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

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 interactions 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 being 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 Preliminaries1 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

P r A.1.1 Proof of da = r∈T nadr ...... 172 1 P r A.1.2 Proof of dr = q a∈A nada ...... 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 associate 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 region 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 dimension 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

1 ?jS1 is represented by the red square penetrated by S , 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 −1 ˜ ˜ further multiply U with A1 and U 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

0 0 0 cube of lx × ly × lz with a hole of size lx × ly × lz in it. (3) Region A is

0 a cube of size lx × ly × lz and a small cube of height lz on top of it. (4)

Region A is a cube of size lx × ly × lz carved on top of it a small cube

0 of height lz...... 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 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...... 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 Chapter2, 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

1 system. For instance, a qubit can be a spin- 2 . On the qubit, the operators can be generated by the Pauli X or Pauli Z operators satisfying anti-commutation relations:

    0 1 1 0     XZ = −ZX,X =   ,Z =   . (1.1) 1 0 0 −1

Conventionally the Pauli Z basis is preferred:

    1 0     |0i =   , |1i =   . (1.2) 0 1

Hence, Z|0i = |0i,Z|1i = −|1i,X|0i = |1i,X|1i = |0i. (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|GSi = |GSi,X1X2|GSi = |GSi,X2X3|GSi = |GSi. (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):

1 |GSi = (|000i + |110i + |101i + |011i) (1.9) 2

The ground state wave function is much simpliﬁed if we use Pauli X basis instead:

1 |GSi = √ (|000ix + |111ix) , (1.10) 2

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 Chapter2. 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 |ψi 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 complement A¯:

ρA = TrA¯|ψihψ|. (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:

X |ψi = Mij|iiA ⊗ |jiA¯, ij (1.13) M = UΛV †.

¯ where |iiA and |jiA¯ 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 renormalization 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 heavily used in Chapter6. Suppose we have an MPS with periodic boundary condition whose local tensor is A:

X X As1 As2 ...AsN |s , s , . . . , s i (1.17) i1,i2 i2,i3 iN ,i1 1 2 N s1,s2,...,sN i1,i2,...,iN

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 distinguished 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 entanglement 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:

K L = IJ µνλa ∂ a (1.21) 4π Iµ ν J,ρ

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.

K L = IJ a ∧ da (1.22) 4π I J

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 (S1) = exp i n a , n ∈ (1.23) {n} ˛ I I  I Z S1 where S1 is a closed loop in (2+1)D spacetime and is also the worldline of the corresponding pair of excitations. The linking correlation function of two loop operators can be calculated by the Green function method[34]:

1 1 −1 hU{n}(S1 )U{m}(S2 )i = exp i 2πm · K · n (1.24)

1 1 where loops S1 and S2 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

1 1 braiding statistics of the two excitations represented by U{n}(S1 ) and U{m}(S2 ). In the modular tensor category, this linking correlation is encoded in the modular S matrix. The topological spin for each loop operator is:

−1 Θ U{n} = exp i πn · K · 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:

1 1 hU{K·l}(S1 )U{m}(S2 )i = 1 (1.26) Θ U{K·l} = 1

Moreover, if the charge vectors of two operators diﬀer by K·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:

1 1 1 1 hU{n}(S1 )U{m}(S2 )i =hU{n+K·l}(S1 )U{m}(S2 )i (1.27) Θ U{n} =Θ U{n+K·l}

1 Hence, we only need to consider the loop operators U{n}(S ) 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 excitations of the TQFT, i.e., the anyons. The anyons are usually labeled by objects

c a, b, c, . . . and are supplemented by other data, such as the fusion coeﬃcients Nab. 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

1 analogy can be drawn to the algebra of spins: if we take two spin 2 particles, they can fuse into either spin 0 and spin 1 particle. For this, we would write, in group

9 1 1 representation theory, the fusion rule 2 × 2 = 0 + 1. In general, the fusion of anyons in a TQFT is represented via

X c a × b = Nabc, (1.28) c

c where a, b, c are labels for the anyons, and the fusion coeﬃcients Nab are non-negative integers. The fusion can be represented by a state |a, b; c, µi in the fusion vector space

ab c c Vc . Here µ = 1, ··· ,Nab labels the vectors that form a basis of the Nab-dimensional

ab fusion vector space Vc . 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

c c Nab = Nba. (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

X d e X e d NabNdc = NadNbc. (1.30) d,e d,e

Other important data associated with anyons are their so-called quantum dimensions 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

1 1 1 1 the case of spin 2 , where 2 × 2 = 0 + 1, spin 2 is 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

1 1 of Hilbert space does not change after fusion. The product 2 × 2 has 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

X c dadb = Nabdc. (1.31) c

The above equation can be viewed as an eigenvalue equation of a matrix Na whose

c entries are (Na)bc = Nab. 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

pP 2 of a TQFT D is deﬁned as the norm of the quantum dimension vector, D = a da. 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 P P Frobenius eigenvalue, and hence has to satisfy minb c(Na)bc ≤ da ≤ maxb c(Na)bc. P 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 important concept in a TQFT is braiding. This allows us to determine how a state transforms 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 transforms 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

ab an operator that acts on the Hilbert space Vc which describes states of a and b that

ab c fuse into a ﬁxed anyon c. If we denote a basis of Vc by |a, b; c, µi with µ = 1, ··· ,Nab, 11 then Rab has the representation

X ab Rab|a, b; c, µi = [Rc ]µν|b, a; c, νi. (1.32) ν

In this notation, the topological spin for an anyon a is deﬁned as

1 X θ = d Tr [Raa], (1.33) a d c c c a c

aa where Trc[··· ] is the trace taken in the fusion vector space Vc .

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

X c ab ba Sab = NabTr[Rc Rc ]dc, (1.34a) c 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:

X abc |a, b; e, αi|e, c; d, βi = [Fd ](e,α,β)(f,µ,ν)|b, c; f, µi|a, f; d, νi (1.35) f,µ,ν

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:

X X H = − Av − Bp. (2.1) v p

14 y

Z x x Z Z x x Z x (a) (b)

Figure 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.

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.

Y Y Av = Zi,Bp = Xi. (2.2) i∈v i∈p

These two terms are illustrated in Fig. 2.1. Note that all these operators will commute with each other:

0 [Av,Av0 ] = 0, ∀ v, v ,

0 [Bp,Bp0 ] = 0, ∀ p, p , (2.3)

[Av,Bp] = 0, ∀ v, p.

Hence, the ground states of the 2D toric code model need to satisfy the constraints:

Av|GSi = |GSi,Bp|GSi = |GSi, (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:

Y Y Av = 1, Bp = 1. (2.5) v p

The ground state degeneracy (GSD) satisfying Eq. (2.4) is counted as follows:

2# of qubits 2# of bonds GSD = = = 22−χ (2.6) 2# of indep. GSD constraints 2# of vertices + # of plaquettes - 2 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 Chapter6. The projector g tensor identiﬁes the physical index with the virtual indices

 1 if p = i = j ∈ {0, 1} p  gij = (2.8)  0 otherwise

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:

Z g g Z Z Z g g = Z Z T Z

Z g

. (2.9) x T x x g x x x g g x = g x x x x x

y x

T g

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 the physical 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:

x+¯x+y+¯y Txx,y¯ y¯ = (−1) Txx,y¯ y¯

Txx,y¯ y¯ = T(1−x)¯x,(1−y)¯y = T(1−x)¯x,y(1−y¯) (2.10)

= Tx(1−x¯),(1−y)¯y = Tx(1−x¯),y(1−y¯).

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:

  0 if x +x ¯ + y +y ¯ = 1 mod 2 Txx,y¯ y¯ = (2.11)  1 if x +x ¯ + y +y ¯ = 0 mod 2.

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). 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. observation is that line operators can be constructed as follows:

Y ˜ ˜ Y W (C) = Xi, W (C) = Zi, (2.12) i∈C i∈C˜ 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’ commutation relation. Their braiding statistics is encoded in the modular S matrix in the basis of (1, e, m, f):   1 1 1 1     1 1 1 −1 −1   S =   (2.14) 2 1 −1 1 1      1 −1 −1 1

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:

  1 0 0 0     0 1 0 0    T =   (2.15) 0 0 1 0      0 0 0 −1

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:

N L = µνρb ∂ a . (2.16) 2π µ ν ρ 20 For the reader who are familiar with diﬀerential forms, it is equivalent to write:

N L = b ∧ da. (2.17) 2π

Note that this Lagrangian can be casted in the form of K-matrix Chern-Simons

theory. Restricting ZN to Z2, the Lagrangian reduces to:

2 L = µνρb ∂ a . (2.18) 2π µ ν ρ

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:

1 L = (bt (∂xay − ∂yax) + at (∂xby − ∂ybx) − bx∂tay + by∂tax) π (2.21) 1 = (−b ∂ a + b ∂ a ) . π x t y y t x

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:

0 0 0 0 [bx(x, y), ay(x , y )] = −iπδ (x − x ) δ (y − y ) , (2.23) 0 0 0 0 [by(x, y), ax(x , y )] = iπδ (x − x ) δ (y − y ) .

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:

µ W (C) = exp i aµdx , ˆC (2.24) ˜ ˜ µ W (C) = exp i bµdx . ˆC˜

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 condensation 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 universal 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 theory [81, 82, 83, 84, 85], more concretely the formulation of modular tensor categories (MTCs), is particularly powerful and, to our knowledge, provides a complete characterization 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, several 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 translated 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 condensation. This was later achieved by Eliënset 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 models 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 formulation, 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

r X t a 7−→ nat, ∀a ∈ A (3.1) t∈T

t t t¯ with the coeﬃcients na ∈ Z≥0, where we assume that na = na¯ and bars denote antiparticles (see Sec. 1.4). Equation (3.1) deﬁnes the “restriction map”. We will

P t also use the phrase “a restricts to t nat” 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

P t we say that “a splits into t nat”. Condensed particles (bosons b in the condensate) ϕ have the additional special property that nb 6= 0, where ϕ is the vacuum particle in 26 ϕ T , that is, their restriction contains the identity of the new T theory. If nb 6= 0, then ϕ ϕ nb = n¯b , 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

t of t. The lifting coefficients are the same na that we used in defining the restriction. Formally, lifting is defined by

l X t t 7−→ naa, ∀t ∈ A. (3.2) a∈A

Finally, we deﬁne particles in T whose lifts do not share a common topological spin

θa as conﬁned, that is

t t t : conﬁned ⇔ ∃ a, b such that nanb 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 t t : deconﬁned ⇔ ∀ a, b such that nanb 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

f A ⊗ A / A

r⊗r r T ⊗ T / T f

27 in which f represents fusion and r represents restriction. More explicitly, the commuting diagram can be written in terms of anyon basis

X r s ˜ t X c t nanbNrs = Nabnc, (3.5) r,s∈T c∈A

c ˜ t where Nab and Nrs are the fusion coeﬃcients in A and T , respectively. This elementary 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)

X r da = nadr, ∀ a ∈ A, (3.6a) r∈T 1 X d = nt d , ∀ t ∈ T , (3.6b) t q a a a∈A

P ϕ where q := a na da. Diagrammatically, Eq. (3.6b) is

(3.7)

It will also be useful to deﬁne the quantity

X t βt := θadana, (3.8) a∈A

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 surprising 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

t the lifts a of the particle t (namely na 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 Eliënset 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:

! 0 X t t X c ϕ Mac := nanc = Nabnb , (3.12a) t∈T b∈A as will be

X t t Mac := nanc . (3.12b) t∈U

Notice how the two above definitions of the matrices M 0 and M with nonnegative integer entries differ subtly but crucially: The expression for M 0 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 completely 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

X X t t [M,T ]ac = (MabTbc − TabMbc) = Macθc − Macθa = nanc(θc − θa). (3.13) b∈A t∈U

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 0,T ] 6= 0, with M 0 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])

1 X x θx Scb = Nc¯b dx, (3.16) DA θcθb x∈A

31 ¯ where DA is the total quantum dimension of the A theory and b denotes the antiparticle of b. From the deﬁnition of M, we express the commutator [M,S] as

