Interplay Between Symmetry and Topological Order in Quantum Spin Systems

Interplay between Symmetry and Topological Order in Quantum Spin Systems

Hao Song

B.S., Nanjing University, 2009

M.S., University of Colorado Boulder, 2012

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulﬁllment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2015 This thesis entitled: Interplay between Symmetry and Topological Order in Quantum Spin Systems written by Hao Song has been approved for the Department of Physics

Prof. Michael Hermele

Prof. Victor Gurarie

Date

The ﬁnal copy of this thesis has been examined by the signatories, and we ﬁnd that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii

Song, Hao (Ph.D., Physics)

Interplay between Symmetry and Topological Order in Quantum Spin Systems

Thesis directed by Prof. Michael Hermele

In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.

In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems.

In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reﬂection.

Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classiﬁcation of pgSPT orders in general. Dedication

To my parents. v

Acknowledgements

First and foremost, I am sincerely grateful to my advisor, Prof. Michael Hermele, for his gen- erous ﬁnancial support, insightful guidance and warm encouragement during my graduate studies.

He led me to the exciting ﬁeld studying topological phases of quantum matter and has been making a lot of stimulating suggestions to my research projects. Without him, this thesis would not have been possible. Besides, Prof. Hermele has also provided a lot of professional help and advices on writing paper, preparing presentations and ﬁnding post-doctoral positions.

Next, I want to thank my friends in the department of physics, Dr. Gang Chen, Dr. Andrew

Essin, Sheng-Jie Huang, Yi-Ping Huang, Han Ma, Abhinav Prem, Zhaochuan Shen and Xiao Yin, for useful discussions and comments. Particular thanks are due to Sheng-Jie Huang for our ongoing inspiring collaborations. In addition, it is a pleasure to acknowledge the help of Dr. Hongcheng Ni on ﬁxing a technical LaTex issue during my thesis writing.

Moreover, I would like to thank Prof. Alexander Gorokhovsky, Prof. Victor Gurarie Prof. Leo

Radzihovsky, Prof. Ana Maria Rey and Prof. Jonathan Wise for being in my committee of compre- hensive exam and defense. I would especially appreciate Prof. Gurarie who proofread this thesis as well.

Finally, I want to express my deep gratitude to my parents who always support me no matter how far away I am. Also, I am grateful to all my friends that I have made in Boulder. Thank you all, sincerely. vi

Contents

Chapter

1 Introduction 1

1.1 Chiral Spin Liquids in Cold Atoms ...... 1

1.2 Topological Phases and Symmetries ...... 5

1.3 Anomalous Symmetry Fractionalizations and SPT Phases ...... 8

2 Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms in Three Dimensions 10

2.1 Theoretical Model ...... 11

2.2 Large-N Ground States ...... 14

2.2.1 Summary of the large-N mean-ﬁeld results ...... 14

2.2.2 Detailed descriptions of the mean-ﬁeld ground states ...... 16

2.2.3 Obtaining the mean-ﬁeld results ...... 18

2.2.4 Relation between bilayer states and square lattice saddle points ...... 21

2.3 Discussion ...... 26

3 Crystal Symmetry Fractionalization 28

3.1 Outline and Main Results ...... 29

3.2 Review of Z2 Topological Order ...... 31 3.3 Review: toric code model on the square lattice ...... 34

3.4 Toric Codes on General Two-dimensional Lattices with Space Group Symmetry . . . 37

3.5 Fractionalization and Symmetry Classes ...... 44 vii

3.5.1 Review of fractionalization and symmetry classes ...... 44

3.5.2 Fractionalization and symmetry classes in the solvable models ...... 46

3.6 Symmetry Classes Realized by Toric Code Models ...... 49

3.6.1 Model realizing all e particle fractionalization classes ...... 50

3.6.2 Toric code models without spin-orbit coupling ...... 53

3.6.3 General toric code models ...... 58

3.7 Summary and Beyond Toric Code Models ...... 61

4 Topological Phases Protected by Reﬂections and Anomalous Crystalline Symmetry Frac-

tionalization 65

4.1 Reﬂection Symmetry Protected Topological Phyases ...... 66

4.1.1 With a Single Reﬂection Symmetry ...... 66

4.1.2 With Two Orthogonal Reﬂection Symmetries ...... 68

4.2 Boundary Degrees of Freedom of SPT Phases ...... 69

4.2.1 Dimer model for 1d SPT ...... 69

4.2.2 CZX model for 2d SPT with onsite Z2 symmetry ...... 70 4.3 Model of Fractionalization Anomaly: a Single Reﬂection (ePmP) ...... 72

4.3.1 Spectrum of the model ...... 74

4.3.2 Symmetry behaviour of anyons ...... 77

4.4 Model of Fractionalization Anomaly: Two Orthogonal Reﬂections ...... 77

4.5 All Fractionalization Anomalies Associated with the Square Lattice Symmetries . . 81

5 Summary and Perspective 83

5.1 Summary ...... 83

5.2 Perspetive ...... 84 viii

Bibliography 86

Appendix

A Complete set of commuting observables 92

B Symmetry-invariant ground states 94

C General construction of e and m localizations in toric code models 99

C.1 General constraints on symmetry classes in toric code models ...... 101

C.1.1 Toric codes without spin-orbital coupling ...... 101

C.1.2 Toric codes with spin-orbit coupling ...... 106

D Models in TC (G) 119

E A model of eTmT 125 ix

Tables

Table

2.1 Ground state saddle-point patterns of χrr0 , and the corresponding energies in units

of NJ Ns for k = 2, 3,..., 10. The diﬀerent types of large-N ground states are

described in the text, and depicted in ﬁgures as indicated...... 16

2.2 This table contains information about our SCM numerical study on the cubic lattice

(1st column), as well as the related problems of a single bilayer (2nd column), and

single layer square lattice with k0 = k/2 (3rd column). On the left-hand side of

each entry of the table, the range of unit cell dimensions is shown as an inequality.

For every choice of lx,y,z within the given range, the number of times we ran the

SCM algorithm with distinct random initial conﬁgurations of χrr0 is shown on the

right-hand side of the entry (top). Also on the right-hand side is the minimum linear

system size L (bottom, italics)...... 24 x

2.3 Comparison of energies of a variety of simple saddle points (top four rows), with the

energy of the ground state found by SCM numerics (bottom row). All energies are in

units of NJ Ns. Each row represents a class of saddle points, described below. For

classes including multiple diﬀerent saddle points, the energy shown is the lowest in

the class. We considered the following classes of saddle points: Bilayer (Φ = 2πn/k).

We considered a generalization of the CSL bilayer saddle point described in the main

text, where the ﬂux through each plaquette is Φ = 2πn/k, where n = 0, . . . , k − 1.

k-site cluster. The energy of a cluster with k sites is proportional to the number of

bonds in the cluster,[1, 2] so the lowest-energy such state can be found by ﬁnding a

k-site cluster containing the greatest number of bonds. Uniform real χ. This is the

state where χrr0 is real and spatially constant. (2πnx,y,z/k)-ﬂux. These states have

2πnx/k ﬂux through every plaquette normal to the x-direction, and similarly for y

and z, where 0 ≤ nx,y,z ≤ k − 1. Since most of these states break lattice rotation

symmetry, the magnitude |χrr0 | is allowed to vary depending on bond orientation,

but is ﬁxed to be translation invariant.[3] ...... 25

3.1 Notation used in the chapter...... 32 xi

Figures

Figure

2.1 Ground-state clusters for k = 2, 3, 4. Shaded bonds are those with χrr0 6= 0. Bonds

with diﬀerent shading (or color in online version) may have diﬀerent magnitudes

|χrr0 |. (a) The k = 2 ground state clusters are dimers and square plaquettes. The

square plaquette is pierced by π-ﬂux, and the ratio of |χrr0 | on light (pink online)

and dark (blue online) bonds can be chosen arbitrarily. Setting |χrr0 | = 0 on the

two light (pink) bonds breaks the plaquette into two dimers. (b) The k = 3 ground

state cluster is a 6-site chain pierced by π-ﬂux. On the cubic lattice, such chains

can exist either as a ﬂat rectangular loop (left), or as the same loop bent by 90◦ in

the middle (right). In both cases, χrr0 = 0 on the dashed bond passing through the

middle of the loop. (c) The k = 4 ground state clusters are square plaquettes and

8-site cubes with Φs-ﬂux through the side plaquettes and Φt-ﬂux through top and

bottom plaquettes. There is a continuous one-parameter family of ground states on

an 8-site cube, described in the text...... 19 xii

2.2 Ground-state saddle point conﬁgurations of χrr0 for k = 5, 6,..., 10. The right

column is a three-dimensional view of each conﬁguration, with larger magnitude

|χrr0 | indicated by darker shading. All these saddle points can be viewed as bilayer

structures, with χrr0 identical on top and bottom layers. The left column thus shows

|χrr0 | on a single layer, with ﬂuxes indicated except for k = 9, where the ﬂuxes are

generally non-zero but follow a complicated pattern. Also, for k = 5, 6 the ﬂuxes

and |χrr0 | can be changed continuously within a single cluster without aﬀecting the

energy; only the simplest conﬁgurations are shown...... 20

2.3 Unit cells used for SCM calculations on the cubic lattice (a), single bilayer (b), and

single-layer square lattice (c). In the cubic case the primitive Bravais lattice vectors

are chosen parallel to the edges of the rectangular prismatic unit cell. The analogous

statement is true for the bilayer and single-layer cases, with primitive Bravais lattice

vectors parallel to the lx,y edges of the unit cell...... 22

3.1 Illustration of some geometrical objects important in the square lattice toric code

model. The edges in plaquette p are shown as thick dark bonds (blue online), while

the edges in star(v) are thick gray bonds (pink online). The two strings sx and sy

winding periodically around the system are also shown as thick dark bonds (blue

online)...... 34

3.2 Depiction of e and m strings in the square lattice toric code. s is an open e-string

joining vertices v and v0, denoted with thick dark bonds (blue online). t is an open

cut joining plaquettes p and p0, shown as a dotted line. The cut t contains the thick

gray bonds (pink online) intersected by the dotted line...... 35 xiii

3.3 (a) The lattice on which all 26 = 64 e particle fractionalization classes can be realized.

There are six types of plaquettes not related by symmetries, and the correponding

m plaquette terms are assigned independent coeﬃcients Ki (i = 1, 2, ··· , 6). Nearest- neighbor pairs of vertices are joined by two edges (dark and light; blue and red

online), drawn curved to avoid overlapping and to be clear about their movement

under space group operations. Plaquetes of type i = 1, 2, 3 are each formed by the

two edges joining a nearest-neighbor pair of vertices. Two vertices v1, v2 and two

e edges l1, l2 are labeled to illustrate the calculation of σpx discussed in the main text.

