.23) in-

duces an anomaly for the 3d domain wall, hence there must be nontrivial degrees

of freedom supported on the domain wall to saturate the induced anomaly. It

should be interesting to see how the four siblings of the domain wall theory are

related to each other. This will be discussed in section 6.3.

3. The second scenario is particularly interesting. If this scenario takes place in

dynamics, the non-Abelian SU(2) gauge theory with matter can access a direct

second order quantum phase transition between a U(1) spin liquid and the

trivial vacuum, or more exotically between two U(1) spin liquids, depending

on the further details which we discuss in section 6.6. This exotic scenario

tremendously enlarges the range of possible candidates of phase transitions,

and hence the multi-universality class, between the above phases.

Since the Z2 gauge theory has already been studied in [108], we will not study it in

detail in the present work. We also have little to say about the last scenario.

6.3 Domain Wall from Time Reversal Sponta-

neously Broken

We consider the scenario where time reversal symmetry is spontaneously broken in

the low energy. There are two vacua which are time reversal partners. Furthermore,

there exists a domain wall interpolating the two vacua. Since the two time reversal

breaking vacua are separately trivially gapped, the notion of domain wall theory

is well defined. The anomaly (6.23) implies that the domain wall theory itself has

163

nontrivial anomaly, which enforces that the domain wall supports nontrivial degrees

of freedom.

6.3.1 Domain Wall for (K1, K2) = (0, 0): Semion with U2 = 1

In this subsection, we discuss the domain wall theory for the sibling (K1, K2) =

(0, 0). The anomaly (6.23) reduces to π∫M5BSq1B + Sq2Sq1B. Under time reversal

transformation, the above anomaly implies that the partition function transforms as

Z → Z exp(iπ∫M4P(B)/2), hence induces an anomaly for the time reversal domain

wall,

SDWanom =π

2

∫

M4

P(B). (6.24)

The domain wall theory saturating the anomaly (6.24) was proposed in [143] to be a

SU(2)1 Chern Simons (CS) theory with an action

SU(2)1 CS : SCS =1

4π

∫

M3

Tr

(ada− 2i

3a3

), (6.25)

where a is a one-form SU(2) gauge field. The theory (6.25) is a non-spin theory.

There are two lines: an identity line 1, and a line with semionic topological spin s,

i.e. {1, s}. These lines obey the Abelian fusion rule: 1× 1 = 1, 1× s = s, s× s = 1.

Hence the theory is an Abelian semion theory. Coincidentally SU(2)1 is equivalent to

U(1)2 Chern Simons. 5 See Fig. 6.2.

What is the origin of the deconfined topological line s on the domain wall? We

follow the discussions in [159]. Since s is also the SU(2) Wilson line in fundamental

representation, it is natural to identify s with the SU(2) Wilson line in the funda-

mental representation in the 4d Yang-Mills theory, i.e., W1/2 ↔ s. The subscript

5This should be contrasted to the level rank duality SU(2)1 ←→ U(1)−2 which only holds whenboth sides are regarded as spin TQFTs.

164

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⇡

2

Z

M4

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Figure 6.2: When time reversal is spontaneously broken, there are two vacua. Weconsider a configuration where each vacuum occupies half of the space, and there isa domain wall in between. Time reversal exchanges the two vacua. The anomaly(6.23) in the bulk induces an anomaly (6.24) on the domain wall, which consequentlyconstrains that there is an Abelian semion TQFT on the wall.

1/2 represents the SU(2) isospin. However, W1/2 in the Yang-Mills obeys area law,

in accordance with the confinement. s on the domain wall has perimeter law, in

accordance with the deconfinement on the wall. The behaviors of the SU(2) line in

the bulk and on the wall can be understood from the different condensates in the two

vacua of the bulk[143, 152, 173, 159]. In one vacua, confinement is due to monopole

condensation. In the other vacua, confinement is due to dyon condensation. Thus

although both vacua are trivially gapped, they differ by a Z2,[1] symmetry protected

topological (SPT) phase which is precisely described by (6.24). When W1/2 tunnels

from one vacuum to the other vacuum, due to the condensate changes, W1/2 has to

deconfine on the wall. The phenomena of deconfinement can also occur on the bound-

ary of a confining (e.g. SPT) or deconfining (e.g. SET) bulk in various dimensions,

see Sec.7 of [162] for further discussions.

s also descends from the Z2,[1] generator U in 4d. The U is a surface operator. In

the vacuum where monopole condenses, U is the spacetime trajectory of the ’t Hooft

line, which does not carry one-form symmetry charge itself. In the vacuum where

165

dyons condense, the Z2,[1] generator is the spacetime trajectory of the dyon line, which

carries one-form symmetry charge. We consider a stretched Z2,[1] generator which

extends to both vacua and crosses the domain wall. U and the domain wall intersects

on a line, which carries Z2,[1] charge which is identified as s. Thus U |DW ↔ s. Notice

that semions in 3d see each other as mutual fermions. Because s ↔ W1/2 ↔ U |DW,

the mutual fermionic statistics between s descends from the mutual semionic statistics

between W1/2 and U : 〈W1/2(γ)U(Σ)〉 = (−1)〈Σ,γ〉.

We further discuss the global symmetries of the domain wall theory SU(2)1. There

is a Z2,[1] one form global symmetry, generated by s. Coupling to the background B

leads to the anomaly (6.24).

What about the time reversal symmetry? In the bulk, time reversal is sponta-

neously broken, hence time reversal exchanges the two vacua on the two sides of the

domain wall. Hence time reversal is not a symmetry of the domain wall theory. In

particular, time reversal T acts as

T [SU(2)1 CS] = SU(2)−1 CS. (6.26)

The reversed sign of the Chern Simons level reflects the reversal of the direction of

the anomaly inflow under T . A useful observation[174, 175] is that T can be modified

to be the symmetry of SU(2)1 by multiplying an unbreakable CP⊥T in 4d. (Analogue

phenomenon and more general relation to the Smith Isomorphism have been discussed

by Hason, Komargodski and Thorngren [174] and independently by Cordova, Ohmori,

Shao and Yan [175]. See also the talk[176] by Thorngren. We apply this general idea

to the special context: the domain wall of SU(2) Yang-Mills.) We define

U = T (CP⊥T ), (6.27)

166

where P⊥ is the reflection along the direction perpendicular to the domain wall. Both

T and CP⊥T are not symmetries of SU(2)1, but their combination U is. Since both

T and CP⊥T are anti-unitary, U is unitary.

How does U act on the line operators in SU(2)1? Because both T and CP⊥T

flip the topological spin of anyons, U preserves the spin. Hence U does not permute

the lines. However, similar to the quantum Hall physics where anyons can trans-

form projectively under U(1) charge conservation symmetry, anyons can transform

projectively under U . The symmetry fractionalization is classified by

H2ρ(Z2, {1, s}) = Z2, (6.28)

where ρ = 1 is the identity because Z2 symmetry generated by U does not permute

the anyons. To determine the action of U , we first compute U2. Using the algebra of

T and CP⊥T in the 4d (K1, K2) = (0, 0) Yang-Mills theory, 6

T 2 = 1, (CP⊥T )2 = 1, T CP⊥T = CP⊥T T . (6.29)

Thus

U2 = T CP⊥T T CP⊥T = T 2(CP⊥T )2 = 1. (6.30)

Hence U generates a Z2 unitary symmetry that acts linearly on W1/2. Since the

Wilson line in the bulk does not transform projectively under T , the Wilson line

on the wall s does not transform projectively under U either.7 Thus the state |s〉

associated with the anyon s carries charge one (rather than the fractional charge)

6T 2 = 1 is because the Wilson line is Kramers singlet. The third equality follows from T (CP⊥) =(CP⊥)T which holds when acting on a bosonic line. If acting on a fermionic line, the third equalityshould be modified to T (CP⊥) = −(CP⊥)T . See section 6.3.5 for further details.

7In the next section, we will see that for the sibling (K1,K2) = (1, 0), the Wilson line transformsprojectively under T and accordingly s transforms projectively under U .

167

under U , i.e., U|s〉 = −|s〉. In summary U is realized linearly on the anyons which

corresponds to the trivial element in (D.67).

How does the domain wall theory couple to the background field of U? Denote the

one-form background field of U as Y , satisfying∮Y ∈ 1 mod 2. The action coupled

to the background field is

2

4πudu− Y du, (6.31)

where u is the U(1) gauge field. Here we have used the equivalence SU(2)1 ≡ U(1)2.

One can check that the Wilson line s = exp(i∮u) indeed has charge one under U .

To see this, one inserts into the path integral a Wilson line along γ, which amounts

to add to the action a term∫u ? j where ?j = δ⊥(γ). To find the U charge of the

Wilson line, we need to find the coefficient of the term πY ? j in the response action

where the dynamical fields are integrated out. This is done by solving the equation

of motion of u and plugging back into the action (6.31).

Further coupling (6.31) to Z2,[1] background field B, the action is

∫

M3

(2

4πudu− uB − Y du+ πY B

)+π

2

∫

M4

P(B), (6.32)

where we suppressed the cup product, e.g. Y B = Y ∪ B. The only anomaly is the

self anomaly of Z2,[1]. There is no anomaly involving U . This is also consistent with

the fact that U is not fractionalized on the anyons {1, s}.

6.3.2 Domain Wall for (K1, K2) = (1, 0): Semion with U2 = −1

We proceed to discuss the domain wall theory for the sibling (K1, K2) = (1, 0). Com-

pared with the anomaly for (K1, K2) = (0, 0), the anomaly for (K1, K2) = (1, 0)

contains an additional term K1π∫w2

1Sq1B = K1π∫w3

1B. Hence one may naively

168

conclude that the anomaly for the domain wall theory is π2

∫M4P(B) + π

∫M4w2

1B.

However, there are several apparent puzzles for the above domain wall anomaly:

1. Since the anomaly involves the background field w1, the domain wall theory

should be time reversal symmetric, and can be formulated on an unorientable

manifold. However, since the 4d theory for (K1, K2) = (1, 0) only differs from

(K1, K2) = (0, 0) by Lorentz symmetry fractionalization, one expects that the

domain wall theory for the sibling (1, 0) should be a modification of SU(2)1 by

modifying the way time reversal acts. But SU(2)1 is not time reversal symmet-

ric in the first place and therefore does not make sense to formulate it on an

unorientable manifold.

2. The anomaly itself, regardless of the details of the domain wall theory, is prob-

lematic. The first term π2

∫M4P(B) is not compatible with unorientable mani-

fold. This is because π2

∫M4P(B) is Z4 valued, while any quantity that can be

integrated on an unorientable manifold has to be Z2 valued.

In this section, we propose a domain wall theory by modifying the U symmetry

realization on the domain wall theory SU(2)1 proposed in section 6.3.1, which resolves

the above puzzles.

For the sibling (K1, K2) = (1, 0), the SU(2) Wilson line in the bulk W1/2 is a

Kramers doublet, hence

T 2 = (−1)2j, (6.33)

where j is the SU(2) isospin. For our purposes, we still regard time reversal symmetry

in 4d as a ZT2 symmetry, and (6.33) is interpreted as the Wilson line transforms in

the projective representation of ZT2 symmetry. The algebra between T , CP⊥T is (see

169

section 6.3.5 for further details)

T 2 = (−1)2j, (CP⊥T )2 = 1, T CP⊥T = CP⊥T T . (6.34)

Hence

U2 = T 2 = (−1)2j. (6.35)

Similar to the discussion below (6.33), we still interpret U as a Z2 unitary symmetry,

and (6.35) implies that the anyon s transforms projectively under U . Such a projective

representation is the nontrivial element in (D.67).

The domain wall theory is thus SU(2)1 with Z2,[1] one-form symmetry and Z2

zero-form symmetry generated by U , satisfying (6.35). How does SU(2)1 couple to

U background field? As in section 6.3.1, we still denote the U background as Y

satisfying∮Y ∈ 1 mod 2. The action coupled to the background field is

2

4πudu− 1

2Y du. (6.36)

Using the method discussed below (6.31), we find that the semion s = exp(i∮u)

carries U charge 1/2, i.e., U|s〉 = i|s〉. U is fractionalized on s as expected.

Is the Z2 symmetry generated by U anomalous? First it does not have anomaly

with itself. To see this, we examine that under the background gauge transforma-

tion Y → Y + δy, (6.36) transforms by −δydu/2, which vanishes modulo 2π. We

further check the mixed anomaly between U and Z2,[1]. The mixed anomaly is most

conveniently seen by activating the Z2,[1] background field B,

∫

M3

(2

4πudu− uB − 1

2Y du

)+

∫

M4

(π2P(B) + πY Y B

). (6.37)

170

Indeed, we find two types of anomaly: π2P(B) is the anomaly already appeared in the

domain wall for the sibling (K1, K2) = (0, 0). πY Y B is the mixed anomaly between

U and Z2,[1], due to nontrivial U symmetry fractionalization in (D.67). Consistently,

πY Y B implies that on the domain wall, the Z2,[1] generator s is attached by a surface

operator exp(iπ∫

ΣY Y ) = exp(iπ/2

∫ΣδY ) = exp(iπ/2

∮∂ΣY ) which precisely reflects

the fact that s carries Y charge 1/2. We make several comments:

1. In (6.37), the domain wall theory SU(2)1 is not time reversal symmetric. Con-

sistently, the anomaly does not involve w1, which resolves the two puzzles men-

tioned in the beginning of this subsection.

2. The time reversal symmetry fractionalization in the 4d induces a unitary Z2

symmetry fractionalization on the domain wall. Correspondingly, the mixed

T − Z2,[1] anomaly πw31B induces a mixed U − Z2,[1] anomaly πY Y B on the

domain wall.

3. Since CP⊥T is always an unbreakable symmetry in 4d Yang-Mills, one can freely

modify the T background field w1 to T (CP⊥T ) background field Y . Hence the

anomaly πw31B for Yang-Mills can be equivalently be written as πw1Y Y B.

This rewriting makes the induced anomaly πY Y B of the domain wall natural,

because under time reversal, 4d Yang-Mills partition function transforms as

Z→ Z exp

(iπ

2

∫

M4

P(B) + iπ

∫

M4

Y Y B

), (6.38)

which naturally provides the anomaly inflow of the 3d domain wall theory (6.37).

We emphasize that in (6.37), one can not replace Y by w1 back, because CP⊥T

is no longer the symmetry of the domain wall.

171

6.3.3 Domain Wall for (K1, K2) = (0, 1): Anti-Semion with

U2 = 1

We proceed to discuss the domain wall theory for the sibling (K1, K2) = (0, 1). We

will find that although the anomaly for 4d Yang-Mills does not depend on K2, the

anomaly of the domain wall does! To see this, we rewrite K2 dependent term in (6.23)

as K2πSq1(w2B) = K2πw1w2B which does not vanish on a manifold with boundary.

This term induces an anomaly on the domain wall K2πw2B. The complete anomaly

for the domain wall is

SDWanom =π

2

∫

M4

P(B) + π

∫

M4

w2B. (6.39)

We look for the domain wall theory that saturates such an anomaly.

We start with SU(2)1 ≡ U(1)2 theory for the sibling (K1, K2) = (0, 0). We have

shown in section 6.3.1 that SU(2)1 saturates the first term in (6.39). One needs to

find a proper fractionalization of the Lorentz symmetry (whose background is w2) to

further match the anomaly πw2B. Denote the topological spin of the Z2,[1] generator

s in SU(2)1 as h(s). The additional anomaly π∫w2B modifies the topological spin

of s by[166]

h(s)→ h(s) +1

2mod 1. (6.40)

Hence after symmetry fractionalization, h(s) shifts from 1/4 to 3/4 mod 1. In other

words, the semion in the domain wall for the sibling (K1, K2) = (0, 0) becomes an

anti-semion for the domain wall in the sibling (K1, K2) = (0, 1). Thus the domain

wall TQFT for (K1, K2) = (0, 1) contains a trivial anyon and an anti-semion, i.e.

172

{1, s}. Such a TQFT is precisely

SU(2)−1 CS. (6.41)

Apart from using Lorentz symmetry fractionalization, the domain wall theory can

further be obtained by rewriting the anomaly (6.39) as

SDWanom =π

2

∫

M4

P(B) + π

∫

M4

P(B) =3π

2

∫

M4

P(B) = −π2

∫

M4

P(B). (6.42)

Comparing with the anomaly (6.24), the anomaly (6.42) simply changes the sign, i.e.,

the direction of the anomaly inflow is reversed. Consistently, the level of the domain

wall Chern Simons theory is also reversed, from SU(2)1 for the sibling (K1, K2) =

(0, 0) to SU(2)−1 for the sibling (K1, K2) = (0, 1).

The Lorentz symmetry fractionalization can also be viewed from the quantum

number of Wilson line W1/2 in the 4d Yang-Mills. For the sibling (K1, K2) =

(0, 1), W1/2 transforms projectively under the SO(3, 1) Lorentz rotation, hence it

is a fermion. As explained in section 6.3.1, the deconfined line s is obtained from

the Wilson line in the bulk. Hence the Lorentz symmetry fractionalization (the shift

of statistics by 1/2) for W1/2 naturally induces a Lorentz symmetry fractionalization

(the shift of statistics by 1/2) for s on the domain wall, which yields s, consistent

with the additional anomaly πw2B for the domain wall.

It is instructive to consider the fractionalization of the unitary Z2 symmetry U

on s. In the 4d Yang-Mills of the sibling (K1, K2) = (0, 1), the Wilson line W1/2

is a Kramers doublet, i.e., T 2 = −1. More generally, T 2 = (−1)2j where j is the

SU(2) isospin. Hence using the algebra of T and CP⊥T , (see section 6.3.5 for further

details)

T 2 = (−1)2j, (CP⊥T )2 = 1, T (CP⊥T ) = (−1)2j(CP⊥T )T , (6.43)

173

we find

U2 = (−1)2jT 2 = (−1)4j = 1. (6.44)

It is ramarkable that although the time reversal symmetry is fractionalized on Wilson

line W1/2 in the bulk, U is not fractionalized on the anyon s ! Hence similar to the

case in section 6.3.1, the anti-semion s transforms linearly under U , i.e., U2(s) = s.

We further couple SU(2)−1 to both Z2,[1] background field B and the U background

field Y ,

∫

M3

(− 2

4πudu+ uB + Y du− πY B

)+

∫

M4

(π2P(B) + πw2B

). (6.45)

The fact that U is not fractionalized on s is in accord with the fact that there is no

anomaly involve U on the wall.

6.3.4 Domain Wall for (K1, K2) = (1, 1): Anti-Semion with

U2 = −1

We finally discuss the domain wall theory for the sibling (K1, K2) = (1, 1). From the

discussion in section 6.3.2 and 6.3.3, we find that the anomaly for the domain wall

theory is

SDWanom =π

2

∫

M4

P(B) + π

∫

M4

(Y Y + w2)B, (6.46)

where Y is the background field for the unitary symmetry U = T (CP⊥T ). The

domain wall theory is SU(2)−1 properly coupled to background fields Y and w2:

∫

M3

(− 2

4πudu+ uB +

1

2Y du

)+

∫

M4

(π2P(B) + πw2B + πY Y B

). (6.47)

174

We emphasize that although time reversal is not fractionalized on the W1/2 in

the bulk, i.e., T 2 = 1, the U unitary symmetry is fractionalized on the anyon s.

Furthermore, we again observe that domain wall carries nontrivial anomaly related

to w2, although the bulk does not.

6.3.5 Remarks On CP⊥ and T , and Summary

We provide some additional remarks on the 4d symmetries CP⊥ and T . The purpose

is to further explain the algebra between T and CP⊥T , i.e. (6.29), (6.34) and (6.43).

As mentioned in section 6.2.3, for the sibling (K1, K2), the Wilson line W1/2 has

spin h(W ) = K2/2, which is explained below (6.16). However, the fact that time

reversal squares to be T 2 = (−1)K1+K2 , rather than T 2 = (−1)K1 , needs further

explanation, which we provide below. (See [169] for similar explanation in 4d Maxwell

theory. ) For K2 = 0, T 2 = (CP⊥)2 = (−1)K1+K2 = (−1)K1 , hence K1 = 0, 1

represents Kramers singlet and doublet respectively. However for K2 = 1, suppose

when we move from a Minkowski spacetime to a Euclidean spacetime, T becomes a

Euclidean reflection R via a Wick rotation. Then T 2 differs by a sign from R2, i.e.

T 2 = −R2. Such a minus sign only occurs when acting on a fermion. Notice that

in Minkowski spacetime, CP⊥ is a still a Euclidean reflection, so T 2 = −(CP⊥)2. To

synthesize, we have

T 2 = (−1)K2(CP⊥)2. (6.48)

If T 2 = (−1)K1+K2 , then (CP⊥)2 = (−1)K1 , hence

T (CP⊥) = CP⊥T CP⊥T T (CP⊥) = (CP⊥T )T 2(CP⊥)2 = CP⊥T (−1)K2 , (6.49)

175

where we used (CP⊥T )2 = 1. This further gives rise to the commutation relation

between T and CP⊥T as

T (CP⊥T ) = (−1)K2(CP⊥T )T . (6.50)

This is precisely the relation in (6.29), (6.34) and (6.43).

We summarize the symmetry properties of the Wilson lines of isospin j in the bulk,

the domain wall theory, their symmetry fractinoalization pattern and the anomalies

in table 6.1.

(K1, K2) (h mod 1, T 2) DW Theory U2 DW Anomaly

(0, 0) (0, 1) SU(2)1 = {1, s} 1 π2

∫M4P(B)

(1, 0) (0, (−1)2j) SU(2)1 = {1, s} (−1)2j π2

∫M4P(B) + π

∫M4Y Y B

(0, 1) (j, (−1)2j) SU(2)−1 = {1, s} 1 π2

∫M4P(B) + π

∫M4w2B

(1, 1) (j, 1) SU(2)−1 = {1, s} (−1)2j π2

∫M4P(B) + π

∫M4

(Y Y + w2)B

Table 6.1: Symmetry fractionalization and anomalies on the domain wall theory for

four siblings of Yang-Mills.

6.4 Application I: Domain Wall Theory Nf < NCFT

We start by considering the domain wall theory for SU(2) QCD with Nf fermions.

The theory depends on the mass and the theta parameter via mNf eiθ. In this section,

we assume m to be real and non-negative, and keep θ in the Lagrangian. (In section

6.6, we will adopt the different assumption.) We exclusively focus on θ = π, which is

time reversal symmetric. We denote Λ as the strong coupling scale.

When m � Λ, one can integrate out the massive fermions, and the low energy

effective theory is the SU(2) Yang-Mills theory with θ = π. Assuming the scenario

176

where the time reversal is spontaneously broken, there are two vacua which are time

reversal partners. Between the two vacua, there is a time reversal domain wall. We

further assume that Nf is below the conformal window, i.e. Nf < NCFT , the domain

wall theory has been conjectured to be[177]

SU(2)1−

Nf2

+Nfψ ←→ U(1)−2 +Nfφ. (6.51)

In the large mass limit, the domain wall theory (6.51) flows to SU(2)1 ≡ U(1)2,

corresponding to the domain wall theory of the pure SU(2) Yang-Mills at θ = π. As

discussed in section 6.3, there are multiple versions of SU(2)1 theories, distinguished

by the enrichments of the unitary symmetry U and the Lorentz symmetry. In this

section, we determine the symmetry enriched versions of SU(2)1−

Nf2

+Nfψ, and how

the symmetry enrichments match across the duality (6.51).

6.4.1 Lorentz Symmetry Fractionalization, K2 = 1

We first show that domain wall theory realized in SU(2) QCD requires K2 = 1. In the

bulk, since the SU(2) gauge field is coupled to fermions, the 2π Lorentz rotation, which

multiplies the fermions by −1, can be compensated by a SU(2) gauge transformation.