1 X X t t x t t x [M,S]ab = θxdx nancNc¯b − nbncNac¯ DAθaθb c,x∈A t∈U " ! !# 1 X X t X t x t X t x = θxdx na ncNc¯b − nb ncNac¯ (3.17) DAθaθb x∈A t∈U c∈A c∈A " ! !# 1 X X t X t c t X t c = θxdx na ncNbx − nb ncNax¯ . DAθaθb x∈A t∈U c∈A c∈A

t t t t In the ﬁrst line we have used that if nanc 6= 0 with t ∈ U, then θc = θa and if ncnb 6= 0 with t ∈ U, then θc = θb which yields the term θaθb in the denominator. To obtain

c ¯b c¯ c c the last line, we have used the equalities Nab = Nac¯ = Na¯¯b and Nab = Nba. We now use Eq. (3.5) to replace the terms in the round brackets and ﬁnd

1 X X s X t r ˜ t t r ˜ t [M,S]ab = θxdxnx nanbNrs − nbnaNrs¯ , (3.18) DAθaθb s∈T x∈A t∈U;r∈T

s s¯ where we have used the equality nx = nx¯ (the assumption that the restriction of x’s antiparticlex ¯ is the antiparticle of the restriction of x) to transfer the antiparticle on

˜ t the Nr,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

X t r ˜ t t r ˜ t X t r ˜ t ˜ r nanbNrs − nbnaNrs¯ = nanb Nrs − Nts¯ (3.19) r,t∈U r,t∈U

˜ r ˜ t¯ ˜ t by exchanging the labels r and t in the second term. Since Nts¯ = Nr¯s¯ = Nrs, 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 P s x∈A θxdxnx = 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˜ = T n, (3.20b)

where S˜ and T˜ are the modular matrices of the U theory and n is the matrix of coef-

t ﬁcients that enter the restriction and lifting maps, (n)at = na, ∀ a ∈ A, t ∈ U. Our assumption Eq. (3.10) will be crucial for these proofs. The second equality Eq. (3.20b)

t is the statement that, component by component, whenever na 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]

X ∗ 2 2 βtβt = q DU (3.21a) t∈U

and

X ∗ 2 βtβt = DA. (3.21b) t∈T

P ∗ P ∗ It then follows from assumption Eq. (3.10), that t∈U βtβt = t∈T βtβt and thus

2 2 2 q DU = DA 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

X ˜ t s X c t X t Nrsnbβr = Nabdaθanc = θbDA Sbcθcnc, (3.23) r,s∈T a,c∈A c∈A

where we have used the deﬁnition of S from Eq. (3.16) and that θa = θa¯. For particles t ∈ U, we furthermore have

X ˜ t s X ˜ t s Nrsnbβr = Nrsnbqθrdr, ∀t ∈ U, (3.24) r,s∈T r,s∈U

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 X ˜ t s X ˜ s Nrsθrdr nb = SstDU θtθsnb. (3.25) r,s∈U s∈U

34 s Since s, t ∈ U, we furthermore have θs = θb if nb 6= 0 which allows us to combine Eq. (3.23) and Eq. (3.25) into

1 DA X t X ˜ s Sbcθcnc = Sstnbθt. (3.26) q DU c∈A s∈U

t Since for all nc 6= 0, we have θc = θt (t ∈ U) and using Eq. (3.22), this expression reduces to

X t X ˜ s Sbcnc = Sstnb. (3.27) c∈A s∈U

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)

P ϕ which follows from the fact that the embedding dimension q = a∈A dana > 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 = φ0, ∀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 J N = 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 {J m|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

ni currents such that Ni are of the form pi , 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 J nφ

[φ] = {φ, Jφ, J 2φ, . . . , J d−1φ},J dφ = φ. (3.30)

Here, d is the smallest positive integer such that J dφ = 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 k = φ k for the action of φk and so the orbit has 2 2 only one element, φ k . Generally, we will simply call the anyon a “ﬁxed point” when 2 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 k is a ﬁxed point under the 2

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

i1 in N1,...,Nn. Any anyon in the condensate can thus be represented as J1 ...Jn , 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

i1 in Ji := J1 ··· Jn . (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 center 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 condensed. 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 nt = δt , we ﬁnd Ji ϕ

1 = nϕ nϕ = N Ji+j nϕ = nϕ . (3.32) Ji Jj Ji,Jj Ji+j Ji+j

As a result nϕ = 1, indicating that J restricts solely to the vacuum, so it must be Ji+j i+j 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 = Sa¯b of the S matrix and the fact that for any anyon θa = θa¯. Choosing a = b = J with J a simple current gives

θJ2 θ1 = ∗ ∗ . (3.33) θJ θJ θJ θJ¯ If J is bosonic, then so is J¯ and the above implies J 2 is also a bosonic simple current,

n i.e., θJ2 = 1. This argument can be iterated by assuming that up to some n0 all J ,

N−n n = 1, . . . , n0, are bosonic (then so are all J , n = 1, . . . , n0, with N the order of

∗ n +1 n n +1 0 J). Solving the equality SJ,J 0 = SJ,JN−n0 for θJ 0 yields that J 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 u u q NxyNpz NyqNxz Y θp Y θq θ θ θ θ p x y q x z r u (3.34) NyzNxr Y θu = θ θ r x r

∀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

θ θa×J θa×J a×Ji j = i+j . (3.35) θa θa θa

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×Ji i1 i2 in = ω1 ω2 ...ωn , (3.36) θa

Ni where ωi’s satisfy ωi = 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 N1−1 N2−1 Nn−1 X θa×Ji X i1 X i2 X in = ω1 ω2 ... ωn θa i1,i2,...,in i1=0 i2=0 in=0 (3.38) = 0.

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

nϕ = ... = nϕ = 1 (3.39) J1 Jn 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.

t We will now proceed to prove a few crucial lemmas for any a, b ∈ A: (i) na = nt , ∀i, for all t ∈ T and (ii) P nt nt 6= 0 if and only if b = a × J for some j. a×Ji t a b 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

N1,N2,...,Nn X X nt nt = N b nϕ a b a,Ji Ji t∈T i ,i ,...i 1 2 n (3.40) N1,N2,...,Nn X = δb,a×Ji . i1,i2,...,in for any a, b ∈ A. To prove (ii), note that:

P t t • If b 6= a × Jj for all j, then t∈T nanb = 0 and particles a, b do not have any common restrictions. Let us write this result as

t t If b∈ / [a] =⇒ nanb = 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

N1,N2,...,Nn X X nt nt = δ a a×Jj a×Jj ,a×Ji t∈T i1,i2,...in (3.42)

=Ra ∈ Z+.

But from (i), nt = nt , so the LHS of this equation is positive. Hence R > 0. a×Jj a a

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

N1,N2,...,Nn X t X βtna = da×Ji θa×Ji t∈T i ,i ,...,i 1 2 n (3.43) N1,N2,...,Nn X θa×Ji = daθa , θa i1,i2,...,in 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)

X t X t βtna = βtna = 0. (3.44) t∈T t∈T /U

42 In fact, the second equality holds even if θa×Ji /θa = 1 for all j, because in that case

t na = 0∀t ∈ T /U – as a result, the second equality holds for all a. Multiplying the

∗ second equality by θada and summing over all particles a we obtain

X X ∗ t X ∗ 0 = βt θadana = βtβt . (3.45) t∈T /U a∈A t∈T /U

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.,

t0 t0 na1 = na2 = 1, (3.46)

t0 otherwise na = 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-

P t0 conﬁned particle case, we prove that βt0 := a∈A na 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 computer, 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 T n = 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 matrix M 0 as in the bracket of Eq. (3.12a);

t (2) factorize it in na; (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, (3.50)

2.M11 =1.

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

ϕ the particles that condense into the vacuum na . (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

X Mab = 1 ∀b. (3.51) a

2 1 P They satisfy M = because of the following reasons: Since a Mab = 1 and all entries of M can only be nonnegative integers, for any b ∈ A, there is only one

0 0 corresponding particle b , such as Mb0b = 1. Further, Mab = 0, ∀a 6= b and M is

0 0 2 P fully-ranked. As a result, if a 6= b then a 6= b . Hence (M )ab = c MacMcb = P c MacMbc = 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

X X MabSbc = SabMbc (3.52) b b we have for c = 1 X X MabSb1 = Sa1 ⇒ Mabdb = da. (3.53) b b

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

X fa ≡ Mab ≥ 1. (3.54) b

On the other hand, summing Eq. (3.53) over a, and using M = M T, gives

X X fbdb = da. (3.55) b a

46 Again, since fa ≥ 1 and the da are strictly positive this equation can only be satisﬁed if X fa ≡ Mab = 1, (3.56) b 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

  1 1 1 1     1 1 1 −1 −1   STC =   (3.57a) 2 1 −1 1 −1     1 −1 −1 1 and

TTC = diag(1, 1, 1, −1). (3.57b)

It admits three nontrivial solutions to Eq. (3.48), one automorphism

  1 0 0 0     0 0 1 0 (1)   M =   (3.58) 0 1 0 0     0 0 0 1

47 that exchanges the e and the m particles and two condensations

    1 1 0 0 1 0 1 0         1 1 0 0 0 0 0 0 (2)   (3)   M =   ,M =   , (3.59) 0 0 0 0 1 0 1 0         0 0 0 0 0 0 0 0 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

t Next we solve for the integer matrix na ≥ 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

 √  1 2 1   1 √ √  iπ/8 SI =  2 0 − 2 ,TI = diag(1, e , −1). (3.61) 2    √  1 − 2 1

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   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     M = 0 0 0 0 2 0 0 0 0 . (3.62)       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

49 It is straightforward to obtain the unique solution n that yields M = nnT

  1 0 0 0 0 0 0 0 1     0 0 0 0 1 0 0 0 0 T   n =   , (3.63) 0 0 0 0 1 0 0 0 0     0 0 1 0 0 0 1 0 0

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˜ = P˜T, then M˜ = nP˜ nT 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 ˜ T triangle equation Maa + Mbb ≥ 2Mab, then M = nP n 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 = nP˜ nT, 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) = nP˜ nT, 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 ˜ T t t¯ also include M = nP n , assumptions such as na = na¯ 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

˜ T example of condensation transitions in SU(2)16, for which two solutions M = nP n and M˜ = nnT to Eq. (3.48) exist.

T The decomposition M = nn 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

t t with na = 1 for each or restrict to one particle with na = 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,˜ T n = 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,

• S˜2 = Θ(S˜T˜)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

˜ ˜ ˜ ˜−1 Nt = SDtS , (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)

    1 1 1 1 ˜ ˜ ˜ ˜ ˜ ˜ 1 S11 S14 S12 + S13 S14 S11  2 2 2 2     1 1 1 1   ˜ ˜ ˜ ˜ ˜ ˜   − 0 −  S21 S24 S22 + S23 S24 S21  2 2 2 2      =   . (3.66)  1 − 1 0 − 1 1  S˜ S˜ S˜ + S˜ S˜ S˜   2 2 2 2   31 34 32 33 34 31     1 1 1 1 ˜ ˜ ˜ ˜ ˜ ˜ 2 2 −1 2 2 S41 S44 S42 + S43 S44 S41

˜ 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 ˜ iπ/4 symmetric.) At the same time, we have from T n = nT that θ1 = 1, θ2 = θ3 = e , 52 ˜2 ˜ ˜ 3 θ4 = −1. Furthermore, the (2,2) component of the equation S = Θ(ST ) reads

1 1 1 + 4S˜2 = 1 + 8iS˜3 (3.67) 2 22 2 22

˜ ˜ 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 symmetric 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 Chapter4. (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

0 0 SA(N) = SA ⊗ · · · ⊗ SA,TA(N) = TA ⊗ · · · ⊗ TA, (3.69) | {z } | {z } N times N times

for the modular matrices and

N N N Y Y Y N 0c = N ci , d0 = d , θ0 = θ , (3.70) a,b ai,bi a ai a ai i=1 i=1 i=1

c for the fusion matrices, quantum dimensions, and topological spins. Here, Na,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

(N) T in A by a vector a = (a1, ··· , aN ) 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)

i2π/3 and topological spins θ3 = θ3¯ = e . 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 10 and all other bosons conﬁned. Besides the vacuum, two more particles are deconﬁned: 30 with lifts (3, 3¯, 1), (1, 3, 3),¯ (3¯, 1, 3),

¯0 ¯ ¯ ¯ 0 0 ¯0 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 (30, 3) and (3¯0, 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 confined, 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 10.

