Topological Phases, Entanglement and Boson
Condensation
Huan He
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 2019
c© Copyright by Huan He, 2019.
All rights reserved.
Abstract
This dissertation investigates the boson condensation of topological phases and the
entanglement entropies of exactly solvable models.
First, the bosons in a “parent” (2+1)D topological phase can be condensed to
obtain a “child” topological phase. We prove that the boson condensation formalism
necessarily has a pair of modular matrix conditions: the modular matrices of the
parent and the child topological phases are connected by an integer matrix. These
two modular matrix conditions serve as a numerical tool to search for all possible
boson condensation transitions from the parent topological phase, and predict the
child topological phases. As applications of the modular matrix conditions, (1) we
recover the Kitaev’s 16-fold way, which classifies 16 different chiral superconductors
in (2+1)D; (2) we prove that in any layers of topological theories SO(3)k with odd
k, there do not exist condensable bosons.
Second, an Abelian boson is always condensable. The condensation formalism in
this scenario can be easily implemented by introducing higher form gauge symmetry.
As an application in (2+1)D, the higher form gauge symmetry formalism recovers
the same results of previous studies: bosons and only bosons can be condensed in
an Abelian topological phase, and the deconfined particles braid trivially with the
condensed bosons while the confined ones braid nontrivially. We emphasize again
that the there exist non-Abelian bosons that cannot be condensed.
Third, the ground states of stabilizer codes can be written as tensor network
states. The entanglement entropy of such tensor network states can be calculated
exactly. The 3D fracton models, as exotic stabilizer codes, are known to have several
features which exceed the 3D topological phases: (i) the ground state degeneracy
generally increase with the system size; (ii) the gapped excitations are immobile or
only mobile in certain sub-manifolds. In our work, we calculate, for the first time,
the entanglement entropy for the fracton models, and show that the entanglement
iii
entropy has a topological term linear to the subregions’ sizes, whereas the topological
phases only have constant topological entanglement entropies.
iv
Acknowledgements
First and foremost, I would like to thank my advisor Prof. B. Andrei Bernevig for
guidance and support. I have benefited enormously from his curiosity, dedication to
physics and perseverance. He provides me with a complete freedom to work on the
problems and topics that I am interested in, and more importantly, he never allows
me to give up easily until we all find a satisfying answer. It is fair to say that rigorous
and solid research is the trademark of our group, which I now completely agree is
truly valuable. I will bear this spirit in mind for the rest of my career.
During the journey, I am indebted to the help from many professors and friends
who make me a better physicist. Among all of them, I owe the most to Prof Nicolas
Regnault, Prof German Sierra and Yunqin Zheng. Many papers were not possible
without their help and efforts. In fact, Nicolas and German are the two aspects of
modern physicists. Nicolas’s numerical technologies are unbelievable. I still remember
how I was amazed when he finished a new program and computation before I finished
speaking my speculations. It turned out that I was wrong and he was correct. I am
lucky to meet and learn from German. Every time I discussed with German, I would
enjoy the aesthetic of his physics and formulas, which I have almost forgotten over
the years of research. His formulas reminded me of my initial motivation of becoming
a physicist: the simple and beautiful laws of nature. Yunqin has been my major
collaborator since the end of my first year in Princeton. In fact, we have known each
other since undergraduate (10 years ago from now). I would never forget all the time
we spent together with chalks on blackboards, pens on papers, and Mathematica on
screens.
The PCTS program and our department certainly bring in brilliant and optimistic
young physicists globally. I am grateful for the collaborations with the postdoctoral
researchers in Princeton: Prof Curt von Keyserlingk, Prof Titus Neupert, Prof Barry
Bradlyn, Prof Jennifer Cano, Prof Yi Li, Dr Abhinav Prem, Dr Lian Biao, Dr Yizhi
v
You and Dr Kiryl Pakrouski. I hope that they also enjoy discussing and working with
me.
It was joyful and also surprisingly efficient to work with Prof Jia-Wei Mei and Dr.
Ji-Yao Chen. A powerful tensor network MATLAB program was delivered in only
a few weeks over the summer. I need to thank Xue-Da Wen, Apoorv Tiwari, Peng
Ye for analytically calculating entanglement entropies, and being reliable and helpful
collaborators for this project. Particularly, Xue-Da is the person who I would like to
discuss all the time because of his enthusiastic passion.
It would be my fault if I did not mention my group members, who I have in-
teractions with, for many enlightening and refreshing conversations and discussions:
Yang-Le Wu, Aris Alexandradinata, Sanjay Moudgalya, Zhi-Jun Wang, Jian Li, Zhi-
Da Song, Fang Xie. I also would like to thank the physicists that I met during confer-
ences, visit, emails and Skype: Rui-Xing Zhang, Chen Fang, Hong Yao, Yuan-Ming
Lu, Meng Cheng, Yang Qi, Yuan Wan, Xie Chen, Giuseppe Carleo, Alan Morningstar,
Yi-Chen Huang, Juven Wang, Yidun Wan, Heidar Moradi, Wen Wei Ho, Tian Lan
and many others.
My life would be miserable if without my friends in Princeton: Jing-Yu Luo, Jie
Wang, Zheng Ma, Jing-Jing Lin, Yu Shen, Wu-Di Wang, Zhen-Bin Yang, Jun Xiong,
Liang-Sheng Zhang, Bin Xu, Zhao-Qi Leng, Tong Gao, Suerfu, Yin-Yu Liu, Jia-Qi
Jiang, Si-Hang Liang, Xin-Ran Li, Xiao-Wen Chen, Bo Zhao, Jun-Yi Zhang, Xue
Song, Xin-Wei Yu, Hao Zheng, Zi-Ming Ji, Ho Tat Lam, Wen-Li Zhao, Po-Shen Hsin,
Xiao-Jie Shi, Chaney Lin and many many others.
I appreciate Prof Duncan Haldane, Prof Waseem Bakr and Prof Silviu Pufu for be-
ing my committee members, and Prof Robert Austin for leading me the experimental
project.
vi
Last but not the least, I appreciate the support from my parents during all these
years. And most importantly, it is beyond my language to describe my gratitude for
many years’ accompany with my love, Yina. This dissertation is dedicated to you.
vii
To my love Yina
viii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Preliminaries 1
1.1 Stabilizer Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Entanglement Entropy and Tensor Network States . . . . . . . . . . . 4
1.3 Abelian Topological Field Theory . . . . . . . . . . . . . . . . . . . . 7
1.4 Modular Tensor Category . . . . . . . . . . . . . . . . . . . . . . . . 9
2 An Example: Toric Code Model 14
2.1 Stabilizer Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Tensor Network State . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Modular Tensor Category . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 BF Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Boson Condensation 23
3.1 Definitions and Assumptions . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 The Condensation Matrix Mab . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Proof that M commutes with T matrix of the A theory . . . . 31
3.2.2 Proof that M commutes with S matrix of the A theory . . . . 31
ix
3.3 The Modular Tensor Category after condensation . . . . . . . . . . . 33
3.4 Simple currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.1 Introduction to simple currents . . . . . . . . . . . . . . . . . 36
3.4.2 Simple current condensation . . . . . . . . . . . . . . . . . . . 38
3.5 One confined particle theories . . . . . . . . . . . . . . . . . . . . . . 43
3.6 Formalism and implementation . . . . . . . . . . . . . . . . . . . . . 44
3.6.1 Solutions for M . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6.2 Solutions for n . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.6.3 The modular matrices of the new theory . . . . . . . . . . . . 51
3.7 Layer constructions and uncondensable bosons . . . . . . . . . . . . . 54
3.7.1 Theories with Zm-graded condensations . . . . . . . . . . . . . 55
3.7.2 Theories with Z-fold way: Fibonacci TQFT . . . . . . . . . . 60
3.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 No-Go Theorem for Boson Condensation in Topologically Ordered
Quantum Liquids 65
4.1 First No-Go Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Example (i): Multiple layers of Fibonacci . . . . . . . . . . . . . . . . 69
4.3 Example (ii): Single layer of SO(3)k . . . . . . . . . . . . . . . . . . . 71
4.4 Example (iii): Multiple layers of SO(3)k . . . . . . . . . . . . . . . . 74
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Abelian Boson Condensation in Field Theory 76
5.1 Abelian Boson Condensation Formalism . . . . . . . . . . . . . . . . 77
5.2 Condensations in K-Matrix Chern-Simons Theories . . . . . . . . . . 81
5.2.1 Condensable Condition . . . . . . . . . . . . . . . . . . . . . . 82
5.2.2 Confinement/Deconfinement . . . . . . . . . . . . . . . . . . . 84
x
6 Fracton Models, Tensor Network States and Their Entanglement
Entropies 87
6.1 Stabilizer Code Tensor Network States . . . . . . . . . . . . . . . . . 91
6.1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.2 Stabilizer Code and TNS Construction . . . . . . . . . . . . . 94
6.1.3 TNS Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1.4 Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Entanglement properties of the stabilizer code TNS . . . . . . . . . . 102
6.2.1 TNS as an exact SVD . . . . . . . . . . . . . . . . . . . . . . 102
6.2.2 Summary of the results . . . . . . . . . . . . . . . . . . . . . . 107
6.3 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.1 Hamiltonian of 3D Toric Code Model . . . . . . . . . . . . . . 109
6.3.2 TNS for 3D Toric Code . . . . . . . . . . . . . . . . . . . . . . 111
6.3.3 Concatenation Lemma . . . . . . . . . . . . . . . . . . . . . . 115
6.3.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.3.5 Transfer Matrix as a Projector . . . . . . . . . . . . . . . . . . 121
6.3.6 GSD and Transfer Matrix . . . . . . . . . . . . . . . . . . . . 124
6.4 X-cube Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.4.1 Hamiltonian of X-cube Model . . . . . . . . . . . . . . . . . . 129
6.4.2 TNS for X-cube Model . . . . . . . . . . . . . . . . . . . . . . 131
6.4.3 Concatenation Lemma . . . . . . . . . . . . . . . . . . . . . . 134
6.4.4 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.4.5 Transfer Matrix as a Projector . . . . . . . . . . . . . . . . . . 141
6.4.6 GSD and Transfer Matrix . . . . . . . . . . . . . . . . . . . . 142
6.5 Haah Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.5.1 Hamiltonian of Haah code . . . . . . . . . . . . . . . . . . . . 146
6.5.2 TNS for Haah Code . . . . . . . . . . . . . . . . . . . . . . . 148
xi
6.5.3 Entanglement Entropy for SVD Cuts . . . . . . . . . . . . . . 154
6.5.4 Entanglement Entropy for Cubic Cuts . . . . . . . . . . . . . 161
6.6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 169
A Appendix for Boson Condensation 172
A.1 Quantum dimensions of A and T . . . . . . . . . . . . . . . . . . . . 172
A.1.1 Proof of da =∑
r∈T nradr . . . . . . . . . . . . . . . . . . . . . 172
A.1.2 Proof of dr = 1q
∑a∈A n
rada . . . . . . . . . . . . . . . . . . . . 173
A.2 Chiral algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
A.3 Condensations in SU(2) CFTs . . . . . . . . . . . . . . . . . . . . . . 181
A.3.1 SU(2)16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
A.3.2 SU(2)28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
B Appendix for Uncondensable Bosons 189
B.1 No-go theorem with Abelian sector . . . . . . . . . . . . . . . . . . . 189
B.2 Proof for Example (iii), Multiple layers of SO(3)k . . . . . . . . . . . 191
B.3 General constraints on boson condensation . . . . . . . . . . . . . . . 194
C Appendix for Tensor Network States and Fracton Models 197
C.1 Proof for the Concatenation Lemma for the 3D Toric Code Model . . 197
C.2 Proof for the Concatenation Lemma for the X-cube Model . . . . . . 200
C.3 GSD for the X-cube Model . . . . . . . . . . . . . . . . . . . . . . . . 204
Bibliography 208
xii
List of Tables
xiii
List of Figures
1.1 A link in (2+1)D space. . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 The Hamiltonian terms of the 2D toric code model. Panel (a) is Av
which is a product of 4 Z operators around the vertex v, and Panel
(b) is Bp which is a product of 4 X operators around the plaquette p. 15
2.2 The TNS for the 2D toric code model on a square lattice. On each
bond, we associate a projector g tensor, and on each vertex, we as-
sociate a local T tensor. The connected lines are contracted virtual
indices. The lines with arrows are the physical indices. . . . . . . . . 17
2.3 Left panel: The line operator in the first equation of Eq. (2.12). The
blue line is the path for C. Right panel: The line operator in the
second equation of Eq. (2.12). The blue line is the path for C on the
dual lattice. The red dots represent the excitation created. W (C)
creates two excitations at the ends of C which have −1 eigenvalues of
Av. W (C) creates two excitations at the ends of C which have −1
eigenvalues of Bp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
xiv
4.1 Tunneling processes mediated by an anyon condensate. The gray re-
gion is a phase in which a boson B is condensed. a) Vertex of a boson
B that localizes a zero mode of anyon ai. In the condensed phase, B
can be converted into an identity particle world line (not shown). By
the axioms of anyon condensation, processes a) and b) are equivalent,
i.e., B can be converted into ai by tunneling through the condensate. 67
4.2 Quantum dimensions and bosons (blue columns) for SO(3)k theories
with a) k = 13 and b) k = 103. These are the smallest k, for which
SO(3)k contains two and four bosons, respectively. Indicated are also
the ranges I–III defined in Eq. (4.10). The maximum quantum dimen-
sion coincides with the boundary between range I and II in Eq. (4.10).
For instance, to apply the no-go theorem to the j = 5 boson in a),
choose Ij=5 = {j = 2} and use that d5 ≈ 3.6 is smaller than d2 ≈ 4.2. 72
5.1 An illustration of integer current and Hodge dual in 3 dimensional
space. In Panel (a), jS1 is the integer which only has support on the
loop S1. It is a 1-form represented by the red arrow. Its Hodge dual
?jS1 is represented by the red square penetrated by S1, which is a 2-
form perpendicular to jS1 . In Panel (b), jS is the integer 2-current
which only has support on the sphere S. In 3 dimensions, ?jS is then
a 1-form going out S denoted by the red dot. . . . . . . . . . . . . . 79
6.1 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. . . . . . . . . . 88
xv
6.2 An illustration of the 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 does not change
as well. Similar TNS gauges also appear in other TNS such as PEPS. 93
6.3 (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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4 Transfer matrix (red dashed square) of a 1D MPS. The connected lines
are the contracted virtual indices. The connected arrow lines are the
contracted physical indices. The MPS norm (or any other quantities)
can be built using the transfer matrix. Higher dimensional transfer
matrices are similarly defined for TNS on a cylinder or a torus, by
contracting in all directions except one. This leads to a 1D MPS with
a bond dimension exponentially larger than the TNS one. . . . . . . . 100
6.5 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. . . . . . . 110
6.6 Contraction of two local T tensors in the z-direction. We emphasize
that there is no projector g tensor in this figure. . . . . . . . . . . . . 115
xvi
6.7 The splitting of tensors near the entanglement cut. . . . . . . . . . . 118
6.8 The Hamiltonian terms of the X-cube model: (a) Av,yz, (b) Av,xy, (c)
Av,xz and (d) Bc. The circled X and Z represent the Pauli matrices
acting on the physical spin-1/2’s. . . . . . . . . . . . . . . . . . . . . 128
6.9 Figures for several regions A for which we calculate the entanglement
entropies. (1) Region A is a cube of size lx × ly × lz. (2) Region A is a
cube of lx× ly × lz with a hole of size l′x× l′y × l′z in it. (3) Region A is
a cube of size lx× ly × lz and a small cube of height l′z on top of it. (4)
Region A is a cube of size lx × ly × lz carved on top of it a small cube
of height l′z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.10 Derivations for the first equation in Eq. (6.107). The rest two equations
can be proved similarly. In the first step, the physical X operators can
be transferred to the virtual level by using Eq. (6.36), and in the third
step, all the virtual X operators are exactly canceled in pairs (dashed
red rectangles in the third figure) due to Eq. (6.87). . . . . . . . . . 141
6.11 Examples for the X-cube TNS in a xy-plane, obtained by acting
WZ [Cz,x] and WZ [Cz,y] on the constructed TNS. The intersection of
one WZ [Cz,x] operator and one WZ [Cz,y] operator with the xy-plane is
only one Pauli Z operator, i.e., the circled blue Z in this figure. . . . 144
6.12 We act WZ [Cx,y] and WZ [Cy,x] operators on Panel (a) in Fig. 6.11.
Hence, we have four TNSs in this xy-plane that can be related to each
other. See the text for the explanations. . . . . . . . . . . . . . . . . 144
6.13 Tensor contraction for the Haah code TNS. (a) The lattice size is 2×
3× 3. (b) The lattice size is 3× 3× 3 . . . . . . . . . . . . . . . . . 148
6.14 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
xvii
6.15 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. . . . . . . . . . . . . . . . . . . . 155
6.16 Transferring the Pauli X operators of the Bc operator from the region
A (a) to the region A (b). . . . . . . . . . . . . . . . . . . . . . . . . 162
xviii
Chapter 1
Preliminaries
The topological phase has been one of the central topics in condensed matter physics
for the past decades since the discovery of integer and fractional quantum Hall
effects[1, 2]. The definition of topological phases has been evolving over the years
of research[3, 4, 5, 6, 7, 8, 9]. It generally refers to the gapped phases of matter
which ground states are different due to the boundary conditions. For instance,
the ground state degeneracy can vary when the system is on torus or sphere[4],
or the boundary can support protected gapless modes when the system is on a
disk[10, 11, 12, 13, 14, 15, 16]. Another significant signature is that the bulk of
the topological phases supports excitations which have nontrivial braiding statistics
and topological spins[17, 18, 19, 20] than those of bosons and fermions. They are
dubbed as “anyons”. In this dissertation, we mostly focus on the topological phases
with anyons.
This dissertation studies two aspects of topological phases in 2 dimensions and
3 dimensions: boson condensations[21] and entanglement entropy[22, 23]. It will be
based on several different languages for topological phases, namely, stabilizer codes,
topological field theories and modular tensor categories. The stabilizer codes and
topological field theories provide the exactly solvable lattice Hamiltonian and field
1
theory models respectively. The modular tensor category, on the other hand, “for-
gets” the specific models and details such as the lattice or the couplings, but only
“remembers” the universal data such as the fusion rules and the braiding statistics
of the gapped excitations. In this chapter, each of the languages will be briefly re-
viewed to provide the foundations for the rest of the dissertation. In Chapter 2, the
2 dimensional toric code model will be described in each of the languages.
1.1 Stabilizer Codes
The stabilizer code refers to a class of exactly solvable Hamiltonian models specified
as follows. The key ingredient is the notion of qubit, which is nothing but a 2-level
system. For instance, a qubit can be a spin-12. On the qubit, the operators can be
generated by the Pauli X or Pauli Z operators satisfying anti-commutation relations:
XZ = −ZX, X =
0 1
1 0
, Z =
1 0
0 −1
. (1.1)
Conventionally the Pauli Z basis is preferred:
|0〉 =
1
0
, |1〉 =
0
1
. (1.2)
Hence,
Z|0〉 = |0〉, Z|1〉 = −|1〉, X|0〉 = |1〉, X|1〉 = |0〉. (1.3)
The stabilizer code is a many-qubit system, and the total Hilbert space is defined as
a tensor product of the Hilbert spaces of each qubit. Hence, the Pauli operators on
different sites commute:
[Xi, Zj] = 0, [Xi, Xj] = 0, [Zi, Zj] = 0, ∀ i 6= j. (1.4)
2
where the subscripts label different qubits. Each of the Hamiltonian terms is a product
of Pauli X and Pauli Z operators. The Hamiltonian is exactly solvable because all
local Hamiltonian terms are required to be commutative with each other. One simple
example for the stabilizer codes is a 3-qubit system, and its Hamiltonian is:
H = −Z1Z2Z3 −X1X2 −X2X3 (1.5)
Note that all three terms are commutative:
[Z1Z2Z3, X1X2] = 0, [Z1Z2Z3, X2X3] = 0, [X1X2, X2X3] = 0. (1.6)
These commutation relations imply that:
[H,Z1Z2Z3] = 0, [H,X1X2] = 0, [H,X2X3] = 0. (1.7)
The eigenstates of the Hamiltonian are thus also the eigenstates of the local Hamil-
tonian terms. Particularly, the ground state satisfies:
Z1Z2Z3|GS〉 = |GS〉, X1X2|GS〉 = |GS〉, X2X3|GS〉 = |GS〉. (1.8)
The ground state wave function, in terms of the Pauli Z basis, contains all the con-
figurations satisfying the first equality of Eq. (1.8):
|GS〉 =1
2(|000〉+ |110〉+ |101〉+ |011〉) (1.9)
The ground state wave function is much simplified if we use Pauli X basis instead:
|GS〉 =1√2
(|000〉x + |111〉x) , (1.10)
3
i.e. a GHZ state.[24]
Another example for stabilizer codes, extremely useful in the topological phases,
is the toric code model on a 2 dimensional lattice[19]. We defer the introduction of
this model to Chapter 2. There we will thoroughly describe this model in each of the
languages.
1.2 Entanglement Entropy and Tensor Network
States
The entanglement (von Neumann) entropy for a wave function |ψ〉 is defined as:
S(A) = TrA (ρA ln (ρA)) (1.11)
where ρA is the reduced density matrix for the subregion A by tracing out A’s com-
plement A:
ρA = TrA|ψ〉〈ψ|. (1.12)
The eigenvalues of the ρA are commonly referred as the entanglement spectrum. An
equivalent method to calculate the entanglement entropy is to perform the singular
value decomposition to the wave function coefficients M :
|ψ〉 =∑ij
Mij|i〉A ⊗ |j〉A,
M = UΛV †.
(1.13)
where |i〉A and |j〉A form the orthonormal basis for the subregion A and A respectively.
U , V are unitary matrices and Λ is a diagonal matrix containing the singular values.
It is easy to show that the eigenvalues of the reduced density matrix are the square
of the singular values.
4
A breakthrough was the conjecture of the area law[25]: the ground states of a
local and gapped Hamiltonian has entanglement entropy linear to the subregion’s
perimeter length/area:
S(A) ∼ Area (∂A) (1.14)
where S(A) is the entanglement for the subregion A and Area (∂A) refers to the A’s
perimeter length/area. For instance, in 1 dimension, the entanglement entropy of the
gapped local Hamiltonian is proved to be[26]:
S(A) ∼ Const. (1.15)
since in 1 dimension, the subregion A is a line segment, and ∂A only contains two
end points.
Further, the entanglement entropy and spectrum, measured using the ground state
wave functions, serve as the smoking gun to numerically and theoretically distinguish
2 dimensional topological phases. Two results need to be highlighted:
1. For a topological phase, the entanglement entropy scales as:
S(A) = α Area (∂A)− γ (1.16)
where the constant γ is a universal constant independent of the subregion’s
size[23, 22]. “Universal” means that γ is the same at any point in the gapped
phase. By computing γ from a microscopic model, most of the topological
phases are eliminated, and only few of them can be the candidates which require
further investigation.
2. In fractional quantum Hall systems, the entanglement spectrum degeneracy
provides the “fingerprint” to identify the different quantum Hall states[27]. The
5
entanglement spectrum degeneracy is argued to be the same as the conformal
field theories on the fractional quantum Hall edge[28].
Inspired by the area law, an efficient ansatz for the ground state wave function was
proposed and widely applied both numerically and analytically. This set of ansatz
is known as tensor network states. To name a few, they are matrix product state
(MPS)[29], projected entangled pair states (PEPS)[30], multi-entanglement renor-
malization ansatz[31, 32]. In this section, we briefly introduce MPS in 1 dimension.
PEPS is the immediate generalization of MPS in higher dimensions and will be heav-
ily used in Chapter 6. Suppose we have an MPS with periodic boundary condition
whose local tensor is A:
∑s1,s2,...,sN
∑i1,i2,...,iN
As1i1,i2As2i2,i3
. . . AsNiN ,i1 |s1, s2, . . . , sN〉 (1.17)
where the tensor A has one physical index sn of dimension d and two virtual indices
in of dimension D. The digram representation for the tensor A at the n-th site is:
(1.18)
where each line represents an index of the tensor. The physical index sn is dis-
tinguished from the virtual indices by the arrows. Diagrammatically, the MPS is
represented as:
(1.19)
6
The connected lines imply the contraction over the two indices. Note that the entan-
glement of such an MPS is upper bounded by the virtual bond dimension D:
S ≤ 2 ln(D), (1.20)
because the rank of the reduced density matrix or the rank of the singular value
matrix is upper bounded by D2 where the number 2 comes from the two cuts.
1.3 Abelian Topological Field Theory
In this section, we review a class of topological field theories, K-matrix Chern-Simons
theories[33], which can describe any Abelian topological phases. The Lagrangian of
a K-matrix Chern-Simons theory is:
L =KIJ
4πεµνλaIµ∂νaJ,ρ (1.21)
where gauge fields aI are U(1) gauge fields, and the matrix KIJ is symmetric and all
elements are integers. In a differential geometry language, it can also be written as
differential forms and exterior derivatives.
L =KIJ
4πaI ∧ daJ (1.22)
When the diagonal elements of the matrix K are all even integers, this type of K-
matrix Chern-Simons theories is called “bosonic”. For a K-matrix whose diagonal
elements have odd integers, it is refereed as “fermionic”. In this dissertation, we
mostly focus on the bosonic K-matrix Chern-Simons theories.
7
Figure 1.1: A link in (2+1)D space.
K-matrix Chern Simons theories have gauge invariant loop operators:
U{n}(S1) = exp i
˛S1
nIaI
, nI ∈ Z (1.23)
where S1 is a closed loop in (2+1)D spacetime and is also the worldline of the corre-
sponding pair of excitations. The linking correlation function of two loop operators
can be calculated by the Green function method[34]:
〈U{n}(S11)U{m}(S
12)〉 = exp i
(2πm ·K−1 · n
)(1.24)
where loops S11 and S1
2 form a link as in Fig. 1.1, and “·” is the matrix multiplication.
The physical interpretation of the linking correlation functions is that they are the
braiding statistics of the two excitations represented by U{n}(S11) and U{m}(S
12). In
the modular tensor category, this linking correlation is encoded in the modular S
matrix. The topological spin for each loop operator is:
Θ(U{n}
)= exp i
(πn ·K−1 · n
)(1.25)
In the modular tensor category, the topological spin is encoded in the modular T
matrix.
One observation is that U{K·l} is a topologically trivial operator when l is an
integer vector: it has trivial linking correlation with all other loop operators, and it
8
has trivial topological spin:
〈U{K·l}(S11)U{m}(S
12)〉 = 1
Θ(U{K·l}
)= 1
(1.26)
Moreover, if the charge vectors of two operators differ byK·l, i.e., considering U{n} and
U{n+K·l}, then they are topologically identical, since their linking correlation functions
with all other loop operators are the same and they have the same topological spin:
〈U{n}(S11)U{m}(S
12)〉 =〈U{n+K·l}(S
11)U{m}(S
12)〉
Θ(U{n}
)=Θ
(U{n+K·l}
) (1.27)
Hence, we only need to consider the loop operators U{n}(S1) up to the shift K · l
where l is an arbitrary integer vector.
1.4 Modular Tensor Category
In this section, we present a short review of the modular tensor category description
of a (2+1)-dimensional TQFT. This approach only describes the low energy exci-
tations of the TQFT, i.e., the anyons. The anyons are usually labeled by objects
a, b, c, . . . and are supplemented by other data, such as the fusion coefficients N cab.
For a comprehensive overview of the category theory approach of TQFT, we refer the
reader to Refs. [20, 35, 36]. Here, we only present a brief and simple review of the
important properties that we frequently use in this dissertation.
Fusion rules and quantum dimension The anyons of a TQFT can fuse. When
two anyons come close to each other spatially, they can fuse into other anyons. An
analogy can be drawn to the algebra of spins: if we take two spin 12
particles, they
can fuse into either spin 0 and spin 1 particle. For this, we would write, in group
9
representation theory, the fusion rule 12× 1
2= 0 + 1. In general, the fusion of anyons
in a TQFT is represented via
a× b =∑c
N cabc, (1.28)
where a, b, c are labels for the anyons, and the fusion coefficients N cab are non-negative
integers. The fusion can be represented by a state |a, b; c, µ〉 in the fusion vector space
V abc . Here µ = 1, · · · , N c
ab labels the vectors that form a basis of the N cab-dimensional
fusion vector space V abc .
Just like the fusion of spins, we require that the fusion rules are symmetric or
commutative, that is, a× b is equivalent to b× a. This translates to
N cab = N c
ba. (1.29)
Moreover, fusion rules are also associative. Suppose we take three anyons a, b, c
and try to fuse them. Then we have two ways to do so: (a× b)× c and a× (b× c).
We require the fusion rule to be associative by requiring that the two fusions yield
the same result. In terms of the fusion coefficients, this translates to
∑d,e
NdabN
edc =
∑d,e
N eadN
dbc. (1.30)
Other important data associated with anyons are their so-called quantum dimen-
sions da, db, · · · . This concept appears because anyons are associated with nontrivial
internal Hilbert spaces. Again, we can take the example of spins to illustrate this. In
the case of spin 12, where 1
2× 1
2= 0 + 1, spin 1
2is associated with a two-dimensional
Hilbert space, and meanwhile spin 0 is associated with a one-dimensional Hilbert
space, spin 1 a three-dimensional Hilbert space. As we can see, the total dimension
of Hilbert space does not change after fusion. The product 12× 1
2has a (2× 2 = 4)-
10
dimensional Hilbert space while 0 + 1 has a (1 + 3 = 4)-dimensional Hilbert space.
Similarly, in a TQFT, we also have
dadb =∑c
N cabdc. (1.31)
The above equation can be viewed as an eigenvalue equation of a matrix Na whose
entries are (Na)bc = N cab. The eigenvector is (db), the eigenvalue is da. Equation (1.30)
says that all matrices Na, Nb, . . . commute and thus they have common eigenvectors,
one of which is the vector of all quantum dimensions. The total quantum dimension
of a TQFT D is defined as the norm of the quantum dimension vector, D =√∑
a d2a.
If all anyons of a TQFT have quantum dimension 1, we call such a theory Abelian.
If there exist anyons with quantum dimension larger than 1, we call such a theory non-
Abelian. This is intimately related to the Perron-Frobenius theorem, where da is a
Frobenius eigenvalue, and hence has to satisfy minb∑
c(Na)bc ≤ da ≤ maxb∑
c(Na)bc.
Hence da > 1 implies that there exists a b such that∑
c(Na)bc > 1, so a× b contains
more than one particle.
Braiding, topological spin and modular matrices Another physically impor-
tant concept in a TQFT is braiding. This allows us to determine how a state trans-
forms when its anyons are adiabatically moved around each other. In Abelian theories,
when we adiabatically move an anyon a fully around another anyon b, the state trans-
forms through multiplication by a universal monodromy phase. For example, if we
take a fermion around a π flux, the wave function obtains a topological minus sign
−1. Another special case is when we exchange two identical abelian anyons a. This
process defines the topological spin θa for the particle a.
In non-Abelian theories, the braiding operation Rab between two anyons a and b is
an operator that acts on the Hilbert space V abc which describes states of a and b that
fuse into a fixed anyon c. If we denote a basis of V abc by |a, b; c, µ〉 with µ = 1, · · · , N c
ab,
11
then Rab has the representation
Rab|a, b; c, µ〉 =∑ν
[Rabc ]µν |b, a; c, ν〉. (1.32)
In this notation, the topological spin for an anyon a is defined as
θa =1
da
∑c
dcTrc[Raac ], (1.33)
where Trc[· · · ] is the trace taken in the fusion vector space V aac .