(b), (c) Subgraphs of the lattice in (a), each containing all the vertices and half the

edges. These subgraphs transform into one another under any improper space group

operation (i.e. reﬂections). We draw these subgraphs to illustrate the plaquettes of

type i = 4, 5, 6...... 51

3.4 TC0 (G) models. The shaded square is a unit cell and the origin of our coordinate

system is at the center of the square. Below each ﬁgure of lattice is the corresponding

TC symmetry class in the form (3.72). Here ar is the ground state eigenvalue of Av

for v at special points r = o, o,˜ κ; and b, b1, b2 are the ground state eigenvalues of Bp

for the plaquette p, which in these models is picked to be the smallest cycle made with

black edges where b, b1 or b2 is written, while b3 is for the plaquette made of a pair of

black and grey edges (black and pink online). These edges are drawn curved to avoid

overlapping and to be clear about their movement under space group operations. The

comparison between (a) and (b) gives an explicit example that moving the coordinate

1 1 system origin by 2 , 2 results in a transformation (3.74): Px → TxPx, σpx ↔ σtxpx,

σpxpxy ↔ σpxpxyσtxty. The symmetry class diﬀers from (e) by such a transformation

can be easily got by moving the coordinate system, so we do not bother drawing a

separate lattice for it...... 54 xiv

3.5 Two example models in TC (G) that realize TC symmetry classes not possible in

TC0 (G). The shaded square is a unit cell and the origin of our coordinate system

is at the center of the square. Below each ﬁgure of lattice is the corresponding TC

symmetry class in the form (3.72). Here ar is the ground state eigenvalue of Av

for v at special points r = o, o,˜ κ and b is the ground state eigenvalue of Bp for

the plaquette p, deﬁned here to be the smallest cycle enclosing the letter “b.” We

write α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ). (a) A model i li x i li xy i li x i li xy realizing some TC symmetry classes (and symmetry classes) that cannot be realized

without spin-orbit coupling. Here ,h1, h2 label two positions of a m particle for the

m 6 calculation of σpx = α1 in the main text. (b) A model realizing all 2 = 64 possible

m particle fractionalization classes [ωm]. Here, for simplicity, we make the restriction

γi = δi ≡ ci...... 59

4.1 A quantum spin system that respects a single reﬂection Px :(x, y, z) → (−x, y, z).

The grey plane x = 0 is invariant under the reﬂection. It divides the system into two

halves. If the system is short-range entangled, then there is an adiabatic process U (t)

tuning one half into a tensor product state, while keeping the other half unchanged.

If meanwhile, we also perform another adiabatic process related to the above one

by the reﬂection to the other half of the system, then in total we have an adiabatic

process U (t) ⊗ Px [U (t)] that keeps the reﬂection symmetry and makes the system

into tensor product state except for the spins on (or near) the grey plane. Thus,

the three-dimensional SPT phase protected by a single reﬂection is related to the

two-dimensional SPT phase lying on the mirror plane protected by on-site unitary

Z2 symmetry. Further, if the system have some boundary (like the top surface), then on (or near) the line (like the thickened line here) where boundary and the invariant

plane of reﬂection meet, there are some boundary degrees of freedom anomalous

under the reﬂection...... 67 xv

4.2 A quantum spin system that respects two orthogonal reﬂections Px :(x, y, z) →

(−x, y, z) and Py :(x, y, z) → (x, −y, z). The planes x = 0 (grey) and y = 0

(red) divide the system into four quarters. If the system is short-range entangled,

then there is an adiabatic process U (t) tuning one quarter into a tensor product

state, while keeping the other quarters unchanged. If meanwhile, we also perform

adiabatic processes related by the reﬂections to the other quarters, then in total we

have an adiabatic process U (t) ⊗ Px [U (t)] ⊗ Py [U (t)] ⊗ PxPy [U (t)] that keeps the

reﬂections and makes the system into tensor product state except for the spins on

(or near) the planes x = 0 and y = 0. Thus, the pgSPT phases are classiﬁed by

3 3 2 3 H (Z2,U (1))×H (Z2,U (1))×H (Z2 × Z2,U (1)) = Z2, with factors corresponding to the on-site SPT phases of the planes x = 0, y = 0 and their intersection line. If

the phase corresponds to only the non-trivial element of the last factor, then the

symmetry behaviour of the boundary degrees of freedom near the black dot on the

top surface is twisted by it...... 67

2 4.3 The dimer model for one-dimensional SPT phase labelled by ω ∈ H (G, UT (1)),

where G is the group of on-site symmetries...... 70

3 4.4 The CZX model for the two-dimensional SPT phase labelled by H (Z2,U (1)). (a) Each site (circle) contains four spins (dots) and the on-site symmetry of order 2 is

x x x x generated by UCZX = σ1 σ2 σ3 σ4 · CZ12 · CZ23 · CZ34 · CZ41. (b) The local term

u d l r on each plaquette p in the Hamiltonian is Hp = −Xp ⊗ Pp ⊗ Pp ⊗ Pp ⊗ Pp , where

Xp = |↑↑↑↑i h↓↓↓↓| + |↓↓↓↓i h↑↑↑↑| acts on the four spins on the plaquette p and

u,d,l,r Pp = |↑↑i h↑↑| + |↓↓i h↓↓| acts on the up, down, left and right neighbouring pairs

of spins around the plaquette p respectively. The ground state is ﬁxed within the

|↑↑↑↑i +|↓↓↓↓i bulk; the spins on each plaquette (square) are entangled as p√ p . But some 2 boundary degrees of freedom are not ﬁxed by the Hamiltonian, like those along the

right edge shown here...... 71 xvi

x y z 4.5 The dashed line is the axis of the reﬂection. Applying σ` (resp. σ` and σ` ) is graphically presented by highlighting ` in blue (resp. magenta and red). The vertex

terms Av at v = v0, v1, ··· , v5 are presented by the highlighted links which form a

t-shape. It can be checked that these six vertex terms commute with each other

and that the vertex terms in the two columns are related by the reﬂection; Avi = Px Av5−i , i = 0, 1, ··· , 5. The highlighted red plaquette in (a) presents a Bp lying

on the reﬂection axis, and it is invariant under the reﬂection. Because vertex terms

and plaquette terms away from the reﬂection axis are exactly the same as in the

toric code model and involve spins transform as σµ → σµ , in general it is not hard ` Px`

to see that Px (Av) = APxv, Px (Bp) = BPxp and that all vertex terms and plaquette

0 0 terms commute with each other; [Av,Av0 ] = [Av,Bp] = Bp,Bp0 = 0, ∀v, v , p, p ... 75

z 4.6 The red string stands for a product of σ` on it. The blue string stands for a product

x of σ` on it. (a) When an m is moved to x = 0, it becomes an excitation with y Av = −1 and Bp = −1 due to the factor σ 1 in Av. The excitation is still a (vx,vy− 2 )

boson. (b) When an ε is moved to x = 0, it becomes an excitation with Bp = −1.

The excitation is still a fermion...... 76

4.7 String operators that create a pair of e’s and m’s related by reﬂection. The red string

z x stands for a product of σ` on it. The blue string stands for a product of σ` on it. (a) A pair of e’s. (b) A pair of m’s...... 77

4.8 Generalized toric code model deﬁned on coupled layers of square lattice. Spins are

µ associated with the links, whose Pauli operators are generally denoted by σ` , µ = x, y, z. Explicitly, we use diﬀerent notations for spins on vertical and horizontal links;

µ µ we write σ` = τv for spins on vertical links (green online) at v = (vx, vy) ∈ Z × Z, µ µ and use σ` = σ(`,k) for horizontal links on the layer at z = k with k = 0, 1. In detail the xy-position ` of a horizontal link ` can be given be the xy-coordinates of its both

ends; ` = (`0, `1) with `0, `1 ∈ Z × Z, the xy-coordinates for the ends of ` respectively. 78 xvii

e −1 B.1 Graphical argument that Lc|ψ0i = |ψ0i, for c = sy ∪ Tx sy. The dotted lines show

−1 the L × L grid of primitive cells, and the paths sy and Tx sy are shown. c encloses a

region of area L, which can be broken (dashed lines) into L smaller sub-regions each of

0 e QL−1 e unit area. Let c be the cycle bounding one of the sub-regions, then L = L n 0 . c n=0 Tx c e e In addition, by translation symmetry L 0 |ψ i = L 0 |ψ i = ±|ψ i. Since an even c 0 Tyc 0 0

e number of sub-regions appear in the decomposition of Lc given above, we have

e Lc|ψ0i = |ψ0i...... 96

e B.2 Graphical illustration of the argument that Lc|ψ0i = |ψ0i, for c = sx ∪ Pxsx. It is

important to note that, in the interest of clarity, this ﬁgure is schematic in the sense

that it accurately shows the connectivity of the paths involved, and their properties

under translation symmetry, but not their properties under Px. The various symbols

are deﬁned in the main text. The vertical dashed line is the Px reﬂection axis, and

the vertex v has been chosen to lie near this axis for convenience. c1 and c2 are the

boundaries of the left and right shaded regions, respectively. The most important

L/2 point is that these two regions are related by Tx translation...... 98

m j C.1 The calculation of σpxpxy. Put an m particle at point h0, let hj = R (h0) , j = 1, 2, 3,

let t ∈ W¯ connecting h to h and t = Rjt . Then we have U m U m (h ) = f mLm 0 0 1 j 0 Px Pxy 0 0 t0 with f m ∈ {±1} and U m U m (h ) = Lm U m U m (h ) R Lm for j = 0 Px Pxy j t0···tj−1 Px Pxy 0 t0···tj−1 4 1, 2, 3. With some calculation, U m U m (h ) = Lm with t = t t t t . Thus, Px Pxy 0 t 0 1 2 3

e 2 2 3 σpxpxy = aVt . If P (v) is enclosed by t with R v 6= v, then v, Rv, R v, R v are

four diﬀerent vertices enclosed by t such as the grey vertices shown here. Since

Q3 e i=0 aRiv = 1, we have σpxpxy = aVt = aP−1(o) = aΓ(R2). The above statements are also true in the cases with spin-orbital coupling using the gauge choice described in

Appendix C.1.2. In the case without spin-orbital coupling, since av = aPxv = aPxyv,

m we have σpxpxy = aΓ(Px,Pxy)...... 102 xviii

m C.2 Illustration of the calculation of σtypx in Lemmas 7 and 16. Solid squares denote

the locations of holes h ∈ H, which are chosen so that h0 is arbitrary (but near the

y-axis), and h1 = Pxh0, h2 = Tyh0, h3 = Tyh1, h4 = Pyh0, h5 = Pyh1. h4 and h5

are not used in Lemma 7. Cuts are represented by solid lines, and hg0h1 denotes,

for example, a cut joining h0 to h1. The cuts t0 = hg0h1, t1 = hg0h2, t2 = hg1h3, and

t3 = hg2h3 are labeled, and are chosen to have properties described in the text. The

1 points o = (0, 0) and κ = (0, 2 ) are shown...... 102 xix

D.1 Depiction of the graphical notation used to represent stacking of vertices and edges.

The ﬁrst row shows the connectivity of vertices and edges, and the second row gives

the corresponding two-dimensional presentation. It is convenient to imagine the

graph of the lattice as ﬁrst being embedded in three-dimensional space, and then

projected into the two-dimensional plane. When these structures are present, we

always assume top edges (blue online) are transformed to bottom edges (red online)

under improper space group operations (i.e. reﬂections), while translations do not

swap edges with diﬀerent colors. Edges parallel to the x-axis, y-axis, z-axis are

labeled by symbols , l, ι, respectively. We use ζ and ξ to label diagonal edges.

For a diagonal edge, we can associate a unit vectore ˆ running along the direction of

the edge, always choosinge ˆx > 0. Then ζ (ξ) is used to label edges withe ˆy > 0

(ˆey < 0). Panels (a,d). This conﬁguration is only used in Fig. D.2c. The two

stacking vertices (blue and red online) together with edge ι1 connecting them are

projected into a point, presented as a ring (blue and red online). Edges 2, l3 pass

through the ring but do not end on it. The triple-stacking edges are presented as

double lines. Panels (b,e). A conﬁguration with double-stacking vertices and no

stacking edges. We use a darker point (blue online) to represent the upper vertex,

and a lighter ring (red online) to represent the lower vertex. The edges linked to the

upper vertex are darker (blue online) and the edges linked to the lower vertex are

lighter (red online). Panels (c,f). A situation with double-stacking vertices and

edges. The vertices are represented as in (b,e). The lower edge is represented by

a lighter double line (red online), and the upper edge is a single darker line (blue

online) drawn in the center of the double line...... 121 xx

D.2 TC (G) models (Part I). The shaded square is a unit cell and the TC symmetry classes are

calculated with the origin o at the center of the shaded square. Below each lattice is the

corresponding TC symmetry class in the form (3.72). The edges are labeled by diﬀerent

letters according to their directions as described in the text and in Fig. D.1. Edges that map

to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ with the subscript indicating their

1 1 1 1 position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the unit cell

is 1 × 1. For short, we deﬁne α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ), i εi x i εi xy i εi x i εi xy

where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, er = aP−1(r), and b is the

eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b is written.