• Any particle with odd number of ψ and otherwise 1 restricts to the deconﬁned particle ψ0. Their fusion rule is

ψ0 × ψ0 = 10. (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.

(N) N N−1 We can say more about the restriction of σ . Note DU = DA/q = 2 /2 = 2, because q is equal to the number of condensed bosons i.e., q = 2N−1. As we √ 0 0 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 10, ψ0, 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

0 and could restrict to the ψ and the vacuum, respectively. However, by DU = 2 it is not possible that ψ0 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, √ N 2 nt = d , (3.76) (σ,··· ,σ) 2 t

since we know from the discussion following Eq. (3.74) that t has only one lift, (σ, ··· , σ).

t From the condition that n(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)/2σ0. It follows from the fusion rules of the original theory, i.e., from Eq. (3.5) by choosing a = b = (σ, ··· , σ), that σ0 × σ0 = 10 + ψ0. (3.77)

58 Thus, 10, σ0, ψ0 furnish the same (Ising) fusion algebra as 1, σ, ψ do in every layer. The spin factors of the deconﬁned restrictions are given by

2πiν/16 θ10 = 1, θσ0 = e , θψ0 = −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/2−1a0+ 2N/2−1b0 with equal coeﬃcients. To ﬁnd the fusion rules for a0 and b0, we solve Eq. (3.64). This leaves two possibilities

a0 × a0 = b0 × b0 = 10, a0 × b0 = ψ0, (3.79)

a0 × a0 = b0 × b0 = ψ0, a0 × b0 = 10. (3.80)

Here, Eq. (3.79) are the toric code fusion rules. Which of the two cases applies can be determined from the equation S˜2 = Θ(S˜T˜)3 = C˜, by using the topological spins

2πiN/16 θa0 = θb0 = e . (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 condensation described here leads to the toric code TQFT in which a0 and b0 are bosons. We have shown above that the toric code can be condensed to the trivial TQFT by condensing either a0 or b0 (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 ψ0 is a point-like fermionic excitation in 3D space, while σ0, a0 and b0 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 transitions 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 i4π/5 It has topological spin θτ = e 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,

X ϕ 1 X n (S (5) ) = d M . (3.83) a Fib a,b (2 + φ)5/2 a a,b a a

Observe that the righthand side is nonnegative for any b. Specializing to b = (τ, 1, 1, 1, 1), we ﬁnd the lefthand side

−5/2 ϕ 4 (2 + φ) φ − n(τττττ)φ , (3.84)

ϕ which is negative for any n(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

10 we denote by (10τ), and 252 = 5 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

ϕ 10 X 5 5 ϕ 10 ϕ n1 φ + (−1) φ na + (−1) n(10τ) a∈5τ bosons (3.85) ϕ X 5 10 = n(10τ) + φ Ma,(10τ) + φ M(10τ),(10τ). a∈5τ bosons

61 ϕ Using n1 = 1, it simpliﬁes to

5 X ϕ 0 = φ M(10τ),(10τ) − 1 + na + Ma,(10τ) . (3.86) a∈5τ bosons

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

ϕ nonnegative numbers that can only vanish if na = 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,

t[a] t[a] ˜ choose nb = 1 if b ∈ [a] and nb = 0 otherwise. Further, choose St[a],t[b] = dSa,b and

d t[c] X n N˜ = N c×J , (3.87) t[a],t[b] a,b n=0 for t[a], t[b], and t[c] unconﬁned. In can be readily shown that this choice is a consistent 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 T n = 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

ϕ 1 condensed theory completely characterized by the coeﬃcients na ? 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 ordered system, the anyons may transform in a projective representation. A direct

1Indeed, we cannot exclude the possibility that additional information, like certain vertex lifting coeﬃcients, 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 superconductors, 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 process 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

1 identity if ai =a ¯j,

2. all a are zero modes of b, by which we mean a × b = b + ..., (i.e. N b > 0) i i aib

3. if a particle ai is in Ib then so is its antiparticle.

1 In demanding that ai × aj does not contain condensable bosons, as opposed to not containing any bosons at all (except the identity), we are anticipating a inductive application of the no-go theorem. 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 in the fusion product ai × aj of the set IB0 of another boson B0. 66 a) phase with b) condensed B ai# B ~ a B ai B i

Figure 4.1: Tunneling processes mediated by an anyon condensate. The gray region 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.

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

b the terminology of the set Ib, observe that Nab > 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 =a ¯j,

X X nr nr = δ + N c nϕ, (4.2) ai aj i,j aia¯j c r∈T c6=1

67 where we used nϕ = nϕ and nr¯ = nr . By assumption, there are no condensable c c¯ a¯j aj bosons in a × a¯ , hence N c and nϕ cannot be both nonzero for any c 6= 1. Thus i j aia¯j c P r r ↓ na na = 1, implying a single restriction ai of ai, with d ↓ = dai using (A.4). r i i ai Moreover, P nr nr = 0 if i 6= j, implying that the restrictions of a 6= a are r ai aj i j distinct particles.

With this knowledge about the restrictions of the ai, Eq. (3.5) for t = ϕ, a = ai, b = B¯ evaluates to a¯↓ a↓ X n i = n i = N c¯ nϕ ≥ N B nϕ , (4.3) B¯ B aiB¯ c aiB B c where we used N B¯ = N B Inserting this inequality in Eq. (A.4) for a = B, and aiB¯ aiB using dai = d ↓ , we have ai m X d ≥ nϕ N B d . (4.4) B B aiB ai i=1

ϕ It follows that in a situation where Eq. (4.1) holds, Eq. (4.4) implies nB = 0, i.e., B does not condense. [Note that in the case N B > 1, a stronger form of Eq. (4.1) with aiB d is replaced by N B d holds.] ai aiB ai To follow up with a pictorial representation of these equations, consider the tunneling 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 condensate and converts B into any of the ai. The existence of this process implies that the

↓ distinct restriction ai 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-

c tend. The theorem involves only data obtainable from Nab. 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 explicitly, 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

i4π/5 particle τ with a fusion rule τ ×τ = 1+τ, a topological spin θτ = e , 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

⊗N whether the TQFT formed by N identical layers of AFib i.e., the TQFT AFib , admits

⊗N N a condensation transition. The TQFT AFib contains 2 particles corresponding to all possible distributions of τ-particles over the N layers. For each r = 0,...,N there are

N i4πr/5 r so-called (rτ) particles with τ’s in exactly r layers, each with spin θ(rτ) = e r ⊗N and quantum dimension d(rτ) = φ . The unique r = 0 particle is the identity of AFib .

69 ⊗N From the topological spin, the bosons in AFib 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

5 2 = 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

5 2 X d(5τ) = φ < 10φ = dai (4.5)

ai∈I(5τ)

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

5n0 by replacing any (5n0 − r0) τ’s in the boson (5n0τ) with a 1. There are such r0

(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

5n0 φ5n0 < φ5n0−r0 . (4.6) r0

5n0 5n0/2 p Using that r0 ∼ 5n0/2 and ∼ 4 / π5n0/2 for large n0, we obtain that the 5n0/2 √ 5n0/2 5n0/2 right-hand side of Eq. (4.6) grows like 4 φ / 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

⊗N concludes the induction step and the proof that no boson in AFib 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

2j+1 sin π k+2 2πij j+1 d = , θ = e k+2 . (4.7) j sin [π/(k + 2)] j

We note that for k odd, all particles have distinct quantum dimensions. The fusion rules are   1 |j1 − j2| ≤ j3 ≤ min{j1 + j2, k − j1 − j2} N j3 = . (4.8) j1j2  0 else

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 jp k j0 = k + 9/4 − 1/2 . (4.9)

71 a) k = 13 b) k = 103

dj dj

j j I II III I II III

Figure 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 dimension 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.

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) k − 1 j − 1 II. bk/4c < j ≤ − 0 (4.10b) 2 2 k − 1 j − 1 k − 1 III. − 0 < j ≤ . (4.10c) 2 2 2

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 noncondensable. 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

bjB /2c−1 X 2jB + 1 < (2j + 1) (4.11) j=1

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

min{k−2jB ,bjB /2c−1} X k − 2jB + 1 < (2j + 1) (4.12) j=1

holds, Eq. (4.1) follows. In the case k − 2jB < bjB/2c − 1, Eq. (4.12) reduces to

2 1 < (k − 2jB) + (k − 2jB), which is true for all jB in range II for all k. In the case

2 k −2jB > bjB/2c−1, Eq. (4.12) simpliﬁes to k +2 < 2jB +(bjB/2c) , 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:

1 1 hU(S ) ...ic = h...ic, ∀ S (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, (5.2) ⇔ c = 0 mod 2π, ∀ closed paths S1 ˛ S1

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:

1 L = L − f(a) ∧ dc, (5.3) c 0 2π

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. 1 In Panel (a), jS1 is the integer which only has support on the loop S . It is a 1-form represented by the red arrow. Its Hodge dual ?jS1 is represented by the red square 1 penetrated by S , 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.

Proof for condensation lemma: The derivations for Eq. (5.1) are as follows:

  hexp i f(a, . . .) ...i ˛  c S1   = D[c]D[a] exp i L + f ... ˆ ˆ c ˛  S1 = D[c]D[a] exp i L + f ∧ ?j 1 ... (5.6) ˆ ˆ c S 1 = D[c]D[a] exp i L − f ∧ d(c + 2πλ) ... ˆ ˆ 0 2π = D[c]D[a] exp i L ... ˆ ˆ c

=h...ic

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:

µ f = f j = f ∧ ?j 1 (5.7) ˛ ˆ µ S1 ˆ S S1

? 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, (5.9)

?jS1 = δD(x)δD(y)dx ∧ dy,

where δD is the Dirac δ function. Hence, in this example, Eq. (5.7) is:

f = f δ (x)δ (y)d3r ˛ ˆ t D D t (5.10)

= f ∧ ?j 1 ˆ S

The second equality to the third in Eq. (5.6): We use the 1-form gauge transformation 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:

λ = dλ = − ? j 1 = − 1 ∈ (5.12) ˛ ˆ ˆ S ˆ Z S1 D2 D2 D2∩S1 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

−1 other words, the condensable U{l} requires that l · K · 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:

K 1 L = IJ a ∧ da − l a ∧ dc (5.13) c 4π I J 2π I I 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

−1 change Lc using integration by part and the fact that K and K are symmetric matrices:

l K−1 l K L = − M MN N c ∧ dc + IJ a − l (K−1) c ∧ d a − l (K−1) c (5.14) c 4π 4π I M MI J N NJ

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):

−1 (5.15) c 7→ c + 2πλ, aI 7→ aI + 2πlM KMI λ, ∀ I.

The extra terms in Lc after the gauge transformation in Eq. (5.15) only come from the ﬁrst term in Eq. (5.14):

l K−1 l δL = − M MN N 4π2λ ∧ dλ + 4πλ ∧ dc (5.16) c 4π

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,

−1 l · K · 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

eff effective expression for Zc with an effective Lagrangian Lc (c).

l · K−1 · l Leﬀ (c) = − c ∧ dc (5.18) c 4π

eﬀ Then we only need to examine whether Lc (c) has the higher form symmetry Eq. (5.5).

eﬀ Lc (c) respects it modulo 2π only when Eq. (5.17) holds, which derivations are the

eff same as in Eq. (5.16). For simplicity, Lc (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):

K li li · K−1 · lj L = IJ a ∧ da − I a ∧ dc ⇒ Leﬀ (c) = − c ∧ dc (5.19) c 4π I J 2π I i c 4π i j

eﬀ where i, j indices label diﬀerent condensates, U{li}, and Lc (c) is obtained from Lc by

eﬀ formally integrating out aI ﬁelds. In order to make Lc (c) in Eq. (5.19) respect the higher form gauge symmetry:

ci 7→ ci + 2πλi, ∀ i (5.20) we need the following conditions:

i −1 j i −1 i l · K · l ∈ Z, ∀ i 6= j; l · K · l ∈ 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

1 U{m}(S ) transforms as:

1 1 −1 U{m}(S ) 7→U{m}(S ) exp i 2πm · K · l λ (5.22) ˛S1

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 ¸ 1 U{m}(S ) needs to satisfy:

−1 m · K · 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:

−1 i m · K · l ∈ 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

1 attaching a Wilson loop of gauge ﬁeld c to any deconﬁned loop operators U{m}(S ) will not change the expectation value, i.e.:

      exp i m a + c = exp i m a , ∀ S1 ⇒ exp i c = 1, ∀ S1. ˛ I I  ˛ I I  ˛  S1 S1 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:

  K −(l , l ,...)T  1 2  Kc =   (5.26) −(l1, l2,...) 0

85 The basis of this Kc matrix is (a1, a2, . . . , c). The central charge of Kc is generally different from that of K. 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 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 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 entanglement 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 eigenstates 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 Chapter2 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 entanglement 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 entanglement 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 entropies 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 Chapter2.