Given the braiding Rab, we can construct the modular matrices S and T which
are the same modular matrices encoding the global data of S and T in a CFT. They
are given by
Sab =∑c
N cabTr[Rab
c Rbac ]dc, (1.34a)
Tab = θaδab. (1.34b)
By definition, S is a symmetric matrix. Moreover, in a modular tensor categories, S
and T are unitary matrices satisfying S†S = SS† = 1, T †T = TT † = 1.
In Refs. [37, 38, 39, 40], the S matrix is used as an order parameter to detect
topological phase transitions and anyon condensations. The implicit assumption in
doing so is that the S matrix represents physical, measurable properties of the state,
unlike, say, the gauge-dependent F -symbol.
F -symbol F -symbol is in fact the generalized version of Wigner 6j-symbol, and
guarantees that all the fusion processes in the MTC are all consistent:
|a, b; e, α〉|e, c; d, β〉 =∑f,µ,ν
[F abcd ](e,α,β)(f,µ,ν)|b, c; f, µ〉|a, f ; d, ν〉 (1.35)
12
The F -symbol also needs to satisfy pentagon and hexagon equations, the details of
which are omitted in this dissertation.
13
Chapter 2
An Example: Toric Code Model
The toric code model and its generalizations have been extensively studied in the
literature. We will present the toric code model in different languages including the
stabilizer code, the tensor network state, the topological field theory and the modular
tensor category.
2.1 Stabilizer Code
The 2D toric code model, realizing a discrete Z2 gauge symmetry on lattice[41], is a
stabilizer code defined on any 2D random lattice. It exhibits the topological order
and supports gapped anyonic excitations. For simplicity, the 2D toric code introduced
in this section is defined on a square lattice with physical spins on all bonds of the
lattice, and the Hamiltonian consists of vertex terms and plaquette terms:
H = −∑v
Av −∑p
Bp. (2.1)
14
Z Z
x
x
xx
x
y
Z
Z
(a) (b)
Figure 2.1: The Hamiltonian terms of the 2D toric code model. Panel (a) is Av whichis a product of 4 Z operators around the vertex v, and Panel (b) is Bp which is aproduct of 4 X operators around the plaquette p.
Here, Av is the product of four Pauli Z matrices around a vertex v and Bp is the
product of four Pauli X matrices around a plaquette p.
Av =∏i∈v
Zi, Bp =∏i∈p
Xi. (2.2)
These two terms are illustrated in Fig. 2.1. Note that all these operators will commute
with each other:
[Av, Av′ ] = 0, ∀ v, v′,
[Bp, Bp′ ] = 0, ∀ p, p′,
[Av, Bp] = 0, ∀ v, p.
(2.3)
Hence, the ground states of the 2D toric code model need to satisfy the constraints:
Av|GS〉 = |GS〉, Bp|GS〉 = |GS〉, (2.4)
for all vertices v and plaquettes p. Each of the eigenvalue equations is a constraint
for the ground state wave function. Note that the local Hamiltonian terms satisfy the
15
following redundancy on a closed manifold:
∏v
Av = 1,∏p
Bp = 1. (2.5)
The ground state degeneracy (GSD) satisfying Eq. (2.4) is counted as follows:
GSD =2# of qubits
2# of indep. GSD constraints=
2# of bonds
2# of vertices + # of plaquettes - 2= 22−χ (2.6)
where
χ = # of vertices + # of plaquettes - # of bonds (2.7)
is the Euler characteristic. For a torus, χ = 0. Thus the GSD on a torus is 4. The
number 2 in the exponent comes from the redundancy in Eq. (2.5). The calculation
can certainly be generalized to other manifolds, and the GSD only depends on the
Euler characteristic and hence topological.
2.2 Tensor Network State
In order to conveniently construct the TNS for the 2D toric code model, we introduce
projector g tensors on each bond of the square lattice and local T tensors on each
vertex of the square lattice. The physical or virtual index is of dimension 2 and
labeled as 0, 1. The g tensor has 1 physical index and 2 virtual indices, and the T
tensor has 4 virtual indices. The TNS is depicted in Fig. 2.2. The same construction
will be heavily used in Chapter 6.
The projector g tensor identifies the physical index with the virtual indices
gpij =
1 if p = i = j ∈ {0, 1}
0 otherwise
(2.8)
16
where p is the physical index, and i and j are thr virtual indices. The rest of the
calculations is to construct a T tensor such that the TNS satisfies Eq. (2.4) on the
TNS. We choose to directly act Av and Bp operators on the TNS. Note that g tensor
identifies the physical index and the virtual indices. The actions of X and Z Pauli
matrices on the physical indices are transferred to the virtual indices:
TZ
Z
Z
Z
Z
Z
Z
Z
g
g
g
g
g
=
Tg
gg g
x
x
x x
x xx
xx x
x
x=
. (2.9)
Tg
x
y
Figure 2.2: The TNS for the 2D toric code model on a square lattice. On each bond,we associate a projector g tensor, and on each vertex, we associate a local T tensor.The connected lines are contracted virtual indices. The lines with arrows are thephysical indices.
17
In order to implement Eq. (2.4), we require the four Pauli Z matrices (in the first
line) and two Pauli X matrices (in the second line) acting on the virtual indices in the
dashed red squares to be identity operations. Then we have two (strong) conditions
respectively:
Txx,yy = (−1)x+x+y+yTxx,yy
Txx,yy = T(1−x)x,(1−y)y = T(1−x)x,y(1−y)
= Tx(1−x),(1−y)y = Tx(1−x),y(1−y).
(2.10)
where x, x are the two indices of the T tensor in the x-direction, and the y, y are the
two indices of the T tensor in the y-direction. As a result, the local T tensors are
fixed up to an overall factor:
Txx,yy =
0 if x+ x+ y + y = 1 mod 2
1 if x+ x+ y + y = 0 mod 2.
(2.11)
Hence, we have constructed the TNS for the ground state. It is a natural question
that how to construct all degenerate ground state on a torus, since this method only
gives rise to one tensor. The solution to this puzzle is that the boundary conditions
of the TNS can be twisted in both directions without breaking the conditions in
Eq. (2.4)[30].
2.3 Modular Tensor Category
The excited states are also labeled by the eigenvalues of Av and Bp, due to the
fact that all local Hamiltonian terms commute with each other. The excited state
has eigenvalues −1 for some local Hamiltonian terms, while the ground state has all
eigenvalues 1 for all local terms. To create such excitations from the ground state, an
18
Figure 2.3: Left panel: The line operator in the first equation of Eq. (2.12). Theblue line is the path for C. Right panel: The line operator in the second equation ofEq. (2.12). The blue line is the path for C on the dual lattice. The red dots representthe excitation created. W (C) creates two excitations at the ends of C which have−1 eigenvalues of Av. W (C) creates two excitations at the ends of C which have −1eigenvalues of Bp.
observation is that line operators can be constructed as follows:
W (C) =∏i∈C
Xi, W (C) =∏i∈C
Zi, (2.12)
where C is a path on the lattice while C is a path on the dual lattice. See Fig. 2.3 for
an illustration. When the line operators act on the ground states, the Hamiltonian
terms near the two ends of the line have eigenvalues −1. If C or C is closed, then the
energy remains the same as the ground state energy. Thus the excitations are always
created by pairs. Moreover, if C or C winds around the cycles of the lattice, for
instance the two cycles of a torus, then W (C) or W (C) commutes with all the local
terms in the Hamiltonian, and cannot be generated by the local Hamiltonian terms.
Hence, these winding operators serve as good quantum numbers to distinguish the 4
ground states on torus.
The excitation, measured by an Av operator and created by W (C), is commonly
dubbed as e (charge), while the excitation, measured by an Bp operator and created by
N(NC), is dubbed as m (flux). Their composite particle is dubbed as f . Combining
the trivial particle denoted as 1, there are four excitations: 1, e, m and f . Their
19
fusion rules can be derived from their operators:
e× e = 1, m×m = 1, f × f = 1, e×m = f. (2.13)
The braiding statistics of these excitations can be derived from their operators’ com-
mutation relation. Their braiding statistics is encoded in the modular S matrix in
the basis of (1, e,m, f):
S =1
2
1 1 1 1
1 1 −1 −1
1 −1 1 1
1 −1 −1 1
(2.14)
The matrix element Sab is the braiding statistics of the anyon a and the anyon b.
Their topological spins are encoded in the modular T matrix:
T =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −1
(2.15)
The diagonal matrix element Taa is the topological spin for the anyon a.
The F -symbols is omitted in this dissertation since it is trivial for the toric code
model.
2.4 BF Field Theory
In this section, we show that a Z2 BF theory will reproduce the same physics as that
in the toric code Hamiltonian model. The Lagrangian for a ZN BF theory is:
L =N
2πεµνρbµ∂νaρ. (2.16)
20
For the reader who are familiar with differential forms, it is equivalent to write:
L =N
2πb ∧ da. (2.17)
Note that this Lagrangian can be casted in the form of K-matrix Chern-Simons
theory. Restricting ZN to Z2, the Lagrangian reduces to:
L =2
2πεµνρbµ∂νaρ. (2.18)
This Lagrangian exhibits the gauge symmetry as follows:
aµ 7→ aµ + ∂µα, bµ 7→ bµ + ∂µβ. (2.19)
where α and β can be any functions.
Applying canonical quantization for this gauge theory will reproduce the same
results for the toric code model. We start with gauge fixing. Tuning α and β can fix
the temporal gauge:
at = 0, bt = 0. (2.20)
The Lagrangian becomes:
L =1
π(bt (∂xay − ∂yax) + at (∂xby − ∂ybx)− bx∂tay + by∂tax)
=1
π(−bx∂tay + by∂tax) .
(2.21)
Temporal gauge fixing enforces the Gauss constraints:
∂xay − ∂yax = 0, ∂xby − ∂ybx = 0. (2.22)
21
The rest two terms in the Lagrangian imply the commutation relations:
[bx(x, y), ay(x′, y′)] = −iπδ (x− x′) δ (y − y′) ,
[by(x, y), ax(x′, y′)] = iπδ (x− x′) δ (y − y′) .
(2.23)
Therefore, the gauge fields bµ serve as the electric fields for the gauge fields aµ.
The lattice Hamiltonian language is in fact exactly the same as the field theory
language. The observation is that when the gauge fields aµ is identified with the
Pauli X operators on the lattice, then using Eq. (2.23) the gauge fields bµ will be
identified with Pauli Z operators on the dual lattice. The two Gauss constraints
in Eq. (2.22) are in fact the two constraints for the ground states of the toric code
model in Eq. (2.4): the first one of Gauss constraints being the Bp while the second
one being Av.
The line operators of Eq. (2.12) in the lattice Hamiltonian language are the Wilson
operators in the field theory language:
W (C) = exp
(i
ˆC
aµdxµ
),
W (C) = exp
(i
ˆC
bµdxµ
).
(2.24)
Note that these two operators are not gauge invariant when C and C are open, which
indicates that there are additional degrees of freedom attaching to the ends of C or C.
This corresponds to the lattice Hamiltonian language that the open line operators will
cost energy. When C and C are closed, then the two operators are gauge invariant in
the field theory language. This corresponds to the lattice Hamiltonian language that
the closed line operators will not cost energy.
22
Chapter 3
Boson Condensation
Prior to the discovery of topological order, it was well known that bosons can macro-
scopically occupy a single quantum state, a fact which allows for the possibility of a
Bose-Einstein condensation phase transition. In a topologically ordered phase, bosons
are more complicated particles: They can have nontrivial braiding behavior with other
anyons[20, 35], and even more exotically they can carry nonlocal internal degrees of
freedom[20], in which case they are called non-Abelian bosons. Notwithstanding,
such bosons can sometimes condense[42, 21, 37, 43, 44, 45, 46, 47, 48, 49, 50, 51]. It
is then natural to ask how this condensation affects the topological order, namely,
what is the fate of the other anyons in the phase. The answer is that anyon con-
densation induces transitions between different topologically ordered phases in such
a way that universal properties of the anyons of the condensed phase can be in-
ferred from those of the initial phase, together with a list of condensed bosons. This
framework of anyon condensation transitions found many applications in the study
of topological order, [52, 53, 54, 55, 56, 57] in particular in solving the question of
bulk-boundary correspondence [58, 59, 59, 60, 61, 62], or in deducing the univer-
sal properties of domain walls [63, 64, 65, 66, 67, 68, 69, 70, 71] and other external
defects [72, 73, 74, 75, 76, 77, 78, 79].
23
The universal aspects of topologically ordered phases are captured by topological
quantum field theories[80]. Among these, the axiomatic approach of category the-
ory [81, 82, 83, 84, 85], more concretely the formulation of modular tensor categories
(MTCs), is particularly powerful and, to our knowledge, provides a complete char-
acterization of topological order in two-dimensional space. [20, 35] At a basic level,
MTC’s are characterized by the types of anyons that appear in the phase as well as
their interrelations in the form of fusion and braiding information, the so-called “F
moves” and “R moves”.
In correspondence with the different descriptions of topological order itself, sev-
eral formulations of anyon condensation were developed. In the context of MTCs,
the phase after condensation is found by studying commutative separable Frobenius
algebras [86, 87, 88, 89, 90] of the initial theory. [91, 62] Bais and Slingerland trans-
lated this procedure into the language of anyon models [42], but their formulation
did not give a systematic method for determining properties of the phase after con-
densation. This was later achieved by Eliens et al. in Ref. [21] via a diagrammatic
formulation of the condensation problem that makes use of the so-called vertex lifting
coefficients. These allowed them to embed the fusion and braiding processes of the
condensed phase in the initial anyon model. However, all these approaches fall short
of providing an algorithmic formulation of boson condensation in a way that could,
for example, be implemented in a computer algebra program allowing for systematic
studies of possible condensations.
In this chapter, we reformulate the problem of boson condensation in anyon mod-
els axiomatically and purely algebraically. The resulting formalism is based on a small
number of natural assumptions such as the commutativity of fusion and condensation
as well as an assumption about the topological spins of the anyons after condensation.
Our approach puts the modular matrices S and T of the initial anyon model center
stage, instead of focusing on the F and R moves, which are the key objects of interest
24
for the diagrammatic approach. [21] The F and R moves are in general notoriously
hard to compute even for relatively simple theories. Our goal is to find the modular
matrices S and T of the final theory after condensation. Using our algebraic formula-
tion, we propose an algorithm that determines all possible condensation instabilities
of an anyon model and can be efficiently implemented on a computer. We solve for
the condensation via a series of linear algebra problems, involving the factorization
of nonnegative integer matrices.
Besides its utility for computer-aided calculations, our algebraic formulation of
condensation also facilitates analytical derivations. As an example, we discuss layer
constructions of topologically ordered states and easily reproduce the known result
that 5 and 10 layers of the Fibonacci anyon model cannot undergo a condensation
transition.
This chapter is structured as follows. In Sec. 3.1 we formulate the condensation
problem along with the axioms relating to fusion rules. In the following Sec. 3.2, we
present the assumptions that allow us to deduce the braiding properties of the theory
after the condensation transition, and several implications are derived. In Sec. 3.3,
we derive central equations which constrain S and T . Subsequently, we show in
Secs. 3.4 and 3.5 that a weaker set of axioms suffices, if the condensate consists of so-
called simple currents and if only one particle is confined through the condensation
transition, respectively. We formulate an algorithm for solving the condensation
problem in Sec. 3.6. The final Sec. 3.7 gives examples of condensation transitions
in multi-layered anyon models and discusses obstructions to boson condensation in 5
and 10 layers with Fibonacci anyons.
25
3.1 Definitions and Assumptions
In this section we present the formalism underpinning anyon condensation, following
Refs. [42] and [21] closely. Our discussion is self-contained with respect to the previous
literature on anyon condensation, but assumes that the reader is familiar with the
basic concepts of MTCs [19] (see Sec. 1.4 for a short review).
The input for our approach to anyon condensation is a MTC A (the uncondensed
theory), and a set of restriction and lifting coefficients, which relate the particle
excitations in A to those in T (the condensed theory). In general, T is only a
fusion category, because it may contain some excitations which are confined by the
surrounding condensate. Projecting out these confined excitations, we are left with a
deconfined condensed MTC that we denote as U . Our goal is to find possible MTCs
U given A and some basic information about the condensate, such as which bosons
condense.
In what follows, we will consider the Bose condensation of a collection of bosons in
the original theory A. This collection of anyons is called the condensate and becomes
part of the vacuum in the new intermediate fusion category T . In condensing these
bosons, a generic anyon a ∈ A will become (or “restrict to”) a superposition of
particles t ∈ T
ar7−→∑t∈T
ntat, ∀a ∈ A (3.1)
with the coefficients nta ∈ Z≥0, where we assume that nta = nta and bars denote
antiparticles (see Sec. 1.4). Equation (3.1) defines the “restriction map”. We will
also use the phrase “a restricts to∑
t ntat” to describe Eq. (3.1). It is possible that
more than one particle t appears on the righthand side of Eq. (3.1), in which case
we say that “a splits into∑
t ntat”. Condensed particles (bosons b in the condensate)
have the additional special property that nϕb 6= 0, where ϕ is the vacuum particle in
26
T , that is, their restriction contains the identity of the new T theory. If nϕb 6= 0, then
nϕb = nϕb, i.e., both the boson and its antiparticle must condense at the same time.
The reverse (or, more precisely, adjoint) operation to restriction is called “lifting”.
For a particle t ∈ T , all the particles in A which restrict to t are defined to be the lifts
of t. The lifting coefficients are the same nta that we used in defining the restriction.
Formally, lifting is defined by
tl7−→∑a∈A
ntaa, ∀t ∈ A. (3.2)
Finally, we define particles in T whose lifts do not share a common topological spin
θa as confined, that is
t : confined ⇔ ∃ a, b such that ntantb 6= 0 with θa 6= θb. (3.3)
Conversely, the deconfined particles in T are the particles whose liftings do share a
common topological spin, which becomes identified with the spin of the deconfined
particle, that is
t : deconfined ⇔ ∀ a, b such that ntantb 6= 0 then θa = θb. (3.4)
Obviously, any particle t ∈ T is either deconfined (t ∈ U) or confined (t ∈ T /U).
With these definitions in place, we now make a fundamental assumption from which
we will derive the structure of the theory after condensation. We assume that the
restriction A → T commutes with fusion. This is represented by the diagram
A⊗A f //
r⊗r��
Ar��
T ⊗ Tf
// T
27
in which f represents fusion and r represents restriction.
More explicitly, the commuting diagram can be written in terms of anyon basis
∑r,s∈T
nransbN
trs =
∑c∈A
N cabn
tc, (3.5)
where N cab and N t
rs are the fusion coefficients in A and T , respectively. This elemen-
tary constraint is surprisingly restrictive. For instance, it immediately leads us to
two constraints on the quantum dimensions of particles in the A and T theories (see
Appendix A.1)
da =∑r∈T
nradr, ∀ a ∈ A, (3.6a)
dt =1
q
∑a∈A
ntada, ∀ t ∈ T , (3.6b)
where q :=∑
a nϕada. Diagrammatically, Eq. (3.6b) is
(3.7)
It will also be useful to define the quantity
βt :=∑a∈A
θadanta, (3.8)
where θa is the topological spin of a ∈ A. Given a particle t ∈ U , it follows from the
aforementioned definition of a deconfined particle Eq.(3.4)
βt = qdtθt, ∀t ∈ U , (3.9)
as a useful corollary to Eq. (3.6b).
28
3.2 The Condensation Matrix Mab
So far, our formalism does not differ appreciably from that of Refs. [42] and [21]. How-
ever, in what follows we opt to not introduce the so-called “vertex lifting coefficients”
on which the approach of [21] is based. Instead, we find that we can extract a sur-
prising amount of information from supplementing the algebraic relations in Sec. 3.1
with two additional assumptions. First, by assumption, we are only interested in
cases where U is a TQFT, so that its anyons form a braided fusion category. Second,
we assume that
βt = 0, ∀t ∈ T /U , (3.10)
where t ∈ T /U runs over all confined anyons. To motivate this equation let us
pictorially represent the lefthand side of Eq. (3.10) as
(3.11)
where a particle t is braided around itself. This process is equivalent to braiding
the lifts a of the particle t (namely nta 6= 0). Each of these braidings is given by
the phase θa, while the loop with particle a is equal to the quantum dimension da.
The result we obtain is the quantity βt which we assume vanishes when t ∈ T /U as
confined particles cannot form a braided category. This is in contrast with Kirillov-
Ostrik [91], Kong [62] and Eliens et al. [21]. In these works, the authors present
the process of boson condensation as the identification of a commutative separable
subalgebra ϕ of A. The condensed theory T is identified as a module over φ living
in A. Using this formalism, which allows one to relate braiding processes in the
T theory to braiding processes in the original theory A, it is possible to show that
βt vanishes when t ∈ T /U . Using our stripped down algebraic formalism, we are
29
currently unable to interpret braiding processes in T in terms of those in A, and so
we are unable to mimic the procedures in the chapter above. As a result, we are
inclined to simply assume βt vanishes when t ∈ T /U . In certain special cases we
can show that Eq. (3.10) follows from the assumptions: in Sec. 3.1, e.g., we do so for
the so-called simple current condensates (see Sec. 3.4). It is so far unclear how to
generally prove Eq. (3.10) in our framework.
With these additional assumptions in place, we define some useful quantities. The
vacuum component t = ϕ of Eq. (3.5) will be a central object in our analysis, the left
hand side of which reads:
M ′ac :=
∑t∈T
ntantc
(=∑b∈A
N cabn
ϕb
), (3.12a)
as will be
Mac :=∑t∈U
ntantc . (3.12b)
Notice how the two above definitions of the matrices M ′ and M with nonnegative
integer entries differ subtly but crucially: The expression for M ′ involves a summation
over T while that for M involves a summation over the deconfined condensed theory
U .
The matrices, which can be factorized as in Eq. (3.12a) and (3.12b), are called
completely positive matrices over the ring of positive integers. We will discuss com-
pletely positive matrix factorization later in the chapter. In the following sections,
we will demonstrate two important properties of M , namely [M,S] = [M,T ] = 0,
where S and T are modular matrices of the A theory. For a discussion of the role of
the matrix M in CFTs, we refer the reader to Appendix A.2.
30
3.2.1 Proof that M commutes with T matrix of the A theory
In the following, we will prove that the M matrix we defined in Eq. (3.12b) commutes
with the modular T matrix of the A theory. The T matrix of the A theory is
Tab = θaδab. Note that
[M,T ]ac =∑b∈A
(MabTbc − TabMbc) = Macθc −Macθa =∑t∈U
ntantc(θc − θa). (3.13)
Since t ∈ U , the spins of all the lifts of t in A are the same, hence θa = θc and each
term in the final line vanishes identically. It follows that
[M,T ] = 0 . (3.14)
Note that this is not valid if the sum in the last line of Eq. (3.13) was not restricted
to the U theory, i.e., [M ′, T ] 6= 0, with M ′ defined in Eq. (3.12a).
3.2.2 Proof that M commutes with S matrix of the A theory
In this section, we will prove that the M matrix commutes with the modular S matrix
of A theory
[M,S] = 0 . (3.15)
We start from the expression of the S matrix for a braided fusion category A (e.g.,
see Ref. [36])
Scb =1
DA
∑x∈A
Nxcb
θxθcθb
dx, (3.16)
31
where DA is the total quantum dimension of the A theory and b denotes the antipar-
ticle of b. From the definition of M , we express the commutator [M,S] as
[M,S]ab =1
DAθaθb
∑c,x∈A
∑t∈U
θxdx(ntan
tcN
xcb − n
tbn
tcN
xac
)=
1
DAθaθb
∑x∈A
∑t∈U
θxdx
[nta
(∑c∈A
ntcNxcb
)− ntb
(∑c∈A
ntcNxac
)]
=1
DAθaθb
∑x∈A
∑t∈U
θxdx
[nta
(∑c∈A
ntcNcbx
)− ntb
(∑c∈A
ntcNcax
)].
(3.17)
In the first line we have used that if ntantc 6= 0 with t ∈ U , then θc = θa and if ntcn
tb 6= 0
with t ∈ U , then θc = θb which yields the term θaθb in the denominator. To obtain
the last line, we have used the equalities N cab = N b
ac = N cab
and N cab = N c
ba. We now
use Eq. (3.5) to replace the terms in the round brackets and find
[M,S]ab =1
DAθaθb
∑s∈T
∑x∈A
θxdxnsx
∑t∈U ;r∈T
(ntan
rbN
trs − ntbnraN t
rs
), (3.18)
where we have used the equality nsx = nsx (the assumption that the restriction of x’s
antiparticle x is the antiparticle of the restriction of x) to transfer the antiparticle on
the N tr,s. We can now split up the r sum in Eq. (3.18) into a sum over U and a sum
over T /U . For the first contribution, we have
∑r,t∈U
(ntan
rbN
trs − ntbnraN t
rs
)=∑r,t∈U
ntanrb
(N trs − N r
ts
)(3.19)
by exchanging the labels r and t in the second term. Since N rts = N t
rs = N trs,
Eq. (3.19) vanishes identically. Thus, r in Eq. (3.18) can only take values in T /U .
By assumption, U is a closed fusion category. This implies that no trivalent vertex
with a single leg in T /U exists in T . As a result, the s-sum in Eq. (3.18) may only
run over T /U . However, for s ∈ T /U we can use the assumption Eq. (3.10) to find
32
∑x∈A θxdxn
sx = 0 for the remaining terms in Eq. (3.18). We conclude that Eq. (3.18)
vanishes identically and thus [S,M ] = 0.
These equations are essential to the theory of condensation, as they establish
that the condensation matrix Mab is a particular symmetry of the S and T modular
matrices. While there exist other such symmetries, for example automorphisms that
are represented by permutation matrices, these matrices are not ‘completely positive’
integer matrices i.e., they cannot be factorized as nnT in terms of a nonnegative
integer matrix n.
3.3 The Modular Tensor Category after condensa-
tion
In the previous section, we identified a matrix M which commutes with the modular
matrices S and T of the A theory. In this section we prove a stronger pair of results,
namely that
nS = Sn, (3.20a)
nT = Tn, (3.20b)
where S and T are the modular matrices of the U theory and n is the matrix of coef-
ficients that enter the restriction and lifting maps, (n)at = nta, ∀ a ∈ A, t ∈ U . Our
assumption Eq. (3.10) will be crucial for these proofs. The second equality Eq. (3.20b)
is the statement that, component by component, whenever nta 6= 0, θt = θa; this is true
by recalling our definition of deconfined particles of the U (⊂ T ) theory, Eq. (3.4).
33
Before starting the proof of the first equality Eq. (3.20a), we note the following
equalities derived in the appendix of Ref. [92]
∑t∈U
βtβ∗t = q2D2
U (3.21a)
and ∑t∈T
βtβ∗t = D2
A. (3.21b)
It then follows from assumption Eq. (3.10), that∑
t∈U βtβ∗t =
∑t∈T βtβ
∗t and thus
q2D2U = D2
A or [91]
q = DA/DU , (3.22)
To prove Eq. (3.20a), we multiply Eq. (3.5) by θada and sum both sides over a ∈ A
to obtain
∑r,s∈T
N trsn
sbβr =
∑a,c∈A
N cabdaθan
tc = θbDA
∑c∈A
Sbcθcntc, (3.23)
where we have used the definition of S from Eq. (3.16) and that θa = θa. For particles
t ∈ U , we furthermore have
∑r,s∈T
N trsn
sbβr =
∑r,s∈U
N trsn
sbqθrdr, ∀t ∈ U , (3.24)
because (i) only r ∈ U contributes to the sum (as βr = 0 if r ∈ T /U) and (ii) only
s ∈ U contributes since U is closed under fusion by assumption. We furthermore
used Eq. (3.9) to rewrite the righthand side of Eq. (3.24). Since we assumed that U
forms a braided fusion category with its S matrix, we use the usual definition of the
S matrix to write ∑r,s∈U
(N trsθrdr
)nsb =
∑s∈U
SstDUθtθsnsb. (3.25)
34
Since s, t ∈ U , we furthermore have θs = θb if nsb 6= 0 which allows us to combine
Eq. (3.23) and Eq. (3.25) into
1
q
DADU
∑c∈A
Sbcθcntc =
∑s∈U
Sstnsbθt. (3.26)
Since for all ntc 6= 0, we have θc = θt (t ∈ U) and using Eq. (3.22), this expression
reduces to
∑c∈A
Sbcntc =
∑s∈U
Sstnsb. (3.27)
We have thus proven Eq. (3.20a) and Eq. (3.20b) within our algebraic formulation
of the condensation transition. These two equations have a well known parallel in
the study of chiral algebra extensions, which we detail in Appendix A.2. As a side
remark, let us derive a consequence of Eq. (3.22), namely
DA > DU , (3.28)
which follows from the fact that the embedding dimension q =∑
a∈A danϕa > 1 and
q = 1 if no condensation is happening. This is always true even if the assumption
Eq. (3.10) is not used. If Eq. (3.10) is used, then Eqs. (3.21a) and (3.21b) imply
DA = qDU . By analogy with the Zamolodchikov c-theorem of Ref. [93] one can call
this result the D-theorem which can be interpreted as the disappearance of some
anyons upon condensation. There is a stronger connection between this result and
the g-theorem, according to which the Affleck-Ludwig boundary entropy of an open
conformal system decreases under the renormalization group flow of the boundary
as long as the bulk theory remains critical throughout the flow.[94, 95, 96] (This
situation is, however, distinct from the case of a condensation transition in which the
bulk is only critical at the transition.) The boundary entropy is in turn related to the
35
quantum dimension of the primary field that characterizes the boundary condition,
which suggests a relation between the g-theorem and Eq. (3.28).
3.4 Simple currents
Simple currents are abelian anyons that, when raised to a certain power by fusion,
equal the identity, see Refs. [97, 98, 99, 100, 101, 102, 103]. The precise definition
follows below. In this section, we consider a condensate that is composed of simple
currents only. In this situation, we can prove that Eq. (3.10), i.e., βt = 0, ∀t ∈ T /U ,
follows from the assumptions in Sec. 3.1.
3.4.1 Introduction to simple currents
There are several equivalent definitions of simple currents in the context of rational
conformal field theory (RCFT). First, a simple current is a primary field J that has
a unique fusion channel with any other primary field of the RCFT
J × φ = φ′, ∀φ. (3.29a)
A second definition is that a simple current is a primary field J that when fused with
its antiparticle or conjugate field J only fuses to the identity (see Ref. [99])
J × J = 1. (3.29b)
A third definition is that the quantum dimension of J is 1,
dJ = 1. (3.29c)
One can show that all these definitions are equivalent. [98]
36
Given two simple currents J1 and J2, their fusion product J1J2 is also a simple
current. The number of primary fields of a RCFT is finite, therefore each simple
current J has an associated integer N such that JN = 1 by using Eq. (3.29a). The
smallest integer N > 0 with this property is called the order of J . A simple current
J generates a set of simple currents {Jm|m = 0, 1, . . . , N − 1}, which is isomorphic
to the abelian group ZN . A RCFT may contain simple currents generated by more
that one primary field. The collection of all of them form an Abelian group which
is isomorphic to the product ZN1 × · · · × ZNr . One can choose a basis of simple
currents such that Ni are of the form pnii , ni ∈ Z, with pi a prime number. This is
the fundamental theorem of finite abelian groups.
As an example, consider the RCFT constructed from the Kac-Moody algebra
SU(2)k. The primary fields are denoted by φ` where ` = 0, 1, . . . , k is twice the
topological spin. The field φk is a simple current because its fusion rule is φk × φ` =
φk−`. Indeed φk is the only non-trivial simple current, and satisfies φk×φk = φ0 = 1.