More precisely, the gauge-spacetime symmetry is

SU(2)× Spin(3, 1)

Z2

, (6.52)

and the constraint of the symmetry bundle is

w2(VSO(3)) = w2. (6.53)

Comparing with (6.16), we find that the Lorentz symmetry SO(3, 1) is always realized

projectively, hence the effective Yang-Mills corresponds to the sibling K2 = 1. As

177

discussed in section 6.3.3, in the large mass limit on the domain wall SU(2)1, the

SU(2) gauge bundle in the domain wall theory is also twisted by the Lorentz symmetry

SO(2,1), i.e. the gauge-spacetime symmetry on the domain wall, as well as the bundle

constraint are

Domain Wall :SU(2)× Spin(2, 1)

Z2

, w2(VSO(3)) = w2. (6.54)

Thus at large mass limit on the wall, the Chern Simons is the K2 = 1 enrichment

of SU(2)1, i.e. SU(2)−1 Chern Simons theory discussed in section 6.3.3. Notice that

this is precisely the large positive mass limit on the bosonic side of (6.51). Hence

the SO(2, 1) Lorentz symmetry fractionalization is matched across the duality on the

wall. See [166] for more examples.

6.4.2 U Unitary Symmetry Fractionalization

We proceed to discuss the fractionalization of Z2 unitary symmetry generated by

U on the domain wall. We first consider the large positive mass limit in the theory

SU(2)1−Nf/2+Nfψ. There are two options of fractionalization of U on the anti-semion

s,8 labeled by K1. Concretely, there is the correspondence

U2 = (−1)K1 on anti-semion s. (6.55)

When the mass of ψ is finite, the Z2 unitary symmetry acts on the fermion ψ. For

K1 = 0, the fermion carries charge 1, while for K1 = 1, the fermion carries charge

1/2 (i.e. fractionalized).

On the other hand, notice that SU(2)1−Nf/2 + Nfψ naturally has the U(1) sym-

metry associated with Baryon conservation, and we adopt the normalization that the

8Notice that Lorentz symmetry fractionalization of the semion results in an anti-semion.

178

Baryon has U(1) charge 2, while the quark ψ has U(1) charge 1. The symmetry is

U(1)× SU(2)× Spin(2, 1)

Z2 × Z2

, (6.56)

where the constraint between the bundles is w2(VSO(3))+c1(VU(1)/Z2)+w2 = 0 mod 2.

How does the Z2 symmetry generated by U relate to U(1)? For K1 = 0, the

quark ψ carries U charge 1, hence the Z2 is embedded in U(1) in the natural way,

i.e. Z2 ⊂ U(1). For K1 = 1, the quark ψ carries U charge 1/2, hence the Z2 is

embedded into U(1) as Z2 ⊂ U(1)/Z2, or equivalently Z4 ⊂ U(1). We enumerate the

total U -gauge-spacetime symmetry and their gauge bundle constraint as follows:

(K1, K2) = (0, 1) :Z2 × SU(2)× Spin(2, 1)

Z2 × Z2

, w2(VSO(3)) + w2 = 0 mod 2,

(K1, K2) = (1, 1) :Z4 × SU(2)× Spin(2, 1)

Z2 × Z2

, Sq1Y + w2(VSO(3)) + w2 = 0 mod 2.

(6.57)

Notice that the gauge bundle constraints for the domain wall theories (6.57) are nicely

in accord with (6.16) in 4d.

Let us consider the dual theory U(1)−2 + Nfφ, and discuss how the U symmetry

is realized. We first consider the large mass limit, where the theory flows to U(1)−2.

The monopole in the bosonic theory is dual to the Baryon in the fermionic theory. In

the fermionic theory, Baryon carries U(1) charge 2. Using the embedding of Z2 into

U(1), we find that Baryon carries U charge K1 mod 2. Thus the monopole carries U

charge K1 mod 2.

The symmetry breaking quantum phase (described by the nonlinear sigma model)

on the domain wall can be easily seen from the bosonic theory. By turning on the

large negative mass squared of the scalar, we land on the symmetry breaking phase

179

described by the nonlinear sigma model with the target space

G =Sp(2)

Sp(1)× Sp(1)=

Sp(2)

Spin(4). (6.58)

In the sigma model, there exists a configuration of skyrmion which also carries the U

charge K1 mod 2.

6.5 Deconfined Gapless U(1) Gauge Theory

In this section, we discuss the scenario where the low energy theory of SU(2) Yang-

Mills at θ = π is a U(1) gauge theory.9 We attempt to find a U(1) gauge theory that

matches the anomaly (6.23).

We consider the time reversal invariant U(1) gauge theory described by the action

S = − 1

4e2

∫

M4

f ∧ ?f +θ

8π2

∫

M4

f ∧ f, θ = 0, 2π, (6.59)

where f = du and u is the U(1) gauge field. The U(1) theory is time reversal

symmetric, where ZT2 acts on the gauge field as

T (u0(t, ~x)) = −u0(−t, ~x), T (ui(t, ~x)) = ui(−t, ~x). (6.60)

This choice of time reversal flips the U(1) gauge charge, while preserves the U(1)

gauge monopole. Hence one can assign monopole Kramers degeneracy, i.e., T 2 to the

lines with charge (qe, qm) = (0, 1).10 In the present case, T 2 = 1 acting on Wilson

lines. Under Lorentz rotation, the Wilson line transforms with integer spin, while the

9We will also comment on θ = 0.10For θ = 0, the dyonic line with charge (qe, qm) = (0, 1) is denoted the ’t Hooft line. However,

for θ = 2π, due to Witten effect, the ’t Hooft line T is has charge (qe, qm) = (1, 1). The dyonic linewith charge (qe, qm) = (0, 1) is W−1T , i.e. ’t Hooft line attached with an anti-Wilson line.

180

’t Hooft line transforms with half integer spin or integer spin depending on θ = 0, 2π,

due to the statistical Witten effect.

(6.59) also has one form symmetries U(1)e,[1]×U(1)m,[1] where the subscripts e and

m represent electric and magnetic respectively. The electric U(1)e,[1] acts on Wilson

lines, and U(1)m,[1] acts on ’t Hooft lines. To make contact with the SU(2) Yang-

Mills, we will focus on the Z2,[1] subgroup of U(1)e,[1]. To couple (6.59) to two-form

background gauge field B, we replace f by f − πB. The action is

S = − 1

4e2

∫

M4

(f − πB) ∧ ?(f − πB) +θ

8π2

∫

M4

(f − πB) ∧ (f − πB), θ = 0, 2π.(6.61)

We further discuss coupling (6.59) to the Lorentz background fields w1, w2.

6.5.1 U(1) Gauge Theory and Spin Liquids at θ = 0

For θ = 0, one can further couple (6.59) to the Lorentz background fields. Changing

B → B + J2w2 modifies the statistics of the U(1) charge. To modify the Lorentz

symmetries of the U(1) monopole, we add to the action a term

1

2(f − πB − J2πw2)(L1w

21 + L2w2). (6.62)

The Lorentz quantum numbers of the U(1) charge E and the U(1) monopole M are

E : h(E) =J2

2mod 1

M : h(M) =L2

2mod 1, T 2

M= (−1)L1+L2 ,

(6.63)

where we use the tilde to emphasize that the time reversal parities of the U(1) charge

and monopole are the opposite compared with the convention in [167], namely the

time reversal flips the charge E other than the monopole M . We will bridge both

conventions at the end of this section.

181

When coupled to all the background fields B,w1, w2 (i.e. by formulating the

theory on an unorientable and non-spin manifold), the U(1) gauge theory with θ = 0

is11

S = − 1

4e2

∫

M4

(f−πB−J2πw2)∧?(f−πB−J2πw2)+1

2

∫

M4

(f−πB−J2πw2)(L1w21+L2w2).

(6.64)

The last term −12

∫M4

(πB + J2πw2) ∧ (L1w21 + L2w2) ⊂ S is not well-defined on an

unorientable manifold. To make sense of it on an unorientable manifold, we need to

promote it to a 5d action,

−π∫

M5

Sq1((B + J2w2)(L1w

21 + L2w2)

). (6.65)

Among the four terms by expanding (6.65), only two terms represent the ’t Hooft

anomalies,

Sanom = −π∫

M5

(L1w21Sq1B + J2L2w2w3), (6.66)

where w3 ≡ w3(TM5) is the Stiefel-Whitney class for the tangent bundle of M5.

When L1 = 1, there is a mixed anomaly between the time reversal and Z2,[1]. When

J2 = L2 = 1, there is an anomaly for the “all fermion electrodynamics”[178, 167, 179].

11The wedge product of the characteristic classes (e.g. Stiefel-Whitney classes) should be under-stood as the cup product. Below, we suppress the cup product for simplicity.

182

We summarize the U(1) gauge theories at θ = 0 and their ’t Hooft anomalies as

(J2, L2, L1) = (0, 0, 0) EbMb 0,

(J2, L2, L1) = (0, 0, 1) EbMbT − π∫

M5

w21Sq1B,

(J2, L2, L1) = (0, 1, 0) EbMfT 0,

(J2, L2, L1) = (0, 1, 1) EbMf − π∫

M5

w21Sq1B,

(J2, L2, L1) = (1, 0, 0) EfMb 0,

(J2, L2, L1) = (1, 0, 1) EfMbT − π∫

M5

w21Sq1B,

(J2, L2, L1) = (1, 1, 0) EfMfT − π∫

M5

w2w3,

(J2, L2, L1) = (1, 1, 1) EfMf − π∫

M5

w21Sq1B + w2w3,

(6.67)

where we used the Lorentz symmetries of the U(1) charge and U(1) monopoles to

label the spin liquid, similar to [167]. However, we emphasize that E is time reversal

odd and M is time reversal even, in contrast to the conventions of [167] where the

time reversal parities are the opposite to ours.

Comparing with the anomalies of SU(2) Yang-Mills with θ = π (6.23), none of

the U(1) spin liquids in (6.67) can be the potential IR candidate phases of SU(2)

Yang-Mills at θ = π. However, we will see in section 6.6 that some of the U(1) spin

liquids in (6.67) can be obtained by Higgsing SU(2) gauge group to U(1) for the

SU(2) Yang-Mills with θ = 0, although it is very unlikely that the deconfined U(1)

spin liquids are dynamically realized by the RG flow.

It is illuminating to connect our identification of the U(1) spin liquids to those in

[167]. In [167], the convention is that U(1) charge E is time reversal even while the

U(1) monopole M is time reversal odd. For θ = 0, the two conventions are related

by S-duality, i.e. E ↔ M,M ↔ E which can be understood as the π/2 rotation of

183

the charge-monopole lattice. Thus we arrive at the following dictionary:

EbMb = EbMb, EbMbT = EbTMb, EbMfT = EfTMb, EbMf = EfMb,

EfMb = EbMf , EfMbT = EbTMf , EfMfT = EfTMf , EfMf = EfMf .

(6.68)

In the dual theory, only M is charged under Z2,[1].

6.5.2 U(1) Gauge theory and Spin Liquids at θ = 2π

We proceed to discuss the U(1) spin liquids with θ = 2π. Similar to section 6.5.1,

one can still modify the statistics of the U(1) charge by replacing B → B + J2w2.

To modify the Lorentz symmetries of the monopole with charge (qe, qm) = (0, 1), we

realize that due to the Witten effect, the ’t Hooft operator (’t Hooft line) carries

θ/2π = 1 electric charge. Thus to form the pure monopole with vanishing electric

charge, one needs to attach a U(1) charge (i.e. a Wilson line). As noted in [169], for

θ = 2π, a dyon with charge (qe, qm) couples to the U(1)e,[1] and U(1)m,[1] background

fields Be and Bm by attaching a surface operator

exp

(i

∫

Σ

(qe − qm)Be + qmBm + (qe − qm)qmπw2

). (6.69)

Applying (6.69) to our case, Be = π(B + J2w2). We demand that when B = 0, the

surface operator for (qe, qm) = (0, 1) should be L1w21 + L2w2. As we will see below,

to match the mixed anomaly between time reversal and Z2,[1], we need to modify the

above expression to L1w21 +L2w2 +B when B is nonvanishing. This implies that both

E and M are charged under Z2,[1], and the mixed T -Z2,[1] anomaly descends from the

mixed anomaly of Z2,[1] ⊂ U(1)e,[1] and Z2,[1] ⊂ U(1)m,[1]. The Lorentz symmetry of

184

the U(1) charge E and U(1) monopole M is

E : h(E) =J2

2mod 1

M : h(M) =L2

2mod 1, T 2

M= (−1)L1+L2 .

(6.70)

Thus we find

Be = π(B + J2w2), Bm = π(L1w

21 + (L2 + J2 + 1)w2

). (6.71)

Notice that the Yang-Mills couples to U(1)e,[1] and U(1)m,[1] background fields Be and

Bm as

S = − 1

4e2

∫

M4

(f−Be)∧?(f−Be)+2π

8π2

∫

M4

(f−Be)∧ (f−Be)+π

2π

∫

M4

(f−Be)Bm.

(6.72)

Substituting (6.71) into (6.72), we obtain the U(1) gauge theory coupled to B,w1, w2

as

S =− 1

4e2

∫

M4

(f − πB − J2πw2) ∧ ?(f − πB − J2πw2)

+2π

8π2

∫

M4

(f − πB − J2πw2)(f − πB − J2πw2)

+π

2π

∫

M4

(f − πB − J2πw2)(L1w

21 + (L2 + J2 + 1)w2

).

(6.73)

The anomaly can be derived by examining the terms in (6.73) that are not well-

defined on an unorientable manifold M4. Such terms are 2π8π2

∫M4

(πB + J2πw2)2 −π2π

∫M4

(πB + J2πw2)(L1w21 + (L2 + J2 + 1)w2) due to the fractional coefficients. To

make sense of these terms, we promote these terms to a 5d integral.

Sanom = π

∫

M5

BSq1B+Sq2Sq1B+(L2+1)Sq1(w2B)+L1w21Sq1B+J2(L2+J2+1)w2w3.

(6.74)

185

As commented in section 6.2, the term Sq1(w2B) is a WZW-like counter term.

We summarize the U(1) spin liquids with θ = 2π and their genuine ’t Hooft

anomalies (i.e. excluding the WZW-like counter terms) as follows:

(J2, L2, L1) = (0, 0, 0) (EbMb)2π π

∫

M5

BSq1B + Sq2Sq1B,

(J2, L2, L1) = (0, 0, 1) (EbMbT )2π π

∫

M5

BSq1B + Sq2Sq1B + w21Sq1B,

(J2, L2, L1) = (0, 1, 0) (EbMfT )2π π

∫

M5

BSq1B + Sq2Sq1B,

(J2, L2, L1) = (0, 1, 1) (EbMf )2π π

∫

M5

BSq1B + Sq2Sq1B + w21Sq1B,

(J2, L2, L1) = (1, 0, 0) (EfMb)2π π

∫

M5

BSq1B + Sq2Sq1B,

(J2, L2, L1) = (1, 0, 1) (EfMbT )2π π

∫

M5

BSq1B + Sq2Sq1B + w21Sq1B,

(J2, L2, L1) = (1, 1, 0) (EfMfT )2π π

∫

M5

BSq1B + Sq2Sq1B + w2w3,

(J2, L2, L1) = (1, 1, 1) (EfMf )2π π

∫

M5

Sq1B + Sq2Sq1B + w21Sq1B + w2w3.

(6.75)

We use the subscript 2π to emphasize that both E and M lines are charged under

Z2,[1]. By rotating the charge-monopole lattice by π/2 (i.e. performing the S-duality),

we are also able to map the U(1) spin liquids in (6.75) to those discussed in [167]. One

simply exchange E ↔ M and M ↔ E. The correspondence has been enumerated in

(6.68). We emphasize that the Z2,[1] one form symmetry background field couples to

both E and M lines in the dual theory.

Notice the WZW-like counter term does not have to be matched along the RG

flow. By matching the genuine ’t Hooft anomalies in (6.75) and the anomalies of

SU(2) Yang-Mills at θ = π, we can enumerate the U(1) spin liquids for each sibling

186

of SU(2) Yang-Mills as follows:

(K1, K2) = (0, 0), (0, 1) : (EbMb)2π, (EbMfT )2π, (EfMb)2π.

(K1, K2) = (1, 0), (1, 1) : (EbMbT )2π, (EbMf )2π, (EfMbT )2π.

(6.76)

The remaining two U(1) spin liquids can not emerge under the RG flow of any sibling

of SU(2) Yang-Mills due to the additional w2w3 anomaly. Merely from matching the

’t Hooft anomalies, we are not able to determine which among the three U(1) spin

liquids in each row of (6.76) is realized for a given (K1, K2). However, by imposing

more physical requirements as we will discuss in section 6.6, we are able to determine

which U(1) spin liquid phase is realized.

6.6 Application II: Gauge Enhanced Quantum

Critical Point Nf ≥ NCFT

In this section, we discuss an application of the deconfinement scenario in section 6.5.

Assuming the SU(2) Yang-Mills at θ = π can flow to a deconfined U(1) gauge theory

which describes the low energy physics of the U(1) quantum spin liquid, it opens up

the possibility of exotic quantum phase transitions between different U(1) spin liquids

and/or trivial paramagnet, where the gauge group is enhanced to SU(2) at and only

at the critical point. We denote such transition as a gauge enhanced quantum critical

point (GEQCP).

187

6.6.1 SU(2) QCD4 and Higher Order Interactions: U(1) Spin

Liquid Phases From Higgsing

We consider the SU(2) QCD4 with Nf fermions, described by the following action

S =

∫

M4

Nf∑

i=1

Ψi(iγµDµ −m)Ψi + Lhigh

− 1

4g2

∫

M4

Tr(F ∧ ?F ). (6.77)

where Ψi is the four component Dirac fermion with a flavor index i = 1, ..., Nf and

a SU(2) color index a = 1, 2 which is suppressed. For the sake of the following dis-

cussion, we have also included a phenomenological four and eight-fermion interaction

term Lhigh,

Lhigh = u3∑

a=1

Nf∑

i=1

ΨiτaΨi

2

+ λ

3∑

a=1

Nf∑

i=1

ΨiτaΨi

2

2

, (6.78)

where τa (a = 1, 2, 3) denotes the generator of the SU(2) gauge group. We will always

take λ > 0 and allow u to be either sign. Throughout, we assume there is a flavor

symmetry Sp(Nf ) or U(Nf ) such that the masses of all the flavors of fermions are

degenerate.

We work in the parameter regime of Nf ≥ NCFT such that the QCD4 with m = 0

flows to a conformal field theory which can describe a second order phase transition

between the two semi-classical phases (which we will discuss in detail below). In

particular, when Nf > 11, the QCD4 with m = 0 is in the infrared free phase and the

coupling constant g flows to zero under RG. At this RG fixed point, the only relevant

perturbation is the fermion mass m, and the terms in Lhigh are irrelevant. Thus for

m = 0, adding the higher order terms Lhigh in (6.77) does not affect the dynamics

in the IR. In particular, u, g and λ all flow to zero, as shown in the middle panel of

figure 6.3.

188

Figure 6.3: Schematic RG flow diagram around the QCD4 fixed point for odd Nf andNf > 11. Possible IR fates are listed for completeness, although some (such as theU(1) SL on the θ = 0 side) may be extremely unlikely.

We proceed to discuss the mass deformation by allowing m to be either positive or

negative.12 We focus on the case when Nf is an odd integer. Then depending on the

sign of m, the QCD flows to the SU(2) Yang-Mills theory with either θ = 0 (for m > 0)

or θ = π (for m < 0). The SU(2) Yang-Mills theory does not describe the ultimate

IR fate of the system. It continues to flow towards different possible IR fixed points

as we have discussed in previous sections. One possibility is that the system enters

the confinement phase, where the coupling g flows large away from the m = 0 QCD4

fixed point. In the confinement phase on the θ = π (m < 0) side, Z2,[1] is unbroken

and time reversal symmetry is spontaneously broken [143, 180]. Another possibility

is that the system remains deconfined with a reduced gauge group, which can lead

to either a U(1) or a Z2 spin liquid phase. The possible U(1) spin liquid phases that

saturate the ’t Hooft anomalies are provided in section 6.5. In the rest of this section,

we provide a potentially possible mechanism for the deconfinement scenario to take

place, and we further determine, if so, which type of U(1) spin liquid (among the

candidates in (6.76)) is indeed realized for a given sibling of SU(2) Yang-Mills.

12In general, the mass parameter in 4d QCD can be complex, which is obvious when we rewritethe Dirac fermions into Weyl fermions with both chirality. In this work, we focus on the real massfor simplicity.

189

Viewing the SU(2) Yang-Mills as the large mass deformation limit of a SU(2)

QCD4 allows us to propose a natural mechanism to realize the deconfinement scenario

in section 6.5. When |m| is nonzero, it is possible that the interaction strength u and

λ in (6.77) and (6.78) could flow strong. Assuming u < 0, the higher order term (6.78)

drives the condensation of SU(2) gauge triplet ΨiτΨi and consequently Higgses the

SU(2) gauge group to its subgroup. If only one component of the gauge triplet acquires

expectation value, e.g. 〈Ψiτ3Ψi〉 6= 0, the SU(2) gauge group will be Higgsed down to

its U(1) subgroup. The remaining low-energy theory will be a U(1) Maxwell theory

that describes the U(1) spin liquid. It will be important in section 6.6.2 that after

Higgsing, each flavor of Ψi gives rise to two types of fermions Ψ1i,Ψ2i which carry

opposite U(1) gauge charge. Ψ1i carries U(1) charge 1, while Ψ2i carries U(1) charge

−1. If more than one components of the gauge triplet acquire expectation values

(depending on the details of higher order interactions), e.g. 〈Ψiτ1Ψi〉, 〈Ψiτ

2Ψi〉 6= 0,

then the remaining gauge group will be Z2, realizing the TQFT description of the

topologically ordered Z2 spin liquid phase.(See [108] for such Z2 spin liquid phases.)

In the following, we will take the U(1) spin liquid as the example to illustrate the

deconfined phase. The schematic RG flow diagram is shown in figure 6.3.

We comment on the possibilities of the signs of m and u in (6.77) and (6.78), and

their consequences.

1. u > 0 for both m > 0 and m < 0: In this scenario, the gauge group SU(2) is

not Higgsed. When m is positive, the theory flows to a trivial gapped phase, in

accord with the standard lore.[148] When m is negative, the theory flows to a

strongly coupled confining phase where time reversal is spontaneously broken.

2. u < 0 for both m > 0 and m < 0: In this scenario, the gauge group SU(2)

is Higgsed for both signs of m, with the only exception at m = 0. The SU(2)

Yang-Mills with both θ = 0 and π flow to certain U(1) spin liquids. We will

determine on the U(1) spin liquid in section 6.6.2. We emphasize that although

190

it is extremely unlikely that SU(2) Yang-Mills with θ = 0 flows to a deconfined

U(1) gauge theory and is beyond the standard lore [148], this scenario is still

not completely ruled out rigorously.

3. u > 0 for m > 0, and u < 0 for m < 0: The signs of m and u are correlated.

In this scenario, the gauge group SU(2) is Higgsed only for θ = π. While for

θ = 0, the SU(2) Yang-Mills flows to a trivial gapped phase, consistent with the

lore [148]. However, the underlying mechanism for the sign correlation between

u and m still needs to be understood.

6.6.2 Symmetries Realizations and Symmetry Enriched U(1)

Spin Liquids in the Infrared

The specific type of the U(1) spin liquid that is realized under the gauge triplet con-

densation depends on how the time-reversal symmetry is implemented in the QCD

theory (6.77). We consider the following two possibilities of time reversal implemen-

tation, where the gauge and global symmetries are

CI :SU(2)× Sp(Nf )× ZT4

Zc2 × Zf2, (6.79)

CII :SU(2)× U(Nf )

Zf2× ZT2 . (6.80)

For (6.79), the SU(2) ≡ Sp(1) gauge transformation and the time reversal sym-

metry act on the fermionic matter field as SU(2) : Ψi → eiθ·τΨi and

CI : T : Ψi → Kγ5γ0Ψ†i . (6.81)

In particular T 2 = −1 on Ψi. Here the Zc2 center of SU(2) is the same as the fermion

parity Zf2 ; we mod out Zc2 = Zf2 twice because SU(2), Sp(Nf ) and ZT4 all share the

same normal subgroup Zc2 = Zf2 . Sp(Nf ) is the flavor symmetry. If we just focus on

191

the SU(2) and time reversal (i.e. ignore the flavor symmetry Sp(Nf )), this symmetry

coincides with the CI symmetry class in the ten fold classification of the fermionic

SPT. This motivates an alternative way to understand the SU(2) QCD4 (6.77): The

SU(2) QCD4 with symmetry class (6.79) can be understood as from gauging the

SU(2) global symmetries of Nf copies of free fermions in symmetry class CI.