The values of er and b are well-deﬁned with respect to any local spin frame system satisfying

Eqs. (C.15-C.17)...... 122

D.3 TC (G) models (Part II). The shaded square is a unit cell and the TC symmetry classes are

calculated with the origin o at the center of the shaded square. Below each lattice is the

corresponding TC symmetry class in the form (3.72). The edges are labeled by diﬀerent

letters according to their directions as described in the text and in Fig. D.1. Edges that

map to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ with the subscript indicating

1 1 1 1 their position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the

unit cell is 1 × 1. For short, we deﬁne α = cx (P ), β = cx (P ), γ = cz (P ) and i εi x i εi xy i εi x

z δ = c (P ), where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, e = a −1 , and i εi xy r P (r)

b is the eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b is

written. In panel (l), b is the eigenvalue of Bp for the top plaquette. The values of er and b

are well-deﬁned with respect to any local spin frame system satisfying Eqs. (C.15-C.17). .. 123 xxi

D.4 TC (G) models (Part III). The shaded square is a unit cell and the TC symmetry classes

are calculated with the origin o at the center of the shaded square. Below each lattice is

the corresponding TC symmetry class in the form (3.72). The edges are labeled by diﬀerent

letters according to their directions as described in the text and in Fig. D.1. Edges that map

to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ with the subscript indicating their

1 1 1 1 position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the unit cell

is 1 × 1. For short, we deﬁne α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ), i εi x i εi xy i εi x i εi xy

where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, er = aP−1(r), and b (or bi) is the

eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b (or bi) is

written. In panels (w) and (x), b1 and b are the eigenvalues of Bp for the top plaquettes.

The values of er and b (or bi) are well-deﬁned with respect to any local spin frame system

satisfying Eqs. (C.15-C.17) except in (w), where a further gauge ﬁxing is needed and we

z z z require c (Pxy) = c 0 (Pxy) = c (Px) = 1...... 124 l1 l1 l2

E.1 The image of the lattice in Fig. 4.8 projected into the xy-plane. The resulting

vertices, links and plaquettes are called xy-vertices, xy-links and xy-plaquettes re-

spectively. Auxiliary dashed lines are added to triangularize the plane. In our eT mT

model, vertices are colored diﬀerently according to the form of vertex term on them.

` ` For each xy-link `, we use T3 to label the black end of `, T0 for the other end, and

` ` ` T1 , T2 for the two vertices next to T3 . In addition, t is a cut of links connecting xy-plaquettes p and p0...... 125 Chapter 1

Introduction

Topological phases of quantum matter are those with an energy gap to all excitations, and host remarkable phenomena such as protected gapless edge states, and anyon quasiparticle excitations with non-trivial braiding statistics. Based on whether symmetries play a crucial role in the stability of a phase, topological phases fall into two subcategories: intrinsic ones and symmetry-protected ones. The intrinsic ones diﬀer from one another due to distinct patterns of quantum entanglement, which are stable even when all symmetries are explicitly broken.

In this thesis, we ﬁrst study Mott insulators of ultracold alkaline earth atoms, which show potential for experimental realization of a kind of topological phase called a chiral spin liquid, thanks to the enhanced spin symmetries in the system. In the reminder of thesis, we study how symmetry may fractionalize on anyons in gapped Z2 spin liquids, which provides a valuable perspective for understanding symmetry protected / enriched topological orders.

1.1 Chiral Spin Liquids in Cold Atoms1

Ultracold atom experiment techniques enable us to vary parameters of quantum many-body systems that can hardly be changed in solid state materials.[5–7] For example, in solid state systems the crystal structure is selected by nature, so it is usually not easy to study the dependence of the system properties on the lattice structure. But in ultracold atom experiments the optical

1 This section has been published as a portion of Hao Song and M. Hermele, Phys. Rev. B 87, 144423 (2013),[4] copyright 2013 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society. 2 lattice can be chosen artificially, and its dimension and geometry can be varied. Also, we have significant freedom to select the constituent particles of a many-body system. They can be atoms or molecules, bosons or fermions, and so on. Different atoms or molecules interact with one another quite differently, and in some cases the interactions can be tuned with electric or magnetic field.

So cold atoms promise to allow us explore systems in new parameter regimes, or even systems that have no analog in solid state materials.

Fermionic[8] ultracold alkaline earth atoms (AEAs) have attracted signiﬁcant interest recently due to their unique properties,[1, 2, 9–27] and experimental progress developing the study of many- body physics in AEA systems has been rapid [28–43]. One key feature of AEAs is the presence, to an excellent approximation, of SU(N) spin rotation symmetry, where N = 2I + 1 and I is the

1 3 nuclear spin.[9, 10] This occurs in both the S0 ground state and a metastable P0 excited state, where the electronic angular momentum Je = 0 and the hyperﬁne interaction is thus quenched.

This leads to the nuclear-spin-independence of the s-wave scattering lengths between AEAs, and to SU(N) spin rotation symmetry. When loaded in optical lattices, AEA systems are described by

SU(N)-symmetric Hubbard models.[9] Since the largest I obtained using AEA is I = 9/2 in the case of 87Sr, N ≤ 10 is the experimentally accessible regime. Diﬀerent setups are possible, and as a result, SU(N) versions of several models, such as the Kugel-Khomskii model, the Kondo lattice model, and the Heisenberg spin model, can be realized with AEAs as special or limiting situations of the more general Hubbard model.

Among these models, we focus in this paper on SU(N) antiferromagnetic Heisenberg models, which describe the Mott insulator phase of fermionic AEAs in optical lattices. More speciﬁcally, we are concerned with such models on three dimensional lattices, which have received much less attention than the one- and two-dimensional cases. Because of the enlarged symmetry, the number of spins needed to make a singlet, denoted by k, is in general larger than two. In the simplest

AEA Heisenberg model with one atom per lattice site, k = N. In addition, in the semiclassical limit of the Heisenberg models that can be realized using AEAs, two neighboring classical spins prefer energetically to be orthogonal rather than anti-parallel.[1] Both these features contrast 3 with SU(2) antiferromagnetic Heisenberg models appropriate for some solid state materials, where neighboring pairs of spins can and tend to form singlet valence bonds, and neighboring classical spins prefer to be anti-parallel. We can thus expect new physics in SU(N) Heisenberg models with k > 2.

Indeed, Ref. [1] argued that the underconstrained nature of the semiclassical limit makes magnetic order unlikely for large enough N on any lattice, and non-magnetic ground states are more likely. While the models of physical interest are challenging to study directly, information about possible non-magnetic ground states can be obtained in a large-N limit designed to address the competition among such states.[44–46] Such a large-N study was carried out for AEA SU(N)

Heisenberg models on the two-dimensional square lattice in Refs. [1, 2]. One possible non-magnetic state is a cluster state, where clusters of k (or a multiple of k) neighboring spins form singlets; this is a generalization of a valence bond state. Another possibility is a spin liquid state, where full

1 translational symmetry is preserved. For the simplest AEA Mott insulators (with S0 ground state atoms only), on the square lattice the large-N study ﬁnds cluster states for k ≤ 4, and a chiral spin liquid (CSL) state for k ≥ 5.[1, 2] The CSL spontaneously breaks time-reversal (T ) and parity (P) symmetries, and can be viewed as a magnetic analog of the fractional quantum Hall eﬀect (FQHE), with similar exciting properties of quasiparticles with anyonic statistics, gapless chiral edge states, and so on.[47–49] CSLs have also been found in a variety of other exactly solvable models.[50–56]

The CSL is, however, intrinsically a two-dimensional phenomenon, so it is natural to ask about non-magnetic ground states of SU(N) antiferromagnetic Heisenberg models in three dimensions.

In this paper, we address this question by a large-N study of a class of SU(N) Heisenberg models on the simple cubic lattice, and find a rich phase diagram as a function of k including cluster states, but also more intricate inhomogenous states. Most strikingly, for k = 7, 10 we find a bilayer CSL state, where the lattice spontaneously breaks into weakly coupled square bilayers (thus breaking rotational symmetry), each of which is a two-dimensional CSL. We thus find that the CSL survives to three dimensions, relying on spontaneous symmetry breaking that results in effective quasi-two- dimensionality. 4

We now deﬁne our model before brieﬂy surveying some related prior work. We consider a

1 fermionic AEA with N spin species, and put m S0 ground state atoms on each site of a simple cubic lattice (see Sec. 2.1 for more details). The atoms form a Mott insulator due to repulsive on- site interactions. For simplicity, we consider the case of dominant on-site interaction, so that the spin degrees are governed by a antiferromagnetic superexchange interaction restricted to nearest neighbors. While m = 1 is the most interesting situation since it best avoids three-body losses,

N we also consider more generally the case where m is an integer. Then, the minimum number of N spins needed to make a SU(N) singlet is k = m . We sometimes refer to k as the ﬁlling parameter. When m = 1, each spin transforms in the fundamental representation of SU(N). In the large-N limit, N is taken large while k is held ﬁxed. Given the physical interpretation of k, we thus view the large-N results for a given k as a guide to the physics of the physically realizable model with m = 1 and N = k.

Our focus is on three spatial dimensions, but we note that one-dimensional SU(N) Heisenberg spin chains have been solved exactly for the case m = 1,[57] and the eﬀective ﬁeld theory of such chains is understood for general m.[58] The latter analysis shows that gapless states with quasi- long-range order, as well as gapless cluster states, occur in one dimension. In two dimensions, early studies of SU(N) antiferromagnets focused on models where two neighboring spins can be combined to form a singlet. This work included the models we consider for the case m = N/2,[44, 45] but also other SU(N) antiferromagnets with spins transforming in two distinct conjugate representations on the two sublattices of a bipartite lattice.[46] Models with k = 2 have also received attention more recently,[13, 25, 59, 60] and two dimensional models with k > 2 have been studied[1, 2, 14,

15, 17, 26, 27, 61–66] (see Ref. [2] for a more detailed discussion of some of these prior works).

The m = 1, N = 3 model on the square lattice is magnetically ordered,[14] and there is also evidence for magnetic order for m = 1, N = 4.[15] Only a little attention has been devoted to the case of three dimensions,[14, 61, 67] but we note the high temperature series study of Ref. [67], where the m = 1 model on the simple cubic lattice was studied for various values of N, and it was found that increasing N led to a decreased tendency toward magnetic order. References [68, 69] 5 studied eﬀective models for four-site singlet clusters on the cubic lattice. Finally, we note that high-spin quantum magnets can also be realized using ultra-cold alkali atoms. While N-component such systems do not generically obey SU(N) spin symmetry, the symmetry is enhanced above

SU(2),[70] and such systems have received signiﬁcant attention.[70–76]

1.2 Topological Phases and Symmetries2

Topological phases of matter are those with an energy gap to all excitations, and host remarkable phenomena such as protected gapless edge states, and anyon quasiparticle excitations with non-trivial braiding statistics. Following the discovery of time-reversal invariant topological band insulators,[78–80] signiﬁcant advances have been made in understanding the role of symmetry in topological phases.

Two broad families of such phases are symmetry protected topological (SPT) phases,[81–

85] and symmetry enriched topological (SET) phases. SPT phases, which include topological band insulators, reduce to the trivial gapped phase if the symmetries present are weakly broken.