The simple currents form a Z2 = {φ0, φk} subcategory.
When acting on a primary field φ, J generates an orbit formed by the fields Jnφ
[φ] = {φ, Jφ, J2φ, . . . , Jd−1φ}, Jdφ = φ. (3.30)
Here, d is the smallest positive integer such that Jdφ = φ. The orbit Eq. (3.30) is
denoted by a representative field φ but one can choose another field belonging to
the orbit. In general d need not equal N , the order of the current J , but d must
divide N . In the example of SU(2)k, if k is odd, all the orbits have two elements,
(d = N = 2), while for k even, φk×φ k2
= φ k2
for the action of φk and so the orbit has
only one element, φ k2. Generally, we will simply call the anyon a “fixed point” when
it is invariant under fusion with a simple current, or equivalently, if its orbit contains
only the anyon itself. In the SU(2)k (k even) example, φ k2
is a fixed point under the
37
fusion with φk. As we shall see below, the existence of fixed points is crucial for the
construction of the condensed theory.
3.4.2 Simple current condensation
We consider a condensation transition, where the condensate consists only of the
set of bosonic simple currents, generated by n simple currents J1, . . . , Jn with orders
N1, . . . , Nn. Any anyon in the condensate can thus be represented as J i11 . . . J inn , where
il = 0, . . . , Nl − 1 and l = 1, . . . , n (note that the fusion product of simple currents is
unique.) We use the shorthand notation i = (i1, . . . , in) and
Ji := J i11 · · · J inn . (3.31)
The initial theory might have simple currents which are not bosons. We do not
consider these, as they cannot condense. We consider the group generated by the
powers of all the bosonic simple currents, which is sometimes called the bosonic cen-
ter C of the RCFT. Powers of a condensed bosonic simple current, or the products of
different condensed bosonic simple currents are also bosonic and condensed. To see
this, examine the Ji, Jj component of Eq.(3.5), where Ji, Jj are assumed to be con-
densed. Recalling that Ji+j := Ji × Jj is automatically a simple current (see above),
and making use of the fact that condensed simple currents like Ji have quantum
dimension 1 so that ntJi = δtϕ, we find
1 = nϕJinϕJj
= NJi+j
Ji,JjnϕJi+j
= nϕJi+j. (3.32)
As a result nϕJi+j= 1, indicating that Ji+j restricts solely to the vacuum, so it must be
a boson. Therefore the product of any two condensed simple current is a condensed
(hence bosonic) simple current.
38
As an aside, we note that for general bosonic currents which are not necessarily
condensed: (i) as before, any power of such a simple current is a bosonic simple
current; (ii) however, the product of two such bosonic simple currents does not have
to be bosonic. For example, in the toric code that we will discuss in detail in Sec. 3.6,
e and m are bosonic simple currents while their product f is actually a fermion (which
cannot condense). To prove (i), one can use the symmetry Sab = S∗ab
of the S matrix
and the fact that for any anyon θa = θa. Choosing a = b = J with J a simple current
gives
θJ2
θJθJ=
θ1
θ∗Jθ∗J
. (3.33)
If J is bosonic, then so is J and the above implies J2 is also a bosonic simple current,
i.e., θJ2 = 1. This argument can be iterated by assuming that up to some n0 all Jn,
n = 1, . . . , n0, are bosonic (then so are all JN−n, n = 1, . . . , n0, with N the order of
J). Solving the equality SJ,Jn0 = S∗J,JN−n0
for θJn0+1 yields that Jn0+1 is also bosonic.
Vafa’s theorem
We first aim to find information about the topological spins of some of the particles
in the theory by analyzing the implications of Vafa’s theorem [104]
∏p
(θpθxθy
)NpxyN
upz∏
q
(θqθxθz
)NuyqN
qxz
=∏r
(θuθxθr
)NryzN
uxr
(3.34)
∀x, y, z, u. For the case of the simple current condensate, we pick a particle x = a, a
particle y = Ji and a particle z = Jj . Note that a can be any particle in the A theory,
not necessarily a simple current. This choice of the anyons uniquely fixes all other
anyons in the equation (p = a × Ji, u = a × Ji+j , q = a × Jj , r = Ji+j). Using the
39
fact that the simple currents and their powers are all bosons, Vafa’s theorem gives
θa×Jiθa
θa×Jjθa
=θa×Ji+j
θa. (3.35)
This equation implies that the fractions θa×Ji/θa are irreducible characters of the
group ZN1 ⊗ ZN2 ⊗ . . . ⊗ ZNn , the bosonic center of RCFT which condenses. The
one-dimensional characters of this group can be written as
θa×Jiθa
= ωi11 ωi22 ...ω
inn , (3.36)
where ωi’s satisfy ωNii = 1. Also note that the ωi’s secretly depend on the subindex
a. There are two cases:
Case 1 : θa×Ji/θa = 1, (3.37a)
Case 2 : θa×Ji/θa 6= 1. (3.37b)
In the latter case, if particles a and a× Ji restrict to the same particle t ∈ T
then this particle is confined (as θa×Ji 6= θa). Moreover, from the orthogonality of
characters Eq. (3.36) we know that in this case
N1,N2,...,Nn∑i1,i2,...,in
θa×Jiθa
=
N1−1∑i1=0
ωi11
N2−1∑i2=0
ωi22 ...
Nn−1∑in=0
ωinn
= 0.
(3.38)
This happens when at least one ωil is not equal to 1.
40
Condensation
Without loss of generality, we will assume J1, J2, . . . , Jn condense. If only a subset of
the simple currents condense then the same analysis applies to just the bosons that
condense (the others factor out). Since dJ1 = . . . = dJn = 1, the bosons restrict only
to the new vacuum ϕ with coefficients unity
nϕJ1 = . . . = nϕJn = 1 (3.39)
and do not split. Using the reasoning in Sec. 3.4.2 it follows that all products of these
simple currents also condense – indeed, all bosonic simple currents Ji condense.
We will now proceed to prove a few crucial lemmas for any a, b ∈ A: (i) nta =
nta×Ji , ∀i, for all t ∈ T and (ii)∑
t ntan
tb 6= 0 if and only if b = a× Jj for some j.
(i) is easily proved by examining the b = Ji component of Eq. (3.5). To show
(ii) we examine the t = ϕ component of Eq. (3.5) and note that all bosonic simple
currents condense giving
∑t∈T
ntantb =
N1,N2,...,Nn∑i1,i2,...in
N ba,Ji
nϕJi
=
N1,N2,...,Nn∑i1,i2,...,in
δb,a×Ji .
(3.40)
for any a, b ∈ A. To prove (ii), note that:
• If b 6= a × Jj for all j, then∑
t∈T ntan
tb = 0 and particles a, b do not have any
common restrictions. Let us write this result as
If b /∈ [a] =⇒ ntantb = 0, ∀t, (3.41)
where [a] = {Jja, Jj ∈ C} is the orbit obtained acting on a with all the bosonic
simple currents.
41
• If b = a× Jj for some j then
∑t∈T
ntanta×Jj =
N1,N2,...,Nn∑i1,i2,...in
δa×Jj ,a×Ji
=Ra ∈ Z+.
(3.42)
But from (i), nta×Jj = nta, so the LHS of this equation is positive. Hence Ra > 0.
For example, for n = 1, Ra = N1/d, with d defined in Eq. (3.30).
Hence we have proved (ii). From (ii) we know that if a and b are in the lift of t then
b = a× Jj for some j. On the other hand from (i), if a is in the lift of t, so is a× Jj .
Hence t is deconfined iff θa = θa×Jj for all j, where a is any particle in the lift of t.
In other words, given an a ∈ A, the character θa×Jj/θa 6= 1 for some j iff a restricts
only to confined particles.
Let us now prove the assumption (3.10). We first multiply Eq. (3.40) by dbθb and
sum over all particles b in the A theory to obtain
∑t∈T
βtnta =
N1,N2,...,Nn∑i1,i2,...,in
da×Jiθa×Ji
= daθa
N1,N2,...,Nn∑i1,i2,...,in
θa×Jiθa
,
(3.43)
where we have used the fact that the quantum dimension of any product of a particle
with simple currents remains the same. Now if θa×Ji/θa is not the identity character
of the trivial representation, then the particle a restricts only to confined particles
and we have from Eq. (3.38)
∑t∈T
βtnta =
∑t∈T /U
βtnta = 0. (3.44)
42
In fact, the second equality holds even if θa×Ji/θa = 1 for all j, because in that case
nta = 0∀t ∈ T /U – as a result, the second equality holds for all a. Multiplying the
second equality by θ∗ada and summing over all particles a we obtain
0 =∑t∈T /U
βt∑a∈A
θ∗adanta =
∑t∈T /U
βtβ∗t . (3.45)
The unique solution is βt = 0 for t confined, coinciding with our assumption (3.10).
3.5 One confined particle theories
In this section we study a simple boson condensation with just one confined particle
t0 in the T theory. Furthermore, assume that the confined particle t0 has only two
lifts a1 and a2 with lifting coefficients both 1, i.e.,
nt0a1 = nt0a2 = 1, (3.46)
otherwise nt0a = 0,∀a 6= a1, a2. (3.47)
With these assumptions, we can prove that the condensate has only one boson
besides vacuum, and this condensed boson has quantum dimension 1. This implies
that the boson is a simple current, so the results of the previous section imply βt0 = 0.
However, we choose to prove this equation through another method which gives more
information about bosonic condensation theories with only one confined particle.
Further we find that dt0 = da1 = da2 , which means that a1 and a2 only restrict to one
particle t0 in the T theory, with no other particles in T . Finally, in this special one-
confined particle case, we prove that βt0 :=∑
a∈A nt0a daθa = 0 which clearly support
the assumption we used in previous sections. The detailed proof can be found in the
appendix of Ref. [92].
43
3.6 Formalism and implementation
We now present an algorithmic prescription, which can be implemented on a com-
puter, and which strongly contrains the possible condensation transitions starting
from a TQFT with given modular matrices S and T . We then apply this procedure
to several example TQFTs. The algorithm is performed in 3 steps:
1. Search for the symmetric matrices M with nonnegative integer entries and
M1,1 = 1 satisfying
[M,S] = [M,T ] = 0. (3.48)
2. For each M , find all nonnegative integer rectangular matrices n such that M =
nnT.
3. For each M and n, find the putative modular matrices S and T of the TQFT
after condensation by solving
Sn = nS and Tn = nT . (3.49)
One subtlety is that we need to make sure that the S, T matrices we obtain
are valid. In this chapter, we use the necessary conditions for a valid S matrix:
it should be symmetric, unitary, and it should generate non-negative fusion
coefficients by Verlinde formula. These are always satisfied if U is a MTC.
This algorithm sidesteps the discussion of the theory T that contains confined anyons
and directly yields the resulting MTC U formed by the remaining deconfined anyons.
The algorithm provides all condensation solutions of theory A. Another algorithm
which does not sidestep T is to (1) build the matrixM ′ as in the bracket of Eq. (3.12a);
(2) factorize it in nta; (3) keep only the deconfined particle t’s; and then apply step (3)
and Eq. (3.49). Whether the two theories are identical hinges on Eq. (3.10), which
we assume to be true. We now address the above steps one by one.
44
3.6.1 Solutions for M
Since T is diagonal, the equation [M,T ] = 0 is satisfied if and only if M is a block
diagonal matrix with nonzero off-diagonal entries only between particles with the same
topological spin. Imposing this block structure, we can solve [M,S] = 0, imposing
that
1. Mab =Mba ≥ 0, M ∈ Z,
2. M11 =1.
(3.50)
The second condition ensures that the A vacuum restricts to the vacuum ϕ of U . In
this case, the first row (or column) of M is equal to the first column of n and describes
the particles that condense into the vacuum nϕa . (From this, it is also clear that only
solutions with M1a ≤ da can lead to a valid theory.)
With conditions 1 and 2 in Eq. (3.50) in place, we obtain two types of solutions for
M , which we call automorphisms and condensations, aside from the trivial solution
M = 1.
Automorphisms are defined by a fully-ranked matrix M satisfying
∑a
Mab = 1 ∀b. (3.51)
They satisfy M2 = 1 because of the following reasons: Since∑
aMab = 1 and all
entries of M can only be nonnegative integers, for any b ∈ A, there is only one
corresponding particle b′, such as Mb′b = 1. Further, Mab = 0, ∀a 6= b′ and M is
fully-ranked. As a result, if a 6= b then a′ 6= b′. Hence (M2)ab =∑
cMacMcb =∑cMacMbc = δab. An automorphism M is thus a permutation matrix of order two –
it is a symmetry of the S, T data under relabeling of particles. All automorphisms of
A form a group under matrix multiplication, which is used to construct “topological
symmetry group” in the presence of a global symmetry. [105] Automorphism, however,
45
still exists even when any other symmetries (e.g. U(1) charge conservation), are
broken.
On the other hand, solutions M that correspond to a condensation have M1a 6= δa,1
for some a, implying that at least one other boson besides the vacuum restricts
to the new vacuum. All the condensations can be superimposed with any of the
automorphisms, yielding a potentially different condensation. In other words, two
condensations can be related via a permutation of A by multiplying the M matrix of
one condensation from both sides with the M matrix of the automorphism – we will
see an example of this below for the toric code TQFT.
We can prove that any M that satisfies Eq. (3.48) and conditions 1 and 2 in
Eq. (3.50) is either an automorphism or a condensation as follows. We first assume
that M is not a condensation solution, that is, the first row and column of M are
all zeros (M1a = Ma1 = 0, ∀a 6= 1) except M11 = 1 . We show that M must be an
automorphism in this case. From
∑b
MabSbc =∑b
SabMbc (3.52)
we have for c = 1 ∑b
MabSb1 = Sa1 ⇒∑b
Mabdb = da. (3.53)
Thus, da is a strictly positive eigenvector of M with eigenvalue 1. Since every Mab is
integer and larger or equal to zero, Eq. (3.53) can only hold if
fa ≡∑b
Mab ≥ 1. (3.54)
On the other hand, summing Eq. (3.53) over a, and using M = MT, gives
∑b
fbdb =∑a
da. (3.55)
46
Again, since fa ≥ 1 and the da are strictly positive this equation can only be satisfied
if
fa ≡∑b
Mab = 1, (3.56)
which, together with the fact that M is symmetric, implies that M has to be an
automorphism (a permutation matrix).
Let us illustrate how automorphism and condensation solutions for M arise from
condition (3.48) for the example of the toric code (TC) TQFT. It contains the anyons
1, e, m, f and has the modular matrices
STC =1
2
1 1 1 1
1 1 −1 −1
1 −1 1 −1
1 −1 −1 1
(3.57a)
and
TTC = diag(1, 1, 1,−1). (3.57b)
It admits three nontrivial solutions to Eq. (3.48), one automorphism
M (1) =
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
(3.58)
47
that exchanges the e and the m particles and two condensations
M (2) =
1 1 0 0
1 1 0 0
0 0 0 0
0 0 0 0
, M (3) =
1 0 1 0
0 0 0 0
1 0 1 0
0 0 0 0
, (3.59)
of either the e or the m boson. They are related by the automorphism M (2) =
M (1)M (3)M (1) [note that (M (1))−1 = M (1)].
3.6.2 Solutions for n
Next we solve for the integer matrix nta ≥ 0, where t labels the deconfined particles in
the MTC U . It is possible that multiple solutions n exist for a given M . However, for
some solutions, it still might not be possible to find a valid condensed MTC: please
refer to our Appendix A.3.2 for an example of 4-layer Ising model condensation. In
that example, we obtain unitary S and T matrices, but they do not correspond, via
Verlinde’s formula, to integer fusion coefficients.
An efficient first step in solving for n is to realize that any column of M that only
contains zeros and ones is equal to a column in n. While the matrix M may contain
several columns with only zeros and ones that are equal, they all correspond to only
a single column in n (there are no duplicate columns in n). After removing from
M all rows and columns that contain only zeros and ones, an actual factorization
routine can be used on the remaining sub-block of the M matrix. (As we will discuss
for an example below, the factorization does not always yield a unique solution for
n in this case.) In the situations we have encountered, this part of the algorithm is
not limited by computational power. In the particularly simple toric code example,
48
deleting duplicate columns directly yields the solution
M (2) = nnT, nT = (1, 1, 0, 0). (3.60)
There is only one particle in the new theory, the vacuum. Thus, condensation of
either the e or the m particle in the toric code yields the trivial TQFT.
As a less trivial example, consider a bilayer of Ising TQFTs. Each layer contains
the anyon types 1, σ, ψ with modular matrices
SI =1
2
1
√2 1
√2 0 −
√2
1 −√
2 1
, TI = diag(1, eiπ/8,−1). (3.61)
The bilayer S and T matrices are direct products SI(2) = SI⊗SI, TI(2) = TI⊗TI, and the
theory supports 9 particle types which we denote 11, 1σ, 1ψ, σ1, σσ, σψ, ψ1, ψσ, ψψ,
where 11 is the vacuum. There is only one nontrivial solution for M , which reads in
this basis
M =
1 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 2 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 1
. (3.62)
49
It is straightforward to obtain the unique solution n that yields M = nnT
nT =
1 0 0 0 0 0 0 0 1
0 0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 1 0 0 0 1 0 0
, (3.63)
which shows that this describes the condensation of the ψψ particle. In this process,
the σσ particle (which has quantum dimension 2) splits into two particles of quantum
dimension 1 and both 1ψ, ψ1 restrict to the same particle. All other particles, except
for the vacuum, become confined.
There exists M that solve Eq. (3.48), but cannot be decomposed as M = nnT
with a nonnegative integer matrix n. Some of them still admit an interpretation in
terms of a condensation in the following sense. If the MTC U that is obtained from a
condensation with matrix M = nnT has an automorphism symmetry P , that is equal
to its transpose P = PT, then M = nPnT is also a symmetric matrix that solves
Eq. (3.48). For instance, one necessary condition for a decomposition M = nnT to
be possible is that Maa +Mbb ≥ 2Mab. If the matrix elements of M do not satisfy the
triangle equation Maa +Mbb ≥ 2Mab, then M = nPnT might be possible instead.
If the TQFT corresponds to a CFT, the possible forms of matrices M that solve
Eq. (3.48) are understood with the help of the “naturality theorem” by Moore and
Seiberg [106, 107]. This theorem implies that all M that solve Eq. (3.48) in a CFT
are either automorphisms of A, condensations of the form M = nnT, or of the form
M = nPnT, with P an automorphism of U . As a corollary, we then conclude that
for any solution to Eq. (3.48) of the from M (2) = nPnT, there is another solution
M (1) = nnT, with the same n, since the identity mass matrix of U always exists. For
the purpose of studying condensations, we thus focused on matrices M that admit
the decomposition M = nnT throughout our analysis. If we relaxed this constraint to
50
also include M = nPnT, assumptions such as nta = nta would not be justified anymore.
We discuss the interpretation of condensation transitions for CFTs in Appendix A.2
and relate it to the “naturality theorem”. Subsequently, in Appendix A.3 we give an
example of condensation transitions in SU(2)16, for which two solutions M = nPnT
and M = nnT to Eq. (3.48) exist.
The decomposition M = nnT is generally not unique. For example, if Maa = 4 for
some particle with quantum dimension 4 or larger, it can either split in 4 particles
with nta = 1 for each or restrict to one particle with nta = 2 (this issue was discussed
previously in Sec. 3.4). However, in all examples we studied, at most one of all
possible decompositions of M lead to a consistent TQFT with valid solutions for S
and T . Thus, the uniqueness of this step in the condensation is an open question.
We mentioned that factorizing M = nnT is a well-known problem in the field of
completely positive matrices. In our cases, the factorization happens over the ring of
positive integers. This problem is known to be NP-hard. With the exception of small
dimension matrices, it has not yet been solved. Some outstanding questions are the
characterization of when a matrix M is completely positive (sufficient and necessary
condition), as well as what is the minimal number of rows in n (called CP rank),
which is translated in our case to the minimal number of particles in U that can be
obtained.
3.6.3 The modular matrices of the new theory
Having obtained the matrix n, we now solve the equations
Sn = nS, Tn = nT (3.64)
for S and T . These equations can have spurious solutions unless we impose a list of
additional constraints. For modular theories, these constraints are
51
• S† = S−1,
• S2 = Θ(ST )3 = C, where C is a permutation matrix that squares to the
identity and Θ = e−iπc/4 with c the chiral central charge of A, which we can
prove remains unchanged (mod 8) during condensation.
• T is a diagonal matrix with complex phases on the diagonal,
• the fusion coefficients obtained from the Verlinde formula
Nt = SDtS−1, (3.65)
with (Dt)rs = δr,sStr/S1r have to be nonnegative integers.
We do not prove that any solution that obeys the above list of conditions is indeed
a valid MTC U . However, any allowed condensation will be a solution to these
conditions. Therefore, if we do not find a solution for a given MTC A, we can
conclude that no condensation transition to a modular U theory out of A exists (we
will discuss a nontrivial example for this situation in Sec. 3.7.2).
For the example of the double layer Ising theory, we have for SI(2)n = nS (skipping
columns of zeros, which correspond to the confined particles)
12
12
1 12
12
12−1
20 −1
212
12−1
20 −1
212
12
12−1 1
212
=
S11 S14 S12 + S13 S14 S11
S21 S24 S22 + S23 S24 S21
S31 S34 S32 + S33 S34 S31
S41 S44 S42 + S43 S44 S41
. (3.66)
Note that all |Sab| = 1/2 since the theory contains only Abelian anyons. Thus,
Eq. (3.66) determines all matrix elements of S, except for S22 = −S23 = −S32 =
S33. (The equality −S23 = −S32 follows from the fact that a modular S matrix is
symmetric.) At the same time, we have from Tn = nT that θ1 = 1, θ2 = θ3 = eiπ/4,
52
θ4 = −1. Furthermore, the (2,2) component of the equation S2 = Θ(ST )3 reads
1
2
(1 + 4S2
22
)=
1
2
(1 + 8iS3
22
)(3.67)
yielding the unique solution S22 = −i/2, that also satisfies |S22| = 1/2. We can use
the thus obtained S matrix to compute the fusion coefficients from Eq. (3.65), and
we find that they are all non-negative integers. The new fusion rules are
2× 2 = 3× 3 = 4, 2× 3 = 1, (3.68)
which are distinct from the toric code fusion rules. The resulting TQFT coincides
with the gauged Chern number 2 superconductor from Kitaev’s 16-fold way [20]. We
have thus shown that this TQFT is obtained in a unique way through condensation
in a double layer of Ising theories (two gauged Chern number 1 superconductors).
In fact, one can iterate this procedure to obtain all TQFTs appearing in Kitaev’s
16-fold way. A natural open question is to find out which TQFTs exhibit such a
closed structure with unique condensations. Using the formalism developed above,
we will show below that another simple non-Abelian TQFT, the Fibonacci category,
does not admit a similar structure, since it does not allow for any condensation.
One may wonder whether Eq. (3.65) needs to be imposed as a separate condition
on the possible solutions for S, or whether it follows from the other conditions in
the above list. To show that Eq. (3.65) is required, we discuss the example of four
layers of Ising TQFTs in Appendix A.3.2, for which there exist a unitary and sym-
metric S matrix, except that the fusion coefficients generated from S by Verlinde’s
formula in Eq. (3.65) are not integer. Therefore, it does not correspond to an allowed
condensation transition and the list of conditions is not complete without Eq. (3.65).
53
3.7 Layer constructions and uncondensable bosons
In this section, we apply the condensation formalism to TQFTs A(N) that are tensor
products of N identical layers of a TQFT A. There are several motivations to study
such a construction:
(1) Some TQFTs are characterized by a Zm grading under layering: N = m
layers can be physically equivalent to the trivial TQFT in the bulk. For a theory to
be condensable to nothing, m is constrained by the fact that the chiral central charge,
which is conserved under condensation, must vanish (mod 8). Condensation provides
a way to determine the grading m as well as all the TQFTs for N = 1, . . . ,m layers.
See the following Sec. 3.7.1 for discussions and details of examples.
(2) The grading of TQFTs has an immediate physical implication: Kitaev’s 16-
fold way, which we discuss below, characterizes 16 different chiral superconductors in
(2+1) dimensions.
(3) Abelian bosons are always condensable in any A. However, there exists a
non-Abelian boson b in an A which cannot be condensed. Furthermore, there exists a
non-Abelian boson b in an A, such that any tensor product of the non-Abelian boson
b(n), ∀ n = 1, 2 . . . , N cannot be condensed in A(N), ∀ N . For instance, the Fibonacci
category studied in Sec. 3.7.2. This phenomenon will be studied more systematically
in Chapter 4.
(4) Layer constructions have been proposed to gain insight into (3+1)-
dimensional phases with topological order, for which there is currently no systematic
understanding[108]. The idea is to couple N layers of a TQFT A by a condensation
transition in such a way that the number of anyons after condensation does not scale
with N . Some of the anyons that restrict to deconfined particles have a nontrivial
particle in every layer. Their restriction is then interpreted as a string excitation of
the (3+1)-dimensional theory. We discuss an example in the following Sec. 3.7.1.
54
Before condensation, the general structure of A(N) is
S ′A(N) = SA ⊗ · · · ⊗ SA︸ ︷︷ ︸N times
, T ′A(N) = TA ⊗ · · · ⊗ TA︸ ︷︷ ︸N times
, (3.69)
for the modular matrices and
N ′ca,b =N∏i=1
N ciai,bi
, d′a =N∏i=1
dai , θ′a =N∏i=1
θai , (3.70)
for the fusion matrices, quantum dimensions, and topological spins. Here, N ca,b, da
and θa, are the fusion coefficients, quantum dimensions, and topological spins of A
and the respective primed quantities belong to A(N). We have labeled the anyons
in A(N) by a vector a = (a1, · · · , aN)T of anyons in each layer, 1, . . . , N , where each
entry ai can be any of the anyons in A.
3.7.1 Theories with Zm-graded condensations
SU(3)1: 4-fold way
As a simple example, let us consider the SU(3)1 TQFT. It has three Abelian anyons
1, 3, 3 with fusion rules
3× 3 = 3, 3× 3 = 3, 3× 3 = 1 (3.71)
and topological spins θ3 = θ3 = ei2π/3. Now, we consider multiple layers of SU(3)1.
Notice that since each layer has a automorphism symmetry 3 ↔ 3, all statements
below should be understood modulo this automorphism symmetry applied to every
layer.
Clearly, the m = 2 layer theory SU(3)1 × SU(3)1 has no bosons and therefore no
condensation transition is possible.
55
The m = 3 layer theory SU(3)1 × SU(3)1 × SU(3)1 has 8 bosons. However, up to
the automorphism, there is a unique condensation corresponding to bosons (1, 1, 1),
(3, 3, 3) and (3, 3, 3) restricting to the vacuum 1′ and all other bosons confined. Besides
the vacuum, two more particles are deconfined: 3′ with lifts (3, 3, 1), (1, 3, 3), (3, 1, 3),
and 3′ with lifts (3, 3, 1), (1, 3, 3), (3, 1, 3). Together, 1′, 3′, and 3′ furnish SU(3)1,
which differs from SU(3)1 by complex conjugation of the topological spins. It might
seem unusual that the condensation of multiple layers of chiral theories results in an
anti-chiral theory, but we remind the reader that the chiral central charge is only
conserved modulo 8 under condensation transitions and hence −2− 2− 2 = 2 mod 8
is allowed.
Then, the m = 4 layer theory is SU(3)1 × SU(3)1 which can be condensed to
the trivial TQFT by condensing simultaneously (3′, 3) and (3′, 3), which confines all
other particles. We have thus shown that condensation induces in a unique way a Z4
grading in the layered SU(3)1 TQFTs.
Ising: Kitaev’s 16-fold way
We want to couple N layers of the Ising TQFT, which is defined in Eq. (3.61). For
condensation, the simplest boson that we can build consists of the ψ-particles in two
consecutive layers n+ 1 and n+ 2,
Bn := (1n, ψ, ψ, 1N−n−2), (3.72)
where 1n stands for the vacuum particle in n consecutive layers. All bosons Bn,
n = 0, ..., N − 2, are condensed. We will identify all bosons of this form with the
vacuum, building a simple current condensate. From
(1n, ψ, ψ, 1N−n−2)× (· · · , ψ, 1, · · · ) = (· · · , 1, ψ, · · · ), (3.73)
56
we see that consistency requires that any pair of anyons (· · · , ψ, 1, · · · ) and
(· · · , 1, ψ, · · · ) restrict to the same anyon after condensation. Here, · · · stands
for any sequence (that agrees between the two particles). Furthermore, by fusion
with the condensate we have
(1n, ψ, ψ, 1N−n−2)× (· · · , σ, 1, · · · ) = (· · · , σ, ψ, · · · ). (3.74)
However, θ(···σ1··· ) = −θ(···σψ··· ), implying that the restrictions of (· · ·σ, 1 · · · ) are con-
fined, because they have another lift (· · ·σ, ψ · · · ) with different topological spin. By
that argument we have shown that the set Q consisting of particles with least one
and at most N − 1 σ’s restricts only to confined particles. On the other hand, we
know that particles containing no σ’s (i.e., only 1’s or ψ’s) restrict to single deconfined
particles:
• By closure of the condensate, any particle with even number of ψ and otherwise
1 restricts to the new vacuum 1′.
• Any particle with odd number of ψ and otherwise 1 restricts to the deconfined
particle ψ′. Their fusion rule is
ψ′ × ψ′ = 1′. (3.75)
The only particle left to consider is σ(N) ≡ (σ, . . . , σ). It is easy to show that
σ(N) × Q ⊆ Q. It then follows from the a = σ(N), b ∈ Q, t = ϕ component of
Eq. (3.5) that σ(N) and particles in Q restrict to disjoint sets of particles, because
the righthand side of Eq. (3.5) is zero in this case, as none of the particles in Q
restrict to the vacuum. But then the restriction of σ(N) cannot possibly contain
confined particles as those confined particles would have just a single lift σ(N), which
57
is impossible from the definition of confined particle. Hence σ(N) restricts only to
deconfined particles, and we can identify Q as the set of lifts of all confined particles.
We can say more about the restriction of σ(N). Note DU = DA/q = 2N/2N−1 =
2, because q is equal to the number of condensed bosons i.e., q = 2N−1. As we
already know 1′, ψ′ are deconfined, DU =√
1 + 1 + . . . = 4, where . . . are additional
contributions from the restriction of σ(N). When N is not a multiple of 8, there are
just two options. Either case (1) (σ, · · · , σ) splits into just two Abelian particles
distinct from 1′, ψ′, or case (2) (σ, · · · , σ) has a single restriction with quantum
dimension√
2. (When N is a multiple of 8 the σ-string is itself a fermion or boson
and could restrict to the ψ′ and the vacuum, respectively. However, by DU = 2 it is
not possible that ψ′ and the σ-string have a common restriction in the case where N
is an odd-integer multiple of 8 (since DU =√
3 in that case). The case where N is a
multiple of 16 will be discussed separately below.) Consider now from Eq. (3.64) the
matrix element that corresponds to any particle t in the restriction of (σ, · · · , σ) and
the identity in A,
nt(σ,··· ,σ) =
√2N
2dt, (3.76)
since we know from the discussion following Eq. (3.74) that t has only one lift,
(σ, · · · , σ).
From the condition that nt(σ,··· ,σ) is integer, we conclude that case (1) applies to
even N and case (2) to odd N . We now analyze the two cases separately.