For (6.80), the SU(2) gauge transformation acts in the same way as in the CI

class. However, time reversal acts on the fermionic matter field differently:

CII : T : Ψi → Kiγ5γ0Ψ†i . (6.82)

Compared with (6.81), there is an additional U(1) ⊂ U(Nf ) flavor transformation.

In particular, T 2 = 1. The quotient in (6.80) is to identify the common normal

subgroup of SU(2) and U(Nf ). The SU(2) QCD4 with symmetry class (6.80) can be

understood as from gauging the SU(2) global symmetries of Nf copies of free fermions

in symmetry class CII.

Under the condensation of 〈Ψiτ3Ψi〉 6= 0, the remaining U(1) gauge group acts as

U(1) : Ψi → eiθτ3Ψi. The U(1) generator commutes with the time reversal transfor-

mation, which forms the AIII symmetry class. The class AIII fermionic SPT state is

Z8 × Z2 classified, where only the phases associated with Z8 can be represented by

the free fermion theories.13 Turning on the fermion mass m effectively put the Ψi

field in the class AIII fermionic SPT states labeled by the topological index ν = 0

(m > 0) or ν = 2Nf (m < 0). Connecting with the U(1) gauge theories in section 6.5,

θ = νπ. If Ψi is in the ν = 0 phase, the U(1) monopole is simply a boson. If Ψi is in

the ν = 2Nf phase, the U(1) monopole will carry will carry 2Nf fermion zero modes.

However, these zero modes carry U(1) gauge charge. To form U(1) gauge invariant

monopole operator, we need to consider only those monopole that are neutral under

13Before gauging, the AIII SPT theory is simply Nf free fermions coupled to U(1) backgroundfields. Hence only the Z8 part is relevant for our purpose.

192

U(1). We note that under Higgsing, both CI and CII classes reduce to AIII classes,

Higgsing : CI→ AIII, CII→ AIII. (6.83)

(6.83) can also be interpreted as different ways of embedding AIII symmetry class into

CI and CII classes. See [164] for extensive discussions of the embedding in (6.83) and

other examples among the ten Cartan symmetry classes. The difference between the

two reduced AIII classes are that the U(1) neutral monopoles have different symmetry

quantum numbers, which we determine below.

We proceed to determine the time reversal properties (Kramers degeneracy) of

the time reversal symmetric monopole operators of charge (qe, qm) = (0, 1). For

illustrative purposes, we first determine the time reversal properties of the monopole

in AIII class ν = 2 (i.e. Nf = 1 copy of AIII system and the topological theta

parameter in the U(1) Mexwell theory is θ = 2π) with the global symmetry

U(1)× ZT4Zf2

. (6.84)

The time reversal properties of the fermion zero modes descend from the time reversal

transformations in (6.81) and (6.82). In (6.81), time reversal maps a fermion to its

conjugate, and only the spinor indices are rotated. Hence the fermion zero mode ca

(for Nf = 1), where a = 1, 2 is the SU(2) index, maps under time reversal as

CI : T : ca → c†a, c†a → ca. (6.85)

In (6.82), time reversal maps a fermion to its conjugate, accompanied by a Z4 ⊂ U(1)

transformation generated by i. Hence the fermion zero mode ca maps under time

193

reversal as

CII : T : ca → ic†a, c†a → −ica. (6.86)

Using the operator-state correspondence, the monopoleM without any fermion zero

mode occupied is mapped to a state |0〉 with ca|0〉 = 0 for a = 1, 2. Under T , the

empty state |0〉 is mapped to a fully occupied state c†1c†2|0〉, i.e. T |0〉 = c†1c

†2|0〉. We

further notice that the two fermion zero modes has opposite gauge charge. c1 carries

U(1) charge 1, while c2 carries U(1) charge −1. Thus the U(1) neutral monopole

operators are associated with the states

|0〉, c†1c†2|0〉 (6.87)

rather than the half filled states c†1|0〉, c†2|0〉. Combined with (6.85) and (6.86), we can

compute T 2 of the empty and full states in (6.87) as

CI : T 2|0〉 = c1c2c†1c†2|0〉 = −|0〉, T 2c†1c

†2|0〉 = c†1c

†2c1c2c

†1c†2|0〉 = −c†1c†2|0〉,

CII : T 2|0〉 = −c1c2c†1c†2|0〉 = |0〉, T 2c†1c

†2|0〉 = −c†1c†2c1c2c

†1c†2|0〉 = c†1c

†2|0〉.

(6.88)

In short, for Nf = 1 (or ν = 2), the (qe, qm) = (0, 1) monopole is Kramers doublet

(T 2 = −1) in the AIII class descended from CI, while Kramers singlet (T 2 = 1)

in the AIII class descended from CII. Moreover, in both cases, the (qe, qm) = (0, 1)

monopole is a boson, which is obvious from (6.70).

Using similar analysis for the monopole quantum numbers in (6.88) for the AIII

class ν = 2, it is straightforward to obtain the monopole quantum numbers for AIII

194

class ν = 2Nf , which is

CI : T 2 = (−1)Nf ,

CII : T 2 = 1,

(6.89)

for monopoles associated with the U(1) neutral states |0〉, c†1ic†2j|0〉, ..., (c†1i1 ...c†1iNf

c†2j1 ...c†2jNf

)|0〉

where the number of 1 and 2 of the SU(2) indices should balance. Since we focus on

the case where Nf is odd, the time reversal Kramers degeneracy for the two cases in

(6.89) are different.

We emphasize that the quantum numbers in (6.89) are for the probe monopoles in

the AIII symmetry classes. Further gauging the U(1) global symmetries of the AIII

fermionic SPTs lead to different U(1) spin liquids. Thus (6.89) also characterizes the

quantum numbers of the dynamical monopoles in the U(1) spin liquids.

We are ready to identify the U(1) spin liquid phases in the IR. We first determine

the candidate U(1) spin liquid for SU(2) Yang-Mills with θ = 0. Since in SU(2) QCD4,

the SU(2) gauge field is coupled to fermions, the SU(2) Yang-Mills theories should

have fermionic Wilson lines, i.e., K2 = 1. (See an similar discussion in section 6.4.1.

) On the other hand, the U(1) charges should also be fermionic because they descend

from Higgsing the fermionic SU(2) charges, i.e. E should be fermionic. Combining

with the U(1) monopole quantum numbers in (6.89), we find that, when m < 0,

the QCD in the CI class flows to (EfMbT )2π, while the QCD in CII class flows to

(EfMb)2π.

The make contact with the siblings of SU(2) Yang-Mills at θ = π, we further

need relate the U(1) spin liquids determined above to the labels of the siblings, i.e.

(K1, K2). As we find above, K2 = 1. Furthermore, K1 can be determined by matching

the anomaly of the U(1) spin liquids in (6.75) with the anomaly of the SU(2) Yang-

Mills (6.23). Thus we we determine the U(1) spin liquids as well as the siblings of

195

the Lorentz symmetry enriched SU(2) Yang-Mills as

ν = 2Nf : CI : (K1, K2) = (1, 1), → AIII : (EfMbT )2π. (6.90)

ν = 2Nf : CII : (K1, K2) = (0, 1), → AIII : (EfMb)2π. (6.91)

The U(1) spin liquids on the m > 0 side is simply

ν = 0 : CI,CII : (K1, K1) = (0, 1), (1, 1) → AIII : EfMb. (6.92)

Thus we have singled out a particular symmetry enriched U(1) spin liquid as the

low energy of SU(2) Yang-Mills from the anomaly matched candidates in (6.76), by

embedding the SU(2) Yang-Mills into a SU(2) QCD4 with the assumed SU(2) triplet

Higgsing pattern. (6.90) and (6.91) are precisely the time reversal CFTs initially

proposed [164].

We finally comment that although Ψi in (6.81) satisfies T 2 = −1, this does not

mean Ψi is Kramers doublet, because Ψi is not mapped to itself under time reversal.

See [103] for an analogue discussion in 2 + 1d. A priori, it seems to be difficult to

determine the (K1, K2) from the symmetry assignment (6.81). Here, we provide a way

to determine it through identifying the U(1) spin liquid (EfMbT )2π and via anomaly

matching. Analogue comments also apply to (6.82).

6.6.3 Gauge Enhanced Quantum Critical Points

From the U(1) spin liquids determined in section 6.6.2, we are able to predict a series

of gauge enhanced quantum critical points (GEQCP) using SU(2) QCD4. We will

focus on the second and third scenarios in section 6.6.1 which involve U(1) spin liquid

phases, and finally comment on the first scenario where no U(1) spin liquid phases

are involved.

196

We first discuss the second scenario in section 6.6.1 where the fermion bilinear

condensation takes place for both m > 0 and m < 0, realizing EfMb and (EfMbT )2π

respectively for the sibling (K1, K2) = (1, 1), while EfMb and (EfMb)2π respectively

for the sibling (K1, K2) = (0, 1). For simplicity, we will mainly discuss the sibling

(K1, K2) = (1, 1) below. The transition between EfMb and (EfMbT )2π spin liquids

can be realized by tuning the mass m in (6.77), assuming the SU(2) Yang-Mills theory

can flow to the deconfined U(1) Maxwell theory on both sides. At m = 0, both the

gauge coupling g and the interaction u are irrelevant (if Nf > 11), such that the

transition is controlled by the IR free QCD fixed point. This provides a novel GEQCP

scenario for the Kramer-changing quantum criticality between EfMb and (EfMbT )2π

spin liquids as a QCD theory, where the gauge group is enhanced from U(1) to

SU(2) at the critical point, which is different from the QED description proposed in

Ref.[167]. Nevertheless, similar to Ref.[167], additional symmetries must be imposed

to guarantee a single direct transition, otherwise the critical point can be interrupted

by other time reversal invariant terms such as the alternating chemical potential term

ψ†iγ5ψi or can be split to multiple transitions if different fermion flavors have different

masses. One simple way is to demand an inversion symmetry I : ψi → γ0ψi,x→ −x

together with the Sp(Nf ) flavor symmetry.

We proceed to the third scenario in in section 6.6.1 where the fermion bilin-

ear condensation takes place only for m < 0, realizing (EfMbT )2π for the sibling

(K1, K2) = (1, 1), while (EfMb)2π for the sibling (K1, K2) = (0, 1). On the m > 0

side, the theory flows to a trivial vacua. For simplicity, we only discuss the sibling

(K1, K2) = (1, 1). The QCD theory also afford a GEQCP scenario for the phase tran-

sition between the (EfMbT )2π U(1) gauge theory and the trivially confined vacuum.

The conventional transition from a EfMb U(1) spin liquid to a trivial paramagnet

can happen by monopole condensation (as a confinement transition). Note that E

is a fermion and can not be condensed, unless condensing in pairs which would lead

197

to a Z2 topological order. However for the (EfMbT )2π spin liquid, if we condense

the monopole, the time reversal symmetry will be spontaneously broken because the

monopole is a Kramers doublet. It seems difficult to drive a direct transition from the

(EfMbT )2π spin liquid to a trivial paramagnet. Nevertheless, our analysis provides

a compelling possibility by first enlarging the gauge group from U(1) to SU(2) and

then allowing the SU(2) to confine trivially by removing tuning to the θ = 0 side.

As shown in the flow diagram Fig.6.3, it is possible to connect the (EfMbT )2π spin

liquid and the trivial paramagnet in the parameter space by going through the plane

of m = 0, which is controlled by the QCD fixed point, where an enlarged SU(2)

gauge group together with gapless fermionic partons will emerge. This constitutes

yet another example of the GEQCP.

We finally comment on the first scenario, where time reversal is spontaneously

broken for m < 0, and a trivial gapped phase is realized for m > 0. If this scenario

takes place, the SU(2) QCD4 with odd Nf fundamental fermions can access as a

second order deconfined phase transition, where deconfinement is realized at and only

at the critical point. This scenario is discussed in [180]. See [181, 182, 183, 184] for

other deconfined quantum critical points (DQCP) between various confining phases.

198

Appendix A

Appendices for Chapter 2

A.1 Conventions for MPS and Canonical MPS

A.1.1 Conventions for MPS and Transfer Matrix

Since each unit cell contains q spins-12’s, it is natural to start with the translational

invariant MPS in Eq. (2.38), i.e.,

|GS〉 =∑

{gri }

Tr

( L−1∏

r=0

T gr1 ...g

rq

)|{gri }〉. (A.1)

For convenience, we introduce the notation of the physical operators acting on the

MPS tensors. Denoting Xri and Zr

i as the Pauli X and Z operators acting on i-th

orbital (i = 1, . . . , q) in the r-th unit cell, their action on the MPS matrices are

defined as:

Xri ◦ T g

r′1 ...g

r′i ...g

r′q =

T gr′1 ...(1−gr

′i )...gr

′q if r′ = r

T gr′1 ...g

r′i ...g

r′q if r′ 6= r,

(A.2)

and

Zri ◦ T g

r′1 ...g

r′i ...g

r′q = (−1)δrr′g

r′i T g

r′1 ...g

r′i ...g

r′q . (A.3)

199

For other, more complex operators, the notation ◦ can be naturally generalized.

To make the equations more compact, let hi ∈ {1, ..., D} be the virtual indices of

the MPS matrices, where D is the bond dimension. Notice that the bold font hi is

different from the Z2 valued virtual indices h’s in the main text. For instance, the

MPS matrix elements of Eq. (2.31) become Tgr1g

r2gr3

h1h2,h3h4≡ (T g

r1gr2gr3)h1,h2 , so we identify

h1 and h2 as the composite of Z2 valued h indices, i.e., h1h2 and h3h4 respectively.

Given the MPS matrix elements (T gr1 ...g

rq )h1,h2 , where h1,h2 ∈ {1, ..., D} are the left

and right virtual indices, we can construct the MPS transfer matrix Th1h3,h2h4 by

contracting over the physical indices,

Th1h3,h2h4 =∑

gr1 ...grq

(T gr1 ...g

rq )h1,h2(T g

r1 ...g

rq )∗h3,h4

. (A.4)

Here, h1h3 is regarded as a composite left virtual index of the transfer matrix, of

dimension D2. The same applies to h2h4. The transfer matrix T is a D2×D2 matrix.

A.1.2 Review of Canonical MPS

We now review the definition and the properties of canonical MPS, and apply the

canonical MPS to stabilizer codes. The MPS matrix T gr1 ...g

rq is called “canonical” if

its transfer matrix satisfies:

∑

h2

(T)h1h3,h2h2

= δh1h3 ,

∑

h1,h3

Λh1h3

(T)h1h3,h2h4

= Λh2h4 ,

(A.5)

where T is the transfer matrix of T gr1 ...g

rq , and Λ is a full-rank diagonal matrix whose

diagonal elements are the entanglement spectrum of a single cut. In Ref. [185], it was

shown that a generic MPS matrix T gr1 ...g

rq on an open chain can be mapped to the

200

canonical form T gr1 ...g

rq via a similarity transformation

T gr1 ...g

rq = S · T gr1 ...grq · S−1, (A.6)

where S is an invertible matrix. We use ˘ to denote the canonical form of the MPS

matrix and the MPS transfer matrix throughout the appendix.

In Ref. [186], it was proved that when there is a non-degenerate ground state on

any compact space, the entanglement spectrum of a stabilizer code ground state is

flat. The reduced density matrices are, in fact, projectors. Their original proof was

formulated in 2 spatial dimensions, but it can be directly generalized to arbitrary

dimensions. See Ref. [87] for the application to 3 spatial dimensions. Here we apply

their conclusion to the case of 1 spatial dimension. Hence, the entanglement spectrum

of a 1D stabilizer code with PBC is flat.

The reduced density matrix on a local and contractible region of a gapped state

should not depend on the boundary condition far away from the local region. Thus

the entanglement spectrum does not depend on the boundary condition either. Thus

for the 1D stabilizer code with OBC, the entanglement spectrum is flat. Hence Λ

in Eq. (A.5) is also flat for one of the ground states with OBC. Since Λ is full-rank,

there are no zero diagonal elements in Λ and Λ is proportional to an identity matrix.

Hence the canonical MPS of a stabilizer code satisfies the following conditions

∑

h2

(T)h1h3,h2h2

= δh1h3 ,

∑

h1

(T)h1h1,h2h4

= δh2h4 .

(A.7)

The two conditions in Eq. (A.7) are graphically represented in Fig. A.1.

Hence we can use Eq. (A.7) to solve for the MPS with OBC. By Assumption 2.0.3,

the MPS matrices for the OBC shall also be the MPS matrices for the PBC.

201

Figure A.1: Graphical representation of Eq. (A.7).

A.2 Correlation Functions and Transfer Matrix

Eigenvalues

In this appendix, we derive the eigenvalue structure of the transfer matrix of a general

translational invariant stabilizer code. As we will prove, there is only one nonzero

eigenvalue of the MPS transfer matrix, obtained by Jordan decomposition. Moreover,

a finite power of the MPS transfer matrix can be decomposed as a tensor product of

two vectors. The lemmas and theorems will be used in App. A.3.

Lemma A.2.1. Suppose an operator O anti-commutes with some of the Hamiltonian

terms in Eq. (2.33), i.e., H = −∑L−1r=0

∑tα=1Orα, its expectation value of the ground

state of Eq. (2.33) satisfies

〈GS|O|GS〉 = 0. (A.8)

Proof. Without loss of generality, suppose O anti-commutes with O01 in Eq. (2.33).

Since the ground state |GS〉 satisfies the stabilizer condition Eq. (2.37), we have

〈GS|O|GS〉 =〈GS|OO01|GS〉

=− 〈GS|O01O|GS〉

=− 〈GS|O|GS〉.

(A.9)

202

Hence

〈GS|O|GS〉 = 0. (A.10)

Consider two operators σi, i = 1, 2. We denote p1 (resp. p2) the support of σ1

(resp. σ2) on the unit cells r1 ≤ r ≤ r1 + p1 − 1 (resp. r2 ≤ r ≤ r2 + p2 − 1). We

define the distance d(σ1, σ2) of the two operators as the number of unit cells between

the two operators plus one, i.e.,

d(σ1, σ2) =

r2 − r1 − p1 + 1 , r2 ≥ r1 + p1

r1 − r2 − p2 + 1 , r1 ≥ r2 + p2

0 , r1 + p1 > r2 > r1 − p2.

(A.11)

In particular, when two operators overlap even only on one site, their distance is zero.

When the distance of two operators σ1 and σ2 are larger than P , where P is the range

of another operator O, then O can not overlap simultaneously with σ1 and σ2.

Lemma A.2.2. Suppose σ1 and σ2 are products of Pauli matrices supported on dif-

ferent regions of distance larger than the maximal interaction range, i.e.:

d(σ1, σ2) > max{P1, . . . , Pt}, (A.12)

where Pα is the support of α-th type of the Hamiltonian term Orα. Then, their expec-

tation values satisfy

〈GS|σ1σ2|GS〉 = 〈GS|σ1|GS〉〈GS|σ2|GS〉. (A.13)

203

Proof. σ1 and σ2 either commute or anti-commute with the Hamiltonian terms, be-

cause σ1, σ2 and stabilizer operators are all products of Pauli matrices. We prove this

lemma case by case:

1. σ1 and σ2 both commute with all stabilizer operators.

[H, σi] = 0, i = 1, 2. (A.14)

Hence for any excited eigenstate |E, k〉 of the Hamiltonian H, i.e., H|E, k〉 =

E|E, k〉 (E is the energy and k labels the degeneracy within the energy

eigenspace), σi|E, k〉 is also an excited eigenstate of H. One can see this from

Eq. (A.14): [H, σi]|E, k〉 = 0 for i = 1, 2, which implies σi|E, k〉 is an energy

eigenstate of H with energy E. So

〈GS|σi|E, k〉 = 0, i = 1, 2. (A.15)

Then

〈GS|σ1|GS〉〈GS|σ2|GS〉

= 〈GS|σ1

(1−

∑

E,k

|E, k〉〈E, k|)σ2|GS〉

= 〈GS|σ1σ2|GS〉 −∑

E,k

〈GS|σ1|E, k〉〈E, k|σ2|GS〉

= 〈GS|σ1σ2|GS〉,

(A.16)

where in the first equality, we have used Assumption 2.0.1, and in the last

equality, we have used Eq. (A.15). Hence Eq. (A.13) holds true in this case.

2. σ1 commutes with all stabilizer operators while σ2 anti-commutes with some of

the stabilizer operators. Hence, σ2 and σ1σ2 both satisfy Lemma A.2.1. Their

204

expectation values are both 0:

〈GS|σ2|GS〉 = 0, 〈GS|σ1σ2|GS〉 = 0. (A.17)

Therefore, Eq. (A.13) holds true in this case.

3. σ1 anti-commutes with some of the stabilizer operators while σ2 commutes with

all stabilizer operators. This is the same situation as the last one. Both sides

of Eq. (A.13) vanish.

4. σ1 and σ2 both anti-commute with some of stabilizer operators. Using Lemma

A.2.1, their expectation values both vanish. There does not exist a stabi-

lizer operator which overlaps simultaneously with σ1 and σ2, because σ1 and

σ2 are separated with a distance larger than the maximal interaction range

max{P1, . . . , Pt}. Hence, σ1σ2 still anti-commutes with some of the stabilizer

operators. So both sides of Eq. (A.13) vanish.

This completes the proof.

Theorem A.2.3. Suppose two arbitrary operators O and O are supported on different

regions separated by a distance larger than max{P1, . . . , Pt}. Then we have

〈GS|OO|GS〉 − 〈GS|O|GS〉〈GS|O|GS〉 = 0. (A.18)

Proof. First we can expand the two operators as the summations of the products of

Pauli matrices:

O =∑

i

φiσi

O =∑

j

θjσj,

(A.19)

205

where the terms σi and σj are products of Pauli matrices supported in two separated

regions, and φi and θj are complex coefficients. Recall our assumption that O and O

are supported on different regions separated by a distance larger than the maximal

interaction range max{P1, . . . , Pt}. Then, σi and σj are also supported on regions

with a distance larger than max{P1, . . . , Pt}. Hence, σi and σj satisfy Lemma A.2.2.

Therefore

〈GS|OO|GS〉 =∑

i,j

φiθj〈GS|σiσj|GS〉

=∑

i,j

φiθj〈GS|σi|GS〉〈GS|σj|GS〉

=〈GS|O|GS〉〈GS|O|GS〉.

(A.20)

This completes the proof.

Theorem A.2.4. Let Tgr1 ...g

rq

h1,h2be the MPS matrix element of a translational invariant

stabilizer code where h1 and h2 are the virtual indices, and Th1h3,h2h4 be the MPS

transfer matrix of T defined in Eq. (A.4). Then T has only 1 nonzero eigenvalue.

Proof. For convenience, we introduce the notation:

T[Or]h1h3,h2h4

=∑

gr1 ...grq

(Or ◦ T gr1 ...grq )h1,h2(T gr1 ...g

rq )?h3,h4

.(A.21)

Moreover, the transfer matrix T can always be decomposed into Jordan blocks:

T = U(Pλ0 + Pλ1 + Pλ2 + · · · )U−1, (A.22)

where |λ0| > |λ1| > |λ2| > · · · are the eigenvalues of T, and Pλi is the corresponding

Jordan block. By a proper scaling of T, we let λ0 = 1. Using this normalization, Pλ0 ≡

P1 is non-degenerate due to the gap and non-degeneracy of the ground state.[187, 4]

206

Without loss of generality, let us consider the special basis of the virtual indices such

that U is an identity matrix, i.e.,

T = Pλ0 + Pλ1 + Pλ2 + · · · . (A.23)

Suppose we have two operators Or and Or+l with a sufficiently large (but finite) l

such that they satisfy Theorem A.2.3. The expectation value of Or and Or+l can be

written in terms of transfer matrices:

〈GS|OrOr+l|GS〉 =Tr[Tr(T[Or])Tl−1(T[Or+l])TL−1−r−l

]

Tr (TL)

=Tr[TL−l−1(T[Or])Tl−1(T[Or+l])

]

Tr (TL).