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:

    0 1 1 0     X =   ,Z =   . (6.1) 1 0 0 −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 andz ¯ 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 ( ) where M is the spatial manifold that the TNS lives on. The correspond-

ing wave function that arises from the contraction is denoted as |TNSiM. When

92 U U-1 A1 A2 A1 A2

(a) (c) U U-1 A1 A2 A1 A2

(b) (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 operator I = UU −1 at the virtual level - it acts on the virtual bonds. The tensor contraction −1 of A1 and A2 does not change. (c) We further multiply U with A1 and U 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.

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 z z y y x x

T g

(a) (b)

Figure 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.

6.1.2 Stabilizer Code and TNS Construction

We now summarize the general idea of constructing TNSs for stabilizer codes. In Chapter2, 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: X X H = − Av − Bp (6.2) v p 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:

0 [Av,Av0 ] = 0, ∀ v, v

0 [Bp,Bp0 ] = 0, ∀ p, p (6.3)

[Av,Bp] = 0, ∀ v, p.

The Hamiltonian eigenstates are the eigenstates of these local terms individually. In particular, any ground state |GSi should satisfy:

Av|GSi = |GSi, ∀ v (6.4)

Bp|GSi = |GSi, ∀ p

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:

 i j 1 s = i = j s  gij = = (6.5) s  0 otherwise

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 |↑i, |↓i respectively where Z|↑i = |↑i, and Z|↓i = −|↓i. 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:

X 3 |TNSi = CR (gs1 gs2 gs3 ...TTT...) |{s}i (6.6) {s} where CR3 denotes the contraction over all virtual indices on R3 as illustrated in Fig. 6.3 (b); |{s}i 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 CR3 to CT 3 and CT 2×R. The TNS for the ground states satisﬁes:

Av|TNSi = |TNSi, ∀ v (6.7)

Bp|TNSi = |TNSi, ∀ p

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 |TNSi 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:

  6= 0 if the indices xxy¯ . . . satisfy  Txxy...¯ some constraints . (6.8)   = 0 otherwise

Applying the second equation of Eq. (6.7) will further restrict the local T tensor to be:

  1 if the indices xxy¯ . . . satisfy  Txxy...¯ = some constraints . (6.9)   0 otherwise

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 degeneracy (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) requires 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:

 ?   X R3 s s s X R3 s s s hTNS|TNSi =  C (g 1 g 2 g 3 ...TTT...) h{s}|  C (g 1 g 2 g 3 ...TTT...) |{s}i {s} {s}

X 3 3 = CR (gs1?gs2?gs3? ...T ?T ?T ? ...) CR (gs1 gs2 gs3 ...TTT...) {s}

X 3 3 = CR (gs1 gs2 gs3 ...T ?T ?T ? ...) CR (gs1 gs2 gs3 ...TTT...) , {s} (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:

i j

X s s gijgmn = s m n  (6.11)  1 i = j = m = n = .  0 otherwise

Thus when computing wave function overlap hTNS|TNSi, 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:

R3 hTNS|TNSi = C (... TTT ...) , (6.12)

3 where CR stands for the contraction of a network of tensors T over the virtual indices on R3. In a slight abuse of notation, CR3 in 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¯ 0x¯0y0... = Txxy...¯ Tx0x¯0y0...δxx0 δx¯x¯0 δyy0 ... (6.13) 2 = |Txxy...¯ | δxx0 δx¯x¯0 δyy0 ..., 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 x0x¯0y0 ... 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,

2 Txxy...,x¯ 0x¯0y0... = |Txxy...¯ | δxx0 δx¯x¯0 δyy0 ... (6.14)

= Txxy...¯ δxx0 δx¯x¯0 δyy0 ....

Then,

R3 hTNS|TNSi =C (... TTT ...) (6.15) =CR3 (...TTT...) .

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 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.

6.1.4 Transfer Matrix

The transfer matrix method is ubiquitous when using MPS (see Fig. 6.4 for an illustration 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

3 cylinder perimeter. In 3D, the TNS norm on T of size Lx × Ly × Lz can be written as an MPS using transfer matrices TMxy in each xy-plane:

hTNS|TNSi = Tr (TMxy,z=1TMxy,z=2 ...) (6.16) Lz = Tr (TMxy,z=1)

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 virtual indices in z-direction which leads to the multiplication and the trace of transfer

100 matrices. The transfer matrix TMxy is deﬁned as:

    2 2 X Txy s1? s2? s3? ? ? ? X Txy s1 s2 s3 TMxy =  C (g g g ...T T T ...) h{s}|  C (g g g ...TTT...) |{s}i , {s} {s} (6.18) where the TNS contraction is performed along the xy-plane with periodic boundary

2 conditions, i.e., the 2D torus Txy. Denoting Txy as the TNS in its ket layer, it can depicted as: T y (pbc) g z x (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:

2 Txy TMxy = C (... 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:

2 Txy TMxy = C (...TTT...) . (6.21)

101 Graphically:

T TTTTT T y (pbc) z x (pbc) TTTTT TT

TMxy = TTTTT TT . (6.22)

TTTTT TT

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: X |{t}i = CM (T T T . . . gs1 gs2 gs3 ...) |{s}i, (6.23) {s}

where M is an open manifold which the TNS lives on, CM stands for the contraction over the virtual indices inside M, but not over the open ones {t} that straddle the boundary of M. In Eq. (6.23), the T tensors and g tensors are the tensors inside M such that the nodes of the local T tensors and the projector g tensors are inside M. For example, when M is a cube, we have a TNS ﬁgure:

|{t}i = , (6.24)

{t}

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}i, the TNS can be written as:

X |TNSi = |{t}iA ⊗ |{t}iA¯ (6.25) {t}

¯ with respect to a region A and its complement A. |{t}iA is the TNS in region A with ¯ open indices {t}, while |{t}iA¯ 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}iA 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}iA¯ form an orthogonal basis. Therefore, Eq. (6.25) is an exact SVD. Proof:

0 We ﬁrst prove the statement for the region A. Suppose that |{t}iA and |{t }iA 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

0 situations respectively, and examine the overlap of two diﬀerent states Ah{t }|{t}iA as a function of the two indices conﬁgurations {t0} and {t}.

(1) If the open virtual index m in the ket layer (i.e. |{t}iA) connects to a projector

0 0 g tensor, then the open virtual index m in the bra layer (i.e. Ah{t }|), at the same place as the index m, also connects to a projector g tensor. If we “zoom in” on the

104 0 0 local area of Ah{t }|{t}iA near the index m and m , we have the following diagram:

Entanglement Cut m ket layer A A (6.26)

bra layer m'

0 0 By using Eq. (6.11), we can conclude that m = m , otherwise Ah{t }|{t}iA = 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.

0 0 Similarly for the index m0 in the bra layer. In terms of a diagram, Ah{t }|{t}iA near

0 the area of the index m0 and m0 can be represented as:

Entanglement Cut ket layer mi T m0 (6.27) A A

m'i T m'0 bra layer

0 where mi and mi with i = 1, 2, 3 ... denote the other virtual indices of the T tensor in

0 the bra and ket layer respectively, except m0 and m0. 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,

0 in the bra layer, the virtual indices mi (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:

0 mi = mi, i = 1, 2,... (6.28)

0 otherwise the overlap would be Ah{t }|{t}iA = 0. The only remaining question is

0 whether the open indices m0 and m0 should be identiﬁed in order to have a non-

0 vanishing overlap Ah{t }|{t}iA.

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,

0 0 mi (i = 1, 2,...) will uniquely determine m0 in order for the T tensor in the bra layer to give a nonzero element. Therefore, Eq. (6.28) implies that:

0 m0 = m0 (6.29)

0 such that the overlap Ah{t }|{t}iA is nonzero. Therefore, both situations (1) and (2) lead to the conclusion that the open indices

0 0 {t} and {t } should be identical in order to have a nonzero overlap Ah{t }|{t}iA. The non-vanishing states |{t}iA are orthogonal basis. A similar proof can be derived for ¯ the region A. The orthogonality of each set |{t}iA and |{t}iA¯ 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}iA and |{t}iA¯ 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}iA. The number of independent Schmidt states |{t}iA 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

Lx+Ly−1 the X-cube model, the transfer matrix is a projector of rank 2 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 construction 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 corresponding 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 generate 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.

2 Speciﬁcally, suppose we consider our TNS on a 3D cylinder Txy × 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-

Lx+Ly−1 cube model, the transfer matrix TMxy has the degeneracy 2 , 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 simplicity, 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: X X H = − Av − Bp. (6.30) v p

109 z

y Z x Z x Z Z Z x x Z

(a) (b) x

Figure 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.

where Av is deﬁned around a vertex v, and Bp is deﬁned on a plaquette p:

Y Y Av = Zi,Bp = Xi, (6.31) i∈v i∈p

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:

0 [Av,Av0 ] = 0, ∀ v, v

0 [Bp,Bp0 ] = 0, ∀ p, p (6.32)

[Av,Bp] = 0, ∀ v, p, and their eigenvalues are ±1:

2 2 Av = 1,Bp = 1. (6.33)

110 The ground states |GSi should satisfy:

Av|GSi = |GSi, ∀ v (6.34)

Bp|GSi = |GSi. ∀ p

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

Y 1 + Av |ψi = |0 i, (6.35) 2 x v

where |0xi 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 = =

. (6.36) x x x =

In terms of algebraic equations, these diagrams correspond to:

s s s i s j gi,j(−1) = gi,j(−1) = gi,j(−1) (6.37) 1−s s gi,j = g1−i,1−j.

111 These two sets of equations are true, because (1) the indices s, i and j are identiﬁed

s s for nonzero gi,j, (2) the nonzero gi,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:

z y

x T x . (6.38) y z

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:

Z g Z g Z Z Z Z g g = Z Z Z T Z g Z Z g . (6.39)

x T x x g x x x g g x = g x x x x x

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:

Z Z = Z Z T Z T Z

x x x x x = =x = = x T T T x T T

x x == = x = (6.40) T x T T T x x x x

x x = x x == x = x T x T T x T

x x = x x == x T T xT

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

i the ﬁrst set of conditions by using Zij = δij(−1) , we have:

x+¯x+y+¯y+z+¯z Txx,y¯ y,z¯ z¯ = (−1) Txx,y¯ y,z¯ z¯

⇔  (6.41)  = 0, if x +x ¯ + y +y ¯ + z +z ¯ = 1 mod 2 Txx,y¯ y,z¯ z¯  6= 0, if x +x ¯ + y +y ¯ + z +z ¯ = 0 mod 2, where x, x,¯ y, y,¯ z, z¯ are the six indices of T in the three directions respectively. We emphasize for clarity thatx ¯ 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,y¯ y,z¯ z¯ =T(1−x)(1−x¯),yy,z¯ z¯

=T(1−x)¯x,(1−y)¯y,zz¯ . (6.42)

=Txx,y¯ y,¯ (1−z)(1−z¯)

= ...

Hence, the nonzero elements of the T tensor are all equal. Up to an overall normalization, we have the unique solution:

  0, if x +x ¯ + y +y ¯ + z +z ¯ = 1 mod 2 Txx,y¯ y,z¯ z¯ = (6.43)  1, if x +x ¯ + y +y ¯ + z +z ¯ = 0 mod 2.

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 introduced 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 topological phases to trivial phases. See Refs. [38, 197, 198, 199, 200] for explanations and examples in the case of 2D TNS.