Case: N odd According to Eq. (3.76), we have (σ, · · · , σ)→ 2(N−1)/2σ′. It follows
from the fusion rules of the original theory, i.e., from Eq. (3.5) by choosing a = b =
(σ, · · · , σ), that
σ′ × σ′ = 1′ + ψ′. (3.77)
58
Thus, 1′, σ′, ψ′ furnish the same (Ising) fusion algebra as 1, σ, ψ do in every layer. The
spin factors of the deconfined restrictions are given by
θ1′ = 1, θσ′ = e2πiν/16, θψ′ = −1, (3.78)
where ν = N mod 16 is an odd integer, for N is odd. We have thus obtained all
TQFTs with Ising fusion rules that appear in Kitaev’s 16-fold way.
Case: N even If N is even, Eq. (3.76) yields the restriction (σ, · · · , σ)→ 2N/2−1a′+
2N/2−1b′ with equal coefficients. To find the fusion rules for a′ and b′, we solve
Eq. (3.64). This leaves two possibilities
a′ × a′ = b′ × b′ = 1′, a′ × b′ = ψ′, (3.79)
a′ × a′ = b′ × b′ = ψ′, a′ × b′ = 1′. (3.80)
Here, Eq. (3.79) are the toric code fusion rules. Which of the two cases applies can
be determined from the equation S2 = Θ(ST )3 = C, by using the topological spins
θa′ = θb′ = e2πiN/16. (3.81)
For N = 2 mod 4 one finds the solution Eq. (3.80) and for N = 4 mod 4 one finds
the solution Eq. (3.79).
The case where N is a multiple of 16 has to be considered separately. The con-
densation described here leads to the toric code TQFT in which a′ and b′ are bosons.
We have shown above that the toric code can be condensed to the trivial TQFT by
condensing either a′ or b′ (which were called e and m before). Thus, in the case where
the σ string is a boson, two condensations are possible: one leads to the toric code
and in the other one, in which the σ string restricts in part to the vacuum, leads to
59
the trivial TQFT. The toric code is also the TQFT that was proposed to describe a
gauged s-wave superconductor without topological edge modes. [109]
Together, this Z16 grading represents Kitaev’s 16-fold way, yielding a (non-
)Abelian fusion category for the vortices of even (odd) layer length. From the point
of view of layer construction[108], we note that ψ′ is a point-like fermionic excitation
in 3D space, while σ′, a′ and b′ are to be interpreted as vortex or line-like excitations
in 3D, because their lift has a nontrivial anyon in each layer.
It is tempting to consider the topological orders that have been proposed in
Refs. [110, 111] as the possible symmetry-preserving gapped surface terminations
of time-reversal symmetric (3+1)-dimensional superconductors as another example
of a theory with Z16 grading under condensation. The topological index ν of the
bulk superconductor has been shown to be only meaningful mod 16 in the presence
of interactions. The ν = 1 surface topological order was proposed to be nonmodular
category SO(3)6, while that for ν = 2 is the so-called T-Pfaffian state. We do not
further elaborate on possible condensations in this theory here, as the focus of the
present work is on condensation in modular categories. However, if we were to apply
the formalism of Eq. (3.64) to this problem, none of the possible condensation tran-
sitions in a double layer SO(3)6×SO(3)6 would lead to the T-Pfaffian. Rather, one
can condense all bosons in SO(3)6×SO(3)6 to obtain the trivial nonmodular TQFT
{1, f} with only one Abelian fermion f .
3.7.2 Theories with Z-fold way: Fibonacci TQFT
Not every TQFT has a Zm-graded structure under condensation. The simplest
counter-example is the Fibonacci TQFT with the single nontrivial anyon τ and the
fusion rule
τ × τ = 1 + τ. (3.82)
60
It has topological spin θτ = ei4π/5 and quantum dimension dτ = φ, where φ =
(1 +√
5)/2 is the golden ratio.
First, we want to show that no condensation is possible in 5 layers of Fibonacci,
despite the presence of the boson (τττττ). We will show that there is no matrix M
that describes a condensation and satisfies Eq. (3.48). To see this, consider the (1,b)
component of the equation MSFib(5) = SFib(5)M ,
∑a
nϕa (SFib(5))a,b =1
(2 + φ)5/2
∑a
daMa,b. (3.83)
Observe that the righthand side is nonnegative for any b. Specializing to b =
(τ, 1, 1, 1, 1), we find the lefthand side
(2 + φ)−5/2(φ− nϕ(τττττ)φ
4), (3.84)
which is negative for any nϕ(τττττ) ≥ 1, i.e., for any condensation. Therefore, no
condensation transition is possible in 5 layers of Fibonacci (see Ref. [21] for an
alternative proof).
Second, let us show further that no condensation is possible in 10 layers of Fi-
bonacci. Besides the vacuum, there is a boson with a τ anyon in every layer, which
we denote by (10τ), and 252 =(
105
)bosons with τ anyons in exactly 5 layers.
Again, we will show that there is no matrix M that describes a condensation and
satisfies Eq. (3.48). To see this, we consider the (1,b) component of the equation
MSFib(10) = SFib(10)M , but this time for the choice b = (10τ). Up to an overall factor
of the total quantum dimension, the equation reads
nϕ1φ10 +
∑a∈5τ bosons
(−1)5φ5nϕa + (−1)10nϕ(10τ)
= nϕ(10τ) +∑
a∈5τ bosons
φ5Ma,(10τ) + φ10M(10τ),(10τ).
(3.85)
61
Using nϕ1 = 1, it simplifies to
0 = φ5(M(10τ),(10τ) − 1
)+
∑a∈5τ bosons
(nϕa +Ma,(10τ)
). (3.86)
We can see that Eq. (3.86) has no nontrivial solution: Since φ5 is irrational, the first
term needs to be zero on its own, which requires M(10τ),(10τ) = 1. This implies that
(10τ) does not condense, as it has noninteger quantum dimension and would therefore
have to split in order to condense. However, the second term in Eq. (3.86) is a sum of
nonnegative numbers that can only vanish if nϕa = 0, ∀a. Hence, none of the bosons
condenses.
In fact, one can show that no condensation is possible for any number of layers N
of the Fibonacci TQFT [112]. We will reformulate this proof much more easily using
the formalism developed in this chapter elsewhere in a way that also generalizes to
other TQFTs.
Obstructions against the condensation of bosons within our formalism can only
ever occur in theories that contain non-Abelian anyons. In Abelian theories, any
potentially condensing boson J is a simple current (of order d), and one can explicitly
construct a theory in wich J is condensed as follows: Form all the orbits [a] with
respect to J , as defined in Eq. (3.30). The orbit of the identity is the condensate. If
all anyons in an orbit [a] have the same topological spin, the orbit labels a particle t[a]
in the theory U , otherwise all particles in the orbit are confined. If t[a] is unconfined,
choose nt[a]b = 1 if b ∈ [a] and n
t[a]b = 0 otherwise. Further, choose St[a],t[b] = dSa,b and
Nt[c]t[a],t[b]
=d∑
n=0
N c×Jna,b , (3.87)
for t[a], t[b], and t[c] unconfined. In can be readily shown that this choice is a consis-
tent solution to Eqs. (3.64) and (3.65) and therefore a valid condensation within our
formalism.
62
3.8 Conclusions
In summary, we derived a framework for the condensation of anyons that is applicable
to modular tensor category models of topological order. Our derivation is based
on a small number of physical assumptions and focuses on the computation of the
modular matrices S and T of the theory after condensation. Based on this, we propose
an algorithm to carry out this computation. This algorithm first seeks symmetric
nonnegative integer matrices M that commute with the modular matrices S and
T of the original theory. It then proceeds by factorizing M = nnT in a product
of a nonnegative integer matrix n with itself. Finally, the equations Sn = nS and
Tn = nT are solved. Our algorithm has proven to be practically useful in all examples
that we studied. We finally demonstrated that the equations that are central to our
derivation are powerful constraints on condensation transitions in general.
This leads us to several open problems that are not answered by the present
work. One concerns the assumption that βt = 0 for all confined particles t. We have
shown in Secs. 3.4 and 3.5 that this relation follows from weaker assumptions for
certain theories. But a general proof of this statement is lacking, so that it remains
an assumption for us. Other questions concern the uniqueness of solutions and the
transitivity of condensation transitions. For example, given an M , is there a unique
n that solves M = nnT and leads to a valid condensed theory? And given such a
solution n, is there a unique consistent solution S and T? In a similar vein, is the
condensed theory completely characterized by the coefficients nϕa?1 At present, we do
not have counterexamples against affirmative answers to these questions.
Another future direction could be the condensations in the presence of global
symmetries[105]. When we have global symmetries on top of a topologically or-
dered system, the anyons may transform in a projective representation. A direct
1Indeed, we cannot exclude the possibility that additional information, like certain vertex liftingcoefficients, are needed to fully determine the topological order of the condensed phase.
63
consequence is that certain condensations may not be able to happen if all global
symmetries are respected.
64
Chapter 4
No-Go Theorem for Boson
Condensation in Topologically
Ordered Quantum Liquids
One motivation to study condensation transitions is to classify topological order. An
important example are the 16 types of gauged chiral superconductors introduced by
Kitaev [20]. Kiteav showed that while two-dimensional superconductors are classified
by an integer Z, only 16 bulk phases are topologically distinct. This construction
can be understood by considering ` layers of initially disconnected chiral p-wave su-
perconductors, i.e., elementary (Ising) TQFTs. Upon introducing generic couplings
between these layers, one obtains a single layer of a chiral `-wave superconductor,
which corresponds to a specific TQFT in Kitaev’s classification. This physical pro-
cess of coupling the layers (by condensing inter-layer cooper pairs), corresponds to
a condensation transition on the level of the TQFTs. For every ` < 16, there is a
unique condensation possible and one obtains exactly 16 distinct TQFTs including
Ising, the toric code and the double semion model. They determine the nature of the
topologically protected excitations in the vortices of each superconductor, including
65
their braiding statistics. In essence, this Z16 classification can be seen as a property
of the Ising TQFT.
It is imperative to ask whether multi-layer systems of other TQFTs show a similar
collapse of the classification from Z to ZN for some integer N . In this chapter,
we derive a criterion for when this is not the case, i.e., when the Z classification
generated by a given TQFT is stable. This criterion is based on the fact that there
exist bosonic anyons that cannot be condensed. An example are the bosons in multi-
layered Fibonacci topological order [112, 21, 113]. In this chapter, we generalize this
observation by formulating a no-go theorem that constitutes a sufficient obstruction
against the condensation of a boson. Our criterion and its proof are given using the
tensor category formulation of topological order [80, 81, 82, 83, 84, 85, 20, 35, 36],
which we can use to describe the condensation transition axiomatically [42, 21, 113].
We apply our no-go theorem to several examples, including the forementioned multi-
layer Fibonacci TQFTs.
4.1 First No-Go Theorem
The following definition is useful for formulating our no-go theorem: For a given
anyon b, a subset Ib = {a1, . . . , am} of anyons is called a set of zero modes localized
by b [105] if for all i, j = 1, . . . ,m:
1. The fusion products ai × aj do not contain condensable bosons, except the
identity if ai = aj,1
2. all ai are zero modes of b, by which we mean ai × b = b+ . . ., (i.e. N baib> 0)
3. if a particle ai is in Ib then so is its antiparticle.
1In demanding that ai × aj does not contain condensable bosons, as opposed to not containingany bosons at all (except the identity), we are anticipating a inductive application of the no-gotheorem. Once we have shown that a boson B, whose set IB is such that ai × aj , with ai, aj ∈ IB ,does not contain any boson (except the identity), is uncondensable, it is allowed that B appears inthe fusion product ai × aj of the set IB′ of another boson B′.
66
Bai
BBai
a) b)
~phase with condensed B a#
i
Figure 4.1: Tunneling processes mediated by an anyon condensate. The gray regionis a phase in which a boson B is condensed. a) Vertex of a boson B that localizes azero mode of anyon ai. In the condensed phase, B can be converted into an identityparticle world line (not shown). By the axioms of anyon condensation, processesa) and b) are equivalent, i.e., B can be converted into ai by tunneling through thecondensate.
Note that the choice of Ib for a given boson b is not unique and that Ib may or
may not contain the identity. (The above conditions are satisfied in both cases.)
Typically, we will be interest to find a set Ib that is as large as possible. To motivate
the terminology of the set Ib, observe that N bab > 0 implies that a anyons can always
be emitted or absorbed by b. Therefore, b must carry a zero-mode excitation of a.
We can now state our first main result, a general condition under which a boson B
cannot condense. It is an obstruction that is sufficient to show that condensation of
B cannot occur.
No-go theorem — A boson B cannot condense if there exists a set IB, such that the
sum of the quantum dimensions of all anyons in IB exceeds the quantum dimension
of B, i.e., if
dB < da1 + da2 + · · ·+ dam . (4.1)
Proof. We start by showing that all particles in IB do not split, and have distinct
restrictions. This follows from inspection of Eq. (3.5) for t = ϕ, a = ai, b = aj,
∑r∈T
nrainraj
= δi,j +∑c6=1
N caiaj
nϕc , (4.2)
67
where we used nϕc = nϕc and nraj = nraj . By assumption, there are no condensable
bosons in ai × aj, hence N caiaj
and nϕc cannot be both nonzero for any c 6= 1. Thus∑r n
rainrai = 1, implying a single restriction a↓i of ai, with da↓i
= dai using (A.4).
Moreover,∑
r nrainraj = 0 if i 6= j, implying that the restrictions of ai 6= aj are
distinct particles.
With this knowledge about the restrictions of the ai, Eq. (3.5) for t = ϕ, a = ai,
b = B evaluates to
na↓iB
= na↓iB =
∑c
N caiBnϕc ≥ NB
aiBnϕB, (4.3)
where we used N BaiB
= NBaiB
Inserting this inequality in Eq. (A.4) for a = B, and
using dai = da↓i, we have
dB ≥ nϕB
m∑i=1
NBaiBdai . (4.4)
It follows that in a situation where Eq. (4.1) holds, Eq. (4.4) implies nϕB = 0, i.e., B
does not condense. [Note that in the case NBaiB
> 1, a stronger form of Eq. (4.1) with
dai is replaced by NBaiBdai holds.]
To follow up with a pictorial representation of these equations, consider the tun-
neling of anyons across the domain wall as shown in Fig. 4.1, where each particle a
in the uncondensed theory is converted into its restriction a↓ in the gray region. Fig-
ure 4.1 (a) shows a vertex allowed by the fusion rule ai×B → B in the uncondensed
phase. The boson B enters the condensed phase, where it can disappear as it is part
of the condensate (one of its restrictions is the vacuum ϕ, the world lines of which can
be removed at will). By the fundamental assumption that fusion and condensation
commute [which is at the heart of Eq. (3.5)], Fig. 4.1 (a) is equivalent to Fig. 4.1 (b).
The latter represents a coherent tunneling process that is mediated by the conden-
sate and converts B into any of the ai. The existence of this process implies that the
distinct restriction a↓i of any ai must be in the restriction of B. Hence, by Eq. (A.4),
the quantum dimension of B must be large enough to accommodate all the distinct
68
restrictions of the ai, if B condenses. Therefore if we find sufficiently many ai such
that Eq. (4.1) holds, B cannot condense.
Note that the no-go theorem does not a priori require knowing the braiding data
of A – although the modular tensor category structure fixes that data to some ex-
tend. The theorem involves only data obtainable from N cab. We remark that the no-go
theorem can only ever yield an obstruction against the condensation of non-Abelian
bosons. For Abelian bosons, the theory after condensation can be constructed explic-
itly, which is a constructive proof that there is no obstruction. [113]
We now demonstrate that the no-go theorem is practically useful by considering
three examples: (i) multiple layers of the Fibonacci TQFT, (ii) single layers of the
SO(3)k TQFT for k odd, and (iii) multiple layers of the latter. We will show that all
these theories, while containing bosons, do not admit condensation transitions. All
the bosons are noncondensable. Additional general results, concerning for instance
TQFTs with a condensing Abelian sector and with only a single boson, are given in
appendix B.1.
4.2 Example (i): Multiple layers of Fibonacci
The Fibonacci category AFib is a non-Abelian TQFT containing just one nontrivial
particle τ with a fusion rule τ×τ = 1+τ , a topological spin θτ = ei4π/5, and a quantum
dimension dτ = φ given by the golden ratio φ = (1+√
5)/2. As AFib does not contain
any nontrivial boson, it cannot undergo a condensation transition. We are interested
whether the TQFT formed by N identical layers of AFib i.e., the TQFT A⊗NFib , admits
a condensation transition. The TQFT A⊗NFib contains 2N particles corresponding to all
possible distributions of τ -particles over the N layers. For each r = 0, . . . , N there are(Nr
)so-called (rτ) particles with τ ’s in exactly r layers, each with spin θ(rτ) = ei4πr/5
and quantum dimension d(rτ) = φr. The unique r = 0 particle is the identity of A⊗NFib .
69
From the topological spin, the bosons in A⊗NFib are (rτ) particles with r = 5n, n ∈ Z.
Using the no-go theorem, we show that none of these bosons can condense.
Using proof by induction on n ≥ 1, we show that for any (5nτ) boson B, there
exists a set I(5nτ) such that Eq. (4.1) holds. We first consider the case n = 1. Given
a (5τ) boson, we must construct a set I(5τ) for this boson. Consider the set formed
by all (2τ) particles obtained by replacing any 3 τ ’s in the boson with a 1. There are(52
)= 10 such (2τ) particles for a given (5τ) boson. They form a set I(5τ) that obeys
point 1–3 from the definition: point 1 holds as any product of two of these particles
has at most 4 τs and is therefore not a (potentially condensable) boson. Points 2 and
3 can be checked by using the Fibonacci fusion rules in each layer. Finally, Eq. (4.1)
holds because
d(5τ) = φ5 < 10φ2 =∑
ai∈I(5τ)
dai (4.5)
evaluates to about 11.1 < 26.2. We conclude that none of the (5τ) bosons condense
for any number N of layers of Fibonacci TQFT.
For the induction step, we assume that none of the (5nτ) bosons can condense
for n < n0, n0 > 1, and we show that the same holds for the (5n0τ) bosons. Define
r0 := b(5n0 − 1)/2c, where bxc is the largest integer smaller than or equal to x. For
a given (5n0τ) boson, form the set I(5n0τ) out of all (r0τ)-particles that are obtained
by replacing any (5n0 − r0) τ ’s in the boson (5n0τ) with a 1. There are(
5n0
r0
)such
(r0τ) particles. They form a set I(5n0τ) for (5n0τ). In particular their fusion products
can only contain (5nτ)-bosons with n < n0, which cannot condense by assumption.
Equation (4.1) reads for this case
φ5n0 <
(5n0
r0
)φ5n0−r0 . (4.6)
Using that r0 ∼ 5n0/2 and(
5n0
5n0/2
)∼ 45n0/2/
√π5n0/2 for large n0, we obtain that the
right-hand side of Eq. (4.6) grows like 45n0/2φ5n0/2/√n0, asymptotically dominating
70
the left-hand side. An explicit evaluation yields that Eq. (4.6) holds for any n0 ≥ 1
in fact. We have thus shown that none of the (5n0τ) bosons can condense. This
concludes the induction step and the proof that no boson in A⊗NFib can condense.
4.3 Example (ii): Single layer of SO(3)k
Our second example focuses on the (single-layer) TQFTs associated with the Lie
group SO(3) at values of odd level k. They contain bosons for an infinite subset of
k. We show that none of these bosons can condense. The SO(3)k TQFTs with k odd
have (k + 1)/2 anyons j = 0, · · · , (k − 1)/2 with
dj =sin(π 2j+1k+2
)sin [π/(k + 2)]
, θj = e2πij j+1k+2 . (4.7)
We note that for k odd, all particles have distinct quantum dimensions. The fusion
rules are
N j3j1j2
=
1 |j1 − j2| ≤ j3 ≤ min{j1 + j2, k − j1 − j2}
0 else
. (4.8)
The smallest odd k for which SO(3)k contains a boson is k = 13, in which j = 5 is a
boson – an uncondensable one, as we shall see.
The topological spins θj yield the condition j(j + 1) = k + 2 for the lowest j
that may correspond to a boson (aside from the vacuum j = 0). (Frequently, this
condition cannot be met with integer j, as in the k = 13 example, and the lowest
boson appears at even higher j.) We conclude that the first boson after j = 0 cannot
occur for j lower than
j0 =⌊√
k + 9/4− 1/2⌋. (4.9)
71
dj dj
j j
I II III I II III
a) b)k = 13 k = 103
Figure 4.2: Quantum dimensions and bosons (blue columns) for SO(3)k theories witha) k = 13 and b) k = 103. These are the smallest k, for which SO(3)k contains twoand four bosons, respectively. Indicated are also the ranges I–III defined in Eq. (4.10).The maximum quantum dimension coincides with the boundary between range I andII in Eq. (4.10). For instance, to apply the no-go theorem to the j = 5 boson in a),choose Ij=5 = {j = 2} and use that d5 ≈ 3.6 is smaller than d2 ≈ 4.2.
We will now discuss separately bosons j in the three ranges (see Fig. 4.2 for two
examples)
I. j0 ≤ j ≤ bk/4c (4.10a)
II. bk/4c < j ≤ k − 1
2−⌊j0 − 1
2
⌋(4.10b)
III.k − 1
2−⌊j0 − 1
2
⌋< j ≤ k − 1
2. (4.10c)
Due to Eq. (4.9), bosons jB in range III have no bosons in their fusion product
jB × jB, other than the identity. Thus, from Eq. (3.5) for t = ϕ, and the fact that B
are their own antiparticles, we conclude that they cannot split. Using Eq. (A.4) and
the fact that they have djB > 1, we conclude that they cannot restrict to the vacuum
i.e., they cannot condense.
We now use our no-go theorem to show that bosons jB in range I are non-
condensable. Specifically, we show that the particles 0 < j < bjB/2c form a set
IjB of jB obeying Eq. (4.1). Before establishing that they satisfy the conditions for a
set IjB , let us show that Eq. (4.1) holds for IjB . For large k, we can rely on the follow-
72
ing asymptotic estimate. Using that the sine function in Eq. (4.7) is monotonously
increasing with negative second derivative for j ≤ bk/4c, the estimate
2jB + 1 <
bjB/2c−1∑j=1
(2j + 1) (4.11)
implies Eq. (4.1) for jB in range I. This inequality holds for all jB ≥ 10. Using
Eq. (4.9) we conclude that it applies to all bosons in range I for k ≥ 109. We
verified explicitly that inequality (4.1) holds (using the exact values of the quantum
dimensions) for all bosons in range I for k < 109. Finally, it is readily verified
using Eq. (4.8) that IjB form a set of zero modes localized by jB provided that all
bosons with j < jB cannot condense. The proof then proceeds straightforwardly by
induction.
We apply our no-go theorem successively to bosons jB in range II in order of
increasing jB. Using the result that all bosons in range I are uncondensable, one
verifies that the particles j with 1 ≤ j ≤ min{k− 2jB, bjB/2c− 1} form a set IjB . As
for range I, we can estimate the quantum dimensions. From the relation sin[π(2jB +
1)/(k+2)] = sin[π(k−2jB+1)/(k+2)] we can estimate the quantum dimension of jB
using sin[π(2jB+1)/(k+2)] < π(k−2jB+1)/(k+2). The quantum dimensions of the
anyons in IjB are estimated as for range I with sin[π(2j+1)/(k+2)] < π(2j+1)/(k+2).
Using these estimates we find that if
k − 2jB + 1 <
min{k−2jB ,bjB/2c−1}∑j=1
(2j + 1) (4.12)
holds, Eq. (4.1) follows. In the case k − 2jB < bjB/2c − 1, Eq. (4.12) reduces to
1 < (k − 2jB)2 + (k − 2jB), which is true for all jB in range II for all k. In the case
k−2jB > bjB/2c−1, Eq. (4.12) simplifies to k+2 < 2jB +(bjB/2c)2, which holds for
all jB in range II if k ≥ 37. We verified explicitly that Eq. (4.1) holds for all bosons
73
in range II if k < 37 (they appear in k = 13, 19, 31). This concludes our proof that
no condensation transition is possible in the SO(3)k TQFT for any odd k.
We note that this result can be readily extended to SU(2)k with k odd, since
SO(3)k is the projection of SU(2)k to anyons with integer j. One simply includes
the half-integer j anyons in the theory (none of which are bosons). The sets Ib as
defined above remain the same and so do all the quantum dimensions. Hence, we
also showed the noncondensability of SU(2)k, with k odd. This is consistent with
the ADE classification of SU(2)k [114]: There are no off-diagonal modular invariant
partition functions for odd k in SU(2)k [115]. Thus, the no-go theorem provides a
proof of this fact that is complementary to the ADE classification.
4.4 Example (iii): Multiple layers of SO(3)k
We can show that any number of layers of SO(3)k, with k odd, does not contain
condensable bosons. Fixing k, the proof proceeds again by induction. As induction
base, we proof that all multi-layer anyons with a nontrivial particle in only a single
layer (and the identity anyon in the other k− 1 layers) cannot condense nor split. To
show that, we can use the single-layer result from Example (ii). For the induction
step, we assume that for a fixed k0 < k all multi-layer anyons with nontrivial particles
in l layers, 1 ≤ l ≤ k0, cannot condense and do not split. We can then show that the
same holds for multilayer anyons with nontrivial particles in k0 +1 layers, completing
the induction. The details of this proof are given in appendix B.2.
4.5 Summary
We have presented a generally applicable no-go theorem against the condensation of a
topological boson and illustrated it with several examples. The proof of our theorem
uses mostly the fusion (as compared to the braiding) information of the TQFT.
74
We showed a connection between our results and the ADE classification of SU(2)k
theories, indicating that the no-go theorem might be useful for the classification of
modular invariant partition functions of conformal field theories more broadly. [113] It
would be interesting to study, whether other obstructions against boson condensation
exist or whether our no-go theorem actually constitutes a necessary condition. In all
examples we know, noncondensability is captured by the no-go theorem.
The no-go theorem can be used to study whether a TQFT is ZN graded under
layering. This provides a way to classify TQFTs depending on whether N is finite or
infinite. As a venue for future work, when restricting the condensations to those that
preserve certain symmetries of the anyon model, one could similarly classify symmetry
enriched topological phases, and with this also symmetry protected topological phases
without intrinsic topological order. The classification of the latter is often related to
the former upon gauging the protecting symmetry. [116, 117]
75
Chapter 5
Abelian Boson Condensation in
Field Theory
In this chapter, we reformulate the boson condensation in a field theory language
when the boson is Abelian. We propose that when an operator is condensed, the
partition function of the condensed phase will be invariant under arbitrarily insertion
of the condensed operators. The Lagrangian can be easily modified to have such an
invariance. The modification of the Lagrangian consists of introducing an integer
gauge field that couples with the condensates.
The organization of this chapter is as follows: In Sec. 5.1, we explain our intuition
and basic formalism of the boson condensation for any TQFT. In Sec. 5.2, we apply
the general formalism for K-matrix Chern-Simons theories. The results obtained
in our formalism are the same as those of the previous studies[60, 61, 11]: bosons
and only bosons can be condensed, and the deconfined operators are those who have
trivial braiding with the condensates.
76
5.1 Abelian Boson Condensation Formalism
In this section, we present the first principle for the boson condensation. For clarity,
we restrict our discussion to the condensation of loop operators. However, it can be
easily generalized to any operators. When an operator U is condensed, we expect
that the correlation functions in the condensed phase stay invariant under arbitrary
insertions of U(S1) for all loops S1 in (2+1)D spacetime:
〈U(S1) . . .〉c = 〈. . .〉c, ∀ S1 (5.1)
where “. . .” represents all other possible operators in TQFT; S1 is an arbitrary loop
in (2+1)D where the operator U(S1) lives; the subindex “c” denotes the expectation
value is taken in the condensed phase with a new yet underived Lagrangian. Our
purpose of the calculations in this section is to derive this Lagrangian. We emphasize
that Eq. (5.1) is required to be true for all possible closed loops.
The intuition for Eq. (5.1) comes from the physical expectation that the operator
U becomes a trivial operator (vacuum) in the condensed phase. Hence it has trivial
correlation functions with all other operators.
In order to realizing Eq. (5.1) at the Lagrangian level, we introduce a dynamical
1-form gauge field “c”, and later couple it with the condensed operator U . More
subtly, we require that gauge field c should be an integer field:
dc = 0 mod 2π,
⇔˛
S1
c = 0 mod 2π, ∀ closed paths S1(5.2)
Otherwise, the gauge field c will introduce more topological operators than we expect
for the condensation, and the central charge of the condensed theory will be generally
changed, both of which are not true in the formalism of the condensation phase
77
transitions. We defer to elaborate on this point when we discuss the deconfined
operators in Eq. (5.25). At the present stage, we take Eq. (5.2) as an assumption.
More explicitly, the condensate U is in the form of exp i(¸
f(a)), where a ab-
stractly represents all fields in TQFT and f is an arbitrary 1-form function of a. The
Lagrangian for the condensed phase is proposed:
Lc = L0 −1
2πf(a) ∧ dc, (5.3)
Before moving on, we point out that it might be confusing to observe that we can
use the gauge symmetry in Eq. (5.5) to gauge fix c = 0. Then the flat gauge field c
no longer appears in the Lagrangian Lc for the condensed phase. It seems that the
flat gauge field c does not play any role and does not change the original theory at
all? We need to explain and emphasize that when we use Eq. (5.5) to gauge away
the gauge field c and canonical quantize the theory in Eq. (5.3), we need to enforce
the corresponding Gauss Law in the Hilbert space:
df(a) = 0 mod 2π (5.4)
Therefore, we can still use the original Lagrangian L0 for the condensed phase, but
only need to enforce such a Gauss Law in the Hilbert space.
We prove the condensation lemma to show that our prescription of Eq. (5.3)
implies the intuition of Eq. (5.1):
Condensation lemma: Eq. (5.1) is true, if Lc has a gauge symmetry:
c 7→ c+ 2πλ, (5.5)
where the gauge parameter λ is an integer 1-form.
78
(a) (b)
Figure 5.1: An illustration of integer current and Hodge dual in 3 dimensional space.In Panel (a), jS1 is the integer which only has support on the loop S1. It is a 1-formrepresented by the red arrow. Its Hodge dual ?jS1 is represented by the red squarepenetrated by S1, which is a 2-form perpendicular to jS1 . In Panel (b), jS is theinteger 2-current which only has support on the sphere S. In 3 dimensions, ?jS isthen a 1-form going out S denoted by the red dot.
Proof for condensation lemma :
The derivations for Eq. (5.1) are as follows:
〈exp i
˛S1
f(a, . . .)
. . .〉c
=
ˆD[c]D[a] exp i
ˆ Lc +
˛
S1
f
. . .
=
ˆD[c]D[a] exp i
(ˆLc + f ∧ ?jS1
). . .
=
ˆD[c]D[a] exp i
(ˆL0 −
1
2πf ∧ d(c+ 2πλ)
). . .
=
ˆD[c]D[a] exp i
(ˆLc
). . .
=〈. . .〉c
(5.6)
The derivations need certain explanations:
79
The first equality to the second in Eq. (5.6): jS1 is the integer current which
only has support on the loop S1, in order to have:
˛
S1
f =
ˆfµj
µS1 =
ˆf ∧ ?jS1 (5.7)
? is 3-dimensional Hodge dual which maps the coordinates of forms to their orthogonal
counterparts. For example ?dt = dx ∧ dy. See Fig. 5.1 for illustrations of the Hodge
dual in 3 dimensions. An example of Eq. (5.7) is when S1 is a loop along t-direction
parametrized in terms of Cartesian coordinates:
S1 : (t, 0, 0) (5.8)
Then, we have:
jS1 = δD(x)δD(y)dt,
?jS1 = δD(x)δD(y)dx ∧ dy,(5.9)
where δD is the Dirac δ function. Hence, in this example, Eq. (5.7) is:
˛t
f =
ˆftδD(x)δD(y)d3r
=
ˆf ∧ ?jS1
(5.10)
The second equality to the third in Eq. (5.6): We use the 1-form gauge trans-
formation with a special gauge parameter satisfying
c 7→ c+ 2πλ, dλ = − ? jS1 (5.11)
where the gauge parameter λ is an integer 1-form. We point out that this equation
of λ and jS1 always has a solution, because it is an equation of the vector potential λ
80
which has a unit of flux in 3 dimensions. Consistently, the solution of λ is an integer
1-form because: ˛
S1
λ =
ˆ
D2
dλ =
ˆ
D2
− ? jS1 = −ˆ
D2∩S1
1 ∈ Z (5.12)
where we have used Stokes’ theorem, D2 is a disk whose boundary is S1, and D2∩S1
is the intersection of D2 and S1 which can only be an integer number of points.