(A.24)

By Assumption 2.0.2 in the beginning of Sec. 2, the MPS matrices is independent of

the system size when L is sufficient large. For simplicity, let us take the limit:

limL→∞

TL−l−1 = limL→∞

TL = P1. (A.25)

Eq. (A.24) then simplifies to

〈GS|OrOr+l|GS〉 =Tr[P1(T[Or])Tl−1(T[Or+l])

]

Tr (P1)

=Tr[P1(T[Or])Tl−1(T[Or+l])

].

(A.26)

Using the Jordan blocks decomposition of T (λ0 = 1), we have

Tl−1 = P1 +∑

|λ|<1

P l−1λ . (A.27)

207

Substituting to Eq. (A.26), we have

〈GS|OrOr+l|GS〉

=Tr

P1(T[Or])(P1 +

∑

|λ|<1

P l−1λ )(T[Or+l])

=Tr[P1(T[Or])P1(T[Or+l])

]+∑

|λ|<1

Tr[P1(T[Or])P l−1

λ (T[Or+l])].

(A.28)

Since P1 is 1 dimensional (unique gapped ground state),

Tr[P1(T[Or])P1(T[Or+l])

]=Tr (P1T[Or]) Tr

(P1T[Or+l]

)

=〈GS|Or|GS〉〈GS|Or+l|GS〉.(A.29)

Hence

〈GS|OrOr+l|GS〉 = 〈GS|Or|GS〉〈GS|Or+l|GS〉+∑

|λ|<1

Tr[P1(T[Or])P l−1

λ (T[Or+l])].

(A.30)

Theorem A.2.3 implies that:

∑

λ 6=1

Tr[P1(T[Or])P l−1

λ (T[Or+l])]

= 0 (A.31)

for any operators Or and Or+l with a sufficiently large but finite l. Then the only

possibility is that for all λ 6= 1, λ = 0. In other words, the only nonzero eigenvalue

of T is 1. This completes the proof.

We numerically checked the Zq−1XZq−1 models with 2 ≤ q ≤ 6 and found that

the transfer matrix indeed has only 1 nonzero eigenvalue.

208

Lemma A.2.5. For a Jordan block P0 of size m ×m with zero diagonal elements,

then

(P0)n = 0, (A.32)

where the integer n ≥ m.

Proof. In terms of matrix elements, P0 is:

P0 =

0 1 0 . . . 0

0 0 1 . . . 0

. . . . . .

0 0 0 . . . 1

0 0 0 . . . 0

(A.33)

Denote ei as the vector of size m whose i-th entry is 1 and 0 otherwise. Then we can

show that:

P0 · e1 = 0,

P0 · ei = ei−1, ∀i = 2, 2, . . . ,m.

(A.34)

Hence, for any vector ei (i = 1, 2, . . . ,m), we can prove that:

(P0)m · ei = (P0)m−1 · ei−1 . . . = (P0)m−i+1 · e1 = 0 (A.35)

Therefore, we conclude that:

(P0)m = 0. (A.36)

For any integer n ≥ m, we also have:

(P0)n = 0. (A.37)

209

Theorem A.2.6. Suppose the transfer matrix T of size D2 × D2 satisfies Theorem

A.2.4. In other words, its nonzero eigenvalues contain a unique 1. Then

(T)D2

= uv (A.38)

for a column vector u of size D2 and a row vector v of size D2 such that

v · u = 1, (A.39)

where · represents the vector multiplication. In terms of matrix elements, Eq. (A.38)

is(

(T)D2)h1h3,h2h4

= uh1h3vh2h4 , (A.40)

and Eq. (A.39) isD∑

h1,h2=1

uh1h2vh1h2 = 1. (A.41)

Proof. Using the fact that T satisfies Theorem A.2.4, its Jordan decomposition is:

T = U(P1 + P0)U−1, (A.42)

where P1 is the projector into the 1 dimensional Jordan block for eigenvalue 1 and

P0 is the projector into the Jordan block for eigenvalue 0. Therefore,

TD2

= U(PD2

1 + PD2

0 )U−1 = UP1U−1, (A.43)

where we have used Lemma A.2.5 and the fact that the size of P0 is smaller than

D2 ×D2:

PD2

0 = 0. (A.44)

210

Since the Jordan block with eigenvalue 1 is 1 dimensional, there is only one nontrivial

matrix element which locates at the diagonal of P1. Without loss of generality, we

assume that the only nonzero element of P1 locates at 1-th row and 1-th column.

Hence, we can write this equation in terms of matrix elements

TD2

h1h3,h2h4=Uh1h3,1

(U−1

)1,h2h4

≡uh1h3vh2h4 ,

(A.45)

where we define

uh1h3 ≡ Uh1h3,1

vh2h4 ≡(U−1

)1,h2h4

.

(A.46)

From these definitions

v · u = (U−1 · U)1,1 = 1. (A.47)

This completes the proof.

Now we explore the properties for the canonical MPS with the tensor T and

Eq. (A.7).

Lemma A.2.7. For a stabilizer code, the transfer matrix of the ground state canonical

MPS satisfies Eq. (A.7). We prove that:

∑

h1

(Tn)h1h1,h2h4

= δh2h4 ,∑

h2

(Tn)h1h3,h2h2

= δh1h3 , (A.48)

for any integer n > 0.

211

Proof. Using the definition of the canonical MPS in Eq. (A.7), we first show that

∑

h1

(Tn)h1h1,h2h4

=∑

h1,h5,h6

Th1h1,h5h6

(Tn−1

)h5h6,h2h4

=∑

h5,h6

δh5h6

(Tn−1

)h5h6,h2h4

=∑

h1

(Tn−1

)h1h1,h2h4

.

(A.49)

Then we repeatedly apply this equation until there is only 1 T matrix.

∑

h1

(Tn)h1h1,h2h4

=∑

h1

(Tn−1

)h1h1,h2h4

=∑

h1

(Tn−2

)h1h1,h2h4

...

=∑

h1

(T)h1h1,h2h4

=δh2h4 .

(A.50)

Similarly, we can prove the other equation. This completes the proof.

Lemma A.2.8. For a stabilizer code, the transfer matrix of its ground state canonical

MPS satisfies Theorem A.2.6. We prove that the elements of u and v are

uh1h2 =δh1h2

Tr(v), vh1h2 =

δh1h2

Tr(u), (A.51)

where

Tr(u) =∑

h

uhh, Tr(v) =∑

h

vhh. (A.52)

In other words,

(TD2

)h1h3,h2h4

=1

Tr(u)Tr(v)δh1h3δh2h4 =

1

Dδh1h3δh2h4 . (A.53)

212

Proof. Using Theorem A.2.6 for a canonical MPS, we have:

(TD2

)h1h3,h2h4

= uh1h3vh2h4 . (A.54)

Applying Lemma A.2.7 with n = D2, we obtain:

δh2h4 =∑

h1

(TD2

)h1h1,h2h4

= Tr(u)vh2h4 . (A.55)

Hence, the second equation of Eq. (A.51) is proved. Similarly, we can prove the first

one. Using Eqs. (A.39) and (A.51), we find that

v · u =D

Tr(u)Tr(v)= 1. (A.56)

This yields

Tr(u)Tr(v) = D. (A.57)

Hence, Eq. (A.53) is proved.

Note that Lemma A.2.8 is not true for a general MPS transfer matrix. Indeed,

using the similarity transformation Eq. (A.6), a general MPS transfer matrix is related

to a canonical one:

Th1h2,h3h4 =∑

h5,6,7,8

Sh1,h5S?h2,h6

Th5h6,h7h8S−1h7,h3

S−1?h8,h4

(A.58)

213

where S is the similarity transformation. Applying Lemma A.2.8, we get:

(TD2

)h1h2,h3h4

=∑

h5,6,7,8

Sh1,h5S?h2,h6

(TD2

)h5h6,h7h8

S−1h7,h3

S−1?h8,h4

=∑

h5,6,7,8

Sh1,h5S?h2,h6

1

Dδh5h6δh7h8S

−1h7,h3

S−1?h8,h4

=1

D

∑

h5,7

Sh1,h5S?h2,h5

S−1h7,h3

S−1?h7,h4

(A.59)

The similarity transformation S is required to be invertible, but does not have to be

unitary. Hence, we conclude that Lemma A.2.8 is not true for a general MPS transfer

matrix.

Lemma A.2.9. For a stabilizer code, the transfer matrix of the ground state canonical

MPS T satisfies:

(Tn)h1h3,h2h4

=1

Dδh1h3δh2h4 , ∀ n > D2 ∈ N. (A.60)

Proof. Using Lemma A.2.8, we have

(Tn)h1h3,h2h4

=(TD2Tn−D2

)h1h3,h2h4

=∑

h5,h6

1

Dδh1h3δh5h6

(Tn−D2

)h5h6,h2h4

=1

Dδh1h3

∑

h5

(Tn−D2

)h5h5,h2h4

.

(A.61)

Using Lemma A.2.7, we obtain

(Tn)h1h3,h2h4

=1

Dδh1h3δh2h4 . (A.62)

This completes the proof.

214

We further remark that the Lemma A.2.9 holds only when n > D2, which is more

restricted than the condition, i.e., n > 0, for the Lemma A.2.7 holds true. However,

when we contract over the two virtual indices h1 and h3 (or h2 and h4) in Eq. (A.62),

we get Eq. (A.48).

A.3 Stabilizer Operator Acts on MPS Locally

In this appendix, we prove that Eq. (2.12) (and its generic case Eq. (2.39)) is a

sufficient and necessary condition satisfied by any MPS description of the 1D stabilizer

codes fulfilling the 3 assumptions of Sec. 2.

Theorem A.3.1. Eq. (2.39) is a necessary and sufficient condition for Eq. (2.37)

when the system size L ≥ D2 + max{P1, . . . , Pt} where P1, P2, . . . , Pt is defined in

Lemma A.2.2.

Proof. By substituting Eq. (2.39) into the left hand side of Eq. (2.37), it is trivial to

show that Eq. (2.39) is a sufficient condition for Eq. (2.37). Hence, our focus in the

rest of the proof is to show that Eq. (2.39) is also a necessary condition for Eq. (2.37).

It suffices to prove this statement for a particular operator O01. The proof generalizes

to other operators.

The strategy of this proof is to first establish this statement for the canonical MPS

T and then for a general MPS T . Typically, we will encounter many long equations

where there are T -matrices with their physical indices uncontracted on both sides.

Using the properties of the canonical MPS, i.e., Eq. (A.7), we are able to shorten the

equations by contracting out those T -matrices. We will use this trick many times

below. Similar to Sec. 2.2, Eq. (2.37) for O01 implies: (Notice that O0

1 is supported

215

from r = 0 to r = P1 − 1)

Tr

(O0

1 ◦(P1−1∏

r=0

T gr1 ...g

rq

)·(

L−1∏

r=P1

T gr1 ...g

rq

))

=Tr

(L−1∏

r=0

T gr1 ...g

rq

).

(A.63)

Multiplying both sides with(∏L−1

r=P1T g

r1 ...g

rq

)?h1h2

and summing over their physical

indices, we obtain:

∑

gP11 ...g

P1q ...gL−1

1 ...gL−1q

Tr

(O0

1 ◦(P1−1∏

r=0

T gr1 ...g

rq

)·(

L−1∏

r=P1

T gr1 ...g

rq

))(L−1∏

r=P1

T gr1 ...g

rq

)?

h1h2

=∑

gP11 ...g

P1q ...gL−1

1 ...gL−1q

Tr

(L−1∏

r=0

T gr1 ...g

rq

)(L−1∏

r=P1

T gr1 ...g

rq

)?

h1h2

.

(A.64)

Summing over the physical indices gives rise to transfer matrices. We rewrite this

equation with explicit virtual indices as follows

∑

h3h4

O01 ◦(P1−1∏

r=0

T gr1 ...g

rq

)

h3h4

(TL−P1

)h4h1,h3h2

=∑

h3h4

(P1−1∏

r=0

T gr1 ...g

rq

)

h3h4

(TL−P1

)h4h1,h3h2

.

(A.65)

Using Lemma A.2.9 and considering L ≥ D2 +max{P1, . . . , Pt} as stated, we simplify

∑

h3h4

O01 ◦(P1−1∏

r=0

T gr1 ...g

rq

)

h3h4

δh4h1δh3h2

=∑

h3h4

(P1−1∏

r=0

T gr1 ...g

rq

)

h3h4

δh4h1δh3h2 .

(A.66)

216

Figure A.2: Graphical representation of (a) Eq. (A.64) and (b) Eq. (A.67).

Equivalently,

O01 ◦(P1−1∏

r=0

T gr1 ...g

rq

)

h2h1

=

(P1−1∏

r=0

T gr1 ...g

rq

)

h2h1

. (A.67)

See Fig. A.2 for the graphical representation of Eqs. (A.64) and (A.67). Notice that

a general MPS tensor T differs from T by a similarity transformation in Eq. (A.6),

then after doing a similarity transformation on both sides of Eq. (A.67), we find that

an analogue equation for non-canonical MPS also holds,

O01 ◦(P1−1∏

r=0

T gr1 ...g

rq

)

h2h1

=

(P1−1∏

r=0

T gr1 ...g

rq

)

h2h1

. (A.68)

This completes the proof.

Applying the theorem A.3.1 to the ZZXZZ model, we find that Eq. (2.12) is a

necessary and sufficient condition for Eq. (2.4) when the system size is large enough,

i.e., L ≥ 16 + 3 = 19.

217

A.4 The Action of L and R Operators on the MPS

Matrices

Theorem A.4.1. Eq. (2.40) is a necessary and sufficient condition of Eq. (2.39).

Proof. It is trivial to show that Eq. (2.40) is a sufficient condition of Eq. (2.39). Our

focus in this proof is to show that it is also a necessary condition. Without loss of

generality, we only need to prove this for a particular pair of L and R operators, Lr1,1and Rr

1,1.

The strategy of this proof is to first establish this statement for the canonical

MPS T and then for a general MPS T . The matrix element Tgr1 ...g

rq

h1,h2of a canonical

MPS satisfies Eq. (A.7). We start with Eq. (2.39), and restore the virtual indices as

follows,

∑

h2

(Lr1,1 ◦ T g

r1 ...g

rq

)h1,h2

Rr

1,1 ◦(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)

h2,h3

=∑

h2

T gr1 ...g

rq

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)

h2,h3

.

(A.69)

Multiplying(∏r+P1−1

r′=r+1 Tgr′

1 ...gr′q

)?h4,h3

on both sides of the Eq. (A.69), and summing

over both the physical indices gr′

1 , . . . , gr′q with r+ 1 ≤ r′ ≤ r+P1− 1 and the virtual

218

index h3, we find that

∑

h2,h3,gr′

1 ,...,gr′q |r+1≤r′≤r+P1−1

(Lr1,1 ◦ T g

r1 ...g

rq

)h1,h2

(Rr

1,1 ◦(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

))

h2,h3

×(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)?

h4,h3

=∑

h2,h3,gr′

1 ,...,gr′q |r+1≤r′≤r+P1−1

Tgr1 ...g

rq

h1,h2

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)

h2,h3

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)?

h4,h3

=∑

h2,h3

Tgr1 ...g

rq

h1,h2

(TP1−1

)h2h4,h3h3

=∑

h2

Tgr1 ...g

rq

h1,h2δh4,h2

=Tgr1 ...g

rq

h1,h4,

(A.70)

where in the third equality, we use Lemma A.2.7. Let us define

(U r1,1)h2,h4 ≡

∑

h3,gr′

1 ,...,gr′q |r+1≤r′≤r+P1−1

(Rr

1,1 ◦(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

))

h2,h3

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)?

h4,h3

.

(A.71)

LHS of Eq. (A.70) becomes

∑

h2

(Lr1,1 ◦ T gr1 ...g

rq )h1,h2(U r

1,1)h2,h4 . (A.72)

Eq. (A.70) and the definition of U r1,1 are graphically represented in (a) and (b) of

Fig. A.3 respectively. Combining Eqs. (A.70), (A.71) and (A.72), we find

∑

h2

(Lr1,1 ◦ T gr1 ...g

rq )h1,h2(U r

1,1)h2,h4 = Tgr1 ...g

rq

h1,h4. (A.73)

219

Figure A.3: Graphical representation of (a) Eq. (A.70) and (b) the virtual operatorU r

1,1.

Applying Lr1,1 on both sides, since (Lr1,1)2 is an identity operator1, we obtain

∑

h2

(T gr1 ...g

rq )h1,h2(U r

1,1)h2,h4 = (Lr1,1 ◦ T gr1 ...g

rq )h1,h4 . (A.74)

This is one of the first set of equations in Eq. (2.40) when the tensors are canonical.

Substituting the RHS of Eq. (A.74) into the LHS of Eq. (A.73), we find

∑

h2

(T gr1 ...g

rq )h1,h2 [(U r

1,1)2]h2,h3 = (T gr1 ...g

rq )h1,h3 . (A.75)

Using the property of the canonical form Eq. (A.7), we obtain that (U r1,1)2 = I is an

identity operator, hence

U r1,1 = (U r

1,1)−1. (A.76)

1This is because the Hamiltonian terms are Hermitian, and should be product of Hermitianoperators, i.e. Pauli operators X,Y and Z.

220

In particular, the U matrices are invertible. Since the product Lr1,1Rr1,1 leaves T g

r1 ...g

rq ·

∏r+P1−1r′=r+1 T

gr′

1 ...gr′q invariant, Rr

1,1 has to transform∏r+P1−1

r′=r+1 Tgr′

1 ...gr′q as

(Rr

1,1 ◦(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

))

h1,h4

=∑

h2

(U r1,1)−1

h1,h2

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)

h2,h4

, (A.77)

which is one of the second set of equations in Eq. (2.40) when the tensors are canonical.

In Eq. (A.77), we use (U r1,1)−1 explicitly to manifest the fact that (Lr1,1Rr

1,1) leaves

the MPS invariant. Similarly, we can prove for other pairs of L and R operators.

Therefore, we have completed the proof for the canonical MPS T .

For a generic MPS T gr1 ...g

rq , it is related to its canonical form via a similarity

transformation, Eq. (A.6). The equations that T obeys can be inferred from those T

obeys in Eqs. (A.74) and (A.77):

∑

h2

(T gr1 ...g

rq )h1,h2(U r

1,1)h2,h4 = (Lr1,1 ◦ T gr1 ...g

rq )h1,h4

∑

h2

(U r1,1)−1

h1,h2

(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

)

h2,h4

=

(Rr

1,1 ◦(r+P1−1∏

r′=r+1

T gr′1 ...g

r′q

))

h1,h4

,

(A.78)

where

U r1,1 = S · U r

1,1 · S−1. (A.79)

where S is the similarity transformation defined in Eq. (A.6). Similarly for other

pairs of L and R operators. Therefore, Eq. (2.40) also holds. This completes the

proof.

Applying Theorem A.4.1 to the ZZXZZ model, we find that Eqs. (2.17), (2.18)

and (2.19) are necessary and sufficient conditions for Eq. (2.12).

221

A.5 Commutation Relations of U Operators

Theorem A.5.1. (Eq. (2.40)) U rα,τ operators have the same commutation/anti-

commutation relation as the Lrα,τ operators or Rrα,τ operators.

Proof. For convenience, we first denote:

Lrα,τLrα′,τ ′ = (−1)trατ,α′τ ′Lrα′,τ ′Lrα,τ . (A.80)

where trατ,α′τ ′ is an integer. Consider these two operators acting on the tensors of the

canonical MPS T :

Lrα,τLrα′,τ ′ ◦(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)

=(−1)trατ,α′τ ′Lrα′,τ ′Lrα,τ ◦

(r+τ−1∏

r′=r

T gr′1 ...g

r′q

).

(A.81)

Apply Eq. (2.40) to both sides of the equation twice when the tensor in Eq. (2.40) is

the canonical one T :

(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)U rα′,τ ′U

rα,τ

=(−1)trατ,α′τ ′

(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)U rα,τ U

rα′,τ ′ .

(A.82)

Multiply both sides with(∏r+τ−1

r′=r T gr′1 ...g

r′q

)†and sum over the physical indices:

∑

gr1 ...grq ...g

r+τ−11 ...gr+τ−1

q

(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)†(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)U rα′,τ ′U

rα,τ

=(−1)trατ,α′τ ′

∑

gr1 ...grq ...g

r+τ−11 ...gr+τ−1

q

(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)†(r+τ−1∏

r′=r

T gr′1 ...g

r′q

)U rα,τ U

rα′,τ ′ .

(A.83)

222

Using the canonical conditions Eq. (A.7), we can find that:

U rα′,τ ′U

rα,τ = (−1)

trατ,α′τ ′ U r

α,τ Urα′,τ ′ . (A.84)

Hence, we have completed the proof that U rα,τ operators form the same commutation

relations as the Lrα,τ does. Similarly, we can prove that the U rα,τ operators form the

same commutation relations as the Rrα,τ does.

We further discuss the case where the MPS matrix T gr1 ...g

rq is not canonical. Since

T gr1 ...g

rq is related to its canonical form via a similarity transformation Eq. (A.6), the

virtual U operator is related to U via the same similarity transformation, S, i.e.,

U rα,τ = S · U r

α,τ · S−1. Hence

U rα′,τ ′U

rα,τ = S · U r

α′,τ ′ · S−1 · S · U rα,τ · S−1

= S · U rα′,τ ′U

rα,τ · S−1

= (−1)trατ,α′τ ′S · U r

α,τ Urα′,τ ′ · S−1

= (−1)trατ,α′τ ′U r

α,τUrα′,τ ′ .

(A.85)

So the virtual U operators (associated to the non-canonical MPS) also satisfy the

same commutation relation as the physical L operators.

A.6 Linear Equations for Local Tensors

In this appendix, we prove that Eq. (2.29) (and its generalization Eq. (2.46)) is a

necessary and sufficient condition of Eqs. (2.17), (2.18) and (2.19) (and their gener-

alization Eq. (2.40)).

Theorem A.6.1. Eq. (2.46) is a necessary and sufficient condition of Eq. (2.40).

223

Figure A.4: Graphical representation of (a) Eq. (A.86) and (b) Eq. (A.88), and (c)Eq. (A.91).

Proof. It is trivial to show that Eq. (2.46) is a sufficient condition for Eq. (2.40).

Our focus in this proof is to show that Eq. (2.46) is also a necessary condition for

Eq. (2.40). We start with the leftmost L operator in Eq. (2.40).

By shifting the positions of Eq. (2.40), we can obtain Eq. (2.41), and we will

mainly use Eq. (2.41). We first consider the case when the MPS is canonical, and

then discuss the general case. To prove that Eq. (2.46) is necessary of Eq. (2.41),

we use an recursive method. In particular, let us focus on the first two equations of

Eq. (2.41) when the tensor is the canonical one T : (See Fig. A.4 (a) for the graphical

224

representation)

Lrα,1 ◦(T g

r1 ...g

rq

)= T g

r1 ...g

rq U r

α,1

Lr−1α,2 ◦

(T g

r−11 ...gr−1

q T gr1 ...g

rq

)= T g

r−11 ...gr−1

q T gr1 ...g

rq U r−1

α,2 .

(A.86)

We can apply (Lr−1α,1 )−1 to the second equation:

(Lr−1α,1 )−1Lr−1

α,2 ◦(T g

r−11 ...gr−1

q T gr1 ...g

rq

)

=(Lr−1α,1 )−1 ◦

(T g

r−11 ...gr−1

q T gr1 ...g

rq U r−1

α,2

).

(A.87)

Using the first equation of Eq. (A.86) at the (r − 1)-th site, we continue to simplify:

(See Fig. A.4 (b) for the graphical representation )

(Lr−1α,1 )−1Lr−1

α,2 ◦(T g

r−11 ...gr−1

q T gr1 ...g

rq

)

=T gr−11 ...gr−1

q (U r−1α,1 )−1T g

r1 ...g

rq U r−1

α,2 .