These phases lack anyon excitations in the bulk, and many characteristic physical properties are conﬁned to edges and surfaces. SET phases, on the other hand, are topologically ordered, with anyon excitations in the bulk. Topological order is robust to arbitrary perturbations provided the gap stays open, and SET phases remain non-trivial even when all symmetries are broken. In the presence of symmetry, there can be an interesting interplay between symmetry and topological order. This interplay is important, because properties tied to symmetry are often easier to observe experimentally. For example, in fractional quantum Hall liquids,[86, 87] quantization of

Hall conductance[86] and fractional charge[88–90] have been directly observed, and arise from the interplay between U(1) charge symmetry and topological order. The example of fractional quantum Hall liquids makes it clear that the study of SET phases has a long history, which cannot be adequately reviewed here; instead, we simply mention two areas of prior work that have close ties

2 This section has been published as a portion of Hao Song and M. Hermele, Phys. Rev. B 91, 014405 (2015),[77] copyright 2015 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society. 6 with the focus and results of the present paper. First, topologically ordered quantum spin liquids are another much-studied class of SET phases.[47, 91–97] Second, a systematic understanding of the role of symmetry in SET phases has recently been developing, including work on classiﬁcation of such phases; some representative studies are found in Refs. [98–121].

Most of the recent work on SPT and SET phases has focused on on-site symmetries such as time reversal, U(1) charge symmetry, and SO(3) spin symmetry. For SPT phases, this restriction makes sense physically, because a generic edge or surface will not have any spatial symmetries, but may have on-site symmetry. Of course, there can be clean edges and surfaces, and some works have examined the role of space group symmetry in SPT phases.[122–131] For SET phases, there is not a good physical justiﬁcation to ignore spatial symmetries; the presence of anyon quasiparticles means that symmetries of the bulk can directly impact characteristic physical properties. Indeed, a number of studies have focused on the role of space group symmetry in SET phases.[98, 101–

105, 107, 114, 117, 120] However, many recent works on SET phases have limited attention to on-site symmetry.

Recently, A. M. Essin and M. Hermele, building on earlier work,[98, 99] introduced a symmetry classification approach to bosonic SET phases in two dimensions, designed to handle both on-site and spatial symmetries.[107] The basic idea is to consider a fixed Abelian topological order and fixed symmetry group G, and establish symmetry classes corresponding to distinct possible actions of symmetry on the anyon quasiparticles, so that two phases in different symmetry classes must be distinct (as long as the symmetry is preserved). Under the simplifying assumption that symmetry does not permute the various anyon species, the approach of Ref. [107] amounts to classifying distinct types of symmetry fractionalization, where this term reflects the fact that the action of symmetry fractionalizes at the operator level when acting on anyons.

Distinct types of symmetry fractionalization are referred to as fractionalization classes, and characterize the projective representations giving the action of the symmetry group on individual anyons. Assigning a fractionalization class to each type of anyon specifies the symmetry class of a SET phase. Ref. [107] focused primarily on the simple case of Z2 topological order, giving a 7 symmetry classification for square lattice space group plus time reversal symmetry, that can easily be generalized to any desired symmetry group. For Z2 topological order with symmetry group G, a symmetry class is specified by fractionalization classes [ωe] and [ωm], for e particle (Z2 charge) and m particle (Z2 flux) excitations, respectively. Mathematically, distinct fractionalization classes are

2 elements of the cohomology group H (G, Z2). In more detail, a symmetry class is an un-ordered pair h[ωe], [ωm]i ' h[ωm], [ωe]i, where the lack of ordering comes from the fact that the distinction between e and m particle excitations is arbitrary, and we are always free to make the relabeling e ↔ m.

A crucial issue left open by the general considerations of Ref. [107] is the realization of symmetry classes in microscopic models (or physically reasonable low-energy eﬀective theories). In this paper, focusing on Z2 topological order and square lattice space group symmetry, we address this issue via a systematic study of a family of exactly solvable lattice models, in which many symmetry classes are realized. This is interesting for several reasons. First, to our knowledge, a general framework to describe SET phases with space group symmetry has not yet emerged, and concrete models for such phases are likely to be useful in developing such a framework. This contrasts with SET phases with on-site symmetry, where powerful tools are available, including approaches based on Chern-Simons theory,[106, 110, 111] on classiﬁcation of topological terms using group cohomology,[108, 109] and on tensor category theory.[132, 133] Second, it is likely that not all symmetry classes are realizable in strictly two-dimensional systems. For on-site symmetry, some symmetry classes can only arise on the surface of a d = 3 SPT phase.[113, 115, 134, 135]

Understanding which space group symmetry classes can be realized in simple models is a step toward addressing the more challenging general question of which classes can (and cannot) occur strictly in two dimensions. Finally, the explicit models we construct can be used as a testing ground for new ideas to probe and detect the characteristic properties of SET phases, in both experiments and numerical studies of more realistic microscopic models.

The models we consider are generalizations of Kitaev’s Z2 toric code[136] to arbitrary two- dimensional lattices with square lattice space group symmetry (a precise deﬁnition appears in 8

Sec. 3.4). By appropriately choosing the lattice geometry, varying the signs of terms in the Hamil- tonian, and allowing symmetry to act non-trivially on spin operators, many but not all symmetry classes can be realized. Varying the signs of terms in the Hamiltonian modulates the pattern of background Z2 fluxes and charges in the ground state, and this in turn affects the symmetry fractionalization of e and m particles, respectively. In addition, non-trivial action of symmetry on the spin degrees of freedom also affects symmetry fractionalization. We have obtained a complete understanding for the specific family of models considered, in the sense that for every symmetry class consistent with the considerations of Ref. [107], we either give an explicit model realizing this symmetry class, or we prove rigorously that it cannot occur within our family of models.

1.3 Anomalous Symmetry Fractionalizations and SPT Phases

The above results raise the question of whether the symmetry classes not found in our family of models can be realized at all in two dimensions. It turns out that some are indeed anomalous; they can only appear on the surface of some three-dimensional system with non-trivial SPT order.

In particular, we make a careful study of the anomalous symmetry fractionalization pattern with a reﬂection squaring to −1 acting on both the bosonic charge (e) and bosonic ﬂux (m) quasiparticles.

We call this situation eP mP , where P comes from the word parity. Some related attempts are made recently to understand this anomaly in the presence of additional U(1) symmetry.[137, 138]

Here we are going to explain the eP mP anomaly without assuming further symmetries but only a single reﬂection.

To understand the anomalous nature, we need first to improve our understanding of SPT orders. During recent years, great efforts have been made to under the SPT phases with on-site symmetries.[85] In this thesis, we focus on bosonic SPT phases protected only by crystalline point group symmetry in three dimensions, which we dub point group SPT (pgSPT) phases. The key insight here is that the pgSPT phase in the present of a reflection is related to the two-dimensional topological phase on the mirror plane with on-site Z2 symmetry. This viewpoint can be extended to give general classifications to pgSPT phases. In addition, 9 it tells us the location of the non-trivial boundary degrees of freedom and how they transform under the point group symmetries. With them, we construct exactly solvable models with Z2 topological order on the surface to realize anomalous symmetry fractionalizations. This logic leads to a bunch of interesting stories showing the connections among SPT phases, non-trivial boundary symmetries and surface topological order with anomalous symmetry fractionalizaton. Chapter 2

Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms in Three Dimensions1

We study a class of SU(N) Heisenberg models, describing Mott insulators of fermionic ultracold alkaline earth atoms on the three-dimensional simple cubic lattice. Based on an earlier semiclassical analysis, magnetic order is unlikely, and we focus instead on a solvable large-N limit designed to address the competition among non-magnetic ground states. We ﬁnd a rich phase diagram as a function of the ﬁlling parameter k, composed of a variety of ground states spontaneously breaking lattice symmetries, and in some cases also time reversal symmetry. One particularly striking example is a state spontaneously breaking lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state.

In Sec. 2.1, we review the large-N solution to our model. This is followed by presentation of the large-N results for k = 2,..., 10 in Sec. 2.2, together with a discussion of how those results are obtained and checked. As part of that discussion, we develop an interesting relation between some cubic lattice saddle points (including the ground state saddle points for k = 5,..., 10) and saddle points on the single-layer square lattice with ﬁlling parameter k0 = k/2. The paper concludes with a discussion of the striking properties of the bilayer CSL state (Sec. 2.3).

1 This chapter has been published as a portion of Hao Song and M. Hermele, Phys. Rev. B 87, 144423 (2013),[4] copyright 2013 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society. 11

2.1 Theoretical Model

The SU(N) Hubbard model

2 X α† X α† HHubbard = −t cr cr0α + h.c. + (U/2) cr crα − m , (2.1) hrr0i r

α† describes the behavior of fermionic AEAs on an optical lattice.[9] Here cr and crα are the creation and annihilation operators for the fermionic atom with spin state α at site r. The sum in the ﬁrst term is over nearest-neighbor pairs of lattice sites. We will primarily consider the simple cubic lattice. We choose the number of atoms so that m is the integer number of atoms per lattice site.

There are N spin states, α, β = 1, 2,...,N, and spin indices are summed over when repeated. The

α† total number of lattice sites is Ns. The operator cr transforms in the fundamental representation of SU(N), while crα transforms in the anti-fundamental representation, which is related to the fundamental by complex conjugation. The upper and lower positions of the Greek indices are used to indicate the distinction between these two representations (they are unitarily equivalent only for

N = 2).

As is well known, the SU(2) Heisenberg model can be obtained as a low energy eﬀective description of the SU(2) Hubbard model when U t. The generalization to the SU(N) version is straightforward. In second order degenerate perturbation theory, one obtains the SU(N) antiferromagnetic Heisenberg model deﬁned by the Hamiltonian

X α† β† H = −J (fr fr0α)(fr0 frβ), (2.2) hrr0i

α† 2 α† with the Hilbert space restricted by fr frα = m, and J = 2t /U > 0. We now use fr rather

α† than cr to denote the fermion creation operator, to emphasize that once we pass to the Heisenberg model, the fermions do not move from site to site. This is important, because the structure of

α† the large-N mean-ﬁeld theory is that of a hopping Hamiltonian for the fr fermions, but it is not correct to interpret this hopping as motion of atoms. Instead, in the large-N mean-ﬁeld theory, the

α† fr fermions are spinons, fractional particles that may be either confined or deconfined depending 12 on the nature of fluctuations about a mean-field saddle point. See Ref. [2] for further discussion of this point.

On each site, there are m atoms that form a SU(N) spin. The Hamiltonian (2.2) deﬁnes an antiferromagnetic interaction, since by rearranging the fermion operators it can be written as

X ˆβ ˆα 0 H = J Sα(r)Sβ (r ), (2.3) hrr0i

β β† where Sˆα(r) = fr frα ﬂips the spin on site r.

We study this model on the simple cubic lattice, the simplest three dimensional case, with varying parameters N and m. While we consider more general parameter values, m = 1 is the case of greatest physical interest because putting only one atom on each site best avoids potential issues due to three body loss. The largest N that can be obtained using alkaline earth atoms is N = 10 in the case of 87Sr.

Based on a semiclassical analysis, Ref. [1] argued that for large enough N, magnetic ordering is unlikely on any lattice. The argument proceeds in the semiclassical limit, where a lower bound on the dimension of the ground state manifold is derived. For N > Nc, where Nc depends on the lattice coordination number, the ground state manifold is extensive, meaning its dimension is proportional to the number of lattice sites. This situation occurs in some geometrically frustrated systems and is likely to lead to a strong or complete suppression of magnetic order[139], even in the semiclassical limit that favors magnetic order by construction. Therefore, non-magnetic ground states are likely when N > Nc. For the square lattice Nc = 3,[1] and the argument is easily extended to ﬁnd Nc = 4 on the cubic lattice.

Ideally, we would like to predict the properties of the SU(N) antiferromagnetic Heisenberg model on cubic lattice for N ≤ 10, m = 1. But this is extremely challenging. Instead, following the work of Refs. [1, 2], we apply a large-N limit in which the model becomes exactly solvable, and which allows us to address the competition among diﬀerent non-magetic ground states. We ﬁx the

N ratio k = m (for integer k), while taking both N → ∞ and m → ∞. We shall sometimes refer to k as the ﬁlling parameter. For each k we thus obtain a sequence of models (N = k, m = 1); 13

(N = 2k, m = 2), and so on. For every model in this sequence, k is the minimum number of spins needed to form a singlet, and it is thus reasonable that the large-N limit may capture the physics of the case N = k, m = 1 of greatest interest.