The fourth equality to the fifth in Eq. (5.6): We use the assumption that
Lc respects the symmetry Eq. (5.11). Hence, from Eq. (5.6), (5.7) and (5.11), the
operator exp i(¸
S1 f(a))
satisfies Eq. (5.1) and is thus condensed in the condensed
phase described by Lc.
Therefore we have completed the proof for the condensation lemma. 2
A related observation was coined “generalized global symmetry” in Ref. [118,
119] when the authors discussed higher form global symmetry for a TQFT. As a
result, the gauge fields in TQFT are shifted by constant 1-forms without changing
the actions. We emphasize that in our situation, it is a gauge symmetry instead of
a global symmetry, and the gauge fields in the condensation formalism are shifted
by local 1-forms (not constant 1-forms), in order to implement Eq. (5.1). Hence, our
starting point Eq. (5.1) implies a 1-form gauge symmetry.
5.2 Condensations in K-Matrix Chern-Simons
Theories
In this section, we apply our formalism (the condensation lemma) to the K-matrix
Chern-Simons theories. This section is divided into two parts. (1) We first derive the
condition under which the operators can be condensed. (2) We find the conditions of
whether operators are confined/deconfined after condensation.
81
5.2.1 Condensable Condition
We begin with a general bosonic K-matrix theory and condense the operator U{l}.
We know from the past studies[60, 61, 11] that only bosons can be condensed. This
is the only requirement for the condensed particle in Abelian topological theories. In
other words, the condensable U{l} requires that l ·K−1 · l ∈ 2Z. We derive this boson
condensation condition from our condensation lemma developed in Sec. 5.1.
The Lagrangian for the condensed phase, according to Eq. (5.3), is:
Lc =KIJ
4πaI ∧ daJ −
1
2πlIaI ∧ dc (5.13)
where K is a symmetric matrix. The repeated indices imply summation. We need to
verify that Lc is invariant under gauge transformation Eq. (5.5). We can first slightly
change Lc using integration by part and the fact that K and K−1 are symmetric
matrices:
Lc = − lMK−1MN lN
4πc ∧ dc+
KIJ
4π
(aI − lM(K−1)MIc
)∧ d(aJ − lN(K−1)NJc
)(5.14)
Therefore, in order for Lc to have the higher form gauge symmetry Eq. (5.5), we
need to introduce the transformations for aI fields, when the gauge field c takes the
transformation Eq. (5.5):
c 7→ c+ 2πλ, aI 7→ aI + 2πlMK−1MIλ, ∀ I. (5.15)
The extra terms in Lc after the gauge transformation in Eq. (5.15) only come from
the first term in Eq. (5.14):
δLc = − lMK−1MN lN
4π
(4π2λ ∧ dλ+ 4πλ ∧ dc
)(5.16)
82
where λ∧ dλ is an integer 3-form, and λ∧ dc is a 2π integer 3-form due to Eq. (5.2).
In order to make the variation of action´δLc to be 0 modulo 2π, we conclude that
the coefficient of λ ∧ dλ has to be an integer multiple of 2π. Hence,
l ·K−1 · l ∈ 2Z, (5.17)
The condition for the higher form gauge invariance in Eq. (5.17) is equivalent to the
statement that U{l} is a boson. As we discussed in Eq. (5.6), the higher form gauge
invariance is equivalent to condensation. Hence Eq. (5.17) is indeed the condition for
condensable operator U{l}. To conclude, when Eq. (5.17) holds, U{l} is condensable.
We can also manifest the higher form gauge invariance condition of Eq. (5.17)
without resorting to a particular gauge transformation (Eq. (5.15)). The basic idea is
that it will be easier to find the higher form gauge invariance, if we have an effective
expression for the partition function Zc in terms of only the gauge field c. We integrate
over all aI fields without any source terms in the path integral, and thus obtain an
effective expression for Zc with an effective Lagrangian Leffc (c).
Leffc (c) = − l ·K
−1 · l4π
c ∧ dc (5.18)
Then we only need to examine whether Leffc (c) has the higher form symmetry Eq. (5.5).
Leffc (c) respects it modulo 2π only when Eq. (5.17) holds, which derivations are the
same as in Eq. (5.16). For simplicity, Leffc (c) will be the notation of the effective
Lagrangian in the condensed phase, which implies the calculations of integrating out
all gauge fields except c.
Therefore, we have proved that the condensed theory Lc has the higher form gauge
symmetry is equivalent to that the condensed operator is a boson (i.e., Eq. (5.17)).
We can directly generalize this statement from one condensed operator to multiple
condensed operators. As a result, the condensation conditions require that each con-
83
densate is a boson and they are mutual bosons. One way to prove such condensation
conditions is to generalize Eq. (5.18):
Lc =KIJ
4πaI ∧ daJ −
liI2πaI ∧ dci ⇒ Leff
c (c) = − li ·K−1 · lj
4πci ∧ dcj (5.19)
where i, j indices label different condensates, U{li}, and Leffc (c) is obtained from Lc by
formally integrating out aI fields. In order to make Leffc (c) in Eq. (5.19) respect the
higher form gauge symmetry:
ci 7→ ci + 2πλi, ∀ i (5.20)
we need the following conditions:
li ·K−1 · lj ∈ Z, ∀ i 6= j; li ·K−1 · li ∈ 2Z, ∀ i. (5.21)
The first condition is the same as the statement that different condensates are mutual
bosons, and the second one states that each condensate U{li} is a boson.
5.2.2 Confinement/Deconfinement
In this part, we examine the confinement/deconfinement after condensing U{l}, using
the higher form gauge symmetry Eq. (5.15). The deconfined operators are invariant
under the higher form gauge symmetry in Eq. (5.15), while the confined ones are not
invariant under the higher form gauge symmetry.
Under the higher form gauge transformation Eq. (5.15), a general loop operator
U{m}(S1) transforms as:
U{m}(S1) 7→U{m}(S1) exp i
(2πm ·K−1 · l
˛S1
λ
)(5.22)
84
We expect that the deconfined operators are invariant under the higher form gauge
symmetry. Because¸S1 λ is generally quantized to an integer, the deconfined operator
U{m}(S1) needs to satisfy:
m ·K−1 · l ∈ Z, (5.23)
in order to stay invariant under the higher form gauge symmetry. This means that
the deconfined U{m} braids trivially with the condensate U{l}.
The generalization of deconfinement to the situation of multiple condensed opera-
tors U{li} is straightforward where the index i denotes different condensed operators.
The deconfined operators should braid trivially with each condensed operator:
m ·K−1 · li ∈ Z, ∀ i . (5.24)
As we promised in Eq. (5.2), we explain that the introduced gauge field “c” must
be a flat gauge field satisfying Eq. (5.2). The reason is that we need to make sure that
attaching a Wilson loop of gauge field c to any deconfined loop operators U{m}(S1)
will not change the expectation value, i.e.:
exp i
˛S1
mIaI + c
= exp i
˛S1
mIaI
, ∀ S1 ⇒ exp i
˛S1
c
= 1, ∀ S1.
(5.25)
This means the introduced gauge field c has to be integer as in Eq. (5.2). On the other
hand, if the gauge field c is a U(1) gauge field without the integer condition Eq. (5.2),
it will also change the chirality when condensation. In this scenario, the condensed
Lagrangian Eq. (5.13) can be written in terms of the K-matrix Chern-Simons theories:
Kc =
K −(l1, l2, . . .)T
−(l1, l2, . . .) 0
(5.26)
85
The basis of this Kc matrix is (a1, a2, . . . , c). The central charge of Kc is generally
different from that ofK. However, according to previous studies based on the modular
tensor categories[92, 120], the boson condensation will not change central charge
modulo 24. This contradiction shows that, Lc, if we treat gauge field c as a U(1)
gauge field without the flatness condition, does not describe the boson condensation.
Therefore, we have justified that the Lagrangian Eq. (5.13) can describe condensed
phase when c is a flat gauge field satisfying Eq. (5.2).
86
Chapter 6
Fracton Models, Tensor Network
States and Their Entanglement
Entropies
TNS have been heavily used in condensed matter physics in the past decade, especially
in the study of 1D and 2D topological phases[121]. Amongst many examples,
1. Numerical simulations of the 1D Haldane chain led to the discovery of symmetry
protected topological phases (SPT)[122].
2. Fractional quantum Hall states can be exactly written as MPS[123, 124, 125,
126, 127, 128, 129, 130, 131, 132, 133, 134] 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[135, 136, 137, 138, 139,
140, 141].
In higher spatial dimensions than 2D, more exotic gapped states of matter exist,
beyond the paradigm of topological phases.[19, 121]. Recently, 3D so-called fracton
87
models[142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157,
158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168] represented by Haah code[142]
and X-cube model have been proposed, attracting the attention of both quantum
information[169] and condensed matter community[170, 171, 172, 173, 174]. They can
be realized by stabilizer code Hamiltonians, 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)
that:
1. Fracton models are gapped, since they can be realized by commuting Hamilto-
nian terms.
2. The ground state degeneracy on the torus changes as the system size changes.
Hence, fracton models seem not to have thermodynamic limits.
3. The low energy excitations can have fractal shapes, other than only points and
loops available in conventional topological phases.
(a)
(b)
Figure 6.1: Examples of TNS lattice wave functions in 1D and 2D. Each node is atensor 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 arethe 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.
88
4. 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).
In this chapter, we obtain a TNS representation for some of the ground states of
three stabilizer codes in 3D: the 3D toric code model[19], the X-cube model and the
Haah code. The two latter ones belong to the catalog of fracton models, while the first
one belongs to the conventional topological phases. For instance, the ground state
degeneracies on the torus of the X-cube model and the Haah code do not converge to a
single number, as the system size increases. In contrast, the ground state degeneracy
(GSD) on the torus for 3D toric code model is 8 for all system sizes. Ref. [175]
treated the X-cube and Haah code models using the idea of lattice gauge theory. The
gauge symmetry is generally generated by part of the commuting Hamiltonian terms;
the rest of the Hamiltonian terms are interpreted as enforcing flat flux conditions.
More explicitly, the authors treated the terms only made of Pauli Z operators as the
gauge symmetry generators, and the terms only made of Pauli X operators as the
flux operators. The gauge symmetries in the X-cube and the Haah code models are
not the conventional Z2 gauge symmetry such as that in the 3D toric code model,
since the gauge symmetry generators, the Pauli Z terms of the X-cube and the Haah
code models, are different from those in the 3D toric code model. Refs. [176, 177]
derived the X-cube model from “isotropically” layered 2D toric code models and
condensations. The caveat is that this condensation is weaker than the conventional
boson condensation in modular tensor category or field theory[21, 62, 92, 120]. The
authors condense “composite flux loop” of coupled layers of 2D toric code model.
The “composite flux loop” refers to a composite of four flux excitations near a bond
of the lattice. See Ref. [177] for explicit explanations.
Using the TNS representations of some of the ground states, we obtain the en-
tanglement entropy upper bounds for all three models. We then derive the reduced
89
density matrix cuts for which the TNS represents the singular value decomposition
(SVD) of the state. For these types of cuts, the entanglement entropy of the three
stabilizer codes can be computed exactly. We find that for the fracton models, the
entanglement entropy has linear corrections to the area law, corresponding to an
exponential degeneracy in the TNS transfer matrix.
The transfer matrices of TNS of 2D toric code[19], whose eigenvalues and eigen-
states dominate the correlation functions, have been studied in Ref. [178, 179]. The
flat entanglement spectra[27] of the 2D toric code were studied in Refs. [180, 181].
Our TNS construction, when restricted to 2D toric code model, gives the exact results
of transfer matrices and entanglement spectra. See Chapter 2 for explicit calculations
and explanations. Beyond the 2D toric code, Refs. [182, 158] prove that the reduced
density matrix of any stabilizer code is a projector. Hence, the corresponding entan-
glement spectrum is flat, a property that we will rederive from our TNS.
We will not discuss cocycle twisted topological phases, including Dijkgraaf-Witten
theories[183, 184, 185] or (generalized) Walker-Wang models[186, 187, 188, 189, 190],
even though they can still be realized by commuting Hamiltonians on lattice[184, 183,
186, 82]. However, the presence of nontrivial cocycles will make the TNS construction
very different, based on the experiences in 2D TNS. Our construction will not work
for these twisted models. For instance, in 2D, the virtual index dimension using
our construction is the same as the physical index dimension. However, when we
consider cocycle twisted topological phases, the “minimal” virtual bond dimension is
generally larger than the physical index dimension[191, 38, 192, 193]. More explicitly,
the minimal virtual bond dimension for 2D toric code model is 2, while the minimal
virtual bond dimension for 2D double semion model (twisted toric code) is 4[191,
38, 192, 193]. The 2D cocycle twisted TNS has been systematically explored in the
literature for bosonic[191] and fermionic[194, 195] systems respectively.
90
The organization of this chapter is as follows: In Sec. 6.1, we set the notations
and provide an overview and the general idea of the TNS construction In Sec. 6.2,
we present the calculation of the entanglement properties using the developed TNS
construction In Sec. 6.3, we present the TNS construction for the toric code model
in 3D. The entanglement entropy is calculated from the obtained TNS. The transfer
matrix is constructed afterwards and is proven to be a projector of rank 2. In Sec. 6.4,
we present the TNS construction for X-cube model. The same calculations for the en-
tanglement entropy and the transfer matrix are presented. They are quickly shown to
be very different from the toric code model. Indeed, the entanglement entropies have
linear corrections to the area law, and the transfer matrix is exponentially degenerate.
In Sec. 6.5, we present the TNS construction for Haah code. The entanglement en-
tropies are calculated for several types of cuts. In Sec. 6.6, we summarize the chapter
and discuss future directions.
6.1 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[19], the X-cube model[175] and
the Haah code[142]. The TNSs 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. For pedagogical purposes, we discuss the 2D toric code in
Chapter 2.
6.1.1 Notations
We first fix some of the notations used in the chapter, to which we will refer throughout
the manuscript:
91
1. The Pauli matrices X and Z are defined as:
X =
0 1
1 0
, Z =
1 0
0 −1
. (6.1)
2. 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
the Haah code have four virtual indices and one physical index. They are
depicted in Eq. (6.5), (6.116) and (6.117).
3. We introduce the T tensor, which denotes the local tensor for each model.
It only has virtual indices and thus no physical indices. The specific tensor
elements are determined by the Hamiltonian terms.
4. 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.
5. Graphically, the physical indices are denoted by arrows, while the virtual indices
are not associated with any arrows. See Fig. 6.1.
6. 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
92
U-1U
A1 A2
A1 A2
(a)
(b)
U-1UA1 A2
A1 A2
(c)
(d)
Figure 6.2: An illustration of the TNS gauge in MPS. (a) A part of an MPS. A1
and A2 are two local tensors contracted together. (b) We insert the identity operatorI = UU−1 at the virtual level - it acts on the virtual bonds. The tensor contractionof 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 andA2 is the same as the tensor contraction of A1 and A2. The TNS does not change aswell. Similar TNS gauges also appear in other TNS such as PEPS.
evaluating the TNS 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 ( ).
7. 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.
8. The 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 local
tensors respectively. The local tensors then change but the wave function stays
invariant. We refer to this gauge degree of freedom as the TNS gauge. The
TNS gauge exists in MPS, PEPS etc. See Fig. 6.2 for an illustration. In our
calculations, we only fix the tensor elements up to the TNS gauge.
93
Tg
T
g
(a) (b)
xzy
xzy
Figure 6.3: (a) A plane of TNS on a cubic lattice. (b) TNS on a cube. The lineswith arrows are the physical indices. The connected lines are the contracted virtualindices, while the open lines are not contracted. On each vertex, there lives a Ttensor, and on each bond, we have a projector g tensor.
6.1.2 Stabilizer Code and TNS Construction
We now summarize the general idea of constructing TNSs for stabilizer codes. In
Chapter 2, we provide the construction of the TNS for the 2D toric code model on a
square lattice. In the following, we assume that the physical spins are defined on the
bonds of the cubic lattice (such as the 3D toric code and the X-cube models). The
cases where the physical spins are defined on vertices can be analyzed similarly. The
generic philosophy of any stabilizer code model is captured by the following exactly
solvable Hamiltonian:
H = −∑v
Av −∑p
Bp (6.2)
where the Hamiltonian is the sum of the Av terms which are products of only Pauli Z
operators, and the Bp terms which are products of only Pauli X operators. v and p
denotes the positions of the Av and Bp operators on the lattice. In the 3D toric code,
v is the vertex of the cubic lattice while p is the plaquette. In the X-cube model, v
is the vertex while p is the cube. In the Haah code, both v and p are cubes. See
Sec. 6.3.1, 6.4.1 and 6.5.1 for the definitions of Hamiltonians of these three models.
94
All these local operators commute with each other:
[Av, Av′ ] = 0, ∀ v, v′
[Bp, Bp′ ] = 0, ∀ p, p′
[Av, Bp] = 0, ∀ v, p.
(6.3)
The Hamiltonian eigenstates are the eigenstates of these local terms individually. In
particular, any ground state |GS〉 should satisfy:
Av|GS〉 = |GS〉, ∀ v
Bp|GS〉 = |GS〉, ∀ p(6.4)
for all positions labeled by v and p. In this chapter, we only consider Hamiltonians
being of a sum of local terms that are either a product of Pauli Z operators, or a
product of Pauli X operators. Thus, we do not include the case of mixed products of
Pauli Z and X operators.
The ground states for the stabilizer codes with Hamiltonian as in Eq. (6.2) 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[192, 193]. In the following, we provide
one possible general construction for such TNSs. 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
(6.5)
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 Pauli Z eigenstates of |↑〉, |↓〉 respectively where Z|↑〉 = |↑〉, and Z|↓〉 = −|↓〉.
The projector g tensor maps the physical spin into the virtual spins exactly. As a
95
result, the virtual index has a bond dimension 2. 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.
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. 6.3 (a) and (b). We define the TNS as:
|TNS〉 =∑{s}
CR3
(gs1gs2gs3 . . . TTT . . .) |{s}〉 (6.6)
where CR3denotes the contraction over all virtual indices on R3 as illustrated in
Fig. 6.3 (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(6.7)
for all positions labeled by v and p.
The actions of Av and Bp operators on the TNS can be transferred to the virtual
indices, using the definition of the g tensor. Since the virtual indices of projector g
tensors are contracted with the virtual indices of T tensors, 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. (6.7), and |TNS〉 belongs to the ground state manifold. For the three models
96
analyzed in this chapter, 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. (6.7) restricts the local T tensor to be:
Txxy...
6= 0 if the indices xxy . . . satisfy
some constraints
= 0 otherwise
. (6.8)
Applying the second equation of Eq. (6.7) will further restrict the local T tensor to
be:
Txxy... =
1 if the indices xxy . . . satisfy
some constraints
0 otherwise
. (6.9)
For simplicity, we calculate the entanglement entropies of the wave function on R3
with respect to some specific entanglement cuts, and compute the ground state de-
generacy (GSD) of the 3D toric code and X-cube model on T 3.
We emphasize that in this chapter, we are only concerned with the bulk wave
functions and their entanglement entropies. In principle, the TNS of Eq. (6.6) 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. (Notice 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.
97
6.1.3 TNS Norm
Evaluating the norm of the TNS given by Eq. (6.6) (or any scalar product between
two TNS) is straightforward. Indeed the g tensors are projectors, and hence greatly
simplify the expression of the tensor network norm when we contract over the physical
indices.
Given the wave function of Eq. (6.6), we can compute its norm as follows:
〈TNS|TNS〉 =
∑{s}
CR3
(gs1gs2gs3 . . . TTT . . .) 〈{s}|
?∑{s}
CR3
(gs1gs2gs3 . . . TTT . . .) |{s}〉
=∑{s}
CR3
(gs1?gs2?gs3? . . . T ?T ?T ? . . .) CR3
(gs1gs2gs3 . . . TTT . . .)
=∑{s}
CR3
(gs1gs2gs3 . . . T ?T ?T ? . . .) CR3
(gs1gs2gs3 . . . TTT . . .) ,
(6.10)
where ? is the complex conjugation, and we have used the fact that the g tensors
are real for our models. Now we specify a contraction order in Eq. (6.10): we first
contract over the physical indices and then we contract over the virtual indices. If the
physical indices of two projector g tensors are contracted over, the four virtual indices
will be enforced to be the same following the definition of the projector g tensor:
∑s
gsijgsmn =
i j
m n
=
1 i = j = m = n
0 otherwise
.
(6.11)
Thus when computing wave function overlap 〈TNS|TNS〉, the virtual indices in the
bra layer and the ket layer at the same place are enforced to be same. As a result,
98
we have:
〈TNS|TNS〉 = CR3
(. . .TTT . . .) , (6.12)
where CR3stands for the contraction of a network of tensors T over the virtual indices
on R3. In a slight abuse of notation, CR3in Eq. (6.12) stands for the contraction
taken over the virtual indices of both the bra and the ket layer, while the contraction
in Eq. (6.6) is taken over the virtual indices in only the ket layer. The double tensor
T is defined as
Txxy...,x′x′y′... = T ∗xxy...Tx′x′y′...δxx′δxx′δyy′ . . .
= |Txxy...|2δxx′δxx′δyy′ . . . ,(6.13)
for all the elements of T and T. The indices are not summed over in the above
equation. The indices xxy . . . come from the bra layer while the indices x′x′y′ . . .
come from the ket layer. In a 2D square lattice, a T tensor usually has 4 virtual
indices x, x, y, y, while in a 3D cubic lattice, a T tensor usually has 6 virtual indices
x, x, y, y, z, z. If the elements of the T tensor are only either 0 or 1, we get,
Txxy...,x′x′y′... = |Txxy...|2δxx′δxx′δyy′ . . .
= Txxy...δxx′δxx′δyy′ . . . .
(6.14)
Then,
〈TNS|TNS〉 =CR3
(. . .TTT . . .)
=CR3
(. . . TTT . . .) .
(6.15)
This result will be frequently used in the following discussions, especially when we
compute wave function overlaps or transfer matrices. Eqs. (6.12) and (6.15) hold true
on other manifolds as well, such as T 3 and T 2 ×R.
99
Figure 6.4: Transfer matrix (red dashed square) of a 1D MPS. The connected linesare the contracted virtual indices. The connected arrow lines are the contractedphysical indices. The MPS norm (or any other quantities) can be built using thetransfer matrix. Higher dimensional transfer matrices are similarly defined for TNSon a cylinder or a torus, by contracting in all directions except one. This leads to a1D MPS with a bond dimension exponentially larger than the TNS one.
6.1.4 Transfer Matrix
The transfer matrix method is ubiquitous when using MPS (see Fig. 6.4 for an il-
lustration of the transfer matrix). It can be generalized to TNS on a 2D cylinder
by contracting tensors along the periodic direction of the cylinder. This implies that
the bond dimension of the transfer matrix is exponentially large with respect to the
cylinder perimeter. In 3D, the TNS norm on T 3 of size Lx × Ly × Lz can be written
as an MPS using transfer matrices TMxy in each xy-plane:
〈TNS|TNS〉 = Tr (TMxy,z=1TMxy,z=2 . . .)
= Tr(
(TMxy,z=1)Lz) (6.16)
where we have assumed that all transfer matrices in each plane are the same:
TMxy,z=1 = TMxy,z=2 = . . . (6.17)
Eq. (6.16) is an alternative way of writing the wave function norm and specifies a
contraction order of the tensors in Eq. (6.12): we first contract the virtual indices
along xy-plane which defines the transfer matrix TMxy, and then contract the vir-
tual indices in z-direction which leads to the multiplication and the trace of transfer
100
matrices. The transfer matrix TMxy is defined as:
TMxy =
∑{s}
CT 2xy (gs1?gs2?gs3? . . . T ?T ?T ? . . .) 〈{s}|
∑{s}
CT 2xy (gs1gs2gs3 . . . TTT . . .) |{s}〉
,
(6.18)
where the TNS contraction is performed along the xy-plane with periodic boundary
conditions, i.e., the 2D torus T 2xy. Denoting Txy as the TNS in its ket layer, it can
depicted as:T
g
x (pbc)z
y (pbc)
(6.19)
TMxy is the overlap of the bra and ket layer of this TNS over the plane with periodic
boundary conditions.
Furthermore, applying Eq. (6.11) to Eq. (6.18), the virtual indices in the bra layer
and the ket layer are identified after the physical indices are contracted in Eq. (6.18).
Hence, we have:
TMxy = CT 2xy (. . .TTT . . .) , (6.20)
where the tensors T, defined in Eq. (6.13), are in the xy-plane with periodic boundary
conditions. The indices in the z-direction are open. By Eq. (6.14) - which is true
when the elements of the T tensor are either 0 or 1 - the transfer matrices is further
simplified to:
TMxy = CT 2xy (. . . TTT . . .) . (6.21)
101
Graphically:
TMxy =
T
x (pbc)z
y (pbc)
T T T T T T
T T T T T T T
T T T T T T T
T T T T T T T
. (6.22)
Suppose the virtual index is of dimension D. Then in Eq. (6.18), the transfer
matrix is of dimension D2LxLy ×D2LxLy . However, in Eq. (6.20), the transfer matrix
reduces to dimension DLxLy × DLxLy , since the indices in the bra layer and the ket
layer are identified due to the contraction over the physical indices of projector g
tensors.
6.2 Entanglement properties of the stabilizer code
TNS
The specific structure of the TNS discussed in the previous section allows us to
derive its entanglement properties. In this section, we show that for a large class
of entanglement cuts, the TNS is already in Schmidt form, i.e. is exactly a singular
value decomposition (SVD). We also summarize the main results for the entanglement
entropies and the transfer matrices that we have obtained for the three stabilizer
codes.
6.2.1 TNS as an exact SVD
We propose a general sufficient condition that the TNS is an SVD with respect to
particular entanglement cuts. Let us denote the TNS with open virtual indices {t}
102
as:
|{t}〉 =∑{s}
CM (TTT . . . gs1gs2gs3 . . .) |{s}〉, (6.23)
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 ofM. In Eq. (6.23), the T tensors and g tensors are the tensors insideM
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}
, (6.24)
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
|{t}〉, the TNS can be written as:
|TNS〉 =∑{t}
|{t}〉A ⊗ |{t}〉A (6.25)
with respect to a region A and its complement A. |{t}〉A is the TNS in region A with
open indices {t}, while |{t}〉A is the TNS 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.
103
We now propose a simple sufficient (but not generally necessary) condition to
determine when Eq. (6.25) is an exact SVD for the TNS constructed in this chapter.
We first have to make an assumption, satisfied by all our TNSs:
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.
This assumption can be easily verified when the local T tensors are obtained for
the three models studied in this chapter, such as the 3D toric code model in Eq. (6.43).
We are now ready to express our SVD condition:
SVD condition: If there are no two open virtual indices in {t} (see Eq. (6.24))
of the region A that connect to the same T tensor in the region A, 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,
then the non-vanishing states |{t}〉A form an orthogonal basis. Therefore, Eq. (6.25)
is an exact SVD.
Proof :
We first prove the statement for the region A. Suppose that |{t}〉A and |{t′}〉A
are two non-vanishing TNSs 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 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
104
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
(6.26)
By using Eq. (6.11), 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
(6.27)
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. Notice that in the ket layer, the
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 and
Eq. (6.11), all the indices except m0 of the T tensor in the ket layer are equal to their
105
respective analogues in the bra layer:
mi = m′i, i = 1, 2, . . . (6.28)
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. (6.28) implies that:
m0 = m′0 (6.29)
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. (6.25)
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, the X-cube 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 and |{t}〉A are not only orthogonal, but also normalized. In particular
for the 3D toric code model and the X-cube model, we can just select the region A to
be a cube which satisfies the SVD condition directly. See respectively Sec. 6.3.4 and
6.4.4 for detailed discussion of these two models and the SVD condition. However,
the Haah code is different: a cubic region A does not fulfill the SVD condition, and
106
in Sec. 6.5.3 we generalize the SVD condition to the Generalized SVD Condition and
apply Bc operators to make the TNS an SVD.
6.2.2 Summary of the results
We now summarize the major results derived in this chapter 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 chapter.
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 chapter, 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 the X-
cube model and Haah code, the correction includes a negative term linear with
the system size, presented in Sec. 6.4.4 and 6.5.4.
107
3. The transfer matrices in Eq. (6.18) of the 3D toric code model and the X-cube
model are shown to be a projector whose eigenvalues are either 0 or 1. For the
3D toric code, the transfer matrix in the xy-plane is a projector of rank 2. For
the X-cube model, the transfer matrix is a projector of rank 2Lx+Ly−1 where Lx
and Ly are the lattice sizes in x- and y- directions respectively.
4. We prove that the TNS ground states obtained on the torus using our construc-
tion are the +1 eigenstates of loop X operators. Hence, our TNS construction
does not include all ground states on the torus. The degeneracy of the corre-
sponding transfer matrix is smaller than the GSD on the torus. We can obtain
all the ground states with loop/surface Z operators on the TNS, which gener-
ate all the wave functions on the torus. We call the TNS with Z operators,
“twisted TNS”. Correspondingly, we also obtain more transfer matrices in the
xy-plane built from the twisted TNS, and these transfer matrices are all the
same projectors. The same TNS phenomenon in the 2D toric code model has
been studied in Ref. [178].
5. In our calculations, both the transfer matrix eigenvalue degeneracies, and the
corrections to the area law of entanglement entropies are rooted in the Con-
catenation lemma. Hence, we believe that the two contributions are related.
Specifically, suppose we consider our TNS on a 3D cylinder T 2xy ×Rz, and the
entanglement cut splits the system in two halves z > 0 and z < 0. Then,
for the toric code model, the transfer matrix TMxy has the degeneracy 2, and
the entanglement entropy correction to the area law is − log(2). For the X-
cube model, the transfer matrix TMxy has the degeneracy 2Lx+Ly−1, and the
entanglement entropy correction to the area law is −(Lx + Ly − 1) log(2) (See
Eq. (6.101)). Moreover, the GSD on T 3 is generally larger than the transfer
matrix degeneracy. Therefore, given these calculations, we conjecture that the
108
negative linear correction to the area law is a signature of the extensive ground
state degeneracy.
6.3 3D Toric Code
In this section, we construct the TNS for the 3D toric code model and then calculate
the entanglement entropy and GSD, both deriving from the Concatenation lemma.
The results are the immediate generalizations of those in the 2D toric code model. We
find a topological entanglement entropy in accordance to that obtained by Ref. [190]
using a field theoretic approach.
This section is organized as follows: In Sec. 6.3.1, we briefly review the toric code
model in a cubic lattice. In Sec. 6.3.2, we construct the TNS for the toric code model.
In Sec. 6.3.3, we prove a Concatenation lemma for toric code TNS, which is useful
in the following calculations. In Sec. 6.3.4, we calculate the entanglement entropies
on R3. In Sec. 6.3.5, we construct the transfer matrix and prove it is a projector of
rank 2. In Sec. 6.3.6, we show how to construct 8 ground states on torus by twisting
the TNS.