(A.88)

Notice that the physical operator (Lr−1α,1 )−1Lr−1

α,2 only acts on the r-th site. We can

rewrite this equation as:

T gr−11 ...gr−1

q

((Lr−1

α,1 )−1Lr−1α,2 ◦ T g

r1 ...g

rq

)

=T gr−11 ...gr−1

q (U r−1α,1 )−1T g

r1 ...g

rq U r−1

α,2 .

(A.89)

Multiply both sides with(T g

r−11 ...gr−1

q

)†and sum over the physical indices:

∑

gr−11 ...gr−1

q

(T g

r−11 ...gr−1

q

)†T g

r−11 ...gr−1

q

((Lr−1

α,1 )−1Lr−1α,2 ◦ T g

r1 ...g

rq

)

=∑

gr−11 ...gr−1

q

(T g

r−11 ...gr−1

q

)†T g

r−11 ...gr−1

q (U r−1α,1 )−1T g

r1 ...g

rq U r−1

α,2 .

(A.90)

225

Now we can apply Eq. (A.7) at the (r− 1)-th site: (See Fig. A.4 (c) for the graphical

representation)

(Lr−1α,1 )−1Lr−1

α,2 ◦ T gr1 ...g

rq = (U r−1

α,1 )−1T gr1 ...g

rq U r−1

α,2 . (A.91)

Hence, we have proved the following equations for the canonical MPS T :

Lrα,1 ◦ T gr1 ...g

rq = T g

r1 ...g

rq · U r

α,1

((Lr−1

α,1 )−1Lr−1α,2

)◦ T gr1 ···grq = (U r−1

α,1 )−1 · T gr1 ···grq · U r−1α,2 .

(A.92)

By iterating the process, we can prove the rest of the equations in Eq. (2.46) for

the canonical MPS with tensor T . The same statement is true for a general MPS

with a tensor T , since the tensor T and T are related by the similarity transformation

in Eq. (A.6). Therefore, we have completed our proof.

Applying the theorem A.6.1 to the ZZXZZ model, we find that Eq. (2.29) is the

necessary and sufficient condition for Eqs. (2.17), (2.18) and (2.19).

Theorem A.6.2. If Lrα,1, U rα,1 and V r

α,1 satisfy

Lrα,1 ◦ T gr1 ...g

rq = T g

r1 ...g

rq · U r

α,1

Lrα,1 ◦ T gr1 ...g

rq = T g

r1 ...g

rq · V r

α,1,

(A.93)

then U rα,1 = V r

α,1.

Proof. We first prove when the T matrix is canonical. Since Lrα,1 and Lrβ,1 are identical

physical operators, LHS of Eq. (A.93) are the same. Hence

T gr1 ...g

rq · U r

α,1 = T gr1 ...g

rq · V r

α,1, (A.94)

226

where T gr1 ...g

rq is the canonical MPS matrix, and U r

α,1 and V rα,1 are the associated virtual

operator. In components,

∑

h2

(T gr1 ...g

rq )h1,h2(U r

α,1)h2,h3 =∑

h2

(T gr1 ...g

rq )h1,h2(V r

α,1)h2,h3 . (A.95)

Multiplying (T gr1 ...g

rq )∗h1,h4

on both sides, and summing over h1 as well as the physical

indices gr1, . . . , grq , and using the canonical condition Eq. (A.7), we find

(U rα,1)h4,h3 = (V r

α,1)h4,h3 . (A.96)

When the MPS is not canonical, we apply the similarity transformation Eq. (A.6):

U rα,1 = S · U r

α,1 · S−1, V rα,1 = S · V r

α,1 · S−1. (A.97)

So

U rα,1 = S · U r

α,1 · S−1 = S · V rα,1 · S−1 = V r

α,1. (A.98)

This completes the proof.

A.7 Virtual U Operators as Tensor Products of

Pauli Matrices

In this appendix, we show that the virtual U operators can be constructed as tensor

products of Pauli matrices.

As discussed in the paragraph before Eq. (2.44) in Sec. 2.4 and proved in Ref. [110],

the anti-symmetric integer matrix t can be block diagonalized by a unimodular integer

matrix V , such that each nontrivial block is a 2×2 anti-symmetric matrix with integer

off-diagonal matrix elements. Consider a general set of operators {Ui} (i = 1, ..., N)

227

which either commute or anti-commute,

UiUj = (−1)tijUjUi. (A.99)

Let us define a new set of operators using the unimodular integer matrix V as follows

Ui = UVi11 UVi2

2 ...UViNN , (A.100)

where Vij are the entries of the unimodular integer matrix V . It is straightforward to

compute the commutation relations of {Ui},

UiUj = (−1)∑k,l Viktkl(V

T )lj UjUi

= (−1)(V ·t·V T )ij UjUi.

(A.101)

Due to Eq.(2.44), V · t · V T is block diagonalized. Since V · t · V T appears on the

exponent of (−1), only the modulo 2 values of the matrix elements matter. Hence the

nontrivial 2× 2 blocks have off-diagonal elements ±1 where we keep the minus signs

to make the anti-symmetry manifest. Suppose n is the number of nontrivial blocks of

the V ·t ·V T . Then one can find the representations of Ui by using the Pauli matrices,

because each 2× 2 block corresponds to a pair of anti-commuting operators. For an

irreducible representation, we can assign for instance

Ui =

I ⊗ ...⊗ I︸ ︷︷ ︸i−1

2

⊗X ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸2n−i−1

2

, i is odd, 1 ≤ i ≤ 2n

I ⊗ ...⊗ I︸ ︷︷ ︸i−2

2

⊗Z ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸2n−i

2

, i is even, 1 ≤ i ≤ 2n

I ⊗ ...⊗ I ⊗ I ⊗ I ⊗ ...⊗ I︸ ︷︷ ︸n

, 2n+ 1 ≤ i ≤ N,

(A.102)

228

where n = rank(t)2

, and each Ui is a tensor product of n Pauli matrices, forming a

2rank(t)

2 = 2n dimensional representation. Since V is unimodular, we can do an inverse

transformation from {Ui} to {Ui}.

Ui = U(V −1)i11 ...U

(V −1)iNN . (A.103)

Since {Ui} are tensor product of Pauli matrices, {Ui} are also tensor product of Pauli

matrices. This generalizes the construction of Sec. 2.2.

229

Appendix B

Appendices for Chapter 3

‘

B.1 Projective Representations and 1D Symmetry

Protected Topological Phases

B.1.1 Projective Representations and Cocycles

In this section, we describe projective representations and cocycles. Suppose G is a

discrete group and ρ(g) is a matrix representation of the group element g ∈ G. ρ is

the projective representation of G if

ρ(g1)ρ(g2) = ω2(g1, g2)ρ(g1g2), ∀ g1, g2 ∈ G, (B.1)

where ω2(g1, g2) is a U(1) phase. As a result of Eq. (B.1) being associative, i.e.,

(ρ(g1)ρ(g2)

)ρ(g3) = ρ(g1)

(ρ(g2)ρ(g3)

). (B.2)

230

ω2(g1, g2) satisfies:

ω2(g1, g2)ω2(g1g2, g3) = ω2(g2, g3)ω2(g1, g2g3). (B.3)

We further require that ρ(g) and ρ(g)µ1(g) belongs to the same class of the projective

representation, where µ1(g) is a U(1) phase. This yields that if two cocycles, ω2 and

ω2, are related by µ1 as follows:

ω2(g1, g2) = µ1(g1)µ1(g2)µ1(g1g2)−1ω2(g1, g2), (B.4)

then they give rise to the same projective representation. The conditions Eqs. (B.3)

and (B.4) require the U(1) phase ω2 belongs to the group cohomology H2(G,U(1))

and is a cocycle.[111, 188, 15].

Throughout the paper, G is an Abelian group of the form (Z2)q, and the group

element g is parametrized by g = (g1, g2, . . . , gq) with gi ∈ Z2 for i = 1, 2, . . . , q. All

the cocycles in H2(G,U(1)) are parametrized as in Eq. (3.20)[112, 111].

B.1.2 Cocycle States

In this subsection, we summarize the construction of a class of short range entangled

states which we dub as the the cocycle states, following Ref. [15]. These states are

interesting because they are the states describing the symmetry protected topological

(SPT) phase, protected by the on-site unitary symmetry G. We first set up the

notations, and then review their results with Abelian groups for simplicity.

Consider a 1D lattice with L unit cells. In each unit cell, the local Hilbert space

basis can be labeled by the elements of G: |gr〉,∀ gr ∈ G, (r = 0, 1, ..., L−1). Besides

the group elements {gr}, Ref. [15] also introduced an auxiliary group element g? ∈ G

which does not belong to the Hilbert space, but nevertheless enables one to cons. The

231

cocycle state is constructed as follows (see Eq. (54) of Ref. [15])

|ψ〉G,ω2 =∑

{gr}

(∑

g?

L−1∏

r=0

ω2(gr − gr−1, g? − gr))|{gr}〉. (B.5)

We further restrict Eq. (B.5) to the (Z2)q group. As introduced in App. B.1.1,

each unit cell contains q number of Z2 group elements/spins, i.e., gr = (gr1, ..., grq). A

generic ω2 is in Eq. (3.20), i.e.,

ω2(gr − gr−1, g? − gr)

= exp

(− iπ

∑

1≤i<j≤q

Pij(grj − gr−1

j )(g?i − gri )).

(B.6)

Plugging Eq. (B.6) to (B.5), the cocycle state of (Z2)q global symmetry becomes:

|ψ〉(Z2)q ,ω2 =∑

{gri }

(∑

{g?i }

exp

(− iπ

L−1∑

r=0

∑

1≤i<j≤q

Pij(grj − gr−1

j )(g?i − gri )))|{gri }〉.

(B.7)

Notice that in the exponent, the coefficient of g?i with fixed j, i.e., −iπ∑L−1r=0 Pij(g

rj −

gr−1j ), vanishes due to PBC. This further simplifies the cocycle state Eq. (B.7) to

|ψ〉(Z2)q ,ω2 =

∑

{gri }

exp

(iπ

L−1∑

r=0

∑

1≤i<j≤q

Pij(grj − gr−1

j )gri

)|{gri }〉.

(B.8)

B.1.3 Cocycle Hamiltonians

We now construct a cocycle Hamiltonian H(Z2)q ,ω2 whose ground state is Eq. (B.8).

The cocycle Hamiltonian has been constructed in Refs. [189, 190]. We present a

simplified construction.

232

Lemma B.1.1. There exist qL operators Orα defined by

Orα ≡∏

1≤k<α

(Zr+1k Zr

k)PkαXr

α

∏

α<l≤q

(Zrl Z

r−1l )Pαl (B.9)

satisfying

Orα|ψ〉(Z2)q ,ω2 = |ψ〉(Z2)q ,ω2 ,∀r ∈ [0, L− 1], α ∈ {1, . . . , q}. (B.10)

In the main text, we adopt a slightly different but equivalent convention to label

all the operators Orα using translation symmetry. See Eq. (3.22). In the main text, the

convention adopted in Eq. (3.22) is consistent with the discussion of general stabilizer

code Eq. (2.36). In this appendix, Orα in Eq. (B.9) shares the same label with Xrα in

its expression. The convention in Eq. (B.9) will simplify the proof without repeating

the same equations for different labels.

Proof. We first act Xrα on |ψ〉(Z2)q ,ω2 (B.8),

Xrα|ψ〉(Z2)q ,ω2 =

∑

{grk}

exp

(iπ

L−1∑

r=0

∑

1≤k<l≤q

Pkl(grl − gr−1

l )grk

)Xrα|{grk}〉

=∑

{grk}

exp

(iπ

L−1∑

r=0

∑

1≤k<l≤q

Pkl(grl − gr−1

l )grk

)|{grk + δrrδkα}〉.

(B.11)

In the second line, we used the fact that since the group element grα is defined mod

2, grk + δrrδkα is equivalent to flipping the value of the spin grα. We further redefine

233

the spins as grk = grk + δrrδkα, and rewrite the equation as

Xrα|ψ〉(Z2)q ,ω2

=∑

{grk}

exp

(iπ

L−1∑

r=0

∑

1≤k<l≤q

Pkl(grl − gr−1

l − δrrδlα + δ(r−1)rδlα)(grk − δrrδkα)

)|{gkr}〉

=∑

{grk}

exp

(iπ

L−1∑

r=0

∑

1≤k<l≤q

Pkl(grl − gr−1

l )grk − iπ∑

1≤k<α

Pkα(grk − gr+1k )

− iπ∑

α<l≤q

Pαl(grl − gr−1

l )

)|{gkr}〉

=∏

1≤k<α

(Zr+1k Zr

k)Pkα

∏

α<l≤q

(Zrl Z

r−1l )Pαl

∑

{grk}

exp

(iπ

L−1∑

r=0

∑

1≤k<l≤q

Pkl(grl − gr−1

l )grk

)|{grk}〉

=∏

1≤k<α

(Zr+1k Zr

k)Pkα

∏

α<l≤q

(Zrl Z

r−1l )Pαl |ψ〉(Z2)q ,ω2 .

(B.12)

In the second line, the first term on the exponent has exactly the same form as the

original |ψ〉(Z2)q ,ω2 , while the second and the third terms on the exponent are extra

terms. They can be reproduced by the acting with the product of Pauli Z operators,∏

1≤k<α(Zkr+1Z

kr )Pkα

∏α<l≤q(Z

lrZ

lr−1)Pαl . This observation directly leads to the third

line. Hence we find the following combination leaves |ψ〉(Z2)q ,ω2 invariant:

Orα ≡∏

1≤k<α

(Zr+1k Zr

k)PkαXr

α

∏

α<l≤q

(Zrl Z

r−1l )Pαl . (B.13)

This completes the proof.

Lemma B.1.2. The operators Orα in Lemma B.1.1 mutually commute, i.e,

[Orα,Or′

α′ ] = 0, ∀r, r′, α, α′. (B.14)

234

Proof. Without loss of generality, we assume r = 1. Then O1α only acts on the

unit cells at 0, 1 and 2. O1α and Or′α′ trivially commute unless X1

α overlap with a

Pauli Z operator of Or′α′ and/or Xr′

α′ overlap with a Pauli Z operator of O1α. It is

straightforward to check that the Pauli X and Z operators overlap when

1. r′ = 2 and α > α′.

2. r′ = 1 and α 6= α′.

3. r′ = 0 and α < α′.

When r′ = 2 and α > α′,

O1αO2

α′ = (−1)Pα′α(−1)Pα′αO2α′O1

α = O2α′O1

α. (B.15)

When r′ = 1 and α > α′,

O1αO1

α′ = (−1)Pα′α(−1)Pα′αO1α′O1

α = O1α′O1

α. (B.16)

When r′ = 1 and α < α′,

O1αO1

α′ = (−1)Pαα′ (−1)Pαα′O1α′O1

α = O1α′O1

α. (B.17)

When r′ = 0 and α < α′,

O1αO0

α′ = (−1)Pαα′ (−1)Pαα′O0α′O1

α = O0α′O1

α. (B.18)

In summary, we have proven that for any r′, α, α′, [O1α,Or

′

α′ ] = 0. By translational

invariance, [Orα,Or′

α′ ] = 0, ∀r, r′, α, α′. This completes the proof.

Lemma B.1.3. The operators Orα in Lemma B.1.1 are all independent for different

r = 0, ..., L− 1 and α = 1, ..., q.

235

Proof. The observation is that each Orα involves only one Pauli X operator, Xrα. Then

all operators Orα are independent.

Lemma B.1.4. The commuting Hamiltonian

H(Z2)q ,ω2 = −L−1∑

r=0

q∑

α=1

Orα. (B.19)

has only one ground state.

Proof. We prove by counting the degrees of freedom and the number of independent

constraints. Since each unit cell contains q spins and there are L unit cells, the total

dimension of the Hilbert space is 2qL. From Lemma B.1.2, all the operators in the

Hamiltonian commute. Thus the ground state |ψ〉(Z2)q ,ω2 must be stabilized by all the

operators satisfying

Orα|ψ〉(Z2)q ,ω2 = |ψ〉(Z2)q ,ω2 . (B.20)

From Lemma B.1.3, all the operators Orα are independent. Hence each Eq. (B.20)

provides one independent constraint for the ground state Hilbert space. Because Orαis a product of Pauli operators, each equation in Eq. (B.20) eliminates half of the

Hilbert space dimension. Since there are qL independent equations, the number of

ground state is 2qL−qL = 1. Hence there is only one ground state.

Summarizing Lemma B.1.1, B.1.2, B.1.3 and B.1.4, we have constructed the co-

cycle Hamiltonian:

Theorem B.1.5. The cocycle state Eq. (B.8) is stabilized by the cocycle Hamiltonian

H(Z2)q ,ω2 = −L−1∑

r=0

q∑

α=1

Orα, (B.21)

236

where

Orα ≡∏

1≤k<α

(Zr+1k Zr

k)PkαXr

α

∏

α<l≤q

(Zrl Z

r−1l )Pαl . (B.22)

The Hamiltonian satisfies

1. All the operators Orα are products of Pauli operators, and mutually commute.

2. There is a unique ground state |ψ〉(Z2)q ,ω2 with PBC.

B.2 Some Useful Identities

In this appendix, we prove that Eq. (3.35) holds. We first prove a Lemma which

turns out to be useful in proving Eq. (3.35).

Lemma B.2.1. If x is an integer, then the following equation holds.

exp

(iπ

1

2x2

)=

1 + exp (iπx)

2+ i

1− exp (iπx)

2. (B.23)

Proof. When x is an even integer, both sides are 1. When x is an odd integer, both

sides are i. Hence Eq. (B.23) holds.

Lemma B.2.2. Eq. (3.35) holds.

Proof. We start with the LHS of Eq. (3.35). Using∑

i<j gigj = 12

((∑

i gi)2 −∑i g

2i ),

we reduce the LHS to

exp

(iπ∑

i<j

gigj

)= exp

(iπ

1

2

((∑

i

gi)2 −

∑

i

g2i

)). (B.24)

237

If we further restrict the value of gi as gi ∈ {0, 1}, we have g2i = gi, hence

∑i g

2i =

∑i gi. Applying Lemma B.2.1 with x =

∑i gi, we further reduce Eq. (B.24) to

(1 + eiπ

∑ni=1 gi

2+ i

1− eiπ∑ni=1 gi

2

)e−

iπ2

∑ni=1 gi

=√

2 cos

(π

2

(n∑

i=1

gi −1

2

)).

(B.25)

Introducing a hidden variable h to write the RHS in the RBM form, we find the RHS

is precisely

1√2

1∑

h=0

exp

(iπ

2(1− 2h)

n∑

i=1

gi − iπ

4(1− 2h)

). (B.26)

This completes the proof.

Two simple examples of Eq. (3.35) are:

exp

(iπg1g2

)

=1√2

1∑

h=0

exp

(iπ

2(1− 2h)(g1 + g2)− iπ

4(1− 2h)

) (B.27)

for n = 2 and

exp

(iπ(g1g2 + g1g3 + g2g3)

)

=1√2

1∑

h=0

exp

(iπ

2(1− 2h)(g1 + g2 + g3)− iπ

4(1− 2h)

) (B.28)

for n = 3.

238

B.3 More Examples of RBM for Cocycle Model

In this appendix, we exemplify the construction of the RBM state in Sec. 3.4.3 by

the cocycle model with P12 = P13 = · · · = P1q = 1 and Pij = 0 with i ≥ 2 and j > i.

The Hamiltonian of the model is

H(Z2)q ,ω2 =−L−1∑

r=0

(q∏

i=2

ZriX

r+11

q∏

i=2

Zr+1i +

q∑

i=2

Zr1Z

r+11 Xr

i

). (B.29)

The ground state is

|GS〉(Z2)q ,ω2 =∑

{gri }

L−1∏

r=0

exp

(iπ

q∑

i=2

(gri − gr−1i )gr1

)|{gri }〉. (B.30)

The q × q Γ matrix (defined in Eq. (3.53)) is

Γ =

0 0 · · · 0 0

1 0 · · · 0 0

1 0 · · · 0 0

......

. . .... 0

1 0 · · · 0 0

. (B.31)

Applying the procedures introduced in the proof of Lemma. 3.4.2, we first use row

operations to set the all the rows of Eq. (B.31) to zero except the first row. Recall

G1 and G2 defined in Eq. (3.47). The row operation is

GT = G2(1, q − 1)G2(1, q − 2) · · ·G2(1, 2)G1(1, q). (B.32)

239

The visible spins transform as

gr1

gr2...

grq−1

grq

→

gr1

gr2...

grq−1

grq

= G−1 ·

gr1

gr2...

grq−1

grq

=

∑qi=2 g

ri

gr2...

grq−1

gr1

. (B.33)

The Γ matrix is transformed to

Γ→ Γ = GT · Γ ·G =

0 0 · · · 0 1

0 0 · · · 0 0

0 0 · · · 0 0

......

. . .... 0

0 0 · · · 0 0

. (B.34)

Hence the rank of the Γ matrix is

rank(Γ) = rank(Γ) = 1. (B.35)

Using the identity Eq. (3.35), we only need to introduce one hidden spin of type h

and type h respectively to express the exponent in Eq. (B.30) in terms of RBM,

L−1∑

r=0

q∑

i=2

(gri − gr−1i )gr1 =

L−1∑

r=0

(− Sym(gr1,

q∑

i=2

gr−1i ) + Sym(gr1,

q∑

i=2

gri )

).

(B.36)

240

The ground state Eq. (B.30) can be written as an RBM state

|GS〉(Z2)q ,ω2 =∑

{gri },{hr1},{hr1}

L−1∏

r=0

exp

(− iπ

2(1− 2hr1)(gr1 +

q∑

i=2

gr−1i ) + i

π

4(1− 2hr1)

+ iπ

2(1− 2hr1)

q∑

i=1

gri − iπ

4(1− 2hr1)

)|{gri }〉.

(B.37)

This RBM can be casted into an MPS with bond dimension 2, and the matrix elements

of the RBM-MPS are:

Tgr1 ,...,g

rq

hr1,hr+11

= exp

(− iπ

2(1− 2hr1)gr1 − i

π

2(1− 2hr+1

1 )

q∑

i=2

gri + iπ

4(1− 2hr1)

)

×1∑

hr1=0

exp

(iπ

2(1− 2hr1)

q∑

i=1

gri − iπ

4(1− 2hr1)

).

(B.38)

We also present the RBM for two examples in Fig. B.1 and B.2 corresponding to

q = 3 and q = 4.

Figure B.1: RBM network for cocycle model with q = 3, P12 = P13 = 1, P23 = 0.

Figure B.2: RBM network for cocycle model with q = 4, P12 = P13 = P14 = 1, P23 =

P24 = P34 = 0.

241

Appendix C

Appendices for Chapter 4

C.1 Proof for the Concatenation Lemma for the

3D Toric Code Model

In this section, we prove the Concatenation lemma for the 3D toric code model

by induction. First of all, we propose and prove two lemmas:

(A) Let Tt1t2t3... be a contraction of a network of local T tensors, whose (i.e. open)

indices {t1t2t3...} are un-contracted virtual indices. If Tt1t2t3... satisfies the Con-

catenation lemma of the 3D toric code model, then the contraction of Tt1t2t3...

over a subset of its open virtual indices, say contracting over t1 and t2, i.e.,∑

t1t2Tt1t2t3...δt1t2 still satisfies the Concatenation lemma of the 3D toric

code model.

(B) If Tt1t2t3... and Tt1 t2 t3... are two networks of contracted local T tensors both

of which satisfying the Concatenation lemma of the 3D toric code model,

then the contraction over one pair of indices, say∑

t1 t1Tt1t2t3...Tt1 t2 t3...δt1 t1 , still

satisfies the Concatenation lemma of the 3D toric code model.