To proceed with the large-N solution, one goes to a functional integral representation, where the partition function is Z Z = DχDχ∗DλDf¯Df e−S, (2.4) where

Z Z 2 X X |χrr0 | S = f¯α∂ f + N r τ rα J τ r τ hrr0i Z X ¯α + χrr0 fr fr0α + h.c. τ hr,r0i Z X ¯α +i λr fr frα − m . (2.5) τ r

The ﬁeld χrr0 is a complex Hubbard-Stratonovich ﬁeld that has been used to decouple the exchange

α† interaction, and λr is a real Lagrange-multiplier ﬁeld enforcing the fr frα = m constraint. The fermion ﬁelds f and f¯ are the usual Grassmann variables. We have introduced J = NJ; J is held

R R β ﬁxed in the large-N limit. Finally, τ ≡ 0 dτ. We shall always be interested in zero temperature, i.e. β → ∞.

When both N and m are large, the eﬀective action for χ and λ (obtained upon integrating out fermions), is proportional to N (since m ∼ N), and therefore the saddle point approximation becomes exact for the χ and λ integrals. We can therefore replace χ and λ by their saddle-point values, χrr0 → χ¯rr0 and λr → iµr. The saddle-point equations are

D α† E m = fr frα , (2.6)

J D α† E χ¯ 0 = − f f . (2.7) rr N r0 rα 14

The above averages are taken in the ground state of the saddle-point (or mean-ﬁeld) Hamiltonian

2 X |χ¯rr0 | X H = N + m µ MFT J r hrr0i r X α† X + χ¯rr0 fr fr0α + h.c. − µrnˆr, (2.8) hrr0i r

α† wheren ˆr ≡ fr frα.

The ground state is determined by finding the global minimum of EMFT ({χrr0 } , {µr}), the ground state energy of HMFT , as a function of the χ’s and µ’s, with the constraint that the saddle point equations must be satisfied. While any solution of the saddle point equations gives an extremum of the energy, in general it is not trivial to find the global minimum. To address this question, we follow Refs. [1, 2] and apply the combination of analytical and numerical techniques developed there, as described below in Sec. 2.2.

2.2 Large-N Ground States

2.2.1 Summary of the large-N mean-ﬁeld results

In the limit N → ∞, the ground states are characterized entirely by the mean-field saddle point values of χrr0 and µr. The most important information is contained in χrr0 , since typically it is possible for a given χrr0 to find µr so that the density constraint Eq. (2.6) is satisfied. For instance, depending on whether two sites are connected (i.e. whether there is a set of nonzero

χrr0 ’s forming a path connecting the two sites), we can tell whether the spins on the two sites are correlated or not. Not all the information contained in χrr0 is physical. The theory has a U(1) gauge redundancy iφ(r) frα → frαe , (2.9) i(φ(r)−φ(r0)) χrr0 → χrr0 e so the physical information is contained in the following gauge-invariant quantities: (1) magnitude

|χrr0 | and (2) ﬂux Φ = a12 + a23 + a34 + a41 through each plaquette, where 1, 2, 3, 4 indicates

ia 0 the four vertices of a plaquette and arr0 is the phase of the χrr0 , i.e. χr0r = e rr |χrr0 |. (Since

∗ χr0r = χrr0 , ar0r = −arr0 .) 15

Based on a combination of analytical and numerical techniques described below, we found the ground state conﬁguration of χrr0 and µr for k = 2,..., 10. These results, which are rigorous for k = 2, 3, 4, are summarized in Table 2.1. Diﬀerent types of ground states are found depending on k. In an n-site cluster pattern of χrr0 , the lattice is partioned into n-site clusters such that

0 χrr0 6= 0 only if r, r lie in the same cluster. We call the corresponding ground state a n-site cluster state, which can be viewed as a generalization of a valence bond state (2-site cluster state, in our terminology). Similarly, a bilayer pattern partitions the lattice into bilayers, and χrr0 is only nonzero for r, r0 in the same bilayer. The corresponding ground states are called bilayer states. In all cases, each bilayer is comprised of two adjacent {100} lattice planes. A CSL bilayer is a special kind of bilayer state, where in each bilayer   0 χ, hrr i lies within either layer; |χrr0 | = (2.10)  J 0  k , hrr i connects the two layers.

Moreover, there is a uniform flux 4π Φ = (2.11) k through each plaquette lying within the two layers, and zero flux through each plaquette perpendicular to the two layers. This situation corresponds to a uniform orbital magnetic field applied perpendicular to the layers. At the mean-field level, a single CSL bilayer exhibits integer quantum

Hall eﬀect with ν = 1 for each spin species of frα fermion.

To fully understand the diﬀerent ground states, one has to go beyond the N = ∞ or mean-

field description. At the mean-field level, the number of ground state arrangements of clusters or bilayers on the cubic lattice diverges with the system size. For example, there are usually many ways to tile the lattice with a given type of n-site cluster. Also, in the CSL bilayer state, the direction of flux can be chosen independently in each bilayer without affecting the N = ∞ ground state energy. Such degeneracies can be resolved by computing the first correction (perturbative in

1/N) to the ground state energy;[46] these calculations are left for future work.

In cluster states, another important effect of fluctuations is to confine the frα fermions; the 16 k Large-N ground state Sketch of χrr0 Energy 2 2/4-site cluster Fig. 2.1a -0.125 3 6-site cluster Fig. 2.1b -0.0833333 4 4/8-site cluster Fig. 2.1c -0.0625 5 20-site cluster Fig. 2.2a, 2.2b -0.0445021 6 12-site cluster Fig. 2.2c, 2.2d -0.0347222 7 CSL bilayer Fig. 2.2e, 2.2f -0.0273888 8 8-site cluster Fig. 2.2g, 2.2h -0.0234375 9 Inhomogeneous bilayer Fig. 2.2i, 2.2j -0.0188265 10 CSL bilayer Fig. 2.2e, 2.2f -0.01577

Table 2.1: Ground state saddle-point patterns of χrr0 , and the corresponding energies in units of NJ Ns for k = 2, 3,..., 10. The diﬀerent types of large-N ground states are described in the text, and depicted in ﬁgures as indicated.

cluster states are thus “ordinary” broken symmetry states, without exotic excitations. A more extensive discussion of ﬂuctuations appears in Ref. [2], and the resulting physical properties of the

CSL bilayer are discussed in Sec. 2.3. We have not considered the eﬀect of ﬂuctuations in the k = 9 inhomogeneous bilayer ground state.

2.2.2 Detailed descriptions of the mean-ﬁeld ground states

We now discuss the mean-ﬁeld ground states for each value of k. We note that, for k ≥ 5, we cannot rule out the possibility that the true ground state is lower in energy than the ground state we found. The ground-state clusters for k = 2, 3, 4 are depicted in Fig. 2.1. These are essentially the same as found in the two-dimensional square lattice,[1, 2] but going to the three-dimensional cubic lattice permits a greater variety of clusters for k = 3, 4.

It was noted in Ref. [46] that for k = 2 there is actually a continuous family of N = ∞ ground states, which can be seen for a single square plaquette as shown in Fig. 2.1a and discussed in the figure caption. This continuous ground state degeneracy is also resolved by the order-1/N corrections to the ground state energy.[46] We found that a similar continuous degeneracy occurs for k = 4 on a single cube (see Fig. 2.1c). As in the figure, consider a single cube with flux Φt through the top and bottom plaquettes (i.e., those lying in the xy-plane), and flux Φs through the side plaquettes (i.e., those lying in the xz- and yz-planes). Flux passing from the center of the cube 17 to the outside is taken positive. In order to reach the ground state we must have 2Φt + 4Φs = ±2π; we choose the positive sign without loss of generality. We let Φt = 4u and Φs = π/2 − 2u; a ground state is obtained if we restrict 0 ≤ u ≤ π/2. In this situation the magnitude |χrr0 | will generally differ on vertical bonds and other bonds [shaded light (pink) and dark (blue), respectively, in Fig. 2.1]. The energy is minimized and saturates the lower bound when

|χ | √ light = 2 cos u sin u. (2.12) |χdark|

The ground-state patterns of χrr0 for 5 ≤ k ≤ 10 are shown in Fig. 2.2. For k = 5, 6, 8 we again ﬁnd cluster ground states. The case k = 8 is particularly simple; there, each cluster is a fully symmetric cube with |χrr0 | constant on every bond, and no ﬂux through the cube faces.

The k = 5 and k = 6 clusters are conveniently thought of as obtained by stacking two single-layer clusters vertically, and connecting them via the vertical bonds. For k = 5 each cluster is a stack of two ten-site T-shaped objects. The k = 6 clusters are obtained by stacking two k = 3 ground state clusters (see Fig. 2.1b). In the k = 5, 6 cases, our numerical calculations ﬁnd evidence for a continuous family of degenerate ground states within each cluster, as for the 4-site k = 2 clusters and 8-site k = 4 clusters (Fig. 2.1). Unlike in those cases, however, we have not been able to ﬁnd a simple parametrization of the degenerate ground states.

For k = 7, 9, 10, we ﬁnd bilayer ground states, with the CSL bilayer saddle point described above occurring for k = 7, 10. The k = 9 ground state is more complicated, spontaneously breaking translation symmetry within each bilayer. Time reversal symmetry is broken as well by a complicated pattern of ﬂuxes. It is interesting to note that all the 5 ≤ k ≤ 10 ground states have a bilayer structure, as the clusters for k = 5, 6, 8 can be arranged into bilayers (see right column of

Fig. 2.2). In addition, the two square lattice layers of each bilayer have identical χrr0 , there is zero

ﬂux on the “vertical” plaquettes connecting the two layers, and the vertical bonds have magnitude

|χrr0 | = J /k. [140] As discussed below, this simple structure allows us to exploit a useful relation with the single-layer square lattice at ﬁlling parameter k0 = k/2. 18

2.2.3 Obtaining the mean-ﬁeld results

We now describe how the large-N ground states were determined. As on the square lattice,[1,

2] the results for k = 2, 3, 4 are rigorous, and are obtained by applying a lower bound on EMFT obtained by Rokhsar for k = 2,[141] and generalized to k > 2 (with a stronger bound holding for bipartite lattices) in Refs. [1, 2]. Cluster states for k = 2, 3, 4 on the square[1, 2] and cubic lattices saturate this lower bound. A necessary condition for saturation on a bipartite lattice is that the mean-field single-particle energy spectrum must be completely flat, with only three energies 0, ± occuring in the spectrum, and with energy − states filled and others empty.[1, 2] We believe that this kind of spectrum can only be produced by a cluster state. Moreover, for larger clusters (and thus with increasing k), it becomes harder to arrange for a spectrum containing only three energies.

While we do not have a rigorous proof, we believe saturation is impossible for k > 4 on the square and cubic lattices.

For k ≥ 5, we resort to a numerical approach to ﬁnd the ground states. We employ the self-consistent minimization (SCM) algorithm developed in Refs. [1, 2], which proceeds as follows

(see Ref. [2] for more details):

1: Start with µr = 0 and a randomly generated conﬁguration of χrr0 .

2: Adjust µr to satisfy the saddle-point equation

α† hfr frαi = m, for all r. (2.13)

µr is determined by a multidimensional Newton’s method.[1, 2, 142] Stop if no solution is found.

3: Generate a new χrr0 using the saddle-point equation

J D α† E χ 0 = − f f . (2.14) rr N r0 rα

4: Go back to step 2 until χrr0 and µr converge.