6.3.1 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. (6.30)
109
Z ZZ
Z
x
x
xx
x
z
y
Z
Z
(a) (b)
Figure 6.5: 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 Av is defined around a vertex v, and Bp is defined on a plaquette p:
Av =∏i∈v
Zi, Bp =∏i∈p
Xi, (6.31)
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. 6.5. 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,
(6.32)
and their eigenvalues are ±1:
A2v = 1, B2
p = 1. (6.33)
110
The ground states |GS〉 should satisfy:
Av|GS〉 = |GS〉, ∀ v
Bp|GS〉 = |GS〉. ∀ p(6.34)
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
1 + Av2|0x〉, (6.35)
where |0x〉 is the tensor product of all X = 1 eigenstates defined on each link.
6.3.2 TNS for 3D Toric Code
We first introduce a projector g tensor Eq. (6.5) on each bond of the lattice. Both
the virtual indices and the physical indices take two values, 0 and 1. The projector g
tensor satisfies:
Z = =
x x=x
. (6.36)
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.
(6.37)
111
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 according to Eq. (6.5). We can
use these conditions to transfer the action of the physical operators to the virtual
operators. Now we introduce an additional T tensor on each vertex of the cubic
lattice, which has six virtual indices. Graphically, we represent such a T tensor as:
Txy
z
z
y
x . (6.38)
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 the ground states in
Eq. (6.34). The actions of Av and Bp operators on the 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=
. (6.39)
where we have used Eq. (6.36) to transfer the physical operators to the virtual ones.
We require a strong version of the solution to 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 which guarantees that the tensors form the
112
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
(6.40)
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,
(6.41)
where x, x, y, y, z, z are the six indices of T in the three directions respectively. We
emphasize for clarity that x is not −x; these are notations for different indices. The
second set of conditions in Eq. (6.40) further enforces that an even number of index
flipping of the virtual indices of a tensor does not change the value of the tensor
113
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)
= . . .
. (6.42)
Hence, the nonzero elements of the T tensor are all equal. Up to an overall normal-
ization, 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.
(6.43)
The TNS is then Eq. (6.6) with the local T being Eq. (6.43). 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. (6.40) have been intro-
duced by several other names in tensor network literature: Z2-injectivity[30], MPO-
injectivity[196], Z2 gauge symmetry[38] etc. The previous studies were in 2D, and
our condition is the 3D generalization. Notice that the first equation in Eq. (6.40)
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. (6.40)
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-
logical phases to trivial phases. See Refs. [38, 197, 198, 199, 200] for explanations and
examples in the case of 2D TNS.
114
T
T
x
z
y
Figure 6.6: Contraction of two local T tensors in the z-direction. We emphasize thatthere is no projector g tensor in this figure.
6.3.3 Concatenation Lemma
In this section, we consider the contraction of a network of local T tensors with
open virtual indices. One example of such a contraction is the tensor network norm
Eq. (6.15) or the transfer matrix Eq. (6.21). Since the elements of a local T tensor
are 0 for the odd sector and 1 for the even sector (see Eq. (6.43)), we will show that,
in general, a network of contracted T tensors obeys a similar rule: some elements
are zeros while the others are nonzero and identical. A Concatenation lemma is
proposed to derive the rule for the contraction of several tensors in general and will
be frequently used in the following discussions. For example, we will use this lemma
to show in Sec. 6.3.5 that the transfer matrix TMxy for the 3D toric code model is a
projector of rank 2.
Concatenation Lemma: For a network of contracted T tensors Eq. (6.43)
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. (6.43) and induction.
The proof is in App. C.1. We explain this lemma by a simple example. Suppose we
115
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 . (6.44)
Graphically, the tensor T is represented by Fig. 6.6. 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.
(6.45)
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.
(6.46)
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
(6.47)
where {t} denotes all the indices of the tensor T. We emphasize that the nonzero
constant does not depend on {t} .
116
6.3.4 Entanglement
We now show that Eq. (6.6) is exactly an SVD for the wave function with respect to
the entanglement cut illustrated in Fig. 6.7. For simplicity, suppose that the TNS is
defined on infinite R3. As we have emphasized at the end of Sec. 6.1.2, we do not
specify the boundary conditions of the 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 in Sec. 6.3.3. 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. (6.6) by separating the tensor contractions to a spatial region A
and its complement region A. Region A contains the g tensors near the entanglement
cut as illustrated in Fig. 6.7:
|TNS〉R3 =∑{t}
|{t}〉A ⊗ |{t}〉A (6.48)
where
|{t}〉A =∑{s}∈A
∑{i}∈A
CA(gs1t1i1gs2t2i2
. . . gs3i3i4gs4i5i6Ti7...Ti8... . . .)|{s}〉. (6.49)
117
Indices denoted by s are the physical indices; indices denoted by t are the open virtual
indices straddling 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. 6.7; 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.
Thereby, |{t}〉 is the TNS 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. 6.7. For instance, when the region A is a cube,
we can graphically denote the basis |{t}〉 as Eq. (6.24), 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}〉. (6.50)
Since the TNSs for region A and A share the same boundary virtual indices {t}, then
in Eq. (6.48) the two basis for region A and A have the same label {t}. For the TNS
of Eq. (6.6), the boundary virtual indices {t} of the regions A and A are contracted
over, and thus in Eq. (6.48) {t} are summed over.
Tg
A Acut
Figure 6.7: The splitting of tensors near the entanglement cut.
118
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. (6.48) 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). (6.51)
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
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}〉
.
(6.52)
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. Following the
same derivations in Sec. 6.1.3, we have:
A〈{t}|{t}〉A = CA (. . . TTT . . .) (6.53)
with the open virtual indices {t}. The contraction CA is over the T tensors in the
region A. Applying the Concatenation lemma in Sec. 6.3.3, A〈{t}|{t}〉A is zero if
119
the open indices {t} are summed to be 1 mod 2:
∑i
ti = 1 mod 2 ⇒ A〈{t}|{t}〉A = 0. (6.54)
Moreover,
A〈{t}|{t}〉A = Const, when∑i
ti = 0 mod 2. (6.55)
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. (6.48) is indeed an SVD, and the singular values are all identical. Hence,
for a entanglement cut, we only need to count the number of singular vectors in
Eq. (6.48). For a connected entanglement surface with N open virtual indices, the
number of singular vectors in Eq. (6.48) 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). (6.56)
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). (6.57)
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
120
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) (6.58)
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 straddling
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,
because they have to obey the Concatenation lemma in Sec. 6.3.3. 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). (6.59)
The topological entanglement entropy Stopo[T d−1] is independent of the dimensional-
ity, and it obeys the conjecture presented in Ref. [190]:
exp(−dStopo[T d−1]) = GSD[T d] (6.60)
where GSD[T d] = 2d.
6.3.5 Transfer Matrix as a Projector
The z-direction transfer matrix TMxy in 3D is defined as a tensor network overlap in
the xy-plane, with periodic boundary conditions. The indices in the z direction are
open and not contracted over (see Eq. (6.18) to Eq. (6.22)). In this section, we will
121
show that TMxy for the 3D toric code model is a projector of rank 2. Let us denote
the indices of the transfer matrix as
(TMxy){z},{z} =
x (pbc)z
y (pbc)
z
z
i,j
i,j
, (6.61)
where zi,j and zi,j are the indices at the position (i, j) on the xy-plane. The vector
space of this transfer matrix is of dimension 2LxLy . Suppose the vector space is
spanned by the basis e{z}, where
e{z} =Lx⊗i=1
Ly⊗j=1
ezi,j = ez1,1 ⊗ ez1,2 ⊗ . . . ezLx,Ly . (6.62)
ezi,j is the local “virtual bond Hilbert space” spin |0〉 = |↑〉, |1〉 = |↓〉 basis for
the index zi,j, where i and j are the coordinates of zi,j in the x- and y-directions
respectively. We can consider the matrix multiplication of the transfer matrix TMxy
with an element of the basis e{z}:
TMxy · e{z} =∑{z}
(TMxy){z},{z} (6.63)
where {z} is fixed for the both LHS and RHS. Applying the Concatenation lemma
in Sec. 6.3.3 to (TMxy){z},{z}, we conclude that the terms satisfying
Lx∑i=1
Ly∑j=1
zi,j + zi,j = 0 mod 2 (6.64)
122
contribute equally to the RHS of Eq. (6.63), while the terms satisfying
Lx∑i=1
Ly∑j=1
zi,j + zi,j = 1 mod 2 (6.65)
do not contribute. Therefore, we can rewrite the summation more precisely:
TMxy · e{z} =∑
{z} with∑i zi+zi even
(TMxy){z},{z} ∝∑
{z} with∑i zi+zi even
e{z}. (6.66)
When the {z} satisfies∑
i,j zi,j = 0 mod 2, we have
TMxy · e{z} ∝∑
{z} with∑i zi even
e{z}, (6.67)
while when the {z} satisfies∑
i,j zi,j = 1 mod 2, we have
TMxy · e{z} ∝∑
{z} with∑i zi odd
e{z}. (6.68)
Therefore, TMxy is a projector of rank 2. Hence, it has only two eigenvalues of 1, and
the corresponding unnormalized eigenvectors are:
∑{z} with
∑i zi even
e{z}
∑{z} with
∑i zi odd
e{z}.
(6.69)
123
6.3.6 GSD and Transfer Matrix
We know that the 3D toric code model has three pairs of nonlocal operators along
the cycles of 3D torus:
WX [Cx] =∏i∈Cx
Xi, WZ [Cyz] =∏i∈Cyz
Zi;
WX [Cy] =∏i∈Cy
Xi, WZ [Cxz] =∏i∈Cxz
Zi;
WX [Cz] =∏i∈Cz
Xi, WZ [Cxy] =∏i∈Cxy
Zi.
(6.70)
where Cx is a loop along the cycle of x direction on lattice, Cyz is a closed surface
along the cycles of yz directions on dual lattice, and similarly for the other directions.
The figures for these operators are:
Z
x
x x
ZZ
x
z
y
(6.71)
The commutation relations include:
WX [Cx]WZ [Cyz] = −WZ [Cyz]WX [Cx],
WX [Cy]WZ [Cxz] = −WZ [Cxz]WX [Cy],
WX [Cz]WZ [Cxy] = −WZ [Cxy]WX [Cz].
(6.72)
124
All other combinations of operators commute. Hence, there are 8 degenerate ground
states in total on the torus, assuming that Eq. (6.70) has exhausted all nonlocal
operators.
We can also put our TNS on the 3-torus, i.e., |TNS〉T 3 . It is not hard to verify
using Eq. (6.36) and (6.40) that:
WX [Cx]|TNS〉T 3 = |TNS〉T 3 ,
WX [Cy]|TNS〉T 3 = |TNS〉T 3 ,
WX [Cz]|TNS〉T 3 = |TNS〉T 3 .
(6.73)
As already mentioned in Sec. 6.3.2, |TNS〉T 3 = |ψ〉 where |ψ〉 is defined in Eq. (6.35).
Both |TNS〉T 3 and |ψ〉 are +1 eigenstates of WX operators. However, the transfer
matrix defined by |TNS〉T 3 does not provide 8 fold degenerate eigenvalues, but only
2, as shown in Sec. 6.3.5.
We can act with the WZ [Cyz] and WZ [Cxz] on the TNS by using Eq. (6.36) and
(6.40) to generate all the ground states. The TNSs obtained by this action in terms
125
of a xy-plane of tensors are depicted as below:
Z
Z
Z
Z
Z Z Z Z Z Z Z
T
T
Z Z Z Z Z Z Z
TZ
Z
Z
Z
x (pbc)
y (pbc)
g
g
g
z
(6.74)
The intersection of WZ [Cyz] and WZ [Cxz] with the xy-plane is the line Z operators,
illustrated by the blue circle Z in Eq. (6.74). WZ [Cxy] acts on the xy-plane on the
dual lattice, and thus does not change the transfer matrix at all. We denote these xy-
planes of TNSs in Eq. (6.74) as Tα,βxy (with open indices along the z direction), where
α, β ∈ {0, 1} label whether we have inserted the Z operators in the x and y direction
respectively. The subindex xy in Tα,βxy means that the TNS is on a xy-plane. Clearly,
Tα,βxy are different since they support different holonomies of WX [Cx] operators and
WX [Cy] operators. After obtaining Tα,βxy , we can define four twisted transfer matrices
correspondingly by contracting the physical indices between bra Tα,βxy and ket Tα,β
xy .
The twisted transfer matrices are denoted as TMα,βxy . For instance, TM0,0
xy is the
untwisted transfer matrix in Eq. (6.61).
126
Each of these transfer matrices TMα,βxy is also a projector of rank 2. The reasons
are that (1) the contraction of the projector g tensors between the bra Tα,βxy and ket
Tα,βxy makes the indices in the bra layer and ket layer identical; (2) the Z operators in
the bra and ket layer will cancel each other and produce an identity operator. Hence,
the transfer matrices built from the twisted TNS TM1,0xy , TM0,1
xy and TM1,1xy are equal
to that built from the untwisted TNS TM0,0xy :
TMα,βxy = TM0,0
xy , ∀ α, β ∈ {0, 1} (6.75)
The transfer matrix has degeneracy 2, and it is the same for each of 4 TNSs which
are different. We have 4 × 2 = 8 degenerate eigenstates. We have to emphasize
that Eq. (6.75) holds true, when there are no physical operators in between the bra
and ket layer of Tα,βxy , and the physical indices of projector g tensors are directly
contracted. If a physical operator is inserted, for instance WX [Cx], then the twisted
transfer matrices in the presence of WX [Cx] are NOT the same as the untwisted one
sandwiching the same WX [Cx].
In constructing the TNS on torus, we need to choose the same Tα,βxy for each xy-
plane. If we use different Tα,βxy in each xy-plane to construct a TNS on torus, the
corresponding wave function is no longer a ground state for the 3D toric code model.
More specifically, in the 3D toric code model, if we contract Lz − 1 T0,0xy ’s with one
twisted Tα,βxy Eq. (6.74), then the corresponding wave function will support a pair of
loop excitations because the Bp operators up and below Z operators are violated, and
is no longer a ground state. Graphically, the contraction of these Lz − 1 number of
T0,0xy and one T1,0
xy is:
Z Z Z Z
z
. (6.76)
127
Z ZZ
ZZ Z
Z
Z
Z
Z
Z
Z
x
x
x
x
xx
x xx x
xxx
z
y
(a) (b) (c) (d)
Figure 6.8: The Hamiltonian terms of the X-cube model: (a) Av,yz, (b) Av,xy, (c)Av,xz and (d) Bc. The circled X and Z represent the Pauli matrices acting on thephysical spin-1/2’s.
The reason for this energy costing is that there is a line of Z operators. However,
the nonlocal operators in 3D toric code do not have a line Z operator, but only have
surface Z operators. See Eq. (6.71). On the other hand, the TNS made of all the
same Tα,βxy , for instance T1,0
xy :
Z Z Z Z
z
Z Z Z Z
Z Z Z Z
Z Z Z Z(6.77)
is allowed, because it corresponds to a closed surface operator included in Eq. (6.71).
We will come back to this point in Sec. 6.4.6 where this issue is subtle and makes a
difference when we count GSD from transfer matrices.
6.4 X-cube Model
In this section, we construct the TNS for the X-cube model, one of the simplest frac-
ton models. Using it, we then calculate the entanglement entropy and the GSD of
this model. The results are generally different from those in conventional topologi-
cal phases. The entanglement entropy has a linear correction to the area law, and
the GSD grows exponentially with the size of the system. This section is organized
as follows: In Sec. 6.4.1, we briefly review the X-cube model in a cubic lattice. In
Sec. 6.4.2, we construct the TNS for the X-cube model. In Sec. 6.4.3, we prove a
128
Concatenation lemma for the X-cube TNS. In Sec. 6.4.4, we calculate the entan-
glement entropies on R3. In Sec. 6.4.5, we construct the transfer matrix and prove
that it is a projector of rank 2Lx+Ly−1. In Sec. 6.4.6, we show how the transfer matrix
gives rise to an extensive GSD on torus.
6.4.1 Hamiltonian of X-cube Model
The model is defined on a cubic lattice. All the spin 1/2’s are associated with the
bonds of the cubic lattice. The Hamiltonian is:
H = −∑v
(Av,xy + Av,yz + Av,xz)−∑c
Bc (6.78)
where the summation is taken over all vertices and cubes respectively. Each term is
depicted in Fig. 6.8. More specifically, Av,xy is the product of four Z operators around
the vertex v in the xy plane. Similarly for Av,yz and Av,xz. Bc flips the 12 spins of a
cube c. It is trivial to show that all the Hamiltonian terms commute:
[Av,xy, Av′,xy] = [Av,yz, Av′,yz] = [Av,xz, Av′,xz] = 0,
[Av,xy, Av′,yz] = [Av,xy, Av′,xz] = [Av,yz, Av′,xz] = 0,
[Bc, Av′,xy] = [Bc, Av′,yz] = [Bc, Av′,xz] = 0
[Bc, Bc′ ] =0, ∀ v, v′, c, c′.
(6.79)
Hence, this model can be exactly solved. The ground state |GS〉 (on R3) needs to
satisfy that:
Av,xy|GS〉 =|GS〉,
Av,yz|GS〉 =|GS〉,
Av,xz|GS〉 =|GS〉,
Bc|GS〉 =|GS〉, ∀ v, c.
(6.80)
129
This set of equations will be used to derive the local T tensor for the X-cube model.
The nonlocal operators of the X-cube model which are required to commute with
all Hamiltonian terms on torus include 9 loop operators[175]:
WX [Cx] =∏i∈Cx
Xi,WX [Cy] =∏i∈Cy
Xi,WX [Cz] =∏i∈Cz
Xi,
WZ [Cx,z] =∏i∈Cx
Zi,WZ [Cy,z] =∏i∈Cy
Zi,WZ [Cz,x] =∏i∈Cz
Zi,
WZ [Cx,y] =∏i∈Cx
Zi,WZ [Cy,x] =∏i∈Cy
Zi,WZ [Cz,y] =∏i∈Cz
Zi,
(6.81)
where Cx is a straight line along the cycle of the x direction on lattice, and Cx,z is a
line along the cycle of the x direction on dual lattice while the lattice bonds of Cx,z
are in the z-direction, and similarly for the other directions. The figures for them are:
Z
x
x x
Z
Z
Z
Z
Z
x
z
y
(a) (b) (c)
(e) (d) (f)
(g) (h) (i)
. (6.82)
The algebras of these loop operators are grouped into three independent sets:
1. The operator (a) in Eq. (6.82) anti-commutes with the operator (f) and the
operator (h) when they overlap at one spin. Thus, WX [Cx] anti-commutes with
WZ [Cz,x] and WZ [Cy,x] when they overlap at one spin.
130
2. The operator (b) in Eq. (6.82) anti-commutes with the operator (g) and the
operator (i) when they overlap at one spin. Thus, WX [Cy] anti-commutes with
WZ [Cx,y] and WZ [Cz,y] when they overlap at one spin.
3. The operator (c) in Eq. (6.82) anti-commutes with the operator (d) and the
operator (e) when they overlap at one spin. Thus, WX [Cz] anti-commutes with
WZ [Cx,z] and WZ [Cy,z] when they overlap at one spin.
All other combinations of operators commute, because they do not overlap at the same
physical spin. Each of the three algebras gives a representation of dimension[155]
2Ly+Lz−1, 2Lz+Lx−1, 2Lx+Ly−1. (6.83)
respectively. Hence the total dimension of the ground state space is 22Lx+2Ly+2Lz−3.
The derivations of the GSD in terms of these operator algebras are explained in
App. C.3.
6.4.2 TNS for X-cube Model
Following the same prescription in Sec. 6.3, we can write down the TNS for the X-
cube model. First, we introduce a projector g tensor on the bonds of the lattice (see
Eq. (6.5)). The virtual index is either 0 or 1. The g tensor is a projector which
projects the physical spin to the virtual indices. The tensor g satisfies Eq. (6.36).
The TNS is not only composed of the g tensor on the bonds of the lattice, but also
the T tensors on the vertices. The T tensor has six virtual indices and no physical
index. Unlike the g tensor, the T tensor will be specified by the Hamiltonian terms.
We now implement Eq. (6.80) on the TNS. Using the condition Eq. (6.36), we can
131
transfer the operators in Hamiltonian terms from physical indices to virtual indices:
TZ
Z
Z
Z
Z
Z
ZZ
g
gg
g
g
g
=
ggg
g
gg
g
g
gg
g
T
T T
T
TT
TT
xx
x
xx
x
xxx
xx
xx
xx
xx x
xx
xxx
x xx
x xx
xxx
=
TZ
Z
Z
Z
ZZ
ZZ
g
gg
g
g
g
=
T
ZZ
Z
Z Z
Z Z
Zg
gg
g
g
g
=
x
x x
x
. (6.84)
132
Requiring that the tensors in the dashed red rectangles are invariant will lead to the
following (strong) conditions on the T tensor:
T T T TZ Z
Z
ZZ Z
Z
Z
Z
Z
Z
Z= = =
T T
x xx
T
x xx
Tx
x
x
T
xx
xT
x
xx Tx
x
T
x
xT
xx
x
xx
= = =
= = =
= =
. (6.85)
The first set of conditions is required by the operators Av,xy, Av,yz and Av,xz around
the vertex v. The second set of equations is required by the operators Bc around the
cube c. The X operators acting on the 12 spins of the cube c will be transferred to
the virtual spins of the eight T tensors around the cube c, using Eq. (6.36). The X
operators act on the eight quadrants of a T tensor. Clearly, these two sets of the
conditions are not independent. The solution to these conditions is:
Txx,yy,zz =
1 if
x+ x+ y + y = 0 mod 2,
x+ x+ z + z = 0 mod 2,
y + y + z + z = 0 mod 2.
0 otherwise.
(6.86)
133
A useful consequence of Eq. (6.85) is:
T Tx
T
x
= = =T
xx
xx . (6.87)
We now have fixed the TNS for the X-cube model using the local conditions Eq. (6.80).
The wave function on R3 can also be represented as a tensor contraction of Eq. (6.6).
6.4.3 Concatenation Lemma
In this section, we consider the contraction of a network of local T tensors with open
virtual indices for the X-cube model, similar to the idea developed in Sec. 6.3.3.
However, here the situation is more complicated than the 3D toric code model. The
elements of a local T tensor are either 0 or 1 in Eq. (6.85), depending on the even/odd
sector in three directions. The elements of the contracted T tensors will also be either
0 or 1 with a similar criterion.
Concatenation Lemma: For a network of the contracted T tensors in
Eq. (6.86), the sums of the open indices along each xy, yz and xz planes have
to be even. Otherwise, the tensor element of this network is zero. The nonzero
elements are constants independent of the virtual indices.
This lemma is a consequence of Eq. (6.86). See App. C.2 for an induction proof.
In this section, we explain this result by considering a simple example. Suppose that
we have two T tensors contracted along the z direction:
Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2 =∑z1,z2
Tx1x1,y1y1,z1z1Tx2x2,y2y2,z2z2δz1z2 . (6.88)
134
Graphically, T is the same as depicted in Fig. 6.6. For each of the two T tensors, we
have x1 + x1 + y1 + y1 = 0 mod 2
x1 + x1 + z1 + z1 = 0 mod 2
y1 + y1 + z1 + z1 = 0 mod 2
(6.89)
and x2 + x2 + y2 + y2 = 0 mod 2
x2 + x2 + z2 + z2 = 0 mod 2
y2 + y2 + z2 + z2 = 0 mod 2.
(6.90)
Therefore, setting z1 = z2 due to the tensor contraction, the open indices of T need
to satisfy:
x1 + x1 + y1 + y1 = 0, mod 2
x2 + x2 + y2 + y2 = 0, mod 2
z1 + z2 + x1 + x1 + x2 + x2 = 0, mod 2
z1 + z2 + y1 + y1 + y2 + y2 = 0, mod 2
(6.91)
in order for the elements of the tensor T to be nonzero. The last set of equations
intuitively means that the open indices of the tensor T (Fig. 6.6) in each xy, yz and
xz plane need to have an even summation. Moreover, the elements of the contracted
135
A A A A
(1) (2) (3) (4)
Figure 6.9: Figures for several regions A for which we calculate the entanglemententropies. (1) Region A is a cube of size lx × ly × lz. (2) Region A is a cube oflx× ly× lz with a hole of size l′x× l′y× l′z in it. (3) Region A is a cube of size lx× ly× lzand a small cube of height l′z on top of it. (4) Region A is a cube of size lx × ly × lzcarved on top of it a small cube of height l′z.
T tensor are 1 independent of indices:
Tx1,x1,y1,y1,z1,x2,x2,y2,y2,z2 =
1 if
x1 + x1 + y1 + y1 = 0, mod 2 and
x2 + x2 + y2 + y2 = 0, mod 2 and
z1 + z2 + x1 + x1 + x2 + x2 = 0, mod 2 and
z1 + z2 + y1 + y1 + y2 + y2 = 0, mod 2
0 otherwise.
(6.92)
Generally for a complicated contraction of local T tensors denoted by T, we have:
T{t} =
Const 6= 0 Concatenation lemma is satisfied
0 else.
(6.93)
This is the notation that we will use when computing the entanglement entropies or
the transfer matrix degeneracies.
6.4.4 Entanglement
We can show that a cubic region A and its deformations are the exact SVD of |TNS〉.
See Fig. 6.9 for some examples of deformed A regions. We can read out the singular
136
values of |TNS〉 for these entanglement cut. Suppose we consider the wave function
on R3 for simplicity (i.e., without dealing with the multiple ground states on T 3).
We rewrite the wave function:
|TNS〉 =∑{t}
|{t}〉A ⊗ |{t}〉A. (6.94)
where {t} represent the tensor virtual indices connecting the region A and A. |{t}〉A
is the TNS for the region A with open virtual indices {t}. Similarly for |{t}〉A for
the complement region A. Because every virtual bond is contracted over for |TNS〉,
|{t}〉A and |{t}〉A have to share the same {t}, and {t} is summed over in Eq. (6.94).
Next we show that this set of basis |{t}〉A is orthonormal:
A〈{t′}|{t}〉A ∝ δ{t′},{t}, (6.95)
when the Concatenation lemma is satisfied for both of {t′}, {t}, and the basis
|{t}〉A are not null vectors.
Proof:
The open virtual indices {t} satisfy the SVD condition in Sec. 6.2.1. Hence, we
can conclude that |{t}〉A and |{t}〉A are orthogonal basis for the region A and A, and
thus Eq. (6.94) is exactly SVD. In order to calculate the entanglement entropies, we
need to show that |{t}〉A and |{t}〉A are not only orthogonal, but also orthonormal.
The proof is essentially the same as in Sec. 6.3.4. Here we briefly repeat it. We
use the same conventions of SVD basis |{t}〉A and |{t′}〉A as in Sec. 6.3.4. Suppose
{t′} and {t} both satisfy Concatenation lemma in Sec. 6.4.3. Hence, |{t}〉A and
|{t′}〉A are not null vectors. More specifically, the wave function |{t}〉A is the same
as in Eq. (6.49). All the virtual indices except {t} are contracted over.
137
When {t′} 6= {t}, the basis overlap is zero, because the spins on the boundary
of A are different. When {t′} = {t}, the overlap is nonzero. Moreover, the overlaps
A〈{t}|{t}〉A are constants independent of {t}, due to the Concatenation lemma
in Sec. 6.4.3. Hence, |{t}〉A is an orthonormal basis for the region A, up an overall
normalization factor. 2
Therefore, using the orthonormal basis |{t}〉A and |{t}〉A, Eq. (6.94) is indeed the
SVD. Furthermore, the singular values are all identical. As a result, the entanglement
of the region A is determined by the number of basis states |{t}〉A which are involved
in Eq. (6.94), i.e., the rank of the contracted tensor for the region A. The rank of the
contracted tensor can be counted by the Concatenation lemma in Sec. 6.4.3. We
only need to count the number of indices that satisfy the Concatenation lemma.
We now list a few simple examples of entanglement entropies. The entanglement cuts
are displayed in Fig. 6.9. Correspondingly, their entanglement entropies are:
1. When region A is a cube of size lx × ly × lz:
SAlog 2
= 2lxly + 2lylz + 2lxlz − lx − ly − lz + 1 = Area− lx − ly − lz + 1.
(6.96)
The calculation details are the following:
The number of indices straddling this entanglement cut is 2lxly + 2lylz + 2lxlz.
This is the maximum possible number of basis states in the SVD of Eq. (6.94).
However, these indices are not free. They are subject to certain constraints, in
order for the singular vectors to have non-vanishing norms. Using the Con-
catenation lemma in Sec. 6.4.3, we know that the open indices in each xy,
yz, and xz plane must have even summations. We denote the indices to be ti,j,k
138
where i, j, k are the coordinates of such a index. Then, we have;
∑i,j
ti,j,k = 0 mod 2, ∀ k
∑i,k
ti,j,k = 0 mod 2, ∀ j
∑j,k
ti,j,k = 0 mod 2, ∀ i
(6.97)
where the summation is only taken over open virtual indices near the entangle-
ment cut in each xy, yz and xz plane. Therefore, we have lx, ly, lz number of
constraints respectively. However, these constraints are not independent. Only
lx + ly + lz − 1 of them are independent, because the three sets of constraints
sum to be an even number. Hence, the number of free indices is
2lxly + 2lylz + 2lxlz − lx − ly − lz + 1.
The number of singular vectors in Eq. (6.94) is:
22lxly+2lylz+2lxlz−lx−ly−lz+1,
which leads to the entropy written in Eq. (6.96). 2
2. When the region A is a cube of size lx × ly × lz with an empty hole of size
l′x × l′y × l′z:
SAlog 2
=2lxly + 2lylz + 2lxlz + 2l′xl′y + 2l′yl
′z + 2l′xl
′z
− l′x − l′y − l′z − lx − ly − lz + 2
=Area− l′x − l′y − l′z − lx − ly − lz + 2.
(6.98)
139
3. When the region A is a cube of size lx × ly × lz with an additional convex cube
of height l′z on top (Fig. 6.9 (3)):
SAlog 2
=Area− lx − ly − lz − l′z + 1. (6.99)
4. When the region A is a cube of size lx× ly× lz with an additional concave cube
of height l′z on top (Fig. 6.9 (4)):
SAlog 2
=Area− lx − ly − lz − l′z + 1. (6.100)
The area part of the entanglement is measured by the number of indices straddling
the entanglement cut. The constant contribution to the entanglement entropy is
universal[201], as it counts the number of connected components of the entanglement
surface. As opposed to the toric code case, the constants are positive numbers. We
emphasize that the linear corrections to the area law in the entanglement entropies
states have not been observed in quantum field theories.
Furthermore, if we put the TNS on a cylinder T 2xy×Rz and the entanglement cut
splits the cylinder into two halves z > 0 and z < 0, then the entanglement entropy
of either side is:
SAlog 2
= Area− Lx − Ly + 1. (6.101)
We emphasize that the entanglement spectrum is flat, because all singular values are
identical in Eq. (6.94).
140
Tg
x x x x x x x
Tg
x x x x x x x x x x x x
Tg
x x x x x x x x x x x x
Tg
x (pbc)
z
y (pbc)
Figure 6.10: Derivations for the first equation in Eq. (6.107). The rest two equationscan be proved similarly. In the first step, the physical X operators can be transferredto the virtual level by using Eq. (6.36), and in the third step, all the virtual Xoperators are exactly canceled in pairs (dashed red rectangles in the third figure) dueto Eq. (6.87).