114 z y

x T

Figure 6.6: Contraction of two local T tensors in the z-direction. We emphasize that there 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:

X Tx1,x¯1,y1,y¯1,z1,x2,x¯2,y2,y¯2,z¯2 = Tx1x¯1,y1y¯1,z1z¯1 Tx2x¯2,y2y¯2,z2z¯2 δz¯1z2 . (6.44) z¯1,z2

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 +x ¯1 + y1 +y ¯1 + z1 +z ¯1 = 0, mod 2   x2 +x ¯2 + y2 +y ¯2 + z2 +z ¯2 = 0, mod 2    (6.45)  z¯1 = z2

⇒ x1 +x ¯1 + y1 +y ¯1 + z1 + x2 +x ¯2 + y2 +y ¯2 +z ¯2

=0, mod 2.

Otherwise, the tensor element of T is zero. Moreover, the elements of the contracted tensor are 1, if nonzero:

  0 if x1 +x ¯1 + y1 +y ¯1 + z1 + x2 +x ¯2 + y2 +y ¯2 +z ¯2 = 1, mod 2 Tx1,x¯1,y1,y¯1,z1,x2,x¯2,y2,y¯2,z¯2 =  1 if x1 +x ¯1 + y1 +y ¯1 + z1 + x2 +x ¯2 + y2 +y ¯2 +z ¯2 = 0, mod 2. (6.46) For a more complicated contraction of T tensors, we have:

  P 0 if i ti = 1, mod 2 T{t} = (6.47)  P Const if i ti = 0, mod 2 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

b P b with open virtual indices {t }, is zero when i ti = 1 mod 2 and nonzero when P b i ti = 0 mod 2, according to the Concatenation lemma of the 3D toric code b P b model in Sec. 6.3.3. Hence, we can only ﬁx the boundary indices {t } to be i ti = 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:

X |TNSiR3 = |{t}iA ⊗ |{t}iA¯ (6.48) {t} where

X X |{t}i = CA(gs1 gs2 . . . gs3 gs4 T T ...)|{s}i. A t1i1 t2i2 i3i4 i5i6 i7... i8... (6.49) {s}∈A {i}∈A

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 gs1 and gs2 etc are t1i1 t1i2 the projector g tensors near the entanglement cut on the region A side as illustrated

in Fig. 6.7; gs3 and gs4 are the projector g tensors inside the region A; for this cut, i3i4 i5i6 all the T tensors are inside the region A. The summation is over all physical indices {s} inside the region A. Thereby, |{t}i 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}i 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¯:

X X A¯ s s |{t}i ¯ = C (g 1 g 2 T T ...)|{s}i. A i1i2 i3i4 t1i5... t2i6... (6.50) {s}∈A¯ {i}∈A¯

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.

AAcut

T g

Figure 6.7: The splitting of tensors near the entanglement cut.

118 We now show that |{t}iA and |{t}iA¯ 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.,

0 X Ah{t }|{t}iA ∝ δ{t0},{t}δ( ti = 0 mod 2). (6.51) i

Proof: Applying the SVD condition to the toric code TNS, we can immediately conclude that the |{t}iA 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 following, we show that |{t}iA 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:

  0 0 0 0 0 X X A s1? s2? s3? s4? ? ? 0 Ah{t }|{t}iA = C (g 0 g 0 . . . g g Tj ...Tj ...... )h{s }|  t1j1 t2j2 j3j4 j5j6 7 8  {s0}∈A {j}∈A   (6.52) X X CA(gs1 gs2 . . . gs3 gs4 T T ...)|{s}i .  t1i1 t2i2 i3i4 i5i6 i7... i8...  {s}∈A {i}∈A

When the open virtual indices {t0} 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}iA are orthogonal. Next we show that h{t}|{t}i is zero when P t is odd. Following the A A ti∈{t} i same derivations in Sec. 6.1.3, we have:

A Ah{t}|{t}iA = C (...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, Ah{t}|{t}iA is zero if

119 the open indices {t} are summed to be 1 mod 2:

X ti = 1 mod 2 ⇒ Ah{t}|{t}iA = 0. (6.54) i

Moreover, X Ah{t}|{t}iA = Const, when ti = 0 mod 2. (6.55) i Hence |{t}i is orthonormal basis up to an overall normalization factor that can be obtained by the normalization of |TNSi. 2

¯ The same proof works for the region A and |{t}iA¯. 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

2 we place our TNS ground state on a 3D cylinder Txy × 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

3 to ZK lattice gauge models on R :

S = Area × log(K) − n log(K) (6.58)

2 with the same equation holding on a cylinder Txy × 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 P 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)

d−1 The topological entanglement entropy Stopo[T ] is independent of the dimensional- ity, and it obeys the conjecture presented in Ref. [190]:

d−1 d exp(−dStopo[T ]) = GSD[T ] (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

y (pbc) z x (pbc) zi,j

zi,j (TMxy){z},{z¯} = , (6.61)

where zi,j andz ¯i,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

L Ly Ox O e{z} = ezi,j = ez1,1 ⊗ ez1,2 ⊗ . . . ezLx,Ly . (6.62) i=1 j=1

ezi,j is the local “virtual bond Hilbert space” spin |0i = |↑i, |1i = |↓i 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¯}:

X TMxy · e{z¯} = (TMxy){z},{z¯} (6.63) {z}

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

L Ly Xx X zi,j +z ¯i,j = 0 mod 2 (6.64) i=1 j=1

122 contribute equally to the RHS of Eq. (6.63), while the terms satisfying

L Ly Xx X zi,j +z ¯i,j = 1 mod 2 (6.65) i=1 j=1 do not contribute. Therefore, we can rewrite the summation more precisely:

X X TMxy · e{z¯} = (TMxy){z},{z¯} ∝ e{z}. (6.66) P P {z} with i zi+¯zi even {z} with i zi+¯zi even

P When the {z¯} satisﬁes i,j z¯i,j = 0 mod 2, we have

X TMxy · e{z¯} ∝ e{z}, (6.67) P {z} with i zi even

P while when the {z¯} satisﬁes i,j z¯i,j = 1 mod 2, we have

X TMxy · e{z¯} ∝ e{z}. (6.68) P {z} with i zi odd

Therefore, TMxy is a projector of rank 2. Hence, it has only two eigenvalues of 1, and the corresponding unnormalized eigenvectors are:

X e{z} {z} with P z even i i (6.69) X e{z}. P {z} with i zi odd

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:

Y ˜ Y WX [Cx] = Xi,WZ [Cyz] = Zi;

i∈Cx i∈C˜yz Y ˜ Y WX [Cy] = Xi,WZ [Cxz] = Zi; (6.70)

i∈Cy i∈C˜xz Y ˜ Y WX [Cz] = Xi,WZ [Cxy] = Zi.

i∈Cz i∈C˜xy

˜ 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:

y x

x x x (6.71)

Z Z

The commutation relations include:

˜ ˜ WX [Cx]WZ [Cyz] = −WZ [Cyz]WX [Cx], ˜ ˜ WX [Cy]WZ [Cxz] = −WZ [Cxz]WX [Cy], (6.72) ˜ ˜ WX [Cz]WZ [Cxy] = −WZ [Cxy]WX [Cz].

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., |TNSiT 3 . It is not hard to verify using Eq. (6.36) and (6.40) that:

WX [Cx]|TNSiT 3 = |TNSiT 3 ,

WX [Cy]|TNSiT 3 = |TNSiT 3 , (6.73)

WX [Cz]|TNSiT 3 = |TNSiT 3 .

As already mentioned in Sec. 6.3.2, |TNSiT 3 = |ψi where |ψi is deﬁned in Eq. (6.35).

Both |TNSiT 3 and |ψi are +1 eigenstates of WX operators. However, the transfer matrix deﬁned by |TNSiT 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:

y (pbc)

z x (pbc)

Z T g Z

Z Z Z Z Z Z Z T g (6.74)

Z Z Z Z Z Z Z Z T g Z

˜ ˜ 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-

α,β planes of TNSs in Eq. (6.74) as Txy (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

α,β respectively. The subindex xy in Txy means that the TNS is on a xy-plane. Clearly,

α,β Txy are diﬀerent since they support diﬀerent holonomies of WX [Cx] operators and

α,β WX [Cy] operators. After obtaining Txy , we can deﬁne four twisted transfer matrices

α,β α,β correspondingly by contracting the physical indices between bra Txy and ket Txy . α,β 0,0 The twisted transfer matrices are denoted as TMxy . For instance, TMxy is the untwisted transfer matrix in Eq. (6.61).

126 α,β Each of these transfer matrices TMxy is also a projector of rank 2. The reasons

α,β are that (1) the contraction of the projector g tensors between the bra Txy and ket

α,β Txy 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,

1,0 0,1 1,1 the transfer matrices built from the twisted TNS TMxy , TMxy and TMxy are equal 0,0 to that built from the untwisted TNS TMxy :

α,β 0,0 TMxy = TMxy , ∀ 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

α,β and ket layer of Txy , 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].

α,β In constructing the TNS on torus, we need to choose the same Txy for each xy-

α,β plane. If we use diﬀerent Txy 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.

0,0 More speciﬁcally, in the 3D toric code model, if we contract Lz − 1 Txy ’s with one

α,β twisted Txy 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

0,0 1,0 Txy and one Txy is:

. (6.76)

Z Z Z Z

127 z x

y x x Z x Z x Z Z x x Z Z Z Z Z Z x x x Z Z x x x (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 the physical 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

α,β 1,0 same Txy , for instance Txy :

Z Z Z Z (6.77) Z Z Z Z Z Z Z Z Z Z Z Z 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 fracton models. Using it, we then calculate the entanglement entropy and the GSD of this model. The results are generally diﬀerent from those in conventional topological 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 entanglement 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:

X X H = − (Av,xy + Av,yz + Av,xz) − Bc (6.78) v c 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,Av0,xy] = [Av,yz,Av0,yz] = [Av,xz,Av0,xz] = 0,

[Av,xy,Av0,yz] = [Av,xy,Av0,xz] = [Av,yz,Av0,xz] = 0, (6.79)

[Bc,Av0,xy] = [Bc,Av0,yz] = [Bc,Av0,xz] = 0

0 0 [Bc,Bc0 ] =0, ∀ v, v , c, c .

Hence, this model can be exactly solved. The ground state |GSi (on R3) needs to satisfy that:

Av,xy|GSi =|GSi,

Av,yz|GSi =|GSi, (6.80)

Av,xz|GSi =|GSi,

Bc|GSi =|GSi, ∀ v, c. 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]:

Y Y Y WX [Cx] = Xi,WX [Cy] = Xi,WX [Cz] = Xi,

i∈Cx i∈Cy i∈Cz ˜ Y ˜ Y ˜ Y WZ [Cx,z] = Zi,WZ [Cy,z] = Zi,WZ [Cz,x] = Zi, (6.81) i∈C˜x i∈C˜y i∈C˜z ˜ Y ˜ Y ˜ Y WZ [Cx,y] = Zi,WZ [Cy,x] = Zi,WZ [Cz,y] = Zi,

i∈C˜x i∈C˜y i∈C˜z

˜ 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:

y x (a) (b) (c)

x x x (e) (d) (f) . (6.82) Z

Z Z (g) (h) (i)

Z Z

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:

g Z g Z Z Z g g = Z Z Z T Z g g

Z g g Z Z Z g g = Z Z T Z g Z g

Z g . (6.84) Z g Z Z g g = Z T Z Z g Z g

x x x x x x x T x T x x x gx g x x x T xg g x g T x g x x = x x x g T x x x x x g 132 x T x x g x g x x x x T g T Requiring that the tensors in the dashed red rectangles are invariant will lead to the following (strong) conditions on the T tensor:

Z Z Z Z =Z Z =Z Z = T Z T T Z T Z Z

x x x x =x =x = T T T

x . (6.85) x x =x =x = x T T x T x x

x x x =x = x x T x T T x x

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:

     x +x ¯ + y +y ¯ = 0 mod 2,      1 if x +x ¯ + z +z ¯ = 0 mod 2,  Txx,y¯ y,z¯ z¯ =  (6.86)    y +y ¯ + z +z ¯ = 0 mod 2.    0 otherwise.