As long as step 2 is successful, the energy EMFT is guaranteed to decrease with each iteration of the 19

(a) k = 2

Π Π

(b) k = 3

Ft F s Fs

Figure 2.1: Ground-state clusters for k = 2, 3, 4. Shaded bonds are those with χrr0 6= 0. Bonds with different shading (or color in online version) may have different magnitudes |χrr0 |. (a) The k = 2 ground state clusters are dimers and square plaquettes. The square plaquette is pierced by π-flux, and the ratio of |χrr0 | on light (pink online) and dark (blue online) bonds can be chosen arbitrarily. Setting |χrr0 | = 0 on the two light (pink) bonds breaks the plaquette into two dimers. (b) The k = 3 ground state cluster is a 6-site chain pierced by π-flux. On the cubic lattice, such chains can exist either as a flat rectangular loop (left), or as the same loop bent by 90◦ in the middle (right). In both cases, χrr0 = 0 on the dashed bond passing through the middle of the loop. (c) The k = 4 ground state clusters are square plaquettes and 8-site cubes with Φs-flux through the side plaquettes and Φt-flux through top and bottom plaquettes. There is a continuous one-parameter family of ground states on an 8-site cube, described in the text. 20

(a) k = 5 (b) k = 5

(e) k = 7, 10 (f) k = 7, 10

(g) k = 8 (h) k = 8

(i) k = 9 (j) k = 9

Figure 2.2: Ground-state saddle point configurations of χrr0 for k = 5, 6,..., 10. The right column is a three-dimensional view of each configuration, with larger magnitude |χrr0 | indicated by darker shading. All these saddle points can be viewed as bilayer structures, with χrr0 identical on top and bottom layers. The left column thus shows |χrr0 | on a single layer, with fluxes indicated except for k = 9, where the fluxes are generally non-zero but follow a complicated pattern. Also, for k = 5, 6 the fluxes and |χrr0 | can be changed continuously within a single cluster without affecting the energy; only the simplest configurations are shown. 21

SCM algorithm.[1, 2] But a random initial configuration of χrr0 does not necessarily converge to the ground state, and can instead converge to a local minimum of EMFT . Therefore, in order to find the ground state, we need to try as many independent random initial configurations of χrr0 as possible.

For those random initial conﬁgurations resulting in the lowest energies, we found extremely good convergence in EMFT by the time the SCM procedure is stopped (typically after 300 iterations), and eﬀects of randomness on the reported values of EMFT are thus entirely negligible.

To improve the performance of the SCM algorithm, we deﬁne χrr0 with µr within some

ﬁxed unit cell, which is then repeated periodically to cover a ﬁnite-size Lx × Ly × Lz lattice with periodic boundary conditions. For simplicity, we always choose the unit cell to be a rectangular prism with edge lengths lx,y,z (see Fig. 2.3), with primitive Bravais lattice vectors parallel to the edges of the rectangular prism.[143] For each value of k, we choose the minimum linear system size

L = min(Lx,Ly,Lz) to be as large as possible given the constraints of our available computing resources and the need to try a reasonably large number of diﬀerent random initial conditions.

In some cases we also considered larger system sizes, especially when we found competing saddle points very close in energy. A more careful study of ﬁnite-size eﬀects would be desirable, but due to the above constraints we leave this for future work. Table 2.2 displays the range of unit cell dimensions studied for each value of k, as well as the number of random initial conditions tried for each cell, and the minimum linear system size L.

2.2.4 Relation between bilayer states and square lattice saddle points

As noted above, the ground states for 5 ≤ k ≤ 10 can all be viewed as bilayer states, which means that such saddle points can also be obtained by a studying the large-N Heisenberg model on a single bilayer. We have also carried out SCM numerical calculations in this geometry (see

Table 2.2 and Fig. 2.3 for more information); this is computationally cheaper than the cubic lattice

SCM calculations, and provides a useful check on those results. These bilayer SCM calculations

ﬁnd the same ground states as the corresponding cubic lattice calculations, except for k = 9, where the bilayer calculation ﬁnds a lower-energy state that can then be extended to a cubic lattice saddle 22

lz ly lx

lx (a) (c)

lz=1

(b)

Figure 2.3: Unit cells used for SCM calculations on the cubic lattice (a), single bilayer (b), and single-layer square lattice (c). In the cubic case the primitive Bravais lattice vectors are chosen parallel to the edges of the rectangular prismatic unit cell. The analogous statement is true for the bilayer and single-layer cases, with primitive Bravais lattice vectors parallel to the lx,y edges of the unit cell. 23 point. Presumably, this saddle point would also be found by SCM on the cubic lattice with enough runs using independent random initial conditions.

There is an interesting relation between certain saddle points of a single bilayer, and corresponding saddle points of a single-layer square lattice, but with ﬁlling parameter k0 = k/2. The cubic lattice ground states for 5 ≤ k ≤ 10 are all of this type. We label the sites of a single bilayer by (r, i), where i = 1, 2 is the layer index, and r labels the square lattice sites within each layer.

2d 2d There are Ns = 2Ns lattice sites, where Ns is the number of sites in a single layer. Consider a saddle point where

χr1,r01 = χr2,r02 ≡ χrr0 (2.15)

µr1 = µr2 ≡ µr (2.16)

χr1,r2 ≡ χv. (2.17)

Here, χv is real and positive, and all other inter-layer χ’s are assumed to vanish. We let n label the

2d one-particle eigenstates of a single layer, with energies n . The full one-particle spectrum is then given by

2d n,σ = n + σχv, (2.18) where σ = ±1. We assume that the energy spectrum and ﬁlling are such that only σ = −1 states

2d are occupied by fermions, in which case the two-dimensional spectrum n (shifted in energy by

2d −χv) is ﬁlled by NNs/k = 2NNs /k fermions. This corresponds to a single-layer problem with twice as many fermions, or ﬁlling parameter k0 = k/2. The saddle point energy is then

χ2 2NN 2d E = NN 2d v − s χ (2.19) MFT s J k v 2N X 2 0 X 2d 0 + |χ 0 | + m µ + E (k ). J rr r f hrr0i r

0 2d 0 Here, m = 2m, and Ef (k ) is the ground state energy of the fermionic part of the mean-ﬁeld Hamiltonian [last two terms of Eq. (2.8)], for a single-layer square lattice with ﬁlling parameter k0.

The ﬁrst two terms of Eq. (2.19) are minimized with respect to χv to ﬁnd χv = J /k. The last 24 k Cubic lattice Single bilayer k/2 square lattice

5 1 ≤ lx,y,z ≤ 5 10 1 ≤ lx,y ≤ 5 10 1 ≤ lx,y ≤ 6 30 30 60 60 6 1 ≤ lx,y,z ≤ 6 4 1 ≤ lx,y ≤ 6 4 30 60 7 1 ≤ lx,y,z ≤ 7 4 1 ≤ lx,y ≤ 7 10 1 ≤ lx ≤ 7 20 21 35 1 ≤ ly ≤ 10 42 8 1 ≤ lx,y ≤ 8 4 1 ≤ lx,y ≤ 8 4 1 ≤ lz ≤ 5 24 40 9 1 ≤ lx,y ≤ 9 4 1 ≤ lx ≤ 9 10 1 ≤ lx ≤ 10 10 1 ≤ lz ≤ 4 36 1 ≤ ly ≤ 11 36 1 ≤ ly ≤ 9 36 10 1 ≤ lx,y ≤ 10 4 1 ≤ lx,y ≤ 10 5 1 ≤ lz ≤ 4 30 60

Table 2.2: This table contains information about our SCM numerical study on the cubic lattice (1st column), as well as the related problems of a single bilayer (2nd column), and single layer square lattice with k0 = k/2 (3rd column). On the left-hand side of each entry of the table, the range of unit cell dimensions is shown as an inequality. For every choice of lx,y,z within the given range, the number of times we ran the SCM algorithm with distinct random initial conﬁgurations of χrr0 is shown on the right-hand side of the entry (top). Also on the right-hand side is the minimum linear system size L (bottom, italics).

2d 0 0 three terms combine to EMFT (k , J ), the saddle point energy of a single-layer square lattice with ﬁlling parameter k0 and J 0 = J /2. Noting that

2d 0 2d 0 2d 0 0 EMFT (k ) ≡ EMFT (k , J ) = 2EMFT (k , J ), (2.20) we obtain the following relation between bilayer and single-layer saddle point energies:

2d EMFT J 1 EMFT (k/2) = − 2 + 2d . (2.21) NsN 2k 4 Ns N This relation allows us to study via SCM the single-layer square lattice with ﬁlling parameter k0 = k/2 as a further check on the cubic lattice results. For integer k0, this was already done in

0 5 7 9 Ref. [1]. We carried out SCM calculations for the half-odd integer ﬁlling parameters k = 2 , 2 , 2 (see Table 2.2 and Fig. 2.3). For all values of k, these calculations ﬁnd the same ground states as found by the single-bilayer SCM calculations.

As a further check on our results, we also computed the energies of some simple competing states. Table 2.3 compares the energies of these states to the ground state saddle point energies found by SCM. 25 flux through /k x πn 10 -0.014 -0.01577 -0.01577 -0.0151133 -0.0151134 9 sites is proportional to the number of k -0.0186271 -0.0178326 -0.0177088 -0.0177579 -0.0188265 . These states have 2 1. Since most of these states break lattice − -flux k ) 8 ≤ /k -site cluster containing the greatest number of bonds. x,y,z -0.0223613 k x,y,z -0.0210391 -0.0212772 -0.0234375 -0.0234375 n πn ≤ 7 , where 0 -0.026239 . The energy of a cluster with z -0.0254048 -0.0261299 -0.0273888 -0.0273888 and y . Each row represents a class of saddle points, described below. For classes s 6 N J -site cluster -0.032407 N k -0.0344012 -0.0312776 -0.0330693 -0.0347222 1. − is real and spatially constant. (2 0 rr χ 5 , . . . , k is allowed to vary depending on bond orientation, but is fixed to be translation invariant.[3] -0.04 | 0 = 0 -0.0444916 -0.0394159 -0.0430802 rr -0.0445021 n χ | ) -direction, and similarly for x χ πn/k , where )-flux /k k πn/k x,y,z ). We considered a generalization of the CSL bilayer saddle point described in the main text, where the flux through -site cluster πn k . This is the state where Uniform real (2 χ πn/k SCM ground state Bilayer (Φ = 2 (Φ = 2 rotation symmetry, the magnitude bonds in the cluster,[1, 2]Uniform so real the lowest-energy such state can be found by finding a Table 2.3: Comparison ofnumerics energies (bottom of a row). variety of All simple energies saddle are points in (top four units rows), of with the energy of the ground state found by SCM every plaquette normal to the including multiple different saddle points,Bilayer the energy shown is the lowest in the class. We considered the following classes of saddle points: each plaquette is Φ = 2 26

2.3 Discussion

The large-N results presented here ﬁnd a rich variety of candidate non-magnetic ground states for Mott insulators of ultra-cold fermionic AEA. It would be fascinating to realize any of these states experimentally. In order to achieve this, there still need to be substantial advances in preparation of low-entropy magnetic states of ultra-cold atoms, and our results add to the increasing motivation to pursue such advances speciﬁcally in AEA systems. In addition, if future experiments can enter a regime where any of the states discussed here can be realized, it will be of crucial importance to devise probes of their characteristic properties.