6.4.5 Transfer Matrix as a Projector
Following the same reasoning explained in Sec. 6.1.4, the transfer matrix of the X-cube
model in the xy-plane is:
TMxy = CT 2xy (. . . TTT . . .) (6.102)
with open virtual indices in the z-direction. Graphically, see Eq. (6.22) and Eq. (6.61).
In this section, we will show that the TMxy for the X-cube model is also a projector.
However, the projection is more complicated than in the 3D toric code example.
Using the transfer matrix basis e{z} defined in Eq. (6.62), we show that:
TMxy · e{z} =∑
Concatenation Lemma
(TMxy){z},{z} ∝∑
Concatenation Lemma
e{z} (6.103)
141
where the notation {z} and {z} collectively denote all z indices perpendicular to the
xy-plane. The summation with the Concatenation lemma in Sec. 6.4.3 means that:
∑i
zi,j + zi,j = 0 mod 2, ∀ j
∑j
zi,j + zi,j = 0 mod 2, ∀ i(6.104)
where the subindex i, j of zi,j denote the positions of zi,j in the x- and y-direction
respectively. These two equations mean that in each xz and yz plane, the indices
have an even summation. For instance, the red dashed rectangles below:
x (pbc)
y (pbc)
TT
. (6.105)
Among the Lx+Ly equations in Eq. (6.104), only Lx+Ly−1 are linearly independent,
because the summations of the two sets of constraints are the same:
∑j
(∑i
zi,j + zi,j
)= 0 mod 2⇔
∑i
(∑j
zi,j + zi,j
)= 0 mod 2. (6.106)
The summation in Eq. (6.103) can be separated into Lx +Ly − 1 number of different
“parity” sectors, similar to the 3D toric code case Eq. (6.69). Hence, TMxy is a
projector of rank 2Lx+Ly−1. It has 2Lx+Ly−1 degenerate nonzero eigenvalues.
6.4.6 GSD and Transfer Matrix
The TNS, which gives us the single ground state on R3, has the minimum energy by
construction. If we contract these tensors on the torus T 3 with periodic boundary
142
conditions, then we still yield only one ground state. Moreover, this ground state is
the +1 eigenstate of all WX operators in Eq. (6.81):
WX [Cx]|TNS〉T 3 = |TNS〉T 3 ,
WX [Cy]|TNS〉T 3 = |TNS〉T 3 ,
WX [Cz]|TNS〉T 3 = |TNS〉T 3 .
(6.107)
These three equations can be proved by using Eq. (6.36) and Eq. (6.85); the deriva-
tions are summarized in Fig. 6.10.
Other ground states on the torus can be obtained by acting with the nonlocal
operators WZ [Cz,x] and WZ [Cz,y] in Eq. (6.81) on the TNS |TNS〉T 3 . The physi-
cal operator WZ [Cz,x] and WZ [Cz,y] can be transferred to the virtual indices using
Eq. (6.36).
After applying WZ [Cz,x] and WZ [Cz,y] in Eq. (6.81) on TNS, we can generate
2Lx+Ly TNSs exemplified by Fig. 6.11. The intersections of WZ [Cz,x] and WZ [Cz,y]
with the xy-plane are the blue circled Z operators in Fig. 6.11. We denote these
planes of tensors in Fig. 6.11 as T~α,~βxy where ~α and ~β are binary vectors (values in
{0, 1}) of length Lx and Ly representing the absence or presence of Pauli Z operators.
For instance, the untwisted plane of TNS is T~0,~0xy using this convention. This notation
T~0,~0xy is similar to that in Sec. 6.3.6. Inserting a Z operator at the virtual level will
change the holonomy of WX in the xy-plane. For instance, for Panel (a) in Fig. 6.11,
the WX operator along the first row has a −1 eigenvalue, while the WX operators
along the second, the third and the fourth row have a +1 eigenvalue.
Each of these T~α,~βxy will generate a transfer matrix TM~α,~β
xy which has 2Lx+Ly−1
degenerate eigenvalues. The reasons are that (1) when building the transfer matrices
from the twisted T~α,~βxy , the contraction over the physical indices of the projector g
tensors makes the virtual indices from the bra layer and ket layer identical; (2) the Z
operators in the bra layer cancel their analogues respectively in the ket layer. Hence,
143
Z
Z
x (pbc)
y (pbc)
T
T
Z
Z
T
T
Z
Z
T
T
(a) (b) (c)
(d) (e) (f)
Z
Z
Z
Z
Z
Z Z
Figure 6.11: Examples for the X-cube TNS in a xy-plane, obtained by acting WZ [Cz,x]and WZ [Cz,y] on the constructed TNS. The intersection of one WZ [Cz,x] operator andone WZ [Cz,y] operator with the xy-plane is only one Pauli Z operator, i.e., the circledblue Z in this figure.
Z
x (pbc)
y (pbc)
T
ZT
T
T
Z Z Z Z Z Z Z
(a) (b)
(c) (d)
Z
Z
Z
Z
Z
Z
Z Z Z Z Z Z Z
Figure 6.12: We act WZ [Cx,y] and WZ [Cy,x] operators on Panel (a) in Fig. 6.11. Hence,we have four TNSs in this xy-plane that can be related to each other. See the textfor the explanations.
the transfer matrices TM~α,~βxy obtained from the twisted T~α,~β
xy are equal to that obtained
from the untwisted one T~0,~0xy .
TM~α,~βxy = TM
~0,~0xy , ∀ ~α, ~β (6.108)
Thus, the transfer matrices TM~α,~βxy built from the twisted T~α,~β
xy are also projectors of
dimension 2Lx+Ly−1.
144
The TNS on torus is then constructed as the contraction of T~α,~βxy on each xy-
plane. However, the subtlety of the X-cube model is that in constructing TNS on
the torus, there are still degree of freedom we can play with. In the 3D toric code
case Sec. 6.3.6, as we evolve the state in the z direction, we have to use the same
Tα,βxy in each plane, otherwise the corresponding wave function is no longer a ground
state. However, we have more choices for the X-cube model. In each plane of T~α,~βxy ,
we have four choices that do not change the energy: we can still act WZ [Cx,y] and
WZ [Cy,x] on TNS in the xy-plane without affecting other xy-planes. These choices
do not raise the energy because WZ [Cx,y] and WZ [Cy,x] are the nonlocal operators of
the X-cube model Eq. (6.81): they do not cost any energy. For each of the T~α,~βxy built
from Fig. 6.11, we can find 3 others: all 4 planes of tensors T~α,~βxy can be used in the z
direction. Take Panel (a) of Fig. 6.11 as an example at one point in the z direction.
The 4 T~α,~βxy are depicted in Fig. 6.12. Their expressions are:
1. For Panel (a), we do not apply any operators.
2. For Panel (b), we apply WZ [Cy,x] on TNS.
3. For Panel (c), we apply WZ [Cx,y] on TNS.
4. For Panel (d), we apply both WZ [Cx,y] and WZ [Cy,x] on TNS.
Due to this choice, four twisted T~α,~βxy will be grouped together, and there are
2Lx+Ly
4(6.109)
number of groups of twisted T~α,~βxy . Hence, the total GSD that we can obtain from the
transfer matrices built from T~α,~βxy is:
2Lx+Ly−1 × 2Lx+Ly
4× 4Lz = 22Lx+2Ly+2Lz−3. (6.110)
145
Each factor in the above formula has an explanation:
1. 2Lx+Ly−1 is the degeneracy of each transfer matrix.
2. 2Lx+Ly
4is the number of groups of the twisted T~α,~β
xy due to the “ambiguity”
explained in the paragraph before Eq. (6.109).
3. 4Lz is the number of combinations for T~α,~βxy , since for each xy plane we can pick
any of the 4 T~α,~βxy belonging to the same group.
This is the total GSD of X-cube model on torus.
6.5 Haah Code
In this section, we derive the TNS for the Haah code following a similar prescription
as that in Sec. 6.3.2 and Sec. 6.4.2. We then compute the entanglement entropies
using the TNS for several types of entanglement cuts. This section is organized as
follows. In Sec. 6.5.1, we review the Hamiltonian of the Haah code. In Sec. 6.5.2,
we present the construction of TNS for the Haah code. In Sec. 6.5.3, we discuss the
entanglement cuts for which the TNS is an exact SVD. In Sec. 6.5.4, we discuss the
cubic entanglement cut, where the TNS is not an exact SVD. The calculation of the
entanglement entropies proceeds in the same way as that of the 3D toric code model
or X-cube model: one counts the number of constraints for open indices.
6.5.1 Hamiltonian of Haah code
The Haah code is defined on a cubic lattice. As opposed to the 3D toric code and the
X-cube model discussed in Sec. 6.3 and 6.4, there are two spin-1/2’s defined on each
vertex of a cubic lattice. The Hamiltonian of the Haah code is a sum of commuting
operators where each term is the product of Pauli X or Z operators. Specifically,
146
there are two types of the Hamiltonian terms:
H = −∑a,b,c
Aabc −∑a,b,c
Babc. (6.111)
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 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.
(6.112)
The superscripts L/R represent the left or the right spin on a vertex where the Pauli
operators act on. The subscripts (ijk) ∈ Z2 × Z2 × Z2 represent the coordinate of
vertices (on a cube). The operators Aabc and Babc on all other cubes can be obtained
by translation. Pictorially the two types of terms are:
x
z
y
(6.113)
It is straightforward to check that all the Hamiltonian terms commute.
147
T T
T T
(a)
x
z
y
T T
T T
T T
T T
(b)
Figure 6.13: Tensor contraction for the Haah code TNS. (a) The lattice size is 2×3×3.(b) The lattice size is 3× 3× 3
6.5.2 TNS for Haah Code
The ground state |GS〉 is obtained by requiring
Aabc|GS〉 = |GS〉 (6.114)
Babc|GS〉 = |GS〉 (6.115)
148
for every a, b, c. We can solve these two equations similarly to the 3D toric code model
in Sec. 6.3.2 and the X-cube model in Sec. 6.4.2 to obtain a TNS representation. 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.
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 g tensors, gLsijkl and gRsijkl, are:
gLsijkl =
s
i
j
k l
I
IIIII
IV
V
V IV II
V III
(6.116)
and
gRsijkl =
l
j
ki
s
I
IIIII
IV
V I
V
V II
V III
(6.117)
where s is the physical index in {|0〉 = |↑〉, |1〉 = |↓〉}, and ijkl are virtual indices.
We use a blue dot for the g tensor on the right spin and a red dot for the g tensor on
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
149
the g tensor acts as a projector from the physical index to the four virtual indices:
gLsijkl =
1 i = j = k = l = s
0 otherwise
,
gRsijkl =
1 i = j = k = l = s
0 otherwise
.
(6.118)
The four virtual indices of gLsijkl extend along the III, VIII, VII, VI octants (as shown
in Eq. (6.116)), and the four virtual indices of gRsijkl extend along the II, VII, IV, V
octants (as shown in Eq. (6.117)).
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 . (6.119)
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. (6.114)
and Eq. (6.115). Imposing the condition Eq. (6.114) and transferring the physical Z
150
operators to the virtual level, we find that:
TZ
Z Z
Z Z
Z
ZZ
= T (6.120)
which amounts to
Ti1i2i3i4i5i6i7i8 = (−1)∑8n=1 inTi1i2i3i4i5i6i7i8 , (6.121)
where i1, · · · , i8 are the eight virtual indices of the T tensor defined in Eq. (6.119).
Hence,
Ti1i2i3i4i5i6i7i8 = 0, if8∑
n=1
in = 1 mod 2. (6.122)
151
Imposing the condition Eq. (6.115) 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
.
(6.123)
In terms of components, Eq. (6.123) 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.
152
For example,
Ti1i2i3i4i5i6i7i8 = T(1−i1)(1−i2)i3i4i5i6i7i8
= Ti1(1−i2)(1−i3)i4i5i6i7i8
= Ti1i2(1−i3)(1−i4)i5i6i7i8
= ...
(6.124)
Combining Eq. (6.121) and (6.124), 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.
(6.125)
For simplicity, we consider the space to be R3 where the Haah code has a unique
ground state.
|TNS〉 =∑{s}
CR3 (gL,s1gR,s2gL,s3gR,s4 . . . TTT . . .
)|{s}〉. (6.126)
We emphasize 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 3D toric code and the X-cube model. As an example of contraction, we take
two blocks of size 2×2×1 and 2×2×2 in Fig. 6.13. The T tensors with their virtual
indices are drawn 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
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contracted with T tensors, while other g tensors have open virtual indices (which are
not explicitly drawn).
6.5.3 Entanglement Entropy for SVD Cuts
In this section, we compute the entanglement entropies for two types of cuts for which
the TNS is an SVD.
Two types of SVD Cuts
To compute the entanglement entropy, we use the same convention adopted in the
discussion of the 3D toric code (in Sec. 6.3) and the X-cube model (in Sec. 6.4): the
open virtual indices of the region A connect directly to the g tensors inside A while
the open virtual indices of the region A connect with T tensors inside A. We further
choose a region A such that the TNS is an SVD, and compute the entanglement
entropy. We find two types of entanglement cuts for which the Haah code TNS is
an exact SVD. For the general cubic region A, we need an extra step to perform the
SVD of the TNS. This derivation will be presented in Sec. 6.5.4. We now specify the
two types of regions that the Haah code TNS is an exact SVD.
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 6.14 shows an example with
l− 1 = 3 contracted along the z direction. (Since in Sec. 6.3 and 6.4, we used l
as the number of vertices along each side of region A, there are l− 1 bonds (or
cubes) along each side.)
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
154
leg, then there are 1 + (lx − 2) + (ly − 2) + (lz − 2) = lx + ly + lz − 5 cubes (or
T tensors) region A. Figure 6.15 shows an example with lx = ly = lz = 3.
In the first case and for l = 2, 3, we use brute-force numerics to find that the reduced
density matrix is diagonal, which shows that the TNS is an exact SVD.
T
T
T
Figure 6.14: 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.
T T
T
T
Figure 6.15: 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.
In order to show that the above cuts correspond to an SVD, we follow the argu-
ments developed in Sec. 6.2.1. In Sec. 6.2.1, we proposed a SVD Condition. However,
we find that the region A of both types, shown in Fig. 6.14 and 6.15, do not sat-
155
isfy 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. 6.15 connects to the same T tensor in their upper-left cube which is in the region
A. Here, we propose a Generalized SVD Condition which suffices to prove that the
entanglement cut corresponding to Fig. 6.14 and 6.15 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. (6.118) and
T tensor constraints Eq. (6.125), then |{t}〉A is orthogonal. Since |{t}〉A is or-
thogonal because all the open virtual indices are connected with g tensors, the TNS
|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. 6.2.1. The proof of normalization of the wave function is independent
of {t} is also the same as in Sec. 6.2.1. Furthermore, A〈{t}|{t′}〉A = 0 for {t} 6= {t′}
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 is an SVD.
156
We explain the Generalized SVD Condition in the simplest example, i.e., l = 2 in
case 1. There is only one T tensor, and the 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. 6.13. 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 constrain the open indices. Consider the T tensor in the region A (which
we denote by T ′) which shares two spins i7, i8 with the region A (The T ′ tensor lives
157
in the lower right corner):
T
T'
(6.127)
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, (6.128)
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
158
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
i5 = fixed
i8 = fixed
i1 = fixed
i4 = fixed
i3 = fixed
i7 = fixed. (6.129)
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.
For the first type of the region A with general l, and the second type of region A
with general lx, ly, lz, we can similarly check that the TNS satisfies the Generalized
SVD Condition. Numerically, we checked that the Haah code TNS 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. The numerical procedure for this check is to
159
list all the constraints for the indices in the region A and find how many solutions
exist for these constraints.
Entanglement entropy
We now compute the entanglement entropy for the exact SVD TNSs. We first consider
the case 1 with general l, such as in Fig. 6.14. 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. (6.130)
We further consider the case 2 — the 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. 6.15,
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 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
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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. (6.131)
6.5.4 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 straddling 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 (6.119), all 16 physical spins belong to the region
A. For l = 3 as shown in Fig. 6.13 (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 subscript 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 model: 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.
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. Notice 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
161
T T
Cut
x
x
x
xx
T T
Cut
x
x
(a)
(b)
Figure 6.16: Transferring the Pauli X operators of the Bc operator from the regionA (a) to the region A (b).
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 is to use the |{t}〉A to 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
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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. 6.16. The Bc operator acts on the
right cube Fig. 6.16 (a). The physical spins on the boundary of the region A which
are flipped by Bc are those covered by circled X in Fig. 6.16 (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. 6.16
(b). Notice that the T tensor for the Haah code is invariant under this action (see
the 12th cube in Eq. (6.123)). 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. (6.123) 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.
(6.132)
The left part of the tensor product is an eigenstate of Bc with eigenvalue 1.
Therefore, in the TNS decomposition Eq. (6.25), we can group the basis state |{t}〉A
which 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. Notice 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
163
form
|TNS〉 =∑{t}′
[∏c
(1 +Bc
2
)|{t}〉A
]⊗ |{t}〉A, (6.133)
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 compute the upper bound of the entanglement entropy, we need to find the number
of singular vectors 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 , (6.134)
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
region and Nc is the number of constraints on the open indices coming from the T
tensors within the region A. We thus get
log2(basis(TNS(A))) = Ng −Nc −NB (6.135)
and the upper bound on the entanglement entropy reads
S(A) = (Ng −Nc −NB) log 2. (6.136)
164
Counting Ng and NB
We first count Ng. The number of g tensors can be computed by looking at Fig. 6.13
(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.
(6.137)
We further count NB. As explained in Sec. 6.5.4, NB is the number of Bc operators
inside the region A and adjacent to the boundary of the region A. For a cubic region
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. (6.138)
For l = 2, we just have one Bc operator. Hence we have
NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2. (6.139)
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Counting Nc: Contribution from the T tensors
The open indices may be constrained by the T tensors 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 detail, and then we describe our algorithm to search the number
of linearly independent constraints.
For l = 2, as shown in Eq. (6.119), no g tensor has all virtual indices contracted.
There is only one T tensor within region A. The element of the T tensor is
Ti1i2i3i4i5i6i7i8 (6.140)
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.
(6.141)
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. (6.142)
There is only one constraint from the T tensor. Hence Nc = 1 for l = 2.
For l = 3, as shown in the Fig. 6.13 (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} (6.143)
166
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.
(6.144)
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 the region A. Hence we obtain a set of equations purely
in terms of the open indices. The number of constraints is the rank of this set of
equations. 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(6.145)
Hence, for l ≥ 3, there are
6l − 12 (6.146)
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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. (6.147)
Entanglement entropy
We are ready to collect all the data we have obtained and compute the entanglement
entropy for the cubic cut. For the entanglement cut of size l× l× l, the total number
of g tensors is
Ng = 12l2 − 24l + 16. (6.148)
The number of of T tensor constraints is
Nc = 6l − 12 + δl,2,∀l ≥ 2. (6.149)
The number of Bc operators is
NB = 6l2 − 24l + 26− δl,2,∀l ≥ 2. (6.150)
Therefore the upper bound of the entanglement entropy reads
S
log 2=Ng −Nc −NB
=6l2 − 6l + 2,∀l ≥ 2.
(6.151)
The entanglement entropies also have negative linear corrections.
168
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.
6.6 Conclusion and Discussion
In this chapter, we present our TNS construction for three stabilizer models in 3D.
The ground states of these stabilizer codes are the eigenstates of all local Hamiltonian
terms with +1 eigenvalues. The constructions of these TNSs share the same general
idea and work in other dimensions as well:
1. We introduce a projector g tensor for each physical spin which identifies the
physical index with the virtual indices.
2. The physical operators acting on the TNS can be transferred to the virtual
indices using Eq. (6.36).
3. The local T tensors contracted with the projector g tensors are specified by the
local Hamiltonian terms.
After we obtain the TNS for the ground state, we can prove that the TNS is
an exact SVD for the ground state with some specific entanglement cuts. The en-
tanglement spectra are completely flat for the models studied in this chapter. The
entanglement entropies can be computed by counting the number of singular vectors.
For the 3D toric code model, the entanglement entropies have a constant correction
to the area law, − log(2). For the X-cube model and the Haah code, the entanglement
entropies have linear corrections to the area law as shown in Sec. 6.4.4 and 6.5.4. The
analytical calculation of the entanglement entropies is rooted in the Concatenation
lemma, since the Concatenation lemma is introduced to count the number of
169
singular vectors. The Concatenation lemmas are rooted the symmetry properties
of the local tensors. For instance, Eq. (6.40) and (6.43) for the 3D toric code model.
The transfer matrices can also be constructed. For the 3D toric code and the
X-cube models, we prove that the transfer matrix is a projector whose dimension
is counted by the Concatenation lemma as well. For the 3D toric code model,
the transfer matrix is of dimension 2. For the X-cube model, the transfer matrix
is of dimension 2Lx+Ly−1 where Lx and Ly are the sizes of the torus in the x and y
directions respectively. The GSD on the torus is generally larger than the degeneracy
of the transfer matrix.
Since both the entanglement entropies and the transfer matrix degeneracies are
rooted in the Concatenation lemma (or more fundamentally the symmetry prop-
erties of the local T tensors), we believe that these two phenomena are related. More-
over, we conjecture that the negative linear correction to the area law is a signature
of fracton models. This is similar to the negative constant correction (i.e., the topo-
logical entanglement entropy[22, 23]) in 2D.
In this chapter, the TNSs are all the ground states of some exactly solvable local
models. If we move away from these fine-tuned points without going through phase
transitions, we expect the transfer matrix degeneracies to be still robust, since these
degeneracies give rise to the GSD. In Ref. [179], this statement has been numerically
verified in the 2D toric code model and its phase transitions to the trivial phases.
If we move away from the fine-tuned points, we also expect that the linear term
of the entanglement entropies for the fracton models does not vanish, although the
specific coefficients of the linear terms might change. An important result is about
the topological entanglement entropy. The topological entanglement entropy for the
fracton models was first introduced in Ref. [158], and is defined as the linear combi-
nations of the entanglement entropies of different regions, in order to exactly cancel
the area law. See Ref. [158] for the definition details. Importantly, the topological
170
entanglement entropies of fracton models are linear with respect to the sizes of the
entanglement cuts. Furthermore, Ref. [158] argues using perturbation theories that
the topological entanglement entropies, of the same three models as in our chapter
are robust to adiabatic perturbations. Hence, Ref. [158] indicates that there should
be a linear correction to the area law which does not vanish, even when moving away
from the fine-tuned wave functions.
However, we also have to admit that the rigorous statements, about the entan-
glement spectra, entropies and the transfer matrix degeneracies of a generic fracton
model ground state, need to be verified by the numerical studies for the 3D fracton
models in the future.
171
Appendix A
Appendix for Boson Condensation
A.1 Quantum dimensions of A and T
A.1.1 Proof of da =∑
r∈T nradr
From the Eq. (3.5) we obtain, by multiplying both sides by the quantum dimension
dt of the particle t in the T theory and summing over t
∑r,s,t∈T
nransbN
trsdt =
∑c∈A,t∈T
N cabn
tcdt =
∑r,s∈A
nransbdrds, (A.1)
where we are only considering T theories which are also fusion categories (not braided
ones) and hence satisfy the equivalent Eq. (1.31) for the T theories
∑t∈T
N trsdt = drds. (A.2)
We then have the trivial re-writing
∑c∈A
N cab
(∑t∈T
ntcdt
)=
(∑r∈T
nradr
)(∑s∈T
nsbds
), (A.3)
172
which means that(∑
t∈T ntcdt)
is an eigenvalue of the matrix (Na)bc with eigenvalues
and eigenvector(∑
r∈T nradr)
and(∑
s∈T nsbds), respectively. Since the eigenvector
has positive entries, by the Perron-Frobenius theorem, the eigenvalue is the largest
eigenvalue of the Na matrix, and hence it is indeed da
da =∑r∈T
nradr. (A.4)
A.1.2 Proof of dr = 1q
∑a∈A n
rada
We start with Eq. (3.5), multiply by da and sum over a ∈ A. Using Eq. (1.31), it
follows
∑r,s∈T
∑a∈A
nradaNtrsn
sb =
∑a,c∈A
ntcNcabda
= db∑c∈A
ntcdc.
(A.5)
For the simplicity of notations, let αt ≡∑
c∈A dcntc, which satisfies the eigenvalue
equation
∑r∈T
(∑s∈T
nsbNs
)tr
αr = dbαt. (A.6)
Notice the unorthodox use of the matrix (Ns)tr = N tsr, unlike in the line following
Eq. (1.31). We define the matrix this way in order not to use the equation nta = nta.
This matrix has the vector of quantum dimensions (d1, . . . , dN)T, where N are the
number of particles in the fusion category T , as an eigenvector ∀s in (Ns)tr. Since
we are using the Ns matrix in an unorthodox fashion (it is the transpose of the usual
Ns matrix), we prove the statement
∑t
N rstdr = dsdt =
∑t
N tsrdr = dsdt. (A.7)
173
It follows from the above that∑
r(Ns)trdr = dsdt. Hence (d1, . . . , dN)T is a common
eigenvector of all the Ns, even as defined in the unusual way above.
We now sum Eq. (A.6) over b to get
∑r∈T
(∑b∈A
∑s∈T
nsbNs
)tr
αr =
(∑b∈A
db
)αt. (A.8)
The matrix (∑
b∈A∑
s∈T nsbNs)tr is a completely positive matrix with integer strictly
positive coefficients: for any t, r, there exists s such that N tsr > 0 and for every s
there exists an nsb > 0. As such, it satisfies a stronger version of the Perron-Frobenius
theorem which says that there is a unique eigenvector with all elements positive, and
all other eigenvectors have at least one negative element. As such, since αt is all
positive, we identify it as the unique largest eigenvector. But since the Ns have a
common eigenvector, the quantum dimensions of the condensed theory, we then can
identify this eigenvector with
αt =∑c∈A
dcntc = qdt, (A.9)
where q is a proportionality constant. We now find two expressions for it. First,
multiplying Eq. (A.9) by dt and summing over t gives
q∑t∈T
d2t =
∑c∈A
dc∑t∈T
ntcdt =∑c∈A
d2c , (A.10)
where the last equality follows from Eq. (A.4). This implies
q = D2A/D
2T . (A.11)
174
Furthermore, multiplying Eq. (3.5) by dadb for t = ϕ and summing over a, b reads
∑c∈A
N cabn
ϕc =
∑t∈T
ntantb, (A.12)
which implies
∑a,b,c∈A
dadbNcabn
ϕc =
∑t∈T
∑a∈A
danta
∑b∈A
dbntb
= q2D2T .
(A.13)
On the other hand,
q2D2T =
∑a,b,c∈A
dadbNcabn
ϕc
=∑b,c∈A
d2bdcn
ϕc
= D2A
∑c∈A
dcnϕc ,
(A.14)
hence
q =∑c∈A
dcnϕc . (A.15)
A.2 Chiral algebra
In this section, we review the connection between the above formalism and CFT. As
pointed out by Bais and Slingerland [42], the mathematics of boson condensation has a
parallel in conformal field theories. First, for at least some MTCsA, the particle labels
are in one-to-one correspondence with the conformal families in some (not necessarily
unique) CFT. (The MTC-conformal block correspondence generalizes Witten’s work
[80] on the relationship between Chern-Simons theory and chiral Wess-Zumino-Witten
175
models.) Second, when this correspondence holds, the process of condensation in the
TQFT is closely related to extending the chiral algebra in the CFT [115].
Let us consider a CFT with a chiral algebra A which contains the stress-tensor
T (z) and all locally commuting holomorphic operators in the theory such as currents
Ja(z) associated to Lie groups, etc. The mode expansions of these operators give
rise to infinite dimensional algebras, like Virasoro, Kac-Moody or W -algebras. The
irreducible representation spaces of the chiral algebra A, denoted by Ha, are labelled
by the primary fields a, whose number is finite in a RCFT. The primary fields are
in one-to-one correspondence with the anyons of a TQFT. The TQFT is nothing but
the CFT reduced to its basic topological data like braiding and fusion matrices, etc.
(However, due to this reduction, several distinct CFTs may correspond to the same
TQFT.)
For each representation space Ha there is a character
χa(τ) = TrHae2πiτ(L0−c/24), (A.16)
given by the partition function of the states in Ha propagating along a torus with
modular parameter τ (with Im τ > 0). The constant c is the central charge of the
CFT and L0 is the zero element of the Virasoro algebra. The modular transformations
act on the characters as
χa(τ + 1) = θae− iπc
12 χa(τ), (A.17)
χa
(−1
τ
)=
∑b
Sabχb(τ),
where θa = e2πiha is the topological spin, ha the conformal weight of the primary
field a. The full CFT also contains an anti-chiral algebra, A, which for simplicity we
assume to be isomorphic to A. Correspondingly, the complete Hilbert space is the
176
tensor product H = ⊕aHa ⊗ Ha and the total partition function is
Zdiag(τ, τ) =∑a
χa(τ)χa(τ), (A.18)
which is modular invariant thanks to the S, T unitarity: SS† = TT † = 1. The
pairing between the left and right states of a non-chiral CFT can be more general
than (A.18)
Z(τ, τ) =∑a,b
χa(τ)Mab χb(τ), (A.19)
where M is called the mass matrix which must satisfy [S,M ] = [T,M ] = 0 to guaran-
tee the modular invariance of the partition function (A.19). A fundamental problem
in RCFT is to classify all possible modular invariant partition functions, that is,
mass matrices M . This program has been achieved for theories with simple currents
[100, 101, 102], but it is far from being solved in general.
There are three types of mass matrices: i) Those associated to automorphisms of
the fusion rule algebra, ii) those corresponding to a chiral extension of A, and iii) a
combination of i) and ii). This result is related to the naturality theorem due to Moore
and Seiberg: In a CFT when the left and right chiral algebras are maximally extended
the field content matrix defines an automorphism ω of the fusion rule algebra, i.e.:
Ma,b = δa,ω(b) [107]. A chiral algebra A ⊗ A is called maximally extended when it
includes all the holomorphic and antiholomorphic fields in H (i.e., those with integer
conformal weights). [106]
The mass matrices and the associated naturality theorem have a precise correspon-
dence within the boson condensation encountered in the main text. Let us explain it
in more detail.
An extension of the chiral algebra A can arise if there exists a subset {γi} of pri-
mary fields with integer conformal weights that are mutually local. One can therefore
177
add these holomorphic fields to those already included in A to obtain an extended
chiral algebra U . It is then clear that the representation spaces of the new algebra
U should be a combination of those of the original algebra A. In particular, the
(irreducible) conformal family vector space Hϕ corresponding to the new identity
representation ϕ will be the direct sum of the old identity conformal family H1 plus
the conformal families corresponding to the old primaries γi, that is Hϕ = H1⊕iHγi .
The fields γi correspond to the bosons that condense in the TQFT. The space Hϕ is
the CFT version of the vacuum after condensation.
The irreducible representation spaces of the extended chiral algebra U , denoted by
Hu, break down into the direct sum of irreducible representations Ha of the smaller
algebra A. Such decompositions are called branching rules and are noted as
Hu → ⊕a∈AnuaHa. (A.20)
The branching coefficient nua gives the multiplicity of the irreducible representation
a of A in the decomposition of the irreducible representation u of U . The fields
appearing in the decomposition (A.20) have to be mutually local with respect to the
fields in the chiral algebra U . From Eqs. (A.20) and. (A.16) follows the expression for
the character of the representation u in terms of the characters of the representations
a [recall Eq.(C1)] [115]
χu(τ) =∑a∈A
nua χa(τ), u ∈ U . (A.21)
The primary field u corresponds to a deconfined anyon in the TQFT. The TQFT
Eq. (3.4) means in CFT that the primary fields that built up a representation of
the extended algebra must have the same conformal weights modulo integers. On
the other hand, if a field a is such that the orbit γi × a, ∀i contains fields with
different conformal weights, then they disappear from the representation theory of
178
U . These fields are associated to the confined anyons defined in Eq. (3.3). Given the
characters (A.21) of the extended chiral algebra U , one can construct the diagonal
partition function
Z(τ, τ) =∑u∈U
|χu(τ)|2, (A.22)
which when written in terms of the characters (A.16) of A reads like Eq. (A.19) with
Mab =∑u∈U
nua nub . (A.23)
This equation shows that an extension of the chiral algebra gives rise to an off-diagonal
partition function and in turn to a boson condensation in the TQFT.