133 A useful consequence of Eq. (6.85) is:

x x =x x = = T T T x T . (6.87) x

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:

X Tx1,x¯1,y1,y¯1,z1,x2,x¯2,y2,y¯2,z¯2 = Tx1x¯1,y1y¯1,z1z¯1 Tx2x¯2,y2y¯2,z2z¯2 δz¯1z2 . (6.88) z¯1,z2

134 Graphically, T is the same as depicted in Fig. 6.6. For each of the two T tensors, we

have   x1 +x ¯1 + y1 +y ¯1 = 0 mod 2   x1 +x ¯1 + z1 +z ¯1 = 0 mod 2 (6.89)    y1 +y ¯1 + z1 +z ¯1 = 0 mod 2

and   x2 +x ¯2 + y2 +y ¯2 = 0 mod 2   x2 +x ¯2 + z2 +z ¯2 = 0 mod 2 (6.90)    y2 +y ¯2 + z2 +z ¯2 = 0 mod 2.

Therefore, settingz ¯1 = z2 due to the tensor contraction, the open indices of T need to satisfy:

  x1 +x ¯1 + y1 +y ¯1 = 0, mod 2    x2 +x ¯2 + y2 +y ¯2 = 0, mod 2 (6.91)  z1 +z ¯2 + x1 +x ¯1 + x2 +x ¯2 = 0, mod 2    z1 +z ¯2 + y1 +y ¯1 + y2 +y ¯2 = 0, mod 2 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 entanglement entropies. (1) Region A is a cube of size lx × ly × lz. (2) Region A is a cube of 0 0 0 lx × ly × lz with a hole of size lx × ly × lz in it. (3) Region A is a cube of size lx × ly × lz 0 and a small cube of height lz on top of it. (4) Region A is a cube of size lx × ly × lz 0 carved on top of it a small cube of height lz.

T tensor are 1 independent of indices:

     x1 +x ¯1 + y1 +y ¯1 = 0, mod 2 and        x2 +x ¯2 + y2 +y ¯2 = 0, mod 2 and  1 if T =  x1,x¯1,y1,y¯1,z1,x2,x¯2,y2,y¯2,z¯2 z1 +z ¯2 + x1 +x ¯1 + x2 +x ¯2 = 0, mod 2 and        z1 +z ¯2 + y1 +y ¯1 + y2 +y ¯2 = 0, mod 2    0 otherwise. (6.92) Generally for a complicated contraction of local T tensors denoted by T, we have:

  Const 6= 0 Concatenation lemma is satisﬁed T{t} = (6.93)  0 else.

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 |TNSi. See Fig. 6.9 for some examples of deformed A regions. We can read out the singular

136 values of |TNSi 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:

X |TNSi = |{t}iA ⊗ |{t}iA¯. (6.94) {t}

¯ where {t} represent the tensor virtual indices connecting the region A and A. |{t}iA is the TNS for the region A with open virtual indices {t}. Similarly for |{t}iA¯ for the complement region A¯. Because every virtual bond is contracted over for |TNSi,

|{t}iA and |{t}iA¯ have to share the same {t}, and {t} is summed over in Eq. (6.94).

Next we show that this set of basis |{t}iA is orthonormal:

0 Ah{t }|{t}iA ∝ δ{t0},{t}, (6.95) when the Concatenation lemma is satisﬁed for both of {t0}, {t}, and the basis

|{t}iA 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}iA and |{t}iA¯ 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}iA and |{t}iA¯ 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

0 use the same conventions of SVD basis |{t}iA and |{t }iA¯ as in Sec. 6.3.4. Suppose

0 {t } and {t} both satisfy Concatenation lemma in Sec. 6.4.3. Hence, |{t}iA and

0 |{t }iA are not null vectors. More speciﬁcally, the wave function |{t}iA is the same as in Eq. (6.49). All the virtual indices except {t} are contracted over.

137 When {t0} 6= {t}, the basis overlap is zero, because the spins on the boundary of A are diﬀerent. When {t0} = {t}, the overlap is nonzero. Moreover, the overlaps

Ah{t}|{t}iA are constants independent of {t}, due to the Concatenation lemma

in Sec. 6.4.3. Hence, |{t}iA is an orthonormal basis for the region A, up an overall normalization factor. 2

Therefore, using the orthonormal basis |{t}iA and |{t}iA¯, 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}iA 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:

S A = 2l l + 2l l + 2l l − l − l − l + 1 = Area − l − l − l + 1. log 2 x y y z x z x y z x y z (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;

X ti,j,k = 0 mod 2, ∀ k i,j X ti,j,k = 0 mod 2, ∀ j (6.97) i,k X ti,j,k = 0 mod 2, ∀ i j,k

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

0 0 0 lx × ly × lz:

S A =2l l + 2l l + 2l l + 2l0 l0 + 2l0 l0 + 2l0 l0 log 2 x y y z x z x y y z x z 0 0 0 (6.98) − lx − ly − lz − lx − ly − lz + 2

0 0 0 =Area − lx − ly − lz − lx − ly − lz + 2.

139 3. When the region A is a cube of size lx × ly × lz with an additional convex cube

0 of height lz on top (Fig. 6.9 (3)):

S A =Area − l − l − l − l0 + 1. (6.99) log 2 x y z z

4. When the region A is a cube of size lx × ly × lz with an additional concave cube

0 of height lz on top (Fig. 6.9 (4)):

S A =Area − l − l − l − l0 + 1. (6.100) log 2 x y z z

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.

2 Furthermore, if we put the TNS on a cylinder Txy × Rz and the entanglement cut splits the cylinder into two halves z > 0 and z < 0, then the entanglement entropy of either side is: S A = Area − L − L + 1. (6.101) log 2 x y

We emphasize that the entanglement spectrum is ﬂat, because all singular values are identical in Eq. (6.94).

140 T T g g x x x x x x x

y (pbc) z x (pbc) T T g g x x x x x x x x x x x x x x x x x x x x x x x x

Figure 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).

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:

2 Txy TMxy = C (...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:

X X TMxy · e{z¯} = (TMxy){z},{z¯} ∝ e{z} (6.103) Concatenation Lemma Concatenation Lemma

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:

X zi,j +z ¯i,j = 0 mod 2, ∀ j i (6.104) X zi,j +z ¯i,j = 0 mod 2, ∀ i j

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:

y (pbc)

x (pbc)

T T . (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:

! ! X X X X zi,j +z ¯i,j = 0 mod 2 ⇔ zi,j +z ¯i,j = 0 mod 2. (6.106) j i i j

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]|TNSiT 3 = |TNSiT 3 ,

WX [Cy]|TNSiT 3 = |TNSiT 3 , (6.107)

WX [Cz]|TNSiT 3 = |TNSiT 3 .

These three equations can be proved by using Eq. (6.36) and Eq. (6.85); the derivations 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 |TNSiT 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

Lx+Ly ˜ ˜ 2 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

~α,β~ ~ planes of tensors in Fig. 6.11 as Txy 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.

~0,~0 For instance, the untwisted plane of TNS is Txy using this convention. This notation

~0,~0 Txy 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.

~ ~α,β~ ~α,β Lx+Ly−1 Each of these Txy will generate a transfer matrix TMxy which has 2 degenerate eigenvalues. The reasons are that (1) when building the transfer matrices

~α,β~ from the twisted Txy , 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 y (pbc)

x (pbc)

Z T T T

(a) (b) (c) Z Z Z Z Z Z Z T T T

Z Z

(d) (e) (f) ˜ 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 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.

y (pbc)

x (pbc)

Z T T

(a) (b) Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z T T

˜ ˜ 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 text for the explanations.

~α,β~ ~α,β~ the transfer matrices TMxy obtained from the twisted Txy are equal to that obtained

~0,~0 from the untwisted one Txy .

~α,β~ ~0,~0 ~ TMxy = TMxy , ∀ ~α, β (6.108)

~α,β~ ~α,β~ Thus, the transfer matrices TMxy built from the twisted Txy are also projectors of dimension 2Lx+Ly−1.

144 ~α,β~ The TNS on torus is then constructed as the contraction of Txy 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

α,β Txy in each plane, otherwise the corresponding wave function is no longer a ground

~α,β~ state. However, we have more choices for the X-cube model. In each plane of Txy , ˜ 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

~α,β~ the X-cube model Eq. (6.81): they do not cost any energy. For each of the Txy built

~α,β~ from Fig. 6.11, we can ﬁnd 3 others: all 4 planes of tensors Txy can be used in the z direction. Take Panel (a) of Fig. 6.11 as an example at one point in the z direction.

~α,β~ The 4 Txy 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.

~α,β~ Due to this choice, four twisted Txy will be grouped together, and there are

2Lx+Ly (6.109) 4

~α,β~ number of groups of twisted Txy . Hence, the total GSD that we can obtain from the

~α,β~ transfer matrices built from Txy is:

2Lx+Ly 2Lx+Ly−1 × × 4Lz = 22Lx+2Ly+2Lz−3. (6.110) 4

145 Each factor in the above formula has an explanation:

1.2 Lx+Ly−1 is the degeneracy of each transfer matrix.

2Lx+Ly ~α,β~ 2. 4 is the number of groups of the twisted Txy due to the “ambiguity” explained in the paragraph before Eq. (6.109).

Lz ~α,β~ 3.4 is the number of combinations for Txy , since for each xy plane we can pick

~α,β~ any of the 4 Txy 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:

X X H = − Aabc − Babc. (6.111) a,b,c a,b,c

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

L L L L R R R R A000 = Z110Z101Z011Z111Z100Z010Z001Z111 (6.112) L L L L R R R R B000 = X000X110X101X011X000X100X010X001.

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:

z (6.113) y x

It is straightforward to check that all the Hamiltonian terms commute.

147 (a)

T T

y x (b)

T T T T

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 |GSi is obtained by requiring

Aabc|GSi = |GSi (6.114)

Babc|GSi = |GSi (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

Ls Rs and right g tensors, gijkl and gijkl, are:

III s II

IV i I gLs = (6.116) ijkl VII VI k l

VIII j V

and

III s II i IV k I Rs gijkl = (6.117) VII j VI

VIII V l

where s is the physical index in {|0i = |↑i, |1i = |↓i}, 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:

 1 i = j = k = l = s Ls  gijkl = ,  0 otherwise  (6.118) 1 i = j = k = l = s Rs  gijkl = .  0 otherwise

Ls The four virtual indices of gijkl extend along the III, VIII, VII, VI octants (as shown

Rs in Eq. (6.116)), and the four virtual indices of gijkl 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:

T Ti1i2i3i4i5i6i7i8 = . (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:

Z Z

Z Z Z

T = T (6.120) Z

Z Z

which amounts to

P8 n=1 in Ti1i2i3i4i5i6i7i8 = (−1) Ti1i2i3i4i5i6i7i8 , (6.121)

where i1, ··· , i8 are the eight virtual indices of the T tensor deﬁned in Eq. (6.119). Hence,

8 X Ti1i2i3i4i5i6i7i8 = 0, if in = 1 mod 2. (6.122) n=1

151 Imposing the condition Eq. (6.115) and transferring the physical X operators to the virtual level, we ﬁnd that

x x x x T = T = T

x x x

= T = T x = T x

x x x = T = T = T (6.123)

x x x

x x T T T = x = = x

x x x x x x T T T = = = x . x x x

In terms of components, Eq. (6.123) means that T = T 0 0 0 0 0 0 0 0 where i1i2i3i4i5i6i7i8 i1i2i3i4i5i6i7i8 0 0 0 0 0 0 0 0 i1i2i3i4i5i6i7i8 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 (6.124)

= Ti1i2(1−i3)(1−i4)i5i6i7i8

= ...

Combining Eq. (6.121) and (6.124), we ﬁnd that any conﬁguration of Ti1i2i3i4i5i6i7i8 P8 satisfying the condition k=1 ik = 0 mod 2 are equal. We can rescale the T tensor P8 such that Ti1i2i3i4i5i6i7i8 = 1 for k=1 ik = 0 mod 2, i.e.,

  P8 1 n=1 in = 0 mod 2 Ti1i2i3i4i5i6i7i8 = (6.125)  P8 0 n=1 in = 1 mod 2.

For simplicity, we consider the space to be R3 where the Haah code has a unique ground state.

X 3 |TNSi = CR gL,s1 gR,s2 gL,s3 gR,s4 ...TTT... |{s}i. (6.126) {s}

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

153 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.