We would like to close by highlighting the CSL bilayer state, which has some striking properties that would be fascinating to realize experimentally, and which we now briefly discuss. At the large-N mean-field level the cubic lattice breaks into disconnected bilayers, and one can understand the properties beyond mean-field theory by first focusing on a single bilayer. The effect of fluctuations is to couple the fermions to a dynamical U(1) gauge field. The mean-field fermions are in a gapped integer quantum Hall state, so integrating them out generates a Chern-Simons term for the

U(1) gauge field. Because the mean-field fermions in a single bilayer and in the single-layer square lattice CSL[1, 2] have in both cases a single chiral edge mode per spin species, the coefficient of the

Chern-Simons term and associated topological properties are the same. The spinons are Abelian anyons with statistics angle θ = π ± π/N, and there is a chiral edge mode with gapless excitations carrying SU(N) spin, which is described by a chiral SU(N)1 Wess-Zumino-Witten model.[1, 2]

If adjacent bilayers are coupled weakly, bulk properties are unaffected due to the energy gap. One simply has a many-layer CSL state, with anyonic spinons confined to the the individual bilayers. Due to the gapless edge modes of single bilayers, the physics on the two-dimensional surface is likely more interesting. This depends crucially on whether adjacent bilayers have the same or opposite magnetic flux, as the direction of the flux controls the direction of the chiral edge modes. If the fluxes are aligned oppositely in neighboring bilayers, then edge modes on neighboring bilayers are counterpropagating and an energy gap is possible on the two-dimensional surface. On 27 the other hand, if all fluxes are parallel, then all the chiral edge modes propagate in the same direction, and the two-dimensional surface is expected to remain gapless. The resulting surface state is a kind of two-dimensional chiral “spin metal,” which could be interesting to study in future work. Chapter 3

Crystal Symmetry Fractionalization1

We study square lattice space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types of symmetry fractionalization (symmetry classes) can be realized within this class of models, which are generalizations of Kitaev’s Z2 toric code to arbitrary lattices. This question is motivated by earlier work [107], where the idea of symmetry classiﬁcation was laid out, and which, for square lattice symmetry, produces 2080 symmetry classes consistent with the fusion rules of Z2 topological order. This approach does not produce a physical model for each symmetry class, and indeed there are reasons to believe that some symmetry classes may not be realizable in strictly two-dimensional systems, thus raising the question of which classes are in fact possible. While our understanding is limited to a restricted class of models, it is complete in the sense that for each of the 2080 possible symmetry classes, we either prove rigorously that the class cannot be realized in our family of models, or we give an explicit model realizing the class. We thus ﬁnd that exactly 487 symmetry classes are realized in the family of models considered. With a more restrictive type of symmetry action, where space group operations act trivially in the internal

Hilbert space of each spin degree of freedom, we ﬁnd that exactly 82 symmetry classes are realized.

In addition, we present a single model that realizes all 26 = 64 types of symmetry fractionalization allowed for a single anyon species (Z2 charge excitation), as the parameters in the Hamiltonian are 1 This chapter has been published as a portion of Hao Song and M. Hermele, Phys. Rev. B 91, 014405 (2015),[77] copyright 2015 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society. 29 varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models.

3.1 Outline and Main Results

Due to the length of the chapter, we ﬁrst point out that readers can ﬁnd the main results in Section 3.6. Readers familiar with the necessary background should be able to understand the statements of results in Sec. 3.6, after quickly consulting Sec. 3.5.1, and especially Eqs. (3.39-3.44), to become familiar with notation and conventions used to present symmetry classes.

Now, to overview the main results, the aim of this paper is to explore the possible symmetry classes associated to the space group G of the square lattice within a particular family of local bosonic models with Z2 topological order. We call this family of models TC(G), and it consists of variations of Kitaev’s Z2 toric code[136] obtained by changing the lattice geometry, varying the signs of terms in the Hamiltonian, and allowing symmetry to act non-trivially on spin operators

(referred to as spin-orbit coupling). Section 3.6 studies symmetry fractionalization in these models, beginning with a speciﬁc example and moving towards increasing generality. First, in Sec. 3.6.1 we describe a single model realizing all e particle fractionalization classes while the m particle always has trivial symmetry fractionalization. The constraints that arise when both e and m particles have non-trivial symmetry fractionalization are considered in the following subsections. In Sec. 3.6.2, we examine a subclass of models, TC0(G) ⊂ TC(G), where no spin-orbit coupling is allowed. The main result of Sec. 3.6.2 is Theorem 1, which describes all symmetry classes that are realized by models in TC0(G). Following the statement of the theorem, example models realizing all possible symmetry classes for TC0(G) are presented. Finally, in Sec. 3.6.3, we treat the general case of

TC(G), and state Theorem 2, which describes all symmetry classes that are realized by models in

TC(G); the discussion parallels that of Sec. 3.6.2. The detailed proofs of the theorems are left to the appendices, together with the presentation of models realizing all possible symmetry classes for TC(G). Our results establish that certain symmetry classes are possible in two dimensional models. For symmetry classes that are not realized by models TC(G), a more general understanding 30 of which such symmetry classes are possible strictly in two dimensions is still lacking.

Now we describe how the rest of the paper is organized. Section 3.2 reviews Z2 topological order, and Sec. 3.3 gives a review of the simplest Z2 Kitaev toric code model, on the two-dimensional square lattice. The crucial objects are the e (Z2 charge) and m (Z2 ﬂux) excitations of Z2 topological order, referred to as e and m particles. Readers already familiar with these topics may wish to skim Sections 3.2 and 3.3, and proceed to Section 3.4, where we introduce the family of toric code models on general lattices with square lattice symmetry; some technical details are presented in

Appendices A and B. We actually introduce two families of models; in one of these, square lattice symmetry acts only by moving spin degrees of freedom from one spatial location to another, but all symmetries act trivially within the internal Hilbert space of each spin. This situation is referred to in our paper as that of no spin-orbit coupling, and the resulting family of models is called

TC0(G), where G is the square lattice space group. We also consider a larger family of models,

TC(G), that contains TC0(G). In TC(G), symmetries are allowed to act non-trivially on the spin degrees of freedom, and we refer to this as the presence of spin-orbit coupling. It should be noted that our usage of the term spin-orbit coupling is a generalization of the usual usage; in particular, our spins do not necessarily transform as electron spins do under a given rigid motion of space. Such a generalization is physically reasonable, because there are many ways in which two-component pseudospin degrees of freedom arise in real systems, and such degrees of freedom do not always transform like electron spins under symmetry.

With the models of interest having been introduced, Sec. 3.5.1 follows Ref. [107] and reviews the notions of fractionalization and symmetry classes. It should be noted that, as in Ref. [107], we always make the simplifying assumption that symmetry does not permute the anyon species.

Indeed, the family of models TC(G) is deﬁned so that permutations of anyons under symmetry never occur. Section 3.5.2 proceeds to give a detailed description of how symmetry fractionalization is realized in the solvable toric code models for both e and m particle excitations. The important notions of e and m localizations of the symmetry are introduced and discussed, which provide the means to calculate the fractionalization and symmetry classes for given models in TC(G). In our 31 solvable models, the e and m particle excitations have diﬀerent character, and it is convenient to distinguish them by introducing the notion of toric code (TC) symmetry class, which is an ordered pair ([ωe], [ωm]). While we do not expect TC symmetry classes to have any universal meaning, they are useful in understanding the possibilities for toric code models. Appendix C proves some general results about e and m localizations, and gives a general expression for these localizations that is useful in deriving constraints on which symmetry classes are possible.

The main results of the paper are presented in Section 3.6, in order of increasing generality.

First, in Section 3.6.1 we describe a single model that realizes all 26 = 64 fractionalization classes for e particle excitations, as the parameters in the Hamiltonian are varied. In this model the m particle fractionalization class is trivial. In Section 3.6.2, we discuss models in TC0(G), the family of toric code models with square lattice symmetry and the restriction of no spin-orbit coupling. We state Theorem 1, which gives conditions ruling out most of the 2080 symmetry classes (4096 TC symmetry classes) permitted by the general considerations of Ref. [107]. In particular, only 95 TC symmetry classes, corresponding to 82 symmetry classes, are not ruled out by the constraints of

Theorem 1, which are proved in Appendix C.1.1. In fact, all 95 of these TC symmetry classes are realized by models in TC0(G); these models are exhibited in Sec. 3.6.2. Moving on to the general case of TC(G) where spin-orbit coupling is allowed, Section 3.6.3 states Theorem 2, which gives constraints similar to but less restrictive than those without spin-orbit coupling; these constraints are proved in Appendix C.1.2. In this case, 945 TC symmetry classes, corresponding to 487 symmetry classes, are not ruled out by the constraints, and again all these classes are realized by explicit models in TC(G). Some examples of such models are described in Sec. 3.6.3, with the full catalog of models given in Appendix D.

Some of the notation used frequently in the paper is collected in Table 3.1.

3.2 Review of Z2 Topological Order

In this paper, we focus on Z2 topological order in two dimensions, which is in some sense the simplest type of topological order. Z2 topological order arises in the deconﬁned phase of Z2 lattice 32

Table 3.1: Notation used in the chapter.

Symbol Meaning H Hamiltonian Symmetry group of H G (square lattice space group) G = (V,E) Graph on which the model is deﬁned P : G → T 2 Planar projection map into torus T 2 v ∈ V Vertex v in set of vertices V ` ∈ E Edge ` in set of edges E s ∈ W Path s in set of paths W C Set of cycles (closed paths) C0 Set of contractible cycles p ∈ P Plaquette p in set of plaquettes P t ∈ W¯ Cut t in set of cuts W¯ C¯ Set of closed cuts C¯0 Set of closed, contractible cuts h ∈ H Hole h in set of holes H x z σ` , σ` Pauli matrix spin operators on edge ` e Ls e-string on path s ∈ W m ¯ Lt m-string on cut t ∈ W Vertex operator A , a v v and corresponding eigenvalue Plaquette operator B , b p p and corresponding eigenvalue o = (0, 0) Special points in the plane. 1 1 o˜ = 2 , 2 (Units of length are chosen such that 1 κ = 0, 2 the size of the unit cell is 1 × 1.) 1 κ˜ = 2 , 0 |X| Size of any ﬁnite set X. Family of toric code models considered, TC(G) with spin-orbit coupling allowed Family of toric code models considered, TC (G) 0 no spin-orbit coupling allowed 33

2 gauge theory with gapped bosonic matter carrying the Z2 gauge charge. There is an energy gap to all excitations, which can carry Z2 gauge charge and/or Z2 flux. There is a statistical interaction between charges and fluxes; the wave function acquires a statistical phase factor eiπ when a charge moves around a flux or vice versa. These properties are associated with a four-fold ground state degeneracy on a torus (i.e. with periodic boundary conditions), although in some circumstances special boundary conditions are present that reduce the degeneracy.

Z2 lattice gauge theory provides a particular concrete realization of Z2 topological order, and it is useful to distill the essential features into a slightly more abstract description. Every localized excitation above a ground state can be assigned one of four particle types: 1, e, m, and .

In terms of lattice gauge theory, e particles are bosonic gauge charges, m particles are Z2-ﬂuxes, and -particles are e-m bound states. Excitations carrying neither Z2 charge nor ﬂux are “trivial,” and are labeled by 1.

e, m and excitations obey non-trivial braiding statistics and are thus referred to as anyons. e and m are bosons, while is a fermion. Any two distinct non-trivial particle types (for example, e and m), have θ = π mutual statistics, with the wave function acquiring a phase eiπ when one is brought around the other. 1 excitations are bosonic and have trivial mutual statistics with the other particle types.

When two excitations are brought nearby, the particle type of the resulting composite object is well-deﬁned and is given by the fusion rules:

e × e = m × m = × = 1 × 1 = 1,

e × 1 = e, m × 1 = m, × 1 = , (3.1)

e × m = , e × = m, m × = e.

It is a very important property that only 1 excitations can be locally created; that is, action with local operators cannot produce a single, isolated e, m or (at least away from edges of the system, if there are open boundaries). The fusion rules then tell us that a pair of e, m or excitations can be created locally. An anyon can be moved from one position to another by acting with a non-local

2 Z2 lattice gauge theory with fermionic matter also gives rise to Z2 topological order. 34

p v

Figure 3.1: Illustration of some geometrical objects important in the square lattice toric code model. The edges in plaquette p are shown as thick dark bonds (blue online), while the edges in star(v) are thick gray bonds (pink online). The two strings sx and sy winding periodically around the system are also shown as thick dark bonds (blue online).

string operator connecting the initial and ﬁnal positions. There are distinct string operators for each type of anyon.