Proof:

242

(A): Since T satisfies the Concatenation lemma, its elements T{t} are:

T{t} =

0 if∑

i ti = 1 mod 2

N if∑

i ti = 0 mod 2

(C.1)

where N is a constant independent of the open virtual indices in the Concatenation

lemma. Suppose that we contract two indices of T, tm, tn ∈ {t}, and we denote the

contraction as T′ and the remaining open virtual indices after the contraction {t′}.

Then we have:

T′{t′} =∑

tm,tn

T{t}δtm,tn

=∑

tm,tn

T...tm...tn...δtm,tn

=∑

tm

T...tm...tm...

=T...0...0... + T...1...1...

(C.2)

Hence, the contraction still satisfies the Concatenation lemma:

T′{t′} =

0 if∑

i t′i = 1 mod 2

2N if∑

i t′i = 0 mod 2

(C.3)

243

(B): Since T and T satisfy the Concatenation lemma, their elements T{t} and

T{t} are:

T{t} =

0 if∑

i ti = 1 mod 2

N if∑

i ti = 0 mod 2

T{t} =

0 if∑

i ti = 1 mod 2

N if∑

i ti = 0 mod 2

(C.4)

where N and N are the constants independent of the indices {t} and {t} respectively.

Suppose that we contract two indices tm ∈ {t} and tn ∈ {t}, and we denote the

contraction as T′ and the remaining open virtual indices after the contraction {t′}.

Then we have:

T′{t′} =∑

tm,tn

T{t}T{t}δtm,tn

=∑

tm,tn

T...tm...T...tn...δtm,tn

=T...0...T...0... + T...1...T...1...

(C.5)

The last line is nonzero if and only if∑

i 6=m ti and∑

j 6=n tn have the same parity. If

this parity is even (resp. odd), only T...0...T...0... (resp. T...1...T...1...) is nonzero and

equal to NN . Since∑

i t′i =

∑i 6=m ti +

∑j 6=n tn, we conclude that:

T′{t′} =

0 if∑

i t′i = 1 mod 2

NN if∑

i t′i = 0 mod 2

(C.6)

T′{t′} still satisfies the Concatenation lemma. 2

Having proved Lemma (A) and (B), we can further prove that:

244

(C) If T and T are two networks of contracted local T tensors which both satisfy

the Concatenation lemma of the 3D toric code model, then their contraction

over any pairs of indices still satisfies the Concatenation lemma of the 3D

toric code model.

Proof:

We can decompose the contraction process into two steps: (1) contract T and T

over one pair of indices; (2) contract the rest of the indices. Lemma (B) guarantees

that the outcome tensor of the contraction (1) still satisfies the Concatenation

lemma. Lemma (A) guarantees that the outcome tensor of the contraction (2) also

satisfies the Concatenation lemma. Hence, Lemma (C) is proved. 2

Now we can complete the induction proof for the Concatenation lemma of the

3D toric code model: First of all, we point out the a single local T tensor satisfies

the Concatenation lemma. Next, we assume that two networks of contracted local

T tensors satisfy the Concatenation lemma, and prove that their contraction also

satisfies the Concatenation lemma. This induction step is, in fact, Lemma (C).

Therefore, we have completed the induction proof for the Concatenation lemma

of the 3D toric code model.

C.2 Numerics for Haah Code

In this appendix, we present various numerical evidences for the entanglement entropy

of the Haah code.

In this appendix, we present the results of numerical calculations for the entan-

glement entropies using various cuts.

We start with the TNS wave function of the Haah code defined on T 3 of size

Lx × Ly × Lz. We choose a bipartition of the TNS, |TNS〉 =∑{t} |{t}〉A ⊗ |{t}〉A

where {t} is the set of open virtual indices. For a given choice of region A, we then

245

compute the reduced density matrix (RDM) ρA = TrA|TNS〉〈TNS|, and diagonalize

the RDM. For all the cases we computed, the non-zero eigenvalues of the RDM is

fully degenerate. This degenerate eigenvalue of the normalized RDM is denoted as λ.

The cases No.1 and No.2 in Table. C.1 corresponds to type 1 exact SVD regions

with l = 2 and l = 3 respectively. We see that this is consistent with the general

formula Eq. (4.66). When l = 2, 6l−5 = 6×2−5 = 7; when l = 3, 6l−5 = 6×3−5 =

13. The case No.3 corresponds to the square region A with size 2× 2× 2. The TNS

wave function under such cut is not an SVD (under the TNS basis), however, as we

have shown in Sec. 4.4.4, we can make a change of basis such that in the new basis the

wave function is an SVD. The counting of new basis gives the entanglement entropy of

the cubic cut Eq. (4.87). The brute force numerical calculation in case No.3 of Table.

C.1 is consistent with the formula: S(A)/ log 2 = 6l2−6l+2 = 6×22−6×2+2 = 14.

Note that all these numerical results have been checked using the direct evaluation

of the full GS wave function up to the system sizes 4× 3× 3.

246

System SizeCoordinate of Region A

λ S/ log 2 SVD? No.x y z Left/Right

3× 3× 3

1 1 0 0

1128

7 Yes 1

1 0 1 00 1 1 01 1 1 01 0 0 10 1 0 10 0 1 11 1 1 1

4× 4× 4

1 1 0 0

18192

13 Yes 2

1 0 1 00 1 1 01 1 1 01 2 0 01 2 1 00 2 1 01 0 0 10 1 0 10 0 1 11 1 1 10 2 0 11 2 1 11 1 0 10 1 1 1

3× 3× 3

0 0 0 0

116384

14 No 3

0 0 0 11 0 0 01 0 0 10 1 0 00 1 0 11 1 0 01 1 0 10 0 1 00 0 1 11 0 1 01 0 1 10 1 1 00 1 1 11 1 1 01 1 1 1

Table C.1: Entanglement entropies for various bipartitions of the |TNS〉 of the Haahcode. The second to fourth column list the coordinates of vertices in region A. Thecolumn of ”Left/Right” labels the spin on the left or right position on the vertex(x, y, z), where 0 and 1 corresponds to the left and right position respectively. Weused the coordinate frame as shown in Eq. 4.49 and Fig. 4.6.247

Appendix D

Appendices for Chapter 5

D.1 Review of Entanglement Entropy and Spec-

trum

In this appendix, we review the definition of the entanglement entropy, and review

the notation that we use in this work.

To define the entanglement entropy, we first partition the space into two parts, A,

and its complement, B, via an entanglement surface Σ.1 For a given pure quantum

state |ψ〉, the wave function can be decomposed as

|ψ〉 =∑

ab

Wab|Aa〉|Acb〉, (D.1)

where a labels normalized basis states of the Hilbert space HA localized in region

A and b labels normalized basis states of the Hilbert space HAc localized in region

Ac. We perform a singular value decomposition (SVD) of the matrix W as Wab =

UacDcdV†db and define new bases |A′c〉 = Uac|Aa〉 and |Ac′

d 〉 = V †db|Acb〉. Dcd is a diagonal

matrix with positive entries, but not all the diagonal elements need be nonzero. The

1Because we are interested in (3+1)D systems, the entanglement surface Σ is a two dimensionalsurface.

248

number of nonzero elements is the rank of W , and the nonzero “singular values” are

denoted as e−ξλ/2. ξλ are termed the entanglement energies, and the whole set of

entanglement energies is the entanglement spectrum {ξλ}λ=1,··· ,Rank(W ). Zero singular

values correspond to infinite entanglement energies. Thus,

|ψ〉 =

Rank(W )∑

λ=1

e−ξλ/2|A′λ〉|Ac′

λ 〉. (D.2)

To compute the entanglement entropy, we trace over the states in region Ac to obtain

a reduced density matrix of region A,

ρA = TrHAc |ψ〉〈ψ| =Rank(W )∑

λ=1

e−ξλ|A′λ〉〈A′λ|. (D.3)

The entanglement entropy is defined as the von Neumann entropy of the reduced

density matrix ρA (see Refs. [4] and [191] for a review),

S(A) = −TrHAρA log ρA = −

Rank(W )∑

λ=1

e−ξλ log e−ξλ . (D.4)

Heuristically, the entanglement entropy measures how much the degrees of freedom

in the two regions A and B are correlated.

In this paper, we denote the entanglement entropy of subregion A (whose bound-

ary is Σ) as either S(A) or S[Σ], using either parentheses or square brackets to

highlight the sub region or the entanglement surface, respectively.

D.2 Local Contributions to the Entanglement En-

tropy

In this appendix, we review the general properties of the entanglement entropy. Fol-

lowing the discussions in Ref. [125], we provide some detailed and quantitative analy-

249

ses on how the non-universal and shape dependent terms can enter into the constant

part of the EE.

The simplest property of the EE is S(A) = S(Ac), which says the entropy com-

puted for region A is equal to the entropy computed for its complement Ac. This is

also true for the full entanglement spectrum, and follows directly from Eq. (D.2).

We assume that in a gapped system with finite correlation length, the EE can be

decomposed into a local part and a topological part,

S(A) = Slocal(A) + Stopo(A). (D.5)

The local part Slocal(A) only depends on the local degrees of freedom near the entan-

glement surface, and therefore can be written in the form of an integral over local

variables. Since the only local functions on Σ are the metric hµν , the extrinsic curva-

ture (second fundamental form) Kµν , and the covariant derivatives of Kµν (covariant

derivatives of hµν are zero by definition), Refs. [125, 192, 193] argued that Slocal should

be expressible in terms of local geometric quantities of the entanglement surface Σ,

i.e.,

Slocal(A) =

∫

Σ

d2x√hF (Kµν ,∇ρKµν , ..., hµν), (D.6)

where F is a local function of Kµν and hµν and their covariant derivatives. 2

In contrast, the topological part of the EE, Stopo(A), is precisely the contribution

that cannot be written as an integral of local variables near the entanglement surface.

(In particular, the Euler characteristic term does not contribute to Stopo(A).) Stopo(A)

should be invariant under smooth deformations of the entanglement surface, and

2Suppose the submanifold is given by the embedding φ : Σ → M , concretely, φ : yi → xµ =(z∗, yi) where z∗ is a fixed number specifying the position of hypersurface in the perpendiculardirection of the embedded space. Let the metric in M be gµν , the induced metric therefore ishij ≡ (φ∗g)ij = ∂xµ

∂yi∂xν

∂yj gµν . Let nµ be the normal unit vector of the surface Σ, then the extrinsic

curvature Kµν of Σ is Kµν = ∇µnν − nµnρ∇ρnν . See Appendix D of Ref. [194] for more details.

250

should also be invariant under smooth deformations of the Hamiltonian of the system

(provided the gap does not close). Therefore, reminiscent of two-dimensional systems,

Stopo(A) is expected to be the constant part of the EE. However, in three spatial

dimensions, there are subtleties as we will explore below.

Before moving on, it is important for us to first specify for which systems the EE

separates into a local and a topological part. Systems such as the toric code and its

generalizations (e.g. Dijkgraaf Witten models), as well as the Walker-Wang models

[133] and their generalizations (e.g., the generalized Walker-Wang models which we

study in Sec. 5.2) satisfy this decomposition. There are some systems for which this

decomposition is obviously not valid. For instance, the systems constructed by layer

stacking of two-dimensional systems do not satisfy Eq. (D.5). The constant part of

entropy depends on the thickness Lz of the layered direction, i.e., −γ2DLz, where γ2D

is the topological entropy of a two-dimensional layer. Another class of systems beyond

our discussion are fracton models[195], whose entanglement entropy does not satisfy

Eq. (D.5). Apart from the area law term and the constant term, the entanglement

entropies of these model generically contain a term linearly proportional to the size of

the subregion [196, 47]. Since the decomposition Eq. (D.5) does not lead to a linear

subleading term, its presence in the layered models and the fracton models suggest

the decomposition Eq. (D.5) does not hold.

Since the definition of the EE dictates that S(A) = S(Ac), this should also be true

of the local part of the EE. To compute S(A), one can expand F (Kµν ,∇ρKµν , ..., hµν)

as

F (Kµν ,∇ρKµν , ..., hµν)

= F0 + F1Kµµ + F2[KµνK

µν − (Kµµ)2]

+ F ′2(Kµµ)2 + F3∇µ∇νK

µν + ...,

(D.7)

251

where ∇µ is the covariant derivative induced from hµν , and the indices are raised and

lowered via hµν and its inverse hµν . All indices are contracted so that the formula

Eq. (D.7) is independent of the choice of the coordinates. Demanding that S(A) =

S(Ac) constrains the form of the function F . To see this, we may simply transform

x1 → −x1 and x2 → x2, under which Kµν → −Kµν and hµν → hµν .3 Then

S(A) = S(Ac) implies

F (Kµν ,∇ρKµν , ..., hµν) = F (−Kµν ,∇ρKµν , ..., hµν). (D.8)

After integration, keeping only those terms even under reflection, we find that the

local part of the EE has the form

Slocal(A) = F0|Σ| − F24πχ+ 4F ′2

∫

Σ

d2x√hH2 + ..., (D.9)

where |Σ| is the area of the entanglement surface. The part proportional to F2

gives the Euler characteristic χ(Σ) of the surface Σ, defined by∫

Σd2x√h[KµνK

µν −

(Kµµ)2] = −4πχ(Σ). This term is invariant under any smooth deformation of the

entanglement surface because the Euler characteristic is a topological invariant of Σ.

The part proportional to F ′2 gives the integral of the square of the mean curvature

H = (k1 + k2)/2 (since 2H = Kµµ), where k1, k2 are the two principal curvatures

of Σ, i.e., the eigenvalues of Kµν . This term, though independent of the size of Σ,

depends on its shape. This shows that the local part of the EE has constant terms,

which contrasts with the familiar case in (2+1)D. Therefore, computing the EE and

3x1 → −x1 and x2 → x2 changes the orientation of the entanglement surface Σ. Since theprinciple curvature is an odd function of the orientation of the surface and the eigenvalues of theextrinsic curvature are two principle curvatures, we conclude that the extrinsic curvature is oddunder x1 → −x1 and x2 → x2.

252

extracting the constant part is not a promising way to extract topological information

about the underlying theory.4

The above analysis shows that for a generic gapped system (which is not at an

RG fixed point), the structure of the entanglement entropy is

S(A) = F0|Σ|+ Stopo(A)− 4πF2χ(Σ)

+4F ′2

∫

Σ

d2x√hH2 +O(1/|Σ|). (D.10)

In the main text, we denote the constant part of the EE as Sc(A) = Stopo(A) −

4πF2χ(Σ) + 4F ′2∫

Σd2x√hH2.

The above analysis gives all the possible terms that can exist, but does not require

that they are non-vanishing for a given theory. In Ref. [197], the authors computed

the entanglement entropy for massive bosons and massive fermions in (3+1)D across

S2. Their results show a constant term in the entanglement entropy. For a massive

scalar with mass m and curvature coupling term 12ξRφ2, Sc(A) = (ξ − 1

6) log(mδ),

where δ is the cut off. For a massive Dirac fermion with mass m, Sc(A) = 118

log(mδ).

Obviously, these entropies are not topological (they depend on the cutoff and on mass

parameters), which shows that non-universal contributions to the local term in fact

do exist.

D.3 Derivation of the Reduction Formula

In this appendix we present the complete derivation of the entropy reduction formula

Eq. (5.10). We will use the SSA inequality in two steps. First, in Subsection D.3.1

we derive and solve a recurrence relation for the dependence of STQFTc on the genus of

4In (2 + 1)D, by applying the same analysis, one can show that there is no constant term in theEE which can be written as an integral of local curvature when the space dimension d is even. Thisis because the term of dimension 1/Ld−1 acquires a minus sign when the coordinates of entangl-ment surface are reversed. In particular, in (2+1)D, the constant term in entanglement entropy istopological.

253

CBA

(a)

C’

B’A’

(b)

Figure D.1: Entanglement surfaces used in the application of strong sub-additivityto derive the recurrence relation Eq. (D.17). In (a), A is a general 3-manifold (as anexample, we draw A with 1 genus 3 surface and 2 genus 0 surfaces), B is 3-ball andC is a solid torus. In (b), A′ is a general 3-manifold (as an example, we draw A′ with1 genus 3 surface and 2 genus 0 surfaces), B′ is a solid torus, and C′ is a 3-ball, whichis located exactly at the hole of B′.

the entanglement cut. Second, in Subsection D.3.2 we derive an additional recurrence

relation for the dependence of STQFTc on the number of disconnected components

of the entanglement surface. We solve this recurrence relation to obtain our main

result Eq. (5.10). Our derivation expands upon the discussion in Ref. [125] in that we

obtain explicit formulas for the entropy of arbitrary multiply-connected entanglement

surfaces.

D.3.1 Recurrence for Genus

In order to find the dependence of the TEE on the data {ng}, we need to consider the

configuration of entanglement surfaces as shown in Fig. C.D.1(a): We start with a

general connected 3-manifold with boundary specified by [(0, n0), . . . , (g∗, ng∗)]. The

3-manifold is cut into three regions A, B and C. B is a 3-ball, C is a solid torus and A

occupies the remainder of the manifold. A is connected to B and disconnected from

C. Suppose A connects with B via a disk (shown as a shaded region) which belongs to

a genus (g∗−1)5 boundary of A and also belongs to the genus 0 boundary of B. Then

5Since C ∪ B has a genus 1 surface boundary.

254

the boundary of region A is specified by [(0, n0), . . . , (g∗− 1, ng∗−1 + 1), (g∗, ng∗ − 1)],

where we adopt the labeling scheme defined in Sec. 5.1.1.

We list the constant part of the EE of all regions by their topologies as follows:

STQFTc (A) = STQFT

c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],

STQFTc (B) = STQFT

c [(0, 1)],

STQFTc (C) = STQFT

c [(0, 0), (1, 1)],

STQFTc (AB) = STQFT

c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],

STQFTc (BC) = STQFT

c [(0, 0), (1, 1)],

STQFTc (ABC) = STQFT

c [(0, n0), . . . , (g∗ − 1, ng∗−1), (g∗, ng∗)].

(D.11)

Then the SSA inequality for regions A, B, and C in Eq. (5.4) reads

STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]

≥ STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT

c [(0, 1)]− STQFTc [(0, 0), (1, 1)].

(D.12)

We could have taken A and B to be connected via a disk which belongs to a genus i

(i ≤ g∗− 1) boundary of A and also belongs to the genus 0 boundary of B. Following

an identical procedure, we conclude:

STQFTc [(0, n0), . . . , (i, ni + 1), (i+ 1, ni+1 − 1), . . . , (g∗, ng∗)] + STQFT

c [(0, 0), (1, 1)]

≥ STQFTc [(0, n0), . . . , (i, ni), (i+ 1, ni+1), . . . , (g∗, ng∗)] + STQFT

c [(0, 1)].

(D.13)

For simplicity, we will only need to adopt the choice where i = g∗ − 1.

We proceed to consider another configuration illustrated in Fig. C.D.1(b): We

start with a general 3-manifold with boundary specified by [(0, n0), . . . , (g∗, ng∗)].

The 3-manifold is cut into two regions, A′ and B′. B′ is a solid torus, and A′ is the

rest of the manifold. We assume A′ connects with B′ via a disk (shown as a shaded

255

region) in the genus (g∗ − 1) boundary of A′ and the genus 1 boundary of B′. Hence

the boundary of A′ is labeled by [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]. In

addition, we denote the 3-ball located in the “hole” of B′ as C′.

We list the constant part of the EE of all regions as follows:

STQFTc (A′) =STQFT

c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)],

STQFTc (B′) =STQFT

c [(0, 0), (1, 1)],

STQFTc (C′) =STQFT

c [(0, 1)],

STQFTc (A′B′) =STQFT

c [(0, n0), . . . , (g∗, ng∗)],

STQFTc (B′C′) =STQFT

c [(0, 1)],

STQFTc (A′B′C′) =STQFT

c [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)].

(D.14)

The SSA for A′, B′ and C′ in Fig. C.D.1(b) reads in this case:

STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]

≤ STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT

c [(0, 1)]− STQFTc [(0, 0), (1, 1)].

(D.15)

Combining inequalities Eq. (D.12) and Eq. (D.15), we find the following equality

STQFTc [(0, n0), . . . , (g∗ − 1, ng∗−1 + 1), (g∗, ng∗ − 1)]

= STQFTc [(0, n0), . . . , (g∗, ng∗)] + STQFT

c [(0, 1)]− STQFTc [(0, 0), (1, 1)].

(D.16)

This relates the constant part of the EE of a given subsystem to that of a system

whose boundary has lower genus. Applying Eq. (D.16) repeatedly, we find

STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)] = STQFT

c [(0,

g∗∑

i=0

ni)] +

g∗∑

i=1

ini

(STQFT

c [(0, 0), (1, 1)]− STQFTc [(0, 1)]

).

(D.17)

256

In summary, we can reduce the constant part of the EE of an arbitrary sur-

face STQFTc [(0, n0), (1, n1), ..., (g∗, ng∗)] to a linear combination of STQFT

c [(0, n)] and

STQFTc [(0, 0), (1, 1)].

D.3.2 Recurrence for b0

We can further simplify STQFTc [(0,

∑g∗

i=0 ni)] in Eq. (D.17), by using STQFTc [(0, n)] =

nSTQFTc [(0, 1)]. Here we derive this relation by making use of the SSA in a manner

similar to that of the derivation above.

CBA

(a)

C’

B’A’

(b)

Figure D.2: Entanglement surfaces used in the application of strong sub-additivity

to derive Eq. (D.22). In (a), A is a 3-manifold with multiple genus zero surfaces, B

is a 3-ball, C is a 3-ball with small 3-ball removed. In (b), A′ is an open 3-manifold

with multiple genus zero surfaces, B′ is a 3-ball with a small 3-ball removed and C′

is a 3-ball located exactly in the empty 3-ball inside B′.

We consider the configuration shown in Fig. C.D.2(a), where A is a 3-manifold

with (n − 1) genus zero surfaces, B is a 3-ball and C is a 3-ball with a small 3-ball

257

inside it removed. The constant parts of the EE for these three manifolds are

STQFTc (A) =STQFT

c [(0, n− 1)],

STQFTc (B) =STQFT

c [(0, 1)],

STQFTc (C) =STQFT

c [(0, 2)],

STQFTc (AB) =STQFT

c [(0, n− 1)],

STQFTc (BC) =STQFT

c [(0, 2)],

STQFTc (ABC) =STQFT

c [(0, n)].

(D.18)

The SSA inequality reads

STQFTc [(0, n− 1)] + STQFT

c [(0, 2)] ≥ STQFTc [(0, n)] + STQFT

c [(0, 1)]. (D.19)

We can furthermore consider another configuration shown in Fig. C.D.2(b), where

A′ is a 3-manifold with (n−1) genus-0 surfaces, B′ is a 3-ball with small 3-ball removed,

and C′ is a 3-ball locating exactly in the empty 3-ball inside B′. The constant parts

of the EE for these three manifolds are

STQFTc (A′) = STQFT

c [(0, n− 1)],

STQFTc (B′) = STQFT

c [(0, 2)],

STQFTc (C′) = STQFT

c [(0, 1)],

STQFTc (A′B′) = STQFT

c [(0, n)],

STQFTc (B′C′) = STQFT

c [(0, 1)],

STQFTc (A′B′C′) = STQFT

c [(0, n− 1)].