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

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}iA¯ is orthogonal. Since |{t}iA is orthogonal because all the open virtual indices are connected with g tensors, the TNS P |TNSi = {t} |{t}iA ⊗ |{t}iA¯ is SVD. To prove the Generalized SVD Condition, we notice that if we have two diﬀerent

0 0 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

0 0 same. Otherwise, if {s}A¯ = {s }A¯, since the physical indices {s}A¯ and {s }A¯ in the ¯ 0 0 region A uniquely determine the open virtual indices {t}A¯ and {t }A¯, {t}A¯ = {t }A¯,

0 hence it is in contradiction with our assumption {t}A¯ 6= {t }A¯. Therefore, {t}A¯ 6=

0 0 0 {t }A¯ implies {s}A¯ 6= {s }A¯, and hence A¯h{t}|{t }iA¯ = 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

0 0 of {t} is also the same as in Sec. 6.2.1. Furthermore, Ah{t}|{t }iA = 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

0 0 1. Ah{t}|{t }iA ∝ δ{t},{t0} when |{t}iA and |{t }iA are not null vectors;

0 0 2. A¯h{t}|{t }iA¯ ∝ δ{t},{t0} when |{t}iA¯ and |{t }iA¯ 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.

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

0 0 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 (6.127)

Since six among the eight virtual indices of T 0 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 0 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) = 6l − 5. (6.130) log 2

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

160 open virtual indices is (7lx + 7ly + 7lz − 38) − (lx + ly + lz − 6) = 6lx + 6ly + 6lz − 32. The entanglement entropy is

S(A) = 6l + 6l + 6l − 32. (6.131) log 2 x y z

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}iA, are orthonormal, since the open virtual ¯ indices are connected with g tensors. However, the TNS basis |{t}iA¯ in the region A are not orthogonal. In other words, the basis |{t}iA¯ 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}iA, 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 Cut x

x T T x x

x (a)

Cut

x T x T

(b)

Figure 6.16: Transferring the Pauli X operators of the Bc operator from the region A (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}iA 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}iA, 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}iA 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}iA to construct the eigenvectors of Bc operators by projection. We prove the following statement:

0 If |{t }iA = Bc|{t}iA when Bc is inside the region A and also adjacent to the

0 0 region A’s boundary, then Ah{t }|{t}iA = 0 and |{t }iA¯ = |{t}iA¯. The proof is as follows. The ﬁrst part of the statement is a consequence of the

|{t}iA basis state orthogonality. Indeed, Bc ﬂips physical spins located at the region

162 A’s boundary. Thus the two sets {t} and {t0} 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

0 change the state at all, i.e., |{t }iA¯ = |{t}iA. As a consequence, we can perform the following factorization

0 0 |{t}iA ⊗ |{t}iA¯ + |{t }iA ⊗ |{t }iA¯ (6.132) h i = (1 + Bc)|{t}iA ⊗ |{t}iA¯.

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}iA 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}iA will generate as many unique states as there are Bc’s. The TNS can be brought to the following

163 form

X Y 1 + Bc |TNSi = |{t}i ⊗ |{t}i ¯, (6.133) 2 A A {t}0 c

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}0 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) (6.137)

+ 4(ly − 2)(lz − 2)

=4lxly + 4lxlz + 4lylz − 8lx − 8ly − 8lz + 16.

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

3 3 2 NB = (l − 1) − (l − 3) = 6l − 24l + 26, ∀l ≥ 3. (6.138)

For l = 2, we just have one Bc operator. Hence we have

2 NB = 6l − 24l + 26 − δl,2, ∀l ≥ 2. (6.139)

165 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 contracted 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, (6.141)

i5 = t5, i6 = t6, i7 = t7, i8 = t8.

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 X (x,y,z) in = 0 mod 2, x, y, z ∈ {0, 1} (6.143) n=1

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

000 100 010 001 i4 = i3 = i1 = i7 mod 2 (6.144) 000 110 101 011 i5 = i2 = i8 = i6 mod 2.

Since we only count the number of constraints for the open indices, we need to Gauss-

000 100 010 001 000 110 101 011 eliminate all these eight virtual indices i4 , i3 , i1 , i7 , i5 , i2 , i8 , i6 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 (6.145) Nc 6 12 18 24 30 36 42 48 54 60 66 72 78 84

Hence, for l ≥ 3, there are 6l − 12 (6.146)

167 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

2 Ng = 12l − 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

2 NB = 6l − 24l + 26 − δl,2, ∀l ≥ 2. (6.150)

Therefore the upper bound of the entanglement entropy reads

S =Ng − Nc − NB log 2 (6.151) =6l2 − 6l + 2, ∀l ≥ 2.

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 entanglement 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

Lx+Ly−1 is of dimension 2 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 properties 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 topological 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 combinations 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 entanglement 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

P r A.1.1 Proof of da = r∈T nadr

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

X r s ˜ t X c t X r s nanbNrsdt = Nabncdt = nanbdrds, (A.1) r,s,t∈T c∈A,t∈T r,s∈A

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

X ˜ t Nrsdt = drds. (A.2) t∈T

We then have the trivial re-writing

! ! ! X c X t X r X s Nab ncdt = nadr nbds , (A.3) c∈A t∈T r∈T s∈T

172 P t which means that t∈T ncdt is an eigenvalue of the matrix (Na)bc with eigenvalues P r P s and eigenvector r∈T nadr and s∈T nbds , 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

X r da = nadr. (A.4) r∈T

1 P r A.1.2 Proof of dr = q a∈A nada

We start with Eq. (3.5), multiply by da and sum over a ∈ A. Using Eq. (1.31), it follows

X X r ˜ t s X t c nadaNrsnb = ncNabda r,s∈T a∈A a,c∈A (A.5) X t = db ncdc. c∈A

P t For the simplicity of notations, let αt ≡ c∈A dcnc, which satisﬁes the eigenvalue equation

! X X s ˜ nbNs αr = dbαt. (A.6) r∈T s∈T tr

˜ ˜ t Notice the unorthodox use of the matrix (Ns)tr = Nsr, unlike in the line following

t t¯ Eq. (1.31). We deﬁne the matrix this way in order not to use the equation na = na¯.

T This matrix has the vector of quantum dimensions (d1, . . . , dN ) , 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

X ˜ r X ˜ t¯ Nstdr = dsdt = Nsr¯dr = dsdt¯. (A.7) t t 173 P ˜ T It follows from the above that r(Ns)trdr = dsdt. Hence (d1, . . . , dN ) 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

! ! X X X s ˜ X nbNs αr = db αt. (A.8) r∈T b∈A s∈T tr b∈A

P P s ˜ The matrix ( b∈A s∈T nbNs)tr is a completely positive matrix with integer strictly ˜ t positive coeﬃcients: for any t, r, there exists s such that Nsr > 0 and for every s

s there exists an nb > 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

X t αt = dcnc = qdt, (A.9) c∈A 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

X 2 X X t X 2 q dt = dc ncdt = dc , (A.10) t∈T c∈A t∈T c∈A where the last equality follows from Eq. (A.4). This implies

2 2 q = DA/DT . (A.11)

174 Furthermore, multiplying Eq. (3.5) by dadb for t = ϕ and summing over a, b reads

X c ϕ X t t¯ Nabnc = nanb, (A.12) c∈A t∈T which implies

X c ϕ X X t X t¯ dadbNabnc = dana dbnb a,b,c∈A t∈T a∈A b∈A (A.13) 2 2 = q DT .

On the other hand,

2 2 X c ϕ q DT = dadbNabnc a,b,c∈A

X 2 ϕ = db dcnc (A.14) b,c∈A

2 X ϕ = DA dcnc , c∈A hence

X ϕ q = dcnc . (A.15) c∈A

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 MTCs A, 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 J a(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

2πiτ(L0−c/24) χa(τ) = TrHa e , (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

− iπc χa(τ + 1) = θae 12 χa(τ), (A.17) 1 X χ − = S χ (τ), a τ ab b b

2πiha where θa = e 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

X Zdiag(τ, τ¯) = χ¯a(¯τ)χa(τ), (A.18) a 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)

X Z(τ, τ¯) = χ¯a(¯τ) Mab χb(τ), (A.19) a,b where M is called the mass matrix which must satisfy [S,M] = [T,M] = 0 to guarantee 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 correspondence 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 primary 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 ⊕i Hγi .

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

u Hu → ⊕a∈AnaHa. (A.20)

u The branching coefficient na 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]

X u χ˜u(τ) = na χa(τ), u ∈ U. (A.21) a∈A

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 ˜ X 2 Z(τ, τ¯) = |χ˜u(τ)| , (A.22) u∈U which when written in terms of the characters (A.16) of A reads like Eq. (A.19) with

X u u Mab = na nb . (A.23) u∈U

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

1 X X χ˜ − = S˜ χ˜ (τ) = S˜ nt χ (τ) s τ st t st a a t t,a (A.24) X 1 X = nsχ − = nsS χ (τ), b b τ b ba a b a,b i.e., nS˜ = Sn. (A.25a)

Similarly nT˜ = T n, (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),

s 2. θs = θa, if na 6= 0,

3. nC˜ = Cn,

P ϕ 4. q ≡ a na da = DA/DU ,

1 P t 5. dt = q a∈A nada,

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 = nP˜ nT, 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

r 2 π(a + 1)(b + 1) S = sin , (A.26) ab 2 + k k + 2

and

2πi a(a+2) 3k T = e 4(k+2) δ , c = . (A.27) ab ab k + 2

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

k X 2 ZAk+1 = |χn| , (A.28a) n=0,n∈Z 2`−2 X 2 2 ZD2`+2 = |χn + χ4`−n| + 2|χ2`| , n=0,n∈2Z 4`−2 X 2 2 ZD2`+1 = |χn| + |χ2`−1| n=0,n∈2Z 2`−3 X + (χnχ¯4`−2−n + χ4`−2−nχ¯n), n=1,n∈2Z+1

181 2 2 2 ZE6 = |χ0 + χ6| + |χ3 + χ7| + |χ4 + χ10| ,

2 2 2 ZE7 = |χ0 + χ16| + |χ4 + χ12| + |χ6 + χ10|

2 +|χ8| + χ8(¯χ2 +χ ¯14) + (χ2 + χ14)¯χ8,

2 ZE8 = |χ0 + χ10 + χ18 + χ28|

2 +|χ6 + χ12 + χ16 + χ22| ,

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 P to condensation. The partition function can be written as ZD2`+1 = a χa χ¯ω(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 simple 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 - (A.29)

E6 10 EXT SU(2)10 ⊂ SO(5)1

E7 16 AUT -

E8 28 EXT SU(2)28 ⊂ (G2)1

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)]

2 2 2 ZD10 = |χ0 + χ16| + |χ2 + χ14| + |χ4 + χ12| (A.30) 2 2 + |χ6 + χ10| + 2 |χ8| and

2 ZE7 = |χ0 + χ16| + (χ2 + χ14)¯χ8 + χ8(¯χ2 +χ ¯14) (A.31) 2 2 2 + |χ4 + χ12| + |χ6 + χ10| + |χ8| .

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) = nP˜ nT, (A.32) where

  10000000000000001      00100000000000100       00001000000010000  T   n =   (A.33)    00000010001000000       00000000100000000      00000000100000000

184 and   1 0 0 0 0 0      0 0 0 0 0 1         0 0 1 0 0 0  P˜ =   (A.34)    0 0 0 1 0 0       0 0 0 0 1 0      0 1 0 0 0 0 is an automorphism of the theory U. These two solutions for M are in one-to-one correspondence 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˜0, where

  sin π 1 cos 2π cos π 1 1  18 2 9 9 2 2     1 1 1 − 1 − 1 − 1   2 2 2 2 2     cos 2π 1 − cos π − sin π 1 1  2  9 2 9 18 2 2  S˜ =   (A.35) 3  π 1 π 2π 1 1   cos − − sin cos − −   9 2 18 9 2 2     1 − 1 1 − 1 − 1 1   2 2 2 2 2   1 1 1 1 1  2 − 2 2 − 2 1 − 2

185 and S˜0 is obtained by exchanging the last two rows of S˜, a so-called Galois symme-

˜ ˜0 ˜ try [202]. Both matrices S and S yield the same fusion rules Nt, e.g.,

  0 0 0 1 0 0      0 0 1 1 1 1       0 1 1 2 1 1  ˜   N˜ =   . (A.36) 3    1 1 2 2 1 1       0 1 1 1 0 1      0 1 1 1 1 0

c It is worth noting that this condensation of a theory without multiplicities (all Nab in ˜ s SU(2)16 are 0 or 1) yields a theory with multiplicities: some of the Ntr 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)]

2 2 2 ZD16 = |χ0 + χ28| + |χ2 + χ26| + |χ4 + χ24|

2 2 2 + |χ6 + χ22| + |χ8 + χ20| + |χ10 + χ18| (A.37)

2 2 + |χ12 + χ16| + 2 |χ14|

and

2 ZE8 = |χ0 + χ10 + χ18 + χ28| (A.38) 2 + |χ6 + χ12 + χ16 + χ22| .

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.