The original and chirally extended CFTs are both assumed to be modular theories,
with their characters transforming under modular transformation S and T of the torus
parameter τ as
χs
(−1
τ
)=∑t
Sstχt(τ) =∑t,a
Sstntaχa(τ)
=∑b
nsbχb
(−1
τ
)=∑a,b
nsbSbaχa(τ),
(A.24)
i.e.,
nS = Sn. (A.25a)
Similarly
nT = Tn, (A.25b)
where S and T are modular matrices for the U algebra. Equation (A.25a) and
Eq. (A.25b) also appear as matching conditions in the study of gapped domain walls
between two topological phases [64].
One can easily deduce that [M,S] = [M,T ] = 0. Moreover, through Eq. (A.25a)
and Eq. (A.25b), we can show that
179
1. c = c (mod 24),
2. θs = θa, if nsa 6= 0,
3. nC = Cn,
4. q ≡∑
a nϕada = DA/DU ,
5. dt = 1q
∑a∈A n
tada,
where C and C are the charge conjugation matrices for the U and A theories re-
spectively, and DU and DA are total quantum dimension of the U and A theory,
respectively.
So far we have discussed the mass matrices that correspond to extensions of the
chiral algebra. The other possibility is that the mass matrix is a permutation P of the
irreducible representations of A that corresponds to an automorphism of the fusion
rules [115]. This case does not describe boson condensation. The third possibility is
that the mass matrix describes an off diagonal partition function of the chiral algebra
U , namely M = nPnT, with P a permutation automorphism of the fusion rules of U .
These possibilities were mentioned before in connection with the Moore and Seiberg
naturally theorem [107].
The conclusions we obtain above, including Eq. (A.25a) and Eq. (A.25b), can
be viewed as necessary conditions for boson condensation. So, a solution of the
above consistency equations does not guarantee the existence of a boson condensation
A → U . It could still happen, for example, that the fusion coefficients derived from
such a solution via the Verlinde formula are not integers (see Appendix A.3.2 for an
example). Then, the solution has to be discarded. However, the absence of a solution
does imply that there is no boson condensation A → U .
180
A.3 Condensations in SU(2) CFTs
To illustrate some properties of the condensation transition we consider the family
of CFTs that correspond to SU(2) at level k. These theories have (k + 1) primary
fields in corresponding conformal blocks labelled by integers a = 0, . . . , k that are
denoted as φa, and the corresponding conformal characters are denoted as χa. (In
the corresponding TQFT, the anyon with a = 0 is the vacuum.) The matrix elements
of the modular S and T matrices are given by
Sab =
√2
2 + ksin
π(a+ 1)(b+ 1)
k + 2, (A.26)
and
Tab = e2πia(a+2)4(k+2) δab, c =
3k
k + 2. (A.27)
All the modular invariant partition functions of this CFT were obtained by Cappelli,
Itzykson and Zuber who found a surprising correspondence with the ADE classifica-
tion of Lie groups [114]. The complete list is
ZAk+1=
k∑n=0,n∈Z
|χn|2, (A.28a)
ZD2`+2=
2`−2∑n=0,n∈2Z
|χn + χ4`−n|2 + 2|χ2`|2,
ZD2`+1=
4`−2∑n=0,n∈2Z
|χn|2 + |χ2`−1|2
+2`−3∑
n=1,n∈2Z+1
(χnχ4`−2−n + χ4`−2−nχn),
181
ZE6 = |χ0 + χ6|2 + |χ3 + χ7|2 + |χ4 + χ10|2,
ZE7 = |χ0 + χ16|2 + |χ4 + χ12|2 + |χ6 + χ10|2
+|χ8|2 + χ8(χ2 + χ14) + (χ2 + χ14)χ8,
ZE8 = |χ0 + χ10 + χ18 + χ28|2
+|χ6 + χ12 + χ16 + χ22|2 ,
where k = 4` in ZD2`+2, k = 4` − 2 and in ZD2`+1
, while k = 10 in ZE6 , k = 16 in
ZE7 , and k = 28 in ZE8 . Here, χa are the characters of the irreducible representation
spaces of the chiral algebra of SU(2)k.
The origin of these off-diagonal partition functions is the following:
• D2`+2: J = φ4`, is a bosonic simple current with integer conformal weight
hJ = `. For ` = 1, φ4 is a current that yields a chiral extension corresponding
to the conformal embedding SU(2)4 ⊂ SU(3)1. (Notice that the central charge
of the two CFTs is the same cSU(2)4 = cSU(3)1 .)
• D2`+1: the simple current J = φ4` has half-odd conformal weights, hJ = `−1/2,
so it does not yield an extension of the chiral algebra, i.e., it does not correspond
to condensation. The partition function can be written as ZD2`+1=∑
a χa χω(a),
where ω is the unique automorphism of the fusion rules, namely ω(a) = a for a
even and ω(a) = k − a for a odd.
• E6: chiral extension with the field φ6 with h6 = 1. This is not a sim-
ple current. The chiral extension corresponds to the conformal embedding
SU(2)10 ⊂ SO(5)1, both CFT’s have the same central charge, namely c = 5/2.
The SO(5)1 algebra can be constructed with 5 Majorana fermions (i.e. Ising
models). In the SU(2)10 theory one has h4 = 1/2, h10 = 5/2, h3 = 5/16,
182
h7 = 21/16, so that h10 − h4 = 2 and h7 − h3 = 1. The field φ3 can be built
from the product of 5 spin fields of the Ising model which have hσ = 1/16.
• E7: explained by an exceptional automorphism of the D10 chiral algebra
[107, 115] [see Eq.(A.34)].
• E8: chiral extension with three operators with h10 = 1, h18 = 3, and h28 = 7.
The remaining fields in ZE8 have weights: h6 = 2/5, h12 = 7/5, h16 = 12/5,
h22 = 22/5. The central charge is c = 14/5, which coincides with that of G2 at
level k = 1 [115].
The results explained above can be summarized in the following table:
Type k Z Comments
Ak+1 k - -
D2`+2 4` EXT SU(2)4 ⊂ SU(3)1
D2`+1 4`− 2 AUT -
E6 10 EXT SU(2)10 ⊂ SO(5)1
E7 16 AUT -
E8 28 EXT SU(2)28 ⊂ (G2)1
(A.29)
where E6, E7, E8 and G2 are the exceptional Lie groups, while EXT and AUT stand
for an extension of the chiral algebra and an automorphism of the theory, respectively.
Note that some theories, e.g., k = 16 have a D as well as a E invariant, as case that
we will now discuss in detail.
183
A.3.1 SU(2)16
The SU(2)16 CFT is special in that it has two different off-diagonal partition functions,
given by [recall Eq. (A.28a)]
ZD10 = |χ0 + χ16|2 + |χ2 + χ14|2 + |χ4 + χ12|2
+ |χ6 + χ10|2 + 2 |χ8|2(A.30)
and
ZE7 = |χ0 + χ16|2 + (χ2 + χ14)χ8 + χ8(χ2 + χ14)
+ |χ4 + χ12|2 + |χ6 + χ10|2 + |χ8|2 .(A.31)
Both of these theories correspond to a condensation of the boson a = 16. There are
exactly two distinct solutions to the equation [M,S] = [M,T ] = 0, given by
M (1) = nnT, M (2) = nPnT, (A.32)
where
nT =
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0
0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
(A.33)
184
and
P =
1 0 0 0 0 0
0 0 0 0 0 1
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 1 0 0 0 0
(A.34)
is an automorphism of the theory U . These two solutions for M are in one-to-one cor-
respondence with the two off-diagonal partition functions above. Here, M (1) encodes
the condensation transition itself, while the existence of the additional matrix M (2) is
related to the “naturality theorem” discussed in the main text and in Appendix A.2.
Interestingly, the equation Sn = nS, that yields the S matrix of the theory after
condensation, has two distinct solutions S and S ′, where
S =2
3
sin(π18
)12
cos(
2π9
)cos(π9
)12
12
12
1 12
−12
−12−1
2
cos(
2π9
)12− cos
(π9
)− sin
(π18
)12
12
cos(π9
)−1
2− sin
(π18
)cos(
2π9
)−1
2−1
2
12
−12
12
−12
−12
1
12
−12
12
−12
1 −12
(A.35)
185
and S ′ is obtained by exchanging the last two rows of S, a so-called Galois symme-
try [202]. Both matrices S and S ′ yield the same fusion rules Nt, e.g.,
N3 =
0 0 0 1 0 0
0 0 1 1 1 1
0 1 1 2 1 1
1 1 2 2 1 1
0 1 1 1 0 1
0 1 1 1 1 0
. (A.36)
It is worth noting that this condensation of a theory without multiplicities (all N cab in
SU(2)16 are 0 or 1) yields a theory with multiplicities: some of the N str in Eq. (A.36)
are larger than 1.
A.3.2 SU(2)28
The SU(2)28 CFT is special in that it also has two different off-diagonal partition
functions, given by [recall Eq. (A.28a)]
ZD16 = |χ0 + χ28|2 + |χ2 + χ26|2 + |χ4 + χ24|2
+ |χ6 + χ22|2 + |χ8 + χ20|2 + |χ10 + χ18|2
+ |χ12 + χ16|2 + 2 |χ14|2
(A.37)
and
ZE8 = |χ0 + χ10 + χ18 + χ28|2
+ |χ6 + χ12 + χ16 + χ22|2 .(A.38)
As is clear from ZE8 , the particles 10, 18, and 28 are bosons, besides the vacuum
0. The two partition functions correspond to two distinct condensations possible in
186
SU(2)28. These are the only condensations possible. Their corresponding n matrices
can be read off directly from these partition functions.
The partition function ZE8 stands for a condensation of all bosons with nϕa = 1
each. The resulting theory is the Fibonacci TQFT with particles 6, 12, 16, and 22
each restricting to the τ particle.
The partition function ZD16 corresponds to the condensation of the top-level boson
28 only, which results in a 9-particle non-Abelian TQFT with some multiplicities
larger than 1. For example, the restriction of 10 and 18, which we call 5, obeys the
fusion rule [c.f. Eq. (A.36)]
5× 5 = 0 + 1 + 2 + 3 + 2 · 4 + 2 · 5 + 2 · 6 + 7 + 8. (A.39)
Condensing four layers of Ising TQFT
Here we give some detail on one particular condensation in a theory composed of a
tensor product of 4 layers of the Ising TQFT. We show that, generically, in our algo-
rithm in the main text Sec. 3.6, we do need to check that the resulting S matrix gives
integer fusion matrices using Verlinde’s formula. This is not a complete discussion of
all possible condensations in the 4 layer Ising theory.
We focus on the condensate containing 1111, 11ψψ, 1ψ1ψ, and 1ψψ1, but not
ψψψψ, ψψ11, ψ1ψ1, and ψ11ψ. (Including these other bosons in the condensate
would yield the ν = 4 theory from Kitaev’s 16-fold way.) The corresponding M matrix
has only one nonzero entry on the column (and row) that corresponds to the σσσσ
particle, namely Mσσσσ,σσσσ = 4. Since the quantum dimension dσσσσ = 4, this allows
for two distinct solutions n: one in which σσσσ particle restricts to twice some particle
a and one in which it splits into 4 Abelian particles a1, a2, a3, a4. In both cases, we
can find solutions to Eq. (3.64) that are unitary and satisfy S2 = Θ(ST )3 = C.
187
However, for the theory in which the σσσσ particle splits into 4 particles, the
fusion coefficients Nt obtained from Eq. (3.65) are not all nonnegative integers (some
of them are ±1/2). This establishes by example that we have to impose that Nt is
nonnegative integer valued, in addition to the other conditions in Sec. 3.6. It also
shows that in the example at hand, despite the ambiguity in the possible solutions
for n, the particle content (up to possible automorphisms) of the final theory is fixed
by the choice of condensate. Whether this statement is true in general is presently
not known to us.
The allowed solution, in which σσσσ particle restricts to twice the same particle,
is the one we naively expect, upon inspection of the anyons in the condensates by the
following argument: all condensed anyons have a vacuum particle 1 in the first layer.
Hence the Ising theory in the first layer will be preserved under condensation and the
result is a direct product of the ν = 1 Ising theory and the ν = 3 Ising theory from
Kitaev’s 16-fold way. The particle that σσσσ twice restricts to is the direct product
of a ν = 1 Ising σ and ν = 3 Ising σ, which we have already proved in Sec. 3.7.1 has
nσ′
(σσσ) = 2.
188
Appendix B
Appendix for Uncondensable
Bosons
B.1 No-go theorem with Abelian sector
We have seen from the examples discussed in the main text, that the no-go theorem
can often be used to not only show that individual bosons in a TQFT cannot con-
dense, but that an entire TQFT is not condensable. Here, we extend this discussion
to examples of TQFTs that have noncondensable sub-structures. This problem is
motived by physical examples: in the fractional quantum Hall effect, for instance,
one frequently discusses phases that are described by a direct (or semi-direct) prod-
uct of an Abelian and a non-Abelian TQFT. A simple example is the Z3 Read-Rezayi
state of bosons, which is described by the TQFT AFib × Z2. While such a theory
admits condensations, already in the Z⊗N2 sector, when enough layers are considered,
one has the intuition that the noncondensability of Fibonacci should still constrain
the possible condensations.
Lemma 1. Consider a TQFT
A×X , (B.1)
189
where X is an Abelian TQFT (i.e., all its anyons have quantum dimension 1). Fur-
ther, for all particles b ∈ A (not only for the bosons), except for the vacuum, let there
exist a set Ib = {a1, . . . , am} of zero modes of b, containing anyons from A, such that
the quantum dimensions satisfy
db ≤m∑i=1
dai . (B.2)
Then, any possible condensation transition will lead to a theory of the form
A× Y , (B.3)
where the Abelian TQFT Y can be obtained from X through a condensation.
Proof. This Lemma follows almost directly from the no-go theorem. Let us denote a
particle from A×X by the pair (b, x) where b ∈ A and x ∈ X . If (b, x) is boson, we
can show that it has to be an uncondensable one, except if b = 1. The set
I(b,x) = {(a1, x), · · · , (am, x)} , (B.4)
(where a1, · · · , am form a set Ib of zero modes of b whose existence is guaranteed by
assumption) satisfies all the conditions 1–3 form the definition of a set of (b, x) zero
modes. Since x is an Abelian particle, dx = 1 and Eq. (B.2) directly implies that the
sum of the quantum dimensions of the particles in I(b,x) satisfies the inequality (4) from
the main text. Hence, (b, x) cannot condense. In turn, this implies any condensable
boson in A × X is of the form (1, x). A condensate of this form is transparent to
the anyons in A and will thus leave this sub-TQFT unaffected. It will only induce a
condensation X → Y , so that the final theory is of the from (B.3).
We return to the example of AFib × Z2. Consider N layers of this theory, i.e.,
A⊗NFib × Z⊗N2 . This multi-layer TQFT satisfies all assumptions of Lemma 1: for each
190
anyon b ∈ A⊗NFib , a choice for the set Ib is given by Ib = {1, b}. This is so because
all possible bosons appearing in the fusion product of b× b are uncondensable by the
no-go theorem and the sum of the quantum dimensions of Ib, given by 1 +db is larger
than db. We conclude that the A⊗NFib structure is preserved under any condensation
transition in such a theory.
B.2 Proof for Example (iii), Multiple layers of
SO(3)k
In this section, we show that no condensation is possible in the TQFT SO(3)⊗Nk
comprised of N layers of SO(3)k for any odd k and any integer N . The proof goes
by induction. We denote the particles in SO(3)⊗Nk with a shorthand notation. An
anyon that has the identity particle from SO(3)k in all layers, except for the k0 layers
i1, i2, · · · , ik0 , is denoted by {ji1ji2 · · · jik0}. Here 1 ≤ jil ≤ (k − 1)/2 can stand for
any anyon from SO(3)k (except the identity 0), for all l = 1, · · · , k0.
Induction base
First, consider particles {ji} with just one nontrivial anyon in some layer i. This will
serve as the induction base. By the no-go theorem and our proof in Example (ii),
we know that no bosons of form {ji} can condense. [Use the particles with only one
nontrivial in that same layer i to build the set Iji as elaborated for Example (ii).] As
a corollary, the anyons {ji} do not split: when fused with themselves no condensable
boson appears in the fusion product, which prevents splitting by Eq. (2) from the
main text for t = ϕ.
Induction step
We assume that for any 1 ≤ l ≤ k0 all {ji1ji2 · · · jil}
191
1. do not condense and
2. do not split.
We now show the induction step, namely that all particles with nontrivial anyons in
(k0 + 1) layers {ji1ji2 · · · jik0+1} neither condense nor split.
We begin by showing that {ji1ji2 · · · jik0+1} cannot condense. The particles
{ji1ji2 · · · jik0+1} can be obtained by fusing a {ji1ji2 · · · jik0} with a {jik0+1
}, where
ik0+1 /∈ {i1, · · · , ik0}. In this case, Eq. (2) from the main text reads for t = ϕ
Nϕ{ji1 ···jik0 }
↓,{jik0+1}↓ = nϕ{ji1ji2 ···jik0 jik0+1
}. (B.5)
Now, because of the uniqueness of the antiparticle, Nϕ{ji1 ···jik0 }
↓,{jik0+1}↓ can be either
0 or 1. If it was 1, {ji1ji2 · · · jik0}↓ would be the antiparticle of {jik0+1
}↓. Because
all particles are their own antiparticles, this would imply {ji1ji2 · · · jik0}↓ = {jik0+1
}↓.
However, this is not possible for k0 > 1, because the associativity of fusion would
then also imply that {ji1}↓ is the antiparticle (and coinciding with) {ji2 · · · jik0 jik0+1}↓,
i.e., {ji1}↓ = {ji2 · · · jik0 jik0+1}↓. Remembering that {ji1ji2 · · · jik0}, {ji2ji2 · · · jik0+1}
do not split, and equating the quantum dimensions of the particles for these two
identifications we have
dji1dji2 · · · djik0 = djik0+1,
dji1 = dji2 · · · djik0 djik0+1.
(B.6)
For k0 > 1, this contradicts the fact that all nontrivial particles in this theory have
quantum dimensions d > 1. This rules out the possibility Nϕ{ji1 ···jik0 }
↓,{jik0+1}↓ = 1 and
shows that {ji1ji2 · · · jik0+1} does not condense for k0 > 1.
The case k0 = 1 needs to be considered separately, as both lines in Eq. (B.6)
are identical in this case, and therefore do not lead to a contradiction. Assume that
Nϕ{ji1}↓,{ji2}↓
= 1. In the case ji1 6= ji2 , we can rely on the following argument to
192
disprove this assumption: As all anyons in SO(3)k with k odd have distinct quantum
dimension, it follows that the two anyons {ji1} and {ji2} restrict to distinct particles
and in particular ϕ /∈ j↓i1 × j↓i2
– with Eq. (2) from the main text this implies that
{ji1ji2} neither splits nor condenses. In the case ji1 = ji2 ≡ j, define j ≡ {ji1ji2}.
We want to show that j does not condense. As there are no fermions in SO(3)k with
k odd, j can only be a boson if θj = 1, i.e., if {ji1} and {ji2} are bosons. Our no-go
theorem applies to all bosons {ji1} and {ji2} with zero mode sets I{ji1} and I{ji2}. We
can then use the set Ij = I{ji1}×I{ji2}, containing the fusion product of any particle
in I{ji1} with any particle in I{ji2}, to prove that j cannot condense. To show that
Ij is a set of zero modes of j, the main challenge is to show that the product of any
two elements from Ij cannot condense. The product of any two elements from Ij is
always of the form {ji1ji2}. We have shown that when ji1 6= ji2 such particles cannot
condense. We therefore need only show that non-trivial particles of form {ji1ji2} with
ji1 = ji2 both bosons cannot condense. In order to show they are not condensable, we
can use the proof given for Example (ii). For that, observe that the anyons j have the
same fusion coefficients among themselves as the j anyons in SO(3)k in Example (ii)
have, i.e., N j′′
j,j′= N j′′
j,j′ , where j, j′, j′′ ∈ SO(3)k. Recall that conditions 1–3 from
the definition of a set of zero modes only depend on the fusion coefficients N j′′
j,j′ and
the information, which particles are bosons. Hence, conditions 1–3 are satisfied for
Ij whenever they are satisfied for Ij in Example (ii). It remains to show that Ij is of
large enough quantum dimension to satisfy the fundamental inequality Eq. (4) from
the main text. For j, Eq. (4) from the main text takes the form
dj = d2j <
(∑a∈Ij
da
)2
=∑a∈Ij
da. (B.7)
193
Upon taking the square root, this is equivalent to Eqs. (14) and (15) from the main
text, which were shown to hold in Example (ii). Therefore the j = {ji1ji2} anyons do
not condense and all {ji} have distinct restrictions.
We conclude that for any k0 ≥ 1 only Nϕ{ji1 ···jik0 }
↓,{jik0+1}↓ = 0 is permitted and
hence Eq. (B.5) implies that {ji1ji2 · · · jik0+1} does not restrict to the identity ϕ, i.e.,
it does not condense. This proves the assumption 1 of the induction step for k0 + 1.
To complete the induction step, we need to show that {ji1ji2 · · · jik0+1} does not
split. For that, consider Eq. (2) from the main text for {ji1 · · · jik0+1} with itself and
t = ϕ
∑r
(nr{ji1 ···jik0+1
}
)2
=∑c
N c{ji1 ···jik0+1
},{ji1 ···jik0+1} n
ϕc
=nϕ1 = 1. (B.8)
We have used that none of the {ji1ji2 · · · jil} with 1 ≤ l ≤ k0 + 1 can restrict to
the identity ϕ since they cannot condense. This implies none of {ji1 · · · jik0+1} splits,
which proves the assumption 2 of the induction step for k0 + 1.
We have thus shown inductively that none of the particles (except for the vacuum)
restricts to the vacuum in the N -layer theory SO(3)⊗Nk . Thus, there is no condensate
and with it no condensation in any number N of layers of SO(3)k with k odd.
B.3 General constraints on boson condensation
In this section, we list lemmas that pose other general constraints on condensation
transitions in TQFTs.
Lemma 2. Suppose S = {a1, · · · , am} is a collection of particles in a TQFT A with
ai × ai containing no bosons other than the identity – i.e., ntai = δta↓i
and ai does not
194
split. Moreover assume a↓i 6= a↓j for i 6= j. Then if a boson B appears in the fusion of
ai and aj, ai × aj = B + · · · for any i 6= j, then B is not condensable.
Proof. Using Eq. (2) from the main text for a = ai, b = aj and t = ϕ, we have
δij =∑
t ntaintaj =
∑c n
ϕcN
caiaj
. For i 6= j we get∑
B nϕBN
Baiaj
= 0. So if boson B
appears in ai × aj, we must have nϕB = 0, so that B is not condensable.
Lemma 3. Consider a TQFT A with no fusion multiplicity and just one boson B.
If B is condensed then either B is abelian or B↓ = ϕ+ r where r is a single anyon.
Proof. As there is just a single boson, B = B. Equation (2) from the main text
implies ∑t
ntBntB =
∑c
nϕcNcBB = 1 + nϕBN
BBB. (B.9)
Notice, however, that the left-hand side is greater or equal to nϕBnϕB. For condensation,
this implies nϕB = 1, and tells us that∑
t ntBn
tB = 1 or 2. In the former case, B↓ = ϕ.
This implies dB = 1, and so B is a quantum dimension 1 boson hence must have
NaBB = δa,1. In the latter case, B restricts to just two particles with multiplicity 1
each, so that B↓ = ϕ+ r.
Lemma 4. With the conditions of Lemma 3, and assuming B has dB > 1, conden-
sation of B can only occur if NaBBN
bBB ≤ NB
ab for all anyons a and b of A.
Proof. Lemma 3 shows B↓ = ϕ + r, where r a simple object. Consider a ∈ A where
a 6= 1, B. Equation (2) from the main text for b = B and t = ϕ reads
nranrB = nra = NB
aBnϕB = NB
aB(B.10)
Consider now Eq. (2) from the main text for b 6= 1, B and for a 6= b, which gives
∑t
ntantb = NB
ab ≥ nranrb (B.11)
195
Combining the a,B and b, B and a, b equations gives the inequality
N bBBN
aBB ≤ NB
ab. (B.12)
196
Appendix C
Appendix for Tensor Network
States and Fracton Models
C.1 Proof for the Concatenation Lemma for the
3D Toric Code Model
In this section, we prove the Concatenation lemma in Sec. 6.3.3 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 in Sec. 6.3.3, then the contrac-
tion 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 in Sec. 6.3.3.
(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 in Sec. 6.3.3, then the contraction over one pair of indices, say
197
∑t1 t1
Tt1t2t3...Tt1 t2 t3...δt1 t1 , still satisfies the Concatenation lemma of the
3D toric code model in Sec. 6.3.3.
Proof:
(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)
198
(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:
199
(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 in Sec. 6.3.3, then their
contraction over any pairs of indices still satisfies the Concatenation lemma
of the 3D toric code model in Sec. 6.3.3.
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 Proof for the Concatenation Lemma for the
X-cube Model
In this section, we prove the Concatenation lemma in Sec. 6.4.3 for the X-cube
model by induction. We point out that the subtlety of the Concatenation lemma
of the X-cube model is that the number of constraints for the open virtual indices is
linear with respect to the system size. More precisely, each constraint corresponds to
a set of open virtual indices along a xy, yz or xz plane to have an even summation.
See Eq. (6.86) for an example. Hence, we need to keep track of the constraints when
200
we contract more tensors. For clarity, we denote the sets of constraints of {t} as:
fxy({t}) :
{ ∑i∈xy-plane
ti, for all xy-planes
}
fyz({t}) :
{ ∑i∈yz-plane
ti, for all yz-planes
}
fxz({t}) :
{ ∑i∈xz-plane
ti, for all xz-planes
}.
(C.7)
Notice that these equations are not linearly independent. Their summation is auto-
matically true. Physically, we can view these quantities fxy({t}), fyz({t}) and fxz({t})
as the “parities” in each xy, yz and xz plane. To begin with, we propose and prove
Lemma (D) which is the induction step:
(D) If T is a network of contracted local T tensors of the X-cube model which
satisfies the Concatenation lemma of the X-cube model in Sec. 6.4.3, then
the contraction of T and a T tensor still satisfies the Concatenation lemma
of the X-cube model in Sec. 6.4.3.
Proof:
We first fix the notations: the elements of T are T{t}; {tc} = {tc1, tc2, . . .} ⊂ {t}
are the indices that contract with the indices of the local T tensor; and the outcome
tensor is denoted as T′{t′} where {t′} denotes the open virtual indices after contraction.
Since T is assumed to satisfy the Concatenation lemma, we have:
T{t} =
N if
fxy({t}) = 0 mod 2
fyz({t}) = 0 mod 2
fxz({t}) = 0 mod 2
0 otherwise,
(C.8)
201
where N is the constant independent of the open virtual indices {t}, and fxy({t}),
fyz({t}) and fxz({t}) denote the set of summations over the open virtual indices in
each xy, yz and xz plane. See Eq. (6.92) for an example when T is a contraction of two
local T tensors. Notice that the elements of Txxyyzz also satisfy the Concatenation
lemma for the X-cube model, as shown by Eq. (6.86).
Using these notations, the tensor contraction is just:
T′{t′} =∑{tc}
T...tc1...tc2...T...tc1...tc2.... (C.9)
We now discuss one particular way of contraction: contraction over one pair of indices.
Other contractions can be proved using the exact same method.
Suppose {tc} contains only one index. Without loss of generality, we assume that
this index is the x index of Txxyyzz. Then the tensor contraction is:
T′{t′} =∑x
T...x...Tx... (C.10)
Graphically,
xy
z
z
y
x . (C.11)
202
When T...x... and Tx... are both nonzero, the indices satisfy that:
fxy({t}/x, x) =0 mod 2
fxz({t}/x, x) =0 mod 2
x+ x+ y + y =0 mod 2
x+ x+ z + z =0 mod 2
⇒
fxy({t}/x, x, y, y) =0 mod 2
fxz({t}/x, x, z, z) =0 mod 2,
(C.12)
where {t}/x denotes the t indices excluding the x index. We only list the equations
whose variables include the index x. Then, we include the fyz constraints from T and
T , which can be concatenated into fyz({t′}). We find:
T′{t′} =
N if
fxy({t′}) = 0 mod 2
fyz({t′}) = 0 mod 2
fxz({t′}) = 0 mod 2
0 otherwise.
(C.13)
Hence, T′ still satisfies the Concatenation lemma of the X-cube model. We can
further contract other indices of T tensor with T. For instance, the index y of T
tensor with another index of T in the same plane. Then the outcome tensor still
satisfies the Concatenation lemma of the X-cube model, because (1) the “parities”
fxy, fyz and fxz do not change after contraction, (2) the contraction is the same for
the open indices of the same parities. Therefore, Lemma (D) is proved. 2
Having proven Lemma (D), now we can complete the induction proof for the
Concatenation lemma of the X-cube model: First of all, we point out that a single
203
local T tensor of the X-cube model satisfies the Concatenation lemma. Next, as
the induction step, we assume that one network of contracted local T tensors satisfies
the Concatenation lemma, and prove that contracting one more local T tensor also
satisfies the Concatenation lemma. This induction step is, in fact, Lemma (D).
Therefore, we have completed the induction proof for the Concatenation lemma
of the X-cube model in Sec. 6.4.3.
C.3 GSD for the X-cube Model
In this section, we work out the representation dimension of the operators in
Eq. (6.82). In particular, the first group of algebras is that WX [Cx] anti-commutes
with WZ [Cz,x] and WZ [Cy,x] when they have overlaps. In the projected yz plane, the
204
operators WX [Cx], WZ [Cz,x] and WZ [Cy,x] can be depicted as:
z (pbc)
y (pbc)
, (C.14)
where the blue dot and the blue lines denote the projected operators on the yz-plane.
There are LyLz number of WX operators and Ly +Lz number of WZ operators. The
205
anti-commutation relations happens when they have overlaps:
z (pbc)
y (pbc)
. (C.15)
Other combinations of operators commute. In a more pictorial language, the commu-
tation relations are just that the blue point at the coordinate (y, z) flips the vertical
line and the horizontal line passing the blue point (y, z). In this pictorial language,
we can find that we can flip any pair of lines independently using the points. In the
operator language, we can flip any pair of WZ operators using WX operators. For
206
instance:z (pbc)
y (pbc)
. (C.16)
Therefore, we set a “reference” line and flip all other Ly + Lz − 1 number of lines
using the dots. In the operator language, we set a “reference” WZ operator in this
projected yz-plane, and flip all other Ly +Lz − 1 number of WZ operators using WX
operators. Hence, we can generate 2Ly+Lz−1 dimensional Hilbert space using operators
WX [Cx], WZ [Cz,x] and WZ [Cy,x]. Similarly for other algebras below Eq. (6.82), we can
generate 2Lx+Lz−1 and 2Lx+Ly−1 dimensional Hilbert space respectively. The ground
state degeneracy is then their product: 22Lx+2Ly+2Lz−3.
207
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