(D.20)

Then SSA inequality reads

STQFTc [(0, n)] + STQFT

c [(0, 2)] ≤ STQFTc [(0, n+ 1)] + STQFT

c [(0, 1)]. (D.21)

258

Combining Eq. (D.19) and Eq. (D.21), one obtains

STQFTc [(0, n)] + STQFT

c [(0, 2)] = STQFTc [(0, n+ 1)] + STQFT

c [(0, 1)]. (D.22)

Since STQFTc [(0, 0)] = 0, we have

STQFTc [(0, n)] = nSTQFT

c [(0, 1)]. (D.23)

Combining this result with Eq. (D.17), we have6

STQFTc [(0, n0), . . . , (g∗, ng∗)]

=

g∗∑

i=0

niSTQFTc [(0, 1)] +

g∗∑

i=1

ini

(STQFT

c [(0, 0), (1, 1)]− STQFTc [(0, 1)]

)

=

g∗∑

i=0

(1− i)niSTQFTc [(0, 1)] +

g∗∑

i=1

iniSTQFTc [(0, 0), (1, 1)]

= b0STQFTc [(0, 0), (1, 1)] +

χ

2

(STQFT

c [(0, 1)]− STQFTc [(0, 0), (1, 1)]

)

= b0STQFTc [T 2] +

χ

2

(STQFT

c [S2]− STQFTc [T 2]

),

(D.24)

where χ =∑g∗

i=0(2 − 2i)ni is the Euler characteristic of the entanglement sur-

face, which in the previous examples of this appendix is ∂(ABC). This is precisely

Eq. (5.10) in the main text. In the last line, we have changed the notation for clarity:

S2 is a 2-sphere and T 2 is a 2-torus. We emphasize that Eq. (5.10) gives the constant

part of the EE for a TQFT. In particular, Eq. (5.10) shows that the constant part of

the EE across an arbitrary entanglement surface is reduced to that across the sphere

S2 and that across the torus T 2. 7

6As remarked in Sec. D.1, we use S(A) to denote the EE of region A, and S[Σ] to denote theEE of region with boundary Σ, such as S[S2] when entanglement surface is Σ = S2. Both notationsrefer to the same thing.

7Notice that STQFTc (A) is an additive variable, i.e., STQFT

c (A ∪ A′) = STQFTc (A) + STQFT

c (A′)if A ∩ A′ = ∅. This fact also follows from the vanishing of mutual information, i.e., I(A ∪ A′) =S(A) + S(A′)− S(A∪A′) = 0 if A∩A′ = ∅. This is because the area part cancels out in I(A∪A′),

259

D.4 Vanishing of the Mean Curvature Contribu-

tion in KPLW Prescription

In this appendix, we explain why the mean curvature terms cancel in the KPLW

combination Eq. (5.14), therefore justifying Eq. (5.18) in the main text.

In the main text, we argued that the KPLW combination of the area law term

and the Euler characteristic term vanish separately, hence we only need to consider

the topological term and the mean curvature term, i.e.,

SKPLW[T 2] = Stopo[T 2] + 4F ′2

∫∂A+∂B+∂C−∂AB−∂AC−∂BC+∂ABC

d2x√hH2. (D.25)

Eq. (D.25) suggests that the mean curvature term in the KPLW combination is invari-

ant under deformations of the entanglement surface since, as argued in the main text,

both SKPLW[T 2] and Stopo[T 2] in Eq. (D.25) are topological invariants. Therefore, we

only need to show that Eq. (5.18) vanishes for one particular entanglement surface

that is topologically equivalent to that in Fig. 5.1 in the main text, such as Fig. D.3.

Then by topological invariance, Eq. (5.18) vanishes for general configurations.

C

B

A

h1

h2

r1r2r3

Figure D.3: KPLW prescription of regularized entanglement surface T 2.

and I(A∪A′) = 0 yields exactly the additivity of the constant part of the entanglement entropy fora TQFT STQFT

c (A).

260

For the configuration in Fig. D.3, we can compute the mean curvature straight-

forwardly. The mean curvature is H = (k1 + k2)/2, where k1 and k2 are the two

principal curvatures at each point of the entanglement surface. We distinguish three

types of points on the cylinder in Fig. D.3.

Points on the top/bottom of a cylinder : the surface is locally flat, k1 = k2 = 0.

Hence, H = (k1 + k2)/2 = 0.

Points on the side of a cylinder : k1 = ±1/r, k2 = 0, where r is the radius of the

cylinder, and the ± sign depends on whether it is inner or outer side surface. Hence,

H = (k1 + k2)/2 = ±1/2r. In the following, we will pick the + sign.

Points on the hinge of a cylinder : One of the hinges of the regular cylinders in

Fig. D.3 is shown as the thick green loop. On every point of the hinge, the Gauss

curvature is the same. To find it, we apply the Gauss-Bonnet theorem to a cylinder.

Because the Gauss curvature on the side and top/bottom of the cylinder vanishes,

integration over the entire surface of the cylinder is reduced to the integration over

the hinge. Hence the Gauss-Bonnet theorem dictates

2

∫

hinge

1

r3

kdσ = 2πχ[C] = 4π, (D.26)

where C is the full cylinder, r3 is the radius of the cylinder. 1/r3 is the principle

curvature along the hinge and k is the principal curvature along the direction per-

pendicular to the hinge. In order to perform the two-dimensional surface integral, we

need to regularize the one-dimensional hinge by smoothing it into an arc of infinites-

imal radius, as shown in Fig. D.4. Assuming the length of the arc is l0, Eq. (D.26)

implies∫ l0

0kdl = 1, which reduces to k = 1/l0. The principal curvature for an ideal

hinge (which corresponds to l0 → 0) is infinite, and we regularize it with the small

parameter l0 to handle the computation.

261

To compute the integral of the mean curvature squared over various surfaces in

Fig. D.3, we first introduce some notation. Let r1 be the inner radius of region

B/C, r2 be the outer radius of region B/C, r3 be the outer radius of region A, h1

be the height of region B, and h2 be the height of region C. We adopt the same

finite regularization for every hinge, although this is not essential. For region A, the

integration∫∂AH2 splits into three parts: the top/bottom, the side and the hinges.

Since the top/bottom surface are flat, they do not contribute to the mean curvature

integral. The mean curvature of the outer side surface is 1/2r3, and that of the inner

side surface is −1/2r2. The integration of the mean curvature over the outer and

inner side of ∂A is

l0A

C

B

r1

r2

r3

✓

r

1

8

7

6

5

4

3

2

11

10

9

12

Figure D.4: Left: Regularization of a rectangular hinge with small arcs. Right: One

choice of regularization of each hinge in Fig. D.3. The numbers label various hinges.

2πr3(h1 + h2)

(1

2r3

)2

+ 2πr2(h1 + h2)

(−1

2r2

)2

=π(h1 + h2)

2r3

+π(h1 + h2)

2r2

. (D.27)

The mean curvature of the outer hinge is (1/r3+1/l0)/2, while according to our choice

of regularization in Fig. D.4, the mean curvature of the inner hinge is (1/l0− 1/r2)/2

because the principle curvature along the θ direction (the meaning of θ and r are

262

specified in Fig. D.4) is −1/r2 and the principle curvature along the r direction is

1/l0 (because we evaluate the curvature from the inside). The integration of the mean

curvature over the hinges is

2× 2πr3l0

(1

2r3

+1

2l0

)2

+ 2× 2πr2l0

(−1

2r2

+1

2l0

)2

, (D.28)

where the factor of 2 in the front comes from equal contribution of the hinges from

the top and bottom respectively. Collecting the above results, we have

∫

∂A

H2 =π(h1 + h2)

2r3

+π(h1 + h2)

2r2

+π(r3 + l0)2

r3l0+π(r2 − l0)2

r2l0. (D.29)

For convenience, we list the mean curvature of each hinge in the following table.

Hinge Mean curvature

1 1/2r3 + 1/2l0

2 1/2r3 + 1/2l0

3 −1/2r2 + 1/2l0

4 −1/2r2 + 1/2l0

5 1/2r2 + 1/2l0

6 1/2r2 + 1/2l0

7 1/2r2 + 1/2l0

8 1/2r2 + 1/2l0

9 −1/2r1 + 1/2l0

10 −1/2r1 + 1/2l0

11 −1/2r1 + 1/2l0

12 −1/2r1 + 1/2l0

263

where the labels of hinges are shown in Fig. D.4. For region B, the side surface

contribution is

2πr2h1

(1

2r2

)2

+ 2πr1h1

(−1

2r1

)2

=πh1

2r2

+πh1

2r1

(D.30)

The hinge contribution is

2× 2πr2l0

(1

2r2

+1

2l0

)2

+ 2× 2πr1l0

(−1

2r1

+1

2l0

)2

=π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.31)

Hence the total contribution from region B is

∫

∂B

H2 =πh1

2r2

+πh1

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.32)

For region C, the side surface contribution is

2πr2h2

(1

2r2

)2

+ 2πr1h2

(−1

2r1

)2

=πh2

2r2

+πh2

2r1

(D.33)

The hinge contribution is

2× 2πr2l0

(1

2r2

+1

2l0

)2

+ 2× 2πr1l0

(−1

2r1

+1

2l0

)2

=π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.34)

Hence the total contribution from region C is

∫

∂C

H2 =πh2

2r2

+πh2

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.35)

For region AB, the side surface contribution is

2πr3(h1 + h2)

(1

2r3

)2

+ 2πr1h1

(−1

2r1

)2

+ 2πr2h2

(−1

2r2

)2

=π(h1 + h2)

2r3

+πh1

2r1

+πh2

2r2

(D.36)

264

The hinge contribution is

2× 2πr3l0

(1

2r3

+1

2l0

)2

+ 2× 2πr1l0

(− 1

2r1

+1

2l0

)2

+ 2πr2l0

(− 1

2r2

− 1

2l0

)2

+ 2πr2l0

(− 1

2r2

+1

2l0

)2(D.37)

Notice that the third term corresponds to the opposite of hinge 7 (which is not hinge

6). Hence the total contribution from region AB is

∫

∂AB

H2 =π(h1 + h2)

2r3

+πh1

2r1

+πh2

2r2

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0+π(r2 + l0)2

2r2l0+π(r2 − l0)2

2r2l0(D.38)

For region AC, the side surface contribution is

2πr3(h1 + h2)

(1

2r3

)2

+ 2πr2h1

(−1

2r2

)2

+ 2πr1h2

(−1

2r1

)2

=π(h1 + h2)

2r3

+πh1

2r2

+πh2

2r1

(D.39)

The hinge contribution is

2×2πr3l0

(1

2r3

+1

2l0

)2

+2×2πr1l0

(− 1

2r1

+1

2l0

)2

+2πr2l0

(− 1

2r2

− 1

2l0

)2

+2πr2l0

(− 1

2r2

+1

2l0

)2

(D.40)

Hence the total contribution from region AC is

∫

∂AC

H2 =π(h1 + h2)

2r3

+πh2

2r1

+πh1

2r2

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0+π(r2 + l0)2

2r2l0+π(r2 − l0)2

2r2l0(D.41)

For region BC, the side surface contribution is

2πr2(h1 + h2)

(1

2r2

)2

+ 2πr1(h1 + h2)

(−1

2r1

)2

=π(h1 + h2)

2r2

+π(h1 + h2)

2r1

(D.42)

265

The hinge contribution is

2×2πr2l0

(1

2r2

+1

2l0

)2

+2×2πr1l0

(−1

2r1

+1

2l0

)2

=π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.43)

Hence the total contribution from region BC is

∫

∂BC

H2 =π(h1 + h2)

2r2

+π(h1 + h2)

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0(D.44)

Finally, for region ABC, the side surface contribution is

2πr3(h1 + h2)

(1

2r3

)2

+ 2πr1(h1 + h2)

(−1

2r1

)2

=π(h1 + h2)

2r3

+π(h1 + h2)

2r1

(D.45)

The hinge contribution is

2×2πr3l0

(1

2r3

+1

2l0

)2

+2×2πr1l0

(−1

2r1

+1

2l0

)2

=π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0(D.46)

Hence the total contribution from region ABC is

∫

∂ABC

H2 =π(h1 + h2)

2r3

+π(h1 + h2)

2r1

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0(D.47)

266

In summary, we obtain the contribution of mean curvature squared of seven regions

as follows.

∫

∂A

H2 =π(h1 + h2)

2r3

+π(h1 + h2)

2r2

+π(r3 + l0)2

r3l0+π(r2 − l0)2

r2l0.,

∫

∂B

H2 =πh1

2r2

+πh1

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0,

∫

∂C

H2 =πh2

2r2

+πh2

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0,

∫

∂AB

H2 =π(h1 + h2)

2r3

+πh1

2r1

+πh2

2r2

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0+π(r2 + l0)2

2r2l0+π(r2 − l0)2

2r2l0,

∫

∂AC

H2 =π(h1 + h2)

2r3

+πh2

2r1

+πh1

2r2

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0+π(r2 + l0)2

2r2l0+π(r2 − l0)2

2r2l0,

∫

∂BC

H2 =π(h1 + h2)

2r2

+π(h1 + h2)

2r1

+π(r2 + l0)2

r2l0+π(r1 − l0)2

r1l0,

∫

∂ABC

H2 =π(h1 + h2)

2r3

+π(h1 + h2)

2r1

+π(r3 + l0)2

r3l0+π(r1 − l0)2

r1l0.

(D.48)

It is straightforward to check that the combination Eq. (5.18) vanishes. Hence the

relation Eq. (5.17) in the main text holds.

D.5 Review of Lattice TQFT

In this section, we briefly review the lattice formulation of TQFTs. We begin with a

triangulation of spacetime. The letters i, j, k etc. label the vertices of a spacetime

lattice. Combinations of vertices denote the simplicies of the lattice. For instance,

(ij) is the 1-simplex (bond) whose ends are vertices i and j. (ijk) is a 2-simplex

(triangle) whose vertices are i, j and k. Gauge fields live on these simplicies. In

our paper, 1-form gauge fields A live on 1-simplicies; 2-form gauge fields B live on

2-simplicies; etc. In the language of discrete theories, A(ij), B(ijk) are the 1-cochain

and 2-cochain associated with the indicated 1-simplex and 2-simplex, respectively.

267

Exterior derivatives are defined by:

dA(ijk) =A(jk)− A(ik) + A(ij),

dB(ijkl) =B(jkl)−B(ikl) +B(ijl)−B(ijk).

(D.49)

Note that the vertices are ordered such that i < j < k < l.

We further illustrate the values that the cochains A(ij) and B(ijk) can take using

canonical quantization. Let us first consider the GWW model described by Eq. (5.19)

on a continuous spacetime with U(1) gauge group. It is known that there are n surface

operators exp(is∮

ΣB), s = 0, 1, · · · , n− 1[134, 135], and exp(in

∮ΣB) = 1 is a trivial

operator for an arbitrary closed surface Σ. Hence∮

ΣB = 2πq

n, where q ∈ Zn and Σ

is any closed surface. The fact that exp(in∮

ΣB) is a trivial operator can be verified

via canonical quantization. To perform canonical quantization, we first use the gauge

transformation Eq. (5.20) to fix the gauge At = 0, Btx = 0, Bty = 0, Btz = 0. The

commutation relations from canonical quantization are

[Ax(t, x, y, z), Byz(t, x′, y′, z′)] = −i2π

nδ(x− x′)δ(y − y′)δ(z − z′). (D.50)

and similarly for other components. Using Eq. (D.50), we find that exp(in∮

ΣB)

commutes with all other gauge invariant operators. Specifically, we compute the

commutation relation between the surface operator exp(in∮

ΣB) and the line operator

exp(il∮γA+ ilp

∫Σ2B). Here Σ is a closed surface in a spatial slice, and Σ2 is an open

surface with boundary γ. Both Σ2 and γ are living in the spatial slice. We find

ein∮ΣBe

il∮γ A+ilp

∫Σ2

B= ei

2πnnlNΣ,γe

il∮γ A+ilp

∫Σ2

Bein

∮ΣB

= eil∮γ A+ilp

∫Σ2

Bein

∮ΣB,

(D.51)

where NΣ,γ is the intersection number of the surface Σ and the loop γ. Since the

phase factor coming from the commutation relation is always 1, exp(in∮

ΣB) com-

268

mutes with all line operators. Since it also commutes with exp(il∮

Σ′B) for any l and

Σ′, we conclude that exp(in∮

ΣB) commutes with all the gauge invariant operators.

Therefore, it must be a constant operator, ein∮ΣB = eiθ where θ is a constant number.

We further show that ein∮Σ B = 1. To show this, we act ein

∮Σ B on a state |0〉 where

B = 0 everywhere (more concretely, if the spacetime is discrete, B = 0 on every

2-simplex). Since ein∮ΣB measures the value of B-field of the state, and B-field is

zero everywhere,

eiθ|0〉 = ein∮Σ B|0〉 = |0〉 (D.52)

Hence the constant number eiθ = 1 everywhere. This proves that ein∮ΣB = 1.

Similarly, exp(in∮γA+ inp

∫Σ2B) commutes with all other operators as well.

ein

∮γ A+inp

∫Σ2

Beil

∮Σ B

= e−i2πnnlNΣ,γeil

∮ΣBe

in∮γ A+inp

∫Σ2

B

= eil∮ΣBe

in∮γ A+inp

∫Σ2

B.

(D.53)

and

ein

∮γ A+inp

∫Σ2

Beil∮γ′ A+ilp

∫Σ′2

B

= e−i 2π

nnlp(Nγ,Σ′2

−Nγ′,Σ2)eil∮γ′ A+ilp

∫Σ′2

Bein

∮γ A+inp

∫Σ2

B

= eil∮γ′ A+ilp

∫Σ′2

Bein

∮γ A+inp

∫Σ2

B.

(D.54)

Therefore ein

∮γ A+inp

∫Σ2

Bcommutes with all gauge invariant operators as well, which

implies ein

∮γ A+inp

∫Σ2

B= eiη where eiη is a constant. Using the same analysis for the

operator ein∮ΣB, we find e

in∮γ A+inp

∫Σ2

B= 1.

On a triangulated lattice, since Σ is any two dimensional surface, exp(in∮

ΣB) = 1

implies that exp(in∮

(ijkl)B) = 1 for any 3-simplex (ijkl). Using the Stokes formula,

269

∮(ijkl)

B =∫

(ijkl)dB = (dB)(ijkl) = B(ijk)−B(ijl) +B(ikl)−B(jkl) where we used

the fact that integrating dB over the volume of 3-simplex (ijkl) is just evaluating the

dB on (ijkl) itself. Hence exp(in∮

(ijkl)B) = 1 implies that B(ijk)−B(ijl)+B(ikl)−

B(jkl) ∈ 2πnZn for any 3-simplex (ijkl). Since the choice of (ijkl) is arbitrary, we

conclude that on each 2-simplex (ijk), B(ijk) takes values in 2πnZn. Similarly, on

each 1-simplex (ij), A(ij) takes values in 2πnZn for any i, j.

Next, we comment on the delta functions obtained from integrating out the A

fields as in Eq. (5.24). For simplicity, we work with a level n = 2 BF/GWW theory.

On each 4-simplex with vertices labeled by (i, j, k, l, s), the action is

2

2π(AdB)(ijkls) =

2

2πA(ij)dB(jkls). (D.55)

Integrating over A means summing over all configurations of A(ij) = 0, π. Hence the

path integral is

1

2

∑

A(ij)=0,π

exp

[i

2

2πA(ij)dB(jkls)

]

=1

2

{1 + exp

[idB(jkls)

]}≡ δ[dB(jkls)

]. (D.56)

This explains the meaning of the delta function in the discrete theory, and we refer to

the B field as flat if the above delta function constraint is satisfied, i.e. if dB(jkls) = 0

mod 2π.

Although we write TQFT actions as integrals in the continuum in the main text,

they can actually be translated into lattice actions using the conventions we have

introduced in this appendix. The wave functions defined via the path integral in

Eqs. (5.23) and (5.27) are then wave functions on the lattice.

270

D.6 Surfaces in the dual lattice

In this appendix, we argue that the simplices on which B = π in the dual lattice form

continuous surfaces. Continuous means that connected simplices in the dual lattice

join via edges, rather than via vertices. Specifically,

1. In three-dimensional space, if a real space 2-cochain B(ijk) satisfies the flatness

condition dB(ijkl) = B(jkl)− B(ikl) + B(ijl)− B(ijk) = 0 mod 2π then its

dual B = π on a closed loop in the dual lattice.

2. In (3 + 1)-dimensional spacetime, if a real space 2-cochain B(ijk) satisfies the

flatness condition dB(ijkl) = B(jkl) − B(ikl) + B(ijl) − B(ijk) = 0 mod 2π

then its dual B = π on a continuous and closed surface in the dual lattice.

The first statement is proven in the main text. In the following, we will present a

more algebraic proof of the first statement, which is easier to generalize to (3 + 1)-

dimensions, allowing for a proof of the second statement.

i

s

qrp

l

k

j

a

e

dcb

Figure D.5: Dual lattice of a tetrahedron (ijkl). (ijkp), (ijlq), (iklr), (jkls) are four

adjacent tetrahedra to (ijkl), which are dual to (b), (c), (d), (e), (a) respectively. The

red dots are the intersection between 2-simplices in the real lattice and the 1-simplices

in the dual lattice. For example, the red dot on (ab) is the intersection point of (ab)

and (ijk).

271

We first redraw the simplex in Fig. 5.3 with some additional details, as shown in

Fig. D.5. To construct the duals of simplices in three-dimensional space, we begin by

considering the tetrahedron (ijkl), in addition to its neighbors (ijkp), (ijlq), (iklr),

and (jkls). 3-simplices in the real lattice are dual to points in the dual lattice: for

example (ijkl) is dual to the point (a), and similarly (ijkp) is dual to (b), (ijlq) is

dual to (c), (iklr) is dual to (d), and (jkls) is dual to (e). 2-simplices in the real

lattice are dual to 1-simplices (bonds). For example, (ijk) is the intersection of (ijkl)

and (ijkp), i.e., (ijk) = (ijkl)∩ (ijkp). Therefore, the dual of (ijk) is the bond (ab),

joining the dual of (ijkl) and (ijkp). Similarly, we are able to identify the duals of

all other simplices. We list the result in the following table:

Real Dual

(ijkl) (a)

(ijkp) (b)

(ijlq) (c)

(iklr) (d)

(jkls) (e)

Real Dual

(ijk) (ab)

(ijl) (ac)

(ikl) (ad)

(jkl) (ae)

The flatness condition implies that there are even number of 2-simplices among the

four faces of the tetrahedron (ijkl) on which B = π. It follows that there are an even

number B = π bonds among the four dual lattice bonds (ab), (ac), (ad), (ae). Thus

these form closed loops in the dual lattice. This proves the first statement.

We proceed to prove the second statement. In (3 + 1) dimensions, spacetime is

triangulated into 4-simplices. Let us consider a 4-simplex labeled by the five vertices

(ijklm) where m is in the extra dimension compared with 3D case shown in Fig. D.5.

To find the dual of 2-simplices, we will begin – as above – by considering the 4-

simplices adjacent to (ijklm) which share one 3-simplex with (ijklm). Introducing

272

the additional vertices p, q, r, s, and t8, these 4-simplices are: (ijkmp), (ijlmq),

(iklmr), (jklms), and (ijklt). Dual simplices in (3 + 1) dimensional spacetime are

determined as follows: 4-simplices in the real lattice are dual to points in the dual

lattice; (ijklm) is dual to a point (a), (ijkmp) is dual to (b), (ijlmq) is dual to (c),

(iklmr) is dual to (d), (jklms) is dual to (e), and (ijklt) is dual to (f)9. 3-simplices

in the real lattice are dual to bonds in the dual lattice. For instance, since (ijkm) is

the intersection of (ijklm) and (ijkmp), i.e., (ijkm) = (ijklm)∩ (ijkmp), the dual of

(ijkl) is the bond (ab), joining the dual of (ijklm) and (ijkmp). Similarly, (ijlm) is

dual to (ac), (iklm) is dual to (ad), (jklm) is dual to (ae), and (ijkl) is dual to (af).

We further proceed to consider the dual of 2-simplices, applying the same method.

For instance, since the 2-simplex (ijk) is the common simplex of (ijkm) and (ijkl),

i.e., (ijk) = (ijkl) ∩ (ijkm), the dual of (ijk) is the surface (abf) joining the dual of

(ijkl) and (ijkm). Similarly, we can identify the duals of the remaining 2-simplices.

We list all the results in the following table:

Real Dual

(ijklm) (a)

(ijkmp) (b)

(ijlmq) (c)

(iklmr) (d)

(jklms) (e)

(ijklt) (f)

Real Dual

(ijkm) (ab)

(ijlm) (ac)

(iklm) (ad)

(jklm) (ae)

(ijkl) (af)

Real Dual

(ijk) (abf)

(ijl) (acf)

(ijm) (abc)

(ikl) (adf)

(ikm) (abd)

(ilm) (acd)

(jkl) (aef)

(jkm) (abe)

(jlm) (ace)

(klm) (ade)

8Notice that t is in the additional dimension as well.9Notice that (f) is in the additional dimension of the dual lattice.