ϕ The partition function ZE8 stands for a condensation of all bosons with na = 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 algorithm 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 S˜2 = Θ(S˜T˜)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

σ0 n(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 condense, 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) product 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 ⊗N admits condensations, already in the Z2 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

m X db ≤ dai . (B.2) i=1

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., ⊗N ⊗N AFib × Z2 . This multi-layer TQFT satisﬁes all assumptions of Lemma1: for each 190 ⊗N anyon b ∈ AFib , 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

⊗N than db. We conclude that the AFib structure is preserved under any condensation transition in such a theory.

B.2 Proof for Example (iii), Multiple layers of

SO(3)k

⊗N In this section, we show that no condensation is possible in the TQFT SO(3)k

comprised of N layers of SO(3)k for any odd k and any integer N. The proof goes

⊗N by induction. We denote the particles in SO(3)k with a shorthand notation. An anyon that has the identity particle from SO(3)k in all layers, except for the k0 layers i , i , ··· , i , is denoted by {j j ··· j }. Here 1 ≤ j ≤ (k − 1)/2 can stand for 1 2 k0 i1 i2 ik0 il 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 {ji1 ji2 ··· 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 (k + 1) layers {j j ··· j } neither condense nor split. 0 i1 i2 ik0+1 We begin by showing that {j j ··· j } cannot condense. The particles i1 i2 ik0+1 {j j ··· j } can be obtained by fusing a {j j ··· j } with a {j }, where i1 i2 ik0+1 i1 i2 ik0 ik0+1

ik0+1 ∈/ {i1, ··· , ik0 }. In this case, Eq. (2) from the main text reads for t = ϕ

˜ ϕ ϕ N ↓ ↓ = n . (B.5) {ji ···ji } ,{ji } {ji ji ···ji ji } 1 k0 k0+1 1 2 k0 k0+1

˜ ϕ Now, because of the uniqueness of the antiparticle, N ↓ ↓ can be either {ji ···ji } ,{ji } 1 k0 k0+1 0 or 1. If it was 1, {j j ··· j }↓ would be the antiparticle of {j }↓. Because i1 i2 ik0 ik0+1 all particles are their own antiparticles, this would imply {j j ··· j }↓ = {j }↓. i1 i2 ik0 ik0+1

However, this is not possible for k0 > 1, because the associativity of fusion would then also imply that {j }↓ is the antiparticle (and coinciding with) {j ··· j j }↓, i1 i2 ik0 ik0+1 i.e., {j }↓ = {j ··· j j }↓. Remembering that {j j ··· j }, {j j ··· j } i1 i2 ik0 ik0+1 i1 i2 ik0 i2 i2 ik0 +1 do not split, and equating the quantum dimensions of the particles for these two identiﬁcations we have

dji dji ··· dji = dji , 1 2 k0 k0+1 (B.6)

dji = dji ··· dji dji . 1 2 k0 k0+1

For k0 > 1, this contradicts the fact that all nontrivial particles in this theory have ˜ ϕ quantum dimensions d > 1. This rules out the possibility N ↓ ↓ = 1 and {ji ···ji } ,{ji } 1 k0 k0+1 shows that {j j ··· j } does not condense for k > 1. i1 i2 ik0+1 0

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 ˜ ϕ N ↓ ↓ = 1. In the case ji1 6= ji2 , we can rely on the following argument to {ji1 } ,{ji2 } 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↓ × j↓ – with Eq. (2) from the main text this implies that i1 i2 ˆ {ji1 ji2 } neither splits nor condenses. In the case ji1 = ji2 ≡ j, deﬁne j ≡ {ji1 ji2 }.

We want to show that ˆj does not condense. As there are no fermions in SO(3)k with ˆ 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 {j } and {j } with zero mode sets I and I . We i1 i2 {ji1 } {ji2 } can then use the set Iˆ = I × I , containing the fusion product of any particle j {ji1 } {ji2 } in I with any particle in I , to prove that ˆj cannot condense. To show that {ji1 } {ji2 } ˆ Iˆj is a set of zero modes of j, the main challenge is to show that the product of any two elements from Iˆj cannot condense. The product of any two elements from Iˆj is always of the form {ji1 ji2 }. We have shown that when ji1 6= ji2 such particles cannot condense. We therefore need only show that non-trivial particles of form {ji1 ji2 } 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 ˆj have the same fusion coeﬃcients among themselves as the j anyons in SO(3)k in Example (ii)

ˆj00 j00 0 00 have, i.e., N = N 0 , where j, j , j ∈ SO(3) . Recall that conditions 1–3 from ˆj,ˆj0 j,j k j00 the definition of a set of zero modes only depend on the fusion coefficients Nj,j0 and the information, which particles are bosons. Hence, conditions 1–3 are satisfied for

Iˆj whenever they are satisﬁed for Ij in Example (ii). It remains to show that Iˆj is of large enough quantum dimension to satisfy the fundamental inequality Eq. (4) from the main text. For ˆj, Eq. (4) from the main text takes the form

2 2 X X dˆj = dj < da = daˆ. (B.7)

a∈Ij aˆ∈Iˆj

193 Upon taking the square root, this is equivalent to Eqs. (14) and (15) from the main ˆ text, which were shown to hold in Example (ii). Therefore the j = {ji1 ji2 } anyons do not condense and all {ji} have distinct restrictions. ˜ ϕ We conclude that for any k0 ≥ 1 only N ↓ ↓ = 0 is permitted and {ji ···ji } ,{ji } 1 k0 k0+1 hence Eq. (B.5) implies that {j j ··· j } does not restrict to the identity ϕ, i.e., i1 i2 ik0+1 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 {j j ··· j } does not i1 i2 ik0+1 split. For that, consider Eq. (2) from the main text for {j ··· j } with itself and i1 ik0+1 t = ϕ

X 2 X nr = N c nϕ {ji ···ji } {ji ···ji },{ji ···ji } c 1 k0+1 1 k0+1 1 k0+1 r c

ϕ = n1 = 1. (B.8)

We have used that none of the {ji1 ji2 ··· jil } with 1 ≤ l ≤ k0 + 1 can restrict to the identity ϕ since they cannot condense. This implies none of {j ··· j } splits, i1 ik0+1 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)

⊗N restricts to the vacuum in the N-layer theory SO(3)k . 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 a × a¯ containing no bosons other than the identity – i.e., nt = δt and a does not i i ai ↓ i ai

194 ↓ ↓ split. Moreover assume ai 6= aj for i 6= j. Then if a boson B appears in the fusion of ai and a¯j, ai × a¯j = B + ··· for any i 6= j, then B is not condensable.

Proof. Using Eq. (2) from the main text for a = ai, b =a ¯j and t = ϕ, we have δ = P nt nt = P nϕN c . For i 6= j we get P nϕ N B = 0. So if boson B ij t ai a¯j c c aia¯j B B aia¯j ϕ appears in ai × a¯j, we must have nB = 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

X t t X ϕ c ϕ B nBnB = nc NBB = 1 + nBNBB. (B.9) t c

ϕ ϕ Notice, however, that the left-hand side is greater or equal to nBnB. For condensation, ϕ P t t ↓ this implies nB = 1, and tells us that t nBnB = 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

a NBB = 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 Lemma3, and assuming B has dB > 1, conden-

a b B sation of B can only occur if NBBNBB ≤ Nab for all anyons a and b of A.

Proof. Lemma3 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

r r r B ϕ B (B.10) nanB = na = NaBnB = NaB

Consider now Eq. (2) from the main text for b 6= 1,B and for a 6= b, which gives

X t t B r r nanb = Nab ≥ nanb (B.11) t 195 Combining the a, B and b, B and a, b equations gives the inequality

b a B NBBNBB ≤ Nab. (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 t , i.e., P T δ still satisﬁes the Concatenation lemma of the 2 t1t2 t1t2t3... t1t2 3D toric code model in Sec. 6.3.3.

˜ (B) If Tt1t2t3... and Tt˜1t˜2t˜3... 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 P T T˜ δ , still satisﬁes the Concatenation lemma of the t1t˜1 t1t2t3... t˜1t˜2t˜3... t1t˜1 3D toric code model in Sec. 6.3.3.

Proof:

(A): Since T satisﬁes the Concatenation lemma, its elements T{t} are:

  P 0 if i ti = 1 mod 2 T{t} = (C.1)  P N if i ti = 0 mod 2,

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 T0 and the remaining open virtual indices after the contraction {t0}. Then we have:

0 X T{t0} = T{t}δtm,tn tm,tn X = T...tm...tn...δtm,tn tm,tn (C.2) X = T...tm...tm... tm

=T...0...0... + T...1...1...

Hence, the contraction still satisﬁes the Concatenation lemma:

 0 if P t0 = 1 mod 2 0  i i T {t0} = (C.3)  P 0 2N if i ti = 0 mod 2.

198 ˜ (B): Since T and T satisfy the Concatenation lemma, their elements T{t} and ˜ T{t˜} are:

  P 0 if i ti = 1 mod 2 T{t} =  P N if i ti = 0 mod 2  (C.4) 0 if P t˜ = 1 mod 2 ˜  i i T{t˜} =  ˜ P ˜ N if i ti = 0 mod 2,

where N and N˜ are the constants independent of the indices {t} and {t˜} respectively.

Suppose that we contract two indices tm ∈ {t} and t˜n ∈ {t˜}, and we denote the contraction as T0 and the remaining open virtual indices after the contraction {t0}. Then we have:

0 X 0 ˜ T{t } = T{t}T{t˜}δtm,t˜n

tm,tñ X ˜ (C.5) = T...tm...T...tñ...δtm,tñ

tm,t˜n ˜ ˜ =T...0...T...0... + T...1...T...1...,

P P ˜ The last line is nonzero if and only if i6=m ti and j6=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 ˜ P 0 P P ˜ equal to NN. Since i ti = i6=m ti + j6=n tn, we conclude that:

 0 if P t0 = 1 mod 2 0  i i T{t0} = (C.6)  ˜ P 0 NN if i ti = 0 mod 2.

0 T{t0} 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:

( ) X fxy({t}): ti, for all xy-planes i∈xy-plane ( ) X fyz({t}): ti, for all yz-planes (C.7) i∈yz-plane ( ) X fxz({t}): ti, for all xz-planes . i∈xz-plane

Notice that these equations are not linearly independent. Their summation is automatically 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:

c c c We ﬁrst ﬁx the notations: the elements of T are T{t}; {t } = {t1, t2,...} ⊂ {t} are the indices that contract with the indices of the local T tensor; and the outcome

0 0 tensor is denoted as T{t0} where {t } denotes the open virtual indices after contraction. Since T is assumed to satisfy the Concatenation lemma, we have:

     fxy({t}) = 0 mod 2     N if  fyz({t}) = 0 mod 2  T{t} =  (C.8)    fxz({t}) = 0 mod 2    0 otherwise,

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 Txxy¯ yz¯ z¯ also satisfy the Concatenation lemma for the X-cube model, as shown by Eq. (6.86). Using these notations, the tensor contraction is just:

0 X T 0 = T c c T c c . (C.9) {t } ...t1...t2...... t1...t2... {tc}

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 Txxy¯ yz¯ z¯. Then the tensor contraction is:

0 X T{t0} = T...x...Tx... (C.10) x

Graphically,

z y

x x y . (C.11) z

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 (C.12)

⇒

fxy({t}/x, x,¯ y, y¯) =0 mod 2

fxz({t}/x, x,¯ z, z¯) =0 mod 2, 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

0 T , which can be concatenated into fyz({t }). We ﬁnd:

   f ({t0}) = 0 mod 2   xy     N if 0  fyz({t }) = 0 mod 2 0  T{t0} =  (C.13)   0  fxz({t }) = 0 mod 2    0 otherwise.

Hence, T0 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 commutation 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.

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