273

The four surfaces (abf), (acf), (adf), (aef) are dual to the four faces (ijk), (ijl), (ikl), (jkl)

of the tetrahedron (ijkl). All of these dual surfaces share a common link (af). The

flatness condition dB(ijkl) = B(jkl)−B(ikl) +B(ijl)−B(ijk) = 0 mod 2π implies

that an even number of faces of the tetrahedron (ijkl) are occupied. Thus, there

are an even number of surfaces among (abf), (acf), (adf), (aef) occupied in the dual

lattice. Since all these occupied surfaces in the dual lattice share a common edge

(af), it follows from our definition of continuity (at the beginning of this appendix)

that surfaces in the dual lattice are continuous. Furthermore, the continuous surfaces

formed by the occupied simplices in the dual lattice are closed, because for any

bond in the dual lattice, for example (af), there exist even (among four) number

of occupied dual-lattice 2-simplices adjacent to it. While for an open dual-lattice

surface, there exist at least one dual-lattice bond such that there are only odd

number of the adjacent dual-lattice 2-simplices occupied, which violate the flatness

condition for the B-cochain. Hence the dual-lattice surface is closed. This proves the

second statement.

For completeness, we comment on how two loops can intersect in the dual space

lattice, and how two surfaces can intersect in the dual spacetime lattice. We first

prove by construction that two loops in the dual spatial lattice can intersect at a

vertex: suppose one dual lattice loop includes the occupied bonds (ab), (ac), and the

other dual lattice loop includes the occupied bonds (ad), (ae). Hence these two loops

intersect at the vertex (a). We now argue that if two surfaces in the dual spacetime

lattice contain the same point, then they must share a bond. Let us assume two

surfaces intersect (at least) at (a). Since all the 2-simplices in the dual lattice including

the vertex (a) are (abc), (abd), (acd), (abe), (ace), (ade), (abf), (acf), (adf) and (aef),

by enumerating all possibilities, we find the two surfaces must share at least one

bond. Without loss of generality, suppose one surface includes the 2-simplices (abc)

and (abd) (notice that (abc) and (abd) join via the bond (ab) and therefore form a

274

continuous surface in the dual lattice). The surface thus includes the three bonds

(ab), (ac), and (ad) emanating from (a). Any other surface that contains (a), would

include, just like this surface, three of bonds emanating from (a). Thus, as (a) is

the only shared part of five bonds (ab), (ac), (ad), (ae), and (af), two surfaces that

include (a) have to share at least one of these bonds, as they occupy three bonds

each. In summary, two loops can intersect at vertices in the dual space lattice, and

two surfaces can intersect at bonds (but not vertices) in the dual spacetime lattice.

D.7 Mutual and Self-Linking Numbers

In this section, we provide all details needed to evaluate the integral Eq. (5.29). As

a simple case, we assume a configuration where B = π only at two surfaces S1, S2 in

the dual lattice of M4, with their boundaries given by the loops l1 = ∂S1, l2 = ∂S2

on the dual lattice of ∂M4. We can write this succinctly as

B = π ∗4 Σ(S1) + π ∗4 Σ(S2), (D.57)

where ∗4 is the discretized version of Hodge star in four spacetime dimensions; its

meaning is explained pictorially in Fig. D.6. Let us comment on Eq. (D.57) in detail.

On ∂M4, B is a 2-cochain, which can be 0 or π; while on the dual lattice of ∂M4,

the π-valued 1-cochains Σ(li) (which are the dual of real-space 2-cochains) form loops

li, i = 1, 2. Moreover, on the spacetime M4, B is still a 2-cochain valued in 0 or

π; while on the dual lattice of M4, the π-valued 2-cochains Σ(Si) (which are the

dual of the real spacetime 2-cochains) form surfaces Si, i = 1, 2 whose boundaries are

li, i = 1, 2. Notice that the closed dual-lattice surfaces which do not intersect with the

spatial slice do not contribute to the wavefunction. Further ∗4Σ(Si) is a 2-cochain on

the original lattice (dual to Si), which is 1 on the dual of Si, and 0 elsewhere. Hence,

the role of the Hodge star is to transform the cochain defined on the dual lattice to the

275

cochain defined on the real lattice. In Fig. D.6 we illustrate the geometric meaning

of these notions with an example in lower dimensions. Returning to the integral in

the wavefunction Eq. (5.29), we thus have

l1

l2

Figure D.6: We illustrate the geometric meaning of the Hodge dual in a two-

dimensional space example. Suppose A is a 1-cochain, which equals π on 1-simplices

in the dual lattice and 0 elsewhere. A = π∗2Σ(l1)+π∗2Σ(l2), where l1 and l2 are loops

in the dual lattice drawn in dashed lines. Σ(l1) and Σ(l2) are 1-cochains living on the

1-simplices in the dual lattice. ∗2 is a lattice version of Hodge star, which transforms

the 1-cochain living on the dual lattice (dashed lines) to a 1-cochain living on the

lattice (green and purple bold lines). Correspondingly, A = π ∗2 Σ(l1) + π ∗2 Σ(l2) is

a 1-cochain living on the green and purple bold lines. We use the dual lattice con-

figuration Si, li to label the B,A-cochains because the dual lattice configurations are

easier to visualize. The interpretation of the 2-cochain B can be straightforwardly

generalized to three spatial dimensions.

276

∫

M4

B ∧B

=π2

∫

M4

(∗4 Σ(S1) + ∗4Σ(S2)

)∧(∗4 Σ(S1) + ∗4Σ(S2)

)

=2π2

∫

M4

∗4Σ(S1) ∧ ∗4Σ(S2) + π2

∫

M4

∗4Σ(S1) ∧ ∗4Σ(S1) + π2

∫

M4

∗4Σ(S2) ∧ ∗4Σ(S2)

=2π2link(l1, l2) + π2link(l1, l1) + π2link(l2, l2),

(D.58)

where link(l1, l2) is the linking number between two loops l1 and l2. This leads to

Eq. (5.30) in the main text.

We will derive the last equality of Eq. (D.58) in Appendix D.7.1, and provide a

detailed discussion of the self-linking numbers of one single loop in Appendix D.7.2.

D.7.1 Intersection and Linking

We prove a statement relating the intersection form in the bulk and the linking number

on the boundary, which in turn explains the last equality in Eq. (D.58).

As explained below Eq. (D.57), ∗4Σ(Si) is a 2-cochain in the real spacetime, which

equals 1 if it is evaluated on any triangulation of Si (in the dual spacetime lattice)

and 0 if evaluated elsewhere. Similarly, ∗3Σ(li) is still a 2-cochain in the real space,

which equals 1 if it is evaluated on the li (in the dual space lattice) and 0 if evaluated

elsewhere. Furthermore, if li is on the boundary of Si (notice that both li and Si are

277

in the dual lattice), we have a relation between these two 2-simplices,10

∗4Σ(Si) = ∗3Σ(∂Si) = ∗3Σ(li). (D.59)

We also notice that B is flat, i.e., d ∗4 Σ(Si) = d ∗3 Σ(li) = 0, i = 1, 2 which come

from the Gauss law for B-cochain Eq. (5.25). This means the duals of the B = π 2-

simplices form two-dimensional surfaces in the spacetime, and form one-dimensional

loops (which are the boundary of two-dimensional dual lattice surfaces) in the space,

as shown in Fig. 5.2. We want to prove,

∫

M4

∗4Σ(S1) ∧ ∗4Σ(S2) =

∫

l1∩∂−1l2

1 ≡ link(l1, l2), (D.60)

where ∂−1l2 denotes a surface in the dual lattice of ∂M4 whose boundary is l2. In

the last equality, we used the definition of the linking number between two loops.

The relation (D.60) can be shown as follows. Keeping in mind that ∗3Σ(l) is a

delta function that is nonzero on l only, we find

∫

l1∩∂−1l2

1 =

∫

M3

∗3Σ(l1) ∧ d−1 ∗3 Σ(l2). (D.61)

10We can understand this formula by constructing examples using the method in appendix D.6.Let (abf), (acf) ∈ S be two dual-lattice 2-simplices in the dual-lattice open surface S in 4D, whichjoin via (af). The boundary is along (ab) and (ac) direction, joined via (a). (ab), (ac) ∈ l forma loop in 3D, which is the boundary of S. We need to compare the real space configuration of Sand l by taking their duals. From the correspondence of real simplices and dual simplices listedin appendix D.6, in 3D, (ab), (ac) are dual to (ijk), (ijl) respectively, and in 4D, (abf), (acf) aredual to (ijk), (ijl) respectively. We find that their real lattice configurations are the same, hence∗4Σ(S) = ∗3Σ(l).

278

Noticing that M3 = ∂M4,

∫

M3

∗3Σ(l1) ∧ d−1 ∗3 Σ(l2) =

∫

∂M4

∗3Σ(l1) ∧ d−1 ∗3 Σ(l2)

=

∫

M4

d(∗4 Σ(S1) ∧ d−1 ∗4 Σ(S2)

)

=

∫

M4

∗4Σ(S1) ∧ ∗4Σ(S2). (D.62)

In the second equality, we used ∗4Σ(Si) = ∗3Σ(li), i = 1, 2. To get the last equality,

we used the flatness condition d ∗4 Σ(Si) = d ∗3 Σ(li) = 0, i = 1, 2. Hence

∫

M4

∗4Σ(S1) ∧ ∗4Σ(S2) =

∫

l1∩∂−1l2

1. (D.63)

Combining Eqs. (D.60), (D.62) and (D.63), we find

∫

M4

B ∧B = 2π2link(l1, l2) + π2link(l1, l1) + π2link(l2, l2).

D.7.2 Self-linking Number

In this subsection, we define the self-linking number of a loop l, i.e., the link(l, l). To

define the self-linking number, we need to regularize the loop into two nearby loops.

This can be achieved by point splitting regularization11. We separate each point of

the spatial lattice into two points, for example

(x, y, z)→

(x, y, z)

(x+ ax, y + ay, z + az)

, (D.64)

where (ax, ay, az) is a constant vector in space chosen to be the same for all loops.

The original loop l splits into two loops l and la.

11The point splitting method is widely used in studying lattice systems, such as in Ref. [198, 199].

279

(ax, ay, az)

Figure D.7: Regularization of a spatial lattice. The blue arrow represents the con-

stant vector (ax, ay, az). The dashed lattice is obtained from the solid lattice by the

translation (x, y, z)→ (x+ ax, y + ay, z + az).

la

l

Figure D.8: An example of lattice regularization of a trefoil knot. l is a knot (drawn

in the dual lattice), while la is the knot obtained by lattice regularization. The

underlying lattice is omitted for clarity.

See Fig. D.7 for an illustration of lattice regularization and Fig. D.8 for an il-

lustration of the regularization of a loop. The mutual-linking number between two

loops is well defined, and it is natural to identify the self-linking number of l to be

280

the mutual-linking number between l and la, i.e.,

link(l, l) ≡ link(l, la). (D.65)

We notice that the definition Eq. (D.65) depends on the regularization Eq. (D.64).

But as long as we use the same regularization for all the loops l [i.e., (ax, ay, az) is

a position-independent constant vector], Eq. (D.65) is consistent [i.e., translating l

(without change its shape) does not change the self-linking number link(l, l) of l].

The definition of the self-linking number of a loop (knot) depends on the point

splitting regularization [i.e., changing the constant vector (ax, ay, az) changes the reg-

ularization, and hence changes the self linking number], and so does the wavefunction.

However, the entanglement entropy is independent of the self-linking number, hence

it is independent of the point splitting regularization.

D.8 NA(CE)NAc(CE) is Independent of CE

In this appendix, we give a more detailed derivation of Eq. (5.41). We first show that

NA(CE)NAc(CE) is independent of CE. We further explain the fact that the number of

configurations on the entanglement surface Σ is 2|Σ|−1.

We start by establishing a one-to-one correspondence between a configuration

CE and a configuration with no dual lattice loops across the entanglement surface.

We find that it is more illuminating to demonstrate this using a two-dimensional

square lattice (but similar arguments work for triangular lattice as well), as shown

in Fig. D.9, which is a spatial slice of the (2 + 1)D spacetime. For simplicity, we

consider the n = 2 case only, where each bond12 is either occupied (B = π mod 2π)

or unoccupied (B = 0 mod 2π). In panel (a), we present a general configuration

12In this section, we will use bonds instead of 1-simplices because simplices are not defined on thesquare lattice.

281

with one occupied loop13 in the dual lattice (the dotted line). The corresponding

configuration in the real lattice is given by the red bonds. The entanglement cut Σ

consists of the green bonds, where two are occupied (bonds which are both green and

red). In panel (b), we present a related configuration with no bonds occupied on Σ.

We denote the boundary configuration on the entanglement surface Σ with no bonds

occupied as C0. The configuration in (b) is obtained from the configuration in (a) by

cutting the loop at Σ in the dual lattice and completing the loops along Σ within

the two regions A and Ac separately. Therefore, we have shown that every bulk

configuration with non-trivial boundary CE can be reduced to a bulk configuration

with trivial boundary configuration C0. However, we note that there can be multiple

ways of cutting and completing the loops (which is more obvious in three spatial

dimensions), and the reduction may not be unique. Hence we have shown that

NAc(CE)NA(CE) ≤ NAc(C0)NA(C0). (D.66)

13The loop configuration is given by the flatness condition dB = 0 mod 2π. On a 2D spatiallattice, B is a 1-form and the flatness condition is (dB)(i, i+x, i+ y, i+x+ y) = B(i, i+x) +B(i+x, i+ x+ y)−B(i+ y, i+ x+ y)−B(i, i+ y) = 0 mod 2π. On a 3D spatial lattice, B is a 2-formand the flatness condition is (dB)(i, i+ x, i+ y, i+ z, i+ x+ y, i+ x+ z, i+ y + z, i+ x+ y + z) =B(i, i+x, i+x+y, i+y)−B(i+z, i+z+x, i+z+x+y, i+z+y)+B(i, i+z, i+x+z, i+x)−B(i+y, i+y+z, i+y+x+z, i+y+x)+B(i, i+y, i+y+z, i+z)−B(i+x, i+x+y, i+x+y+z, i+x+z) = 0mod 2π.

282

A

A

⌃

c

(a)

A

A

⌃

c

(b)

Figure D.9: A configuration associated with nontrivial CE (on panel (a)) can be

reduced to a configuration associated with trivial CE (on panel (b)).

A

A

⌃

c

(a)

A

A

⌃

c

(b)

Figure D.10: A configuration associated with trivial CE (on panel (a)) can be reduced

to a configuration associated with a nontrivial CE (on panel (b)).

To complete the one-to-one correspondence, we have to consider the opposite

deformation: every bulk configuration with trivial boundary configuration C0 can be

changed to a bulk configuration with a specified non-trivial boundary configuration

CE. We use Fig. D.10 to illustrate this process. In panel (a), we present a configuration

with no bonds occupied on Σ, corresponding to the trivial boundary configuration

C0. In panel (b), we draw a specific configuration in which two bonds are occupied.

283

The two occupied bonds on Σ are connected via a “thin” loop along the two sides of

Σ. Therefore, a bulk configuration with nontrivial boundary configuration CE can be

obtained from a bulk configuration with trivial boundary configuration C0 by adding

a “thin” loop along the two sides of the entanglement cut. However, we note that

starting from a configuration with C0, there can be multiple ways to add the thin

loops to obtain a corresponding configuration with a nontrivial CE. Hence, we have

shown that

NAc(C0)NA(C0) ≤ NAc(CE)NA(CE). (D.67)

Combining the inequalities (D.66) and (D.67), we obtain

NAc(CE)NA(CE) = NAc(C0)NA(C0). (D.68)

Equation (D.68) shows that NAc(CE)NA(CE) is independent of the configuration CE,

as expected.

284

(a)

(g)(f)(e)

(d)(c)(b)

(h)

Figure D.11: Configurations on a 2 × 2 lattice with periodic boundary conditions.

There are two entanglement cuts, denoted by two green lines. The occupied bonds

in the real lattice are shown in red, and occupied bonds in the dual lattice are shown

as dotted lines. (a), (b), (c), (d) are configurations with no bonds occupied on the

entanglement cut. (e), (f), (g), (h) are configurations with two bonds occupied on

the entanglement cut.

In addition to the general arguments, it is beneficial to consider an example. In

Fig. D.11, we present all the configurations on a 2 × 2 lattice associated with C0

(no bonds occupied on the entanglement surface) and with CE (two bonds in the

middle occupied on the entanglement surface). The configuration such as does

not exist because the configuration in the dual lattice is not a loop. In each case,

there are 4 configurations, which agrees with our general analysis NAc(CE)NA(CE) =

NAc(C0)NA(C0).

We further show that the total number of configurations on CE is 2|Σ|−1 for the

n = 2 theory, where |Σ| is the number of simplices (bonds) on Σ. (The discussion

285

in this paragraph works for both triangular and square lattices, and we will use the

notations simplices and cochains here.) Notice that since each B-cochain can take

2 values, i.e., 0 mod 2π or π mod 2π, the naive counting of configurations of CE

is 2|Σ|. However, since the simplices where B = π mod 2π form loops in the dual

lattice, there must be an even number of simplices occupied on Σ. This reduces

the total number of CE configurations by half. Therefore, there are 2|Σ|−1 possible

configurations on the entanglement surface. Applying the normalization condition

Eq. (5.40), we complete the demonstration of Eq. (5.41).

D.9 A Case Study of the Conjecture Between GSD

and TEE

In this appendix, we examine the conjecture Eq. (5.58) for the BF theory with level

n in (d+ 1)D by explicitly computing both the GSD on d-dimensional torus T d and

the constant part of the EE across T d−1 (which we believe is the topological part for

the BF theory).

The action of the BF theory with level n on the spacetime T d × S1 is

SBF =

∫

T d×S1

n

2πB ∧ dA, (D.69)

where A is a 1-form gauge field and B is a (d − 1)-form gauge field. The gauge

transformations are A → A + dg, B → B + dλ where λ is a u(1) valued (d − 1)-

form gauge field, and g is a compact scalar (i.e., g ' g + 2π). The gauge invariant

operators, which wrap around the non-contractible cycles of the spatial torus T d, are

V kTi1···id−1

= exp(ik

∮

Ti1···id−1

B), k ∈ {0, 1, · · · , n− 1},

W lTi

= exp(il

∮

Ti

A), l ∈ {0, 1, · · · , n− 1}, (D.70)

286

and their combinations. In the first equation Ti1···id−1is a (d−1)-dimensional torus ex-

tending along the i1 · · · id−1 directions and in the second equation Ti is a 1-dimensional

circle extending along the i-th direction. (The fact that V nTi1···id−1

and W nTi

are triv-

ial operators will be explained in the following.) We will use canonical quantization

to determine the commutation relation between these operators, from which we can

determine the ground state degeneracy GSD[T d].

To perform the canonical quantization, we first fix the gauge as A0 =

0, B0i1···id−2= 0 for any i1 · · · id−2 using the gauge transformations A→ A+dg, B →

B+ dλ. Moreover, the Gauss constraints are ε0i1···id−1id∂id−1Aid = 0 for any i1 · · · id−2,

and ε0i1···id−1id∂i1Bi2···id = 0 where summation over repeated indices is implied. We

have used the definition of totally anti-symmetric tensor

εi1···id−1 =

+1, if i1 · · · id−1 is an even permutation of 0 · · · d− 2

−1, if i1 · · · id−1 is an odd permutation of 0 · · · d− 2

0 otherwise.

(D.71)

The Lagrangian, after gauge fixing, is

LBF =n

2π

(−1)d−1

(d− 1)!εi1···idBi1···id−1

∂0Aid , (D.72)

where Bi1···id−1and Aid obey the Gauss constraints. The canonical quantization con-

ditions on the gauge fields are

[(−1)d−1

(d− 1)!εi1···idBi1···id−1

(t, ~x), Ajd(t, ~y)

]=

2πi

nδidjdδ(~x− ~y). (D.73)

From this canonical relation, one can determine the commutation relation of the line

and higher volume operators by applying the Baker-Campbell-Hausdorff formula. We

287

find

V kTi1···id−1

W lTid

= e(−1)di2πkl/nW lTidV kTi1···id−1

. (D.74)

From Eq.(D.74), we can see that exp(in∮Ti1···id−1

B) commutes with any line

operator exp(ik∮TiA), and also trivially commutes with any surface operator

exp(ik∮Tj1···jd−1

B). Therefore, exp(in∮Ti1···id−1

B) commutes with any gauge invari-

ant operator and should be a constant. By using the same argument as in App. D.5,

exp(in∮Ti1···id−1

B) = 1. Similarly, we find that exp(in∮TiA) = 1 as well. The

explains that the charges k and l of the non-local operators V kTi1···id−1

and W lTid

only

take n different values.

We can define the ground states |u1 · · ·ud〉 to be the eigenstates of W li , and choose

V kTi1···id−1

as the raising and lowering operators acting on the ground states. Since

W ni = 1, the eigenvalues of Wi should be n-th root of unity, i.e., e−(−1)di2πui/n, where

ui ∈ {0, 1, · · · , n− 1}. Specifically,

W li |u1 · · ·ud〉 = e−(−1)di2πlui/n|u1 · · ·ud〉,

V kT12···(i−1)(i+1)···d

|u1 · · ·ud〉 = |u1 · · ·ui−1(ui + 1)ui+1 · · ·ud〉,(D.75)

where ui ∈ {0, 1, · · · , n − 1} for all i. Therefore, there are nd ground states on the

d-dimensional spatial torus, GSD[T d] = nd.

To obtain the EE, we generalize the calculations of Sec. 5.2. Since most of the

calculations are similar, we will only present the crucial steps.

We start by formulating the theory on the higher dimensional triangulated space-

time lattice Md+1. The ground state wavefunction is still the equal weight superpo-

sition of loop configurations in the dual of the spatial lattice,

|ψ〉 = C∑

C∈L

|C〉, (D.76)

288

where the sum is taken over the set L of all possible loop configurations C at the

dual lattice of spatial slice Sd = ∂Md+1. We choose the entanglement surface to be

a (d − 1)-dimensional torus, separating the space into two regions A and Ac. The

wavefunction is

|ψ〉 = C∑

CE

NA(CE)∑

a=1

NAc (CE)∑

b=1

|ACEa 〉|AcCEb 〉, (D.77)

from which one can obtain the reduced density matrix by tracing over the degrees of

freedom in region Ac,

ρA = |C|2∑

CE

NAc(CE)

NA(CE)∑

a,a′=1

|ACEa 〉〈ACEa′ |. (D.78)

The normalization constant C is determined by TrHAρA = |C|2NA(CE)NAc(CE)n|Σ|−1 =

1, where |Σ| is the number of (d− 1)-simplices on the entanglement surface. The EE

is

S(A) = −TrHAρA log ρA =

d

dN

(− TrHA

ρNA(TrHA

ρA)N

)∣∣∣∣N=1

= − d

dN

(|C|2N

∑

CE

NAc(CE)NNA(CE)N)∣∣∣∣

N=1

= − d

dN

(∑

CE

n−(|Σ|−1)N

)∣∣∣∣N=1

= − d

dN

(n−(|Σ|−1)(N−1)

)∣∣∣∣N=1

= |Σ| log n− log n.

(D.79)

In the second line, we used the normalization TrHAρA = 1, TrHA

ρNA = |C|2N∑CE NAc(CE)NNA(CE)N .

In the third line, we used |C|2NA(CE)NAc(CE) = n−(|Σ|−1). In the fourth line, since

the summand does not depend on CE, we just multiply the summand by the number

of CE n|Σ|−1. In the last line, we take the differential with respect to N and take

289

N = 1. Therefore, the constant part of the EE across T d−1 is − log n, which we

conjecture to be the TEE across T d−1. Combining the results GSD[T d] = nd and

Stopo[T d−1] = − log n, we expect that the conjecture exp(−dStopo[T d−1]) = GSD[T d]

of Eq. (5.58) holds for the (d+ 1)-dimensional BF theory.

290

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