We remark that the fusion and braiding properties are invariant under the relabeling e ↔ m, which means we are free to make such a relabeling – this is a kind of Z2 electric-magnetic duality. This feature is important for a proper counting of symmetry classes.

3.3 Review: toric code model on the square lattice

We now review Kitaev’s toric code model[136] on the square lattice, which is the simplest model realizing Z2 topological order. We consider a L × L square lattice with periodic boundary conditions (forming a torus), and we label vertices by v, edges by `, and square plaquettes by p.

The degrees of freedom are spin-1/2 spins, residing on the edges. Local operators are then built

µ from Pauli matrices σ` (µ = x, y, z) acting on the spin at `. We introduce operators associated with vertices and plaquettes,

Y x Av = σ` (3.2) `∈star(v) Y z Bp = σ` , (3.3) `∈p where p contains the four edges in the perimeter of a square plaquette, and star(v) is the set of 35

Figure 3.2: Depiction of e and m strings in the square lattice toric code. s is an open e-string joining vertices v and v0, denoted with thick dark bonds (blue online). t is an open cut joining plaquettes p and p0, shown as a dotted line. The cut t contains the thick gray bonds (pink online) intersected by the dotted line.

four edges touching v (see Fig. 3.1). The Hamiltonian is

e X m X H = −K Av − K Bp, (3.4) v p with Ke,Km > 0. It is easy to see that

[Av,Av0 ] = Bp,Bp0 = [Av,Bp] = 0, (3.5) rendering the Hamiltonian exactly solvable. The energy eigenstates can be chosen to satisfy

Av|ψi = av|ψi (3.6)

Bp|ψi = bp|ψi, (3.7)

where av, bp ∈ {±1}.

The Hilbert space has dimension 22L2 , so we need 2L2 independent Hermitian operators with eigenvalues ±1 to form a complete set of commuting observables (CSCO), whose eigenvalues Q Q uniquely label a basis of states. Due to the periodic boundary conditions, v Av = p Bp = 1,

2 and the Av and Bp only give 2L − 2 independent operators. To obtain a CSCO, we need two additional operators, and one choice is given by

e Y z e Y z Lx = σ` ,Ly = σ` , (3.8) `∈sx `∈sy 36

e with eigenvalues lx,y ∈ {±1}, where sx, sy are non-contractible loops winding around the system in

e e the x and y directions, respectively, as shown in Fig. 3.1. The eigenvalues {av, bp, lx, ly} uniquely label a basis of energy eigenstates. In particular, there are four ground states with av = bp = 1, a sign of Z2 topological order.

Excitations above the ground state reside at vertices with av = −1, and plaquettes with bp = −1. These excitations have no dynamics; this is tied to the exact solubility of the model, and adding generic perturbations to the model causes the excitations to become mobile. We identify av = −1 vertices as e particles, and bp = −1 plaquettes as m particles. excitations are e-m

z pairs. Acting on a ground state with σ` creates a pair of e particles, at the two vertices touching

x `. Similarly, acting with σ` creates two m particles in the two plaquettes touching `. Since any operator can be built from products of Pauli matrices, it follows that isolated e and m excitations cannot be created locally.

We now introduce e and m string operators. To deﬁne an e-string operator, let s be a set of edges ` forming a connected path, which may be either closed or open (see Fig. 3.2). Then we deﬁne

e Y z Ls = σ` . (3.9) `∈s e Suppose s is an open path with endpoints v1 and v2. If Ls acts on a ground state, it creates e

e particles at v1 and v2. Alternatively, acting on a state with an e particle at v1 and none at v2, Ls moves the e particle from v1 to v2. On the other hand, if s is a closed path and is contractible (i.e.

e does not wind around the torus), and if |ψ0i is a ground state, then Ls|ψ0i = |ψ0i.

m-strings are deﬁned on a cut t, which contains the set of edges intersected by a path drawn on top of the lattice, running from plaquette to plaquette, as shown in Fig. 3.2. Alternatively, t can be viewed as a set of edges in the dual lattice forming a connected path. The m-string operator is then

m Y x Lt = σ` . (3.10) `∈t m Just as with e-strings, if t is an open cut, with endpoints in two plaquettes p1, p2, Lt can be used 37 to create a pair of m particles or to move a single m particle from one plaquette to another. If t is

m a closed, contractible cut, Lt gives unity acting on a ground state.

If the path s and the cut t cross nc(s, t) times, then

e m nc(s,t) m e LsLt = (−1) Lt Ls. (3.11)

This can be used to verify that the e, m and excitations indeed obey the braiding statistics of Z2 topological order.

3.4 Toric Codes on General Two-dimensional Lattices with Space Group Symmetry

We now introduce the family of models studied in this paper, which are generalizations of the toric code to arbitrary lattices with square lattice space group symmetry. Sometimes it will be convenient to refer to this family of models as TC(G), where in this paper G is always the square lattice space group. We will also introduce a smaller family of models TC0(G) ⊂ TC(G). These two families are distinguished in that “spin-orbit coupling” (as deﬁned below) is allowed for models in TC(G), but is absent in TC0(G).

We begin by deﬁning a toric code model on an arbitrary ﬁnite connected graph G with sets of vertices and edges denoted by V and E, respectively. The number of edges (vertices) is denoted

|E| (|V |). We allow for the possibility that two vertices may be joined by more than one edge.

µ Spin-1/2 degrees of freedom reside on edges, and we again denote work with Pauli matrices σ` (µ = x, y, z) acting on the spin at edge ` ∈ E.

To proceed, it is helpful to introduce some notation and terminology. A path is a sequence of edges s = `1`2 ··· `n joining successive vertices; that is, ì and ì+1 are incident on a common vertex. Paths are considered unoriented, so that `1`2 ··· `n = `n ··· `2`1. The set of all paths is denoted W . A path may either be open with distinct endpoints v1, v2 ∈ V , or closed. Two open paths s and s0 sharing an endpoint can be composed into the path ss0. Since an edge may appear in s more than once, more precisely the definition (3.9) of e-string operator should be understood 38 as

e Y z z z z Ls = σ` = σ`1 σ`2 ··· σ`n , (3.12) `∈s for for s = `1`2 ··· `n. Since operators in the product commute, there is no harm to interpret s as Q multiset of edges as well. In this paper, we use the product notation `∈X for all three cases: X is a set, a multiset or a sequence of edges.

The set of open paths is denoted Wo ⊂ W , while closed paths are called cycles, and the set

e Q z of cycles is C ⊂ W . e-string operators are deﬁned on paths s ∈ W by Ls = `∈s σ` . An important part of the speciﬁcation of a model will be the selection of a subset P ⊂ C, where elements p ∈ P are called plaquettes. The choice of P is not entirely arbitrary, and is required to satisfy certain properties discussed below.

Just as for the square lattice,

Y x Av = σ` (3.13) `∈star(v) Y z Bp = σ` , (3.14) `∈p where p ∈ P , and star(v) is again the set of edges touching v. It is again easy to see that

[Av,Av0 ] = Bp,Bp0 = [Av,Bp] = 0. (3.15)

The Hamiltonian is

X e X m H = − Kv Av − Kp Bp, (3.16) v∈V p∈P e m where now the coeﬃcients Kv , Kp are allowed to depend on the vertex or plaquette. Only the signs

e m of the coeﬃcients will be important, so for convenience of notation we take Kv ,Kp ∈ {±1}. Energy eigenstates can again be labeled by av, bp ∈ {±1}, the eigenvalues of Av and Bp, respectively.

e m Any ground state will satisfy av = Kv and bp = Kp , provided it is possible to ﬁnd such a state. This is not guaranteed, as the couplings in the Hamiltonian could be frustrated. We will assume the Hamiltonian is “frustration-free,” meaning it is possible to ﬁnd at least one ground

e m 3 state with av = Kv , bp = Kp .

3 e m This is the case provided Kv and Kp are compatible with constraints obeyed by Av and Bp operators. More 39

Our discussion so far is for a general graph, but we want to specialize to two-dimensional lattices. Essentially, this just means that we can draw the graph in two-dimensional space (with periodic boundary conditions), so that the resulting drawing has the symmetry of the square lattice.

We do not assume the graph is planar; for instance, edges are allowed to cross or stack on top of each other when the graph is drawn in two dimensions.

In order to make general statements about the family of models considered, it will be useful to be more precise. First, letting G be the square lattice space group, we introduce an action of G on G. Group elements g ∈ G act on vertices and edges of the graph, and we write v 7→ gv, g 7→ g`.

G is generated by translation x → x + 1 (Tx), reﬂection x → −x (Px), and reﬂection x ↔ y (Pxy).

Translation by y → y + 1 is given in terms of the generators by Ty = PxyTxPxy. The group G can be deﬁned in terms of the generators by requiring them to obey the relations,

2 Px = 1, (3.17)

2 Pxy = 1, (3.18)

2 (TxPx) = 1, (3.19)

4 (PxPxy) = 1, (3.20)

−1 −1 TxTyTx Ty = 1, (3.21)

−1 −1 TyPxTy Px = 1. (3.22)

We wish to consider a L × L lattice with periodic boundary conditions, with L the integer number of square primitive cells in the x and y directions. More formally, for all v ∈ V , we assume

nx ny v = (Tx) (Ty) v if and only if nx, ny = 0 mod L, with the same statement holding for all ` ∈ E.

We now introduce the planar projection P : G → T 2, where T 2 is the 2-torus, viewed as a square with dimensions L×L and periodic boundary conditions. P is a continuous map that sends vertices to points and edges to curves. (See Fig. D.1 for an example.) Symmetry operations g ∈ G act on the graph G as described above, and also act naturally on T 2 as rigid motions of space. We

Q e Q e 0 precisely, we have v Av = 1, which implies Kv must satisfy v Kv = 1. In addition, suppose P is a subset of P Q Q m for which p∈P 0 Bp = 1, then we must have p∈P 0 Kp = 1. 40 require

gP = Pg, (3.23) which means the action of G on G is compatible with the action of rigid motions on the planar projection P(G). The additional structure thus introduced ensures that G is truly playing the role of a space group.

The above discussion implies that the planar projection P(G) is an L×L grid of 1×1 square primitive cells. We note that edges in P(G) are allowed to cross at points other than vertices.

Vertices and edges are also allowed to stack on top of one another; that is, it may happen that

P(v1) = P(v2) for v1 6= v2. It is always possible to choose P(`) to be a straight line connecting its endpoints, although sometimes it will be convenient not to do so.

Now we are in a position to discuss the requirements on the set of plaquettes P . First, any plaquette p ∈ P should be in some sense local. This can be achieved by requiring there to be a maximum size (by some measure that does not need to be precisely deﬁned) for all p ∈ P , where the maximum size is independent of L. Second, we require that any contractible cycle can be decomposed into plaquettes. Non-contractible cycles are those that, under the planar projection, wind around either direction of T 2 an odd number of times, and all others are contractible. We let C0 ⊂ C be the set of contractible cycles. The assumption that contractible cycles can be decomposed into plaquettes means that, given s ∈ C0, there exists {p1, . . . , pn} ⊂ P so that

e Qn Ls = i=1 Bpi . The physical reason for this requirement is that it ensures there are no local zero-energy excitations, as there would certainly be if we chose P to be too small.

As in the square lattice, we introduce two large cycles sx and sy that wind around the torus

e e in the x and y directions, respectively. The operators {Av}, {Bp}, Lsx and Lsy form a complete

e e set of commuting observables (Appendix A). Denoting eigenvalues of Lsx,sy by lx,y ∈ {±1}, it is then easy to see that H has a four-fold degenerate ground state, corresponding to the four choices

e e m of lx,y with the other eigenvalues ﬁxed to av = Kv and bp = Kp .

e Just as for the square lattice toric code, e particles lie at vertices where av = −Kv ; that is, 41

e where av diﬀers from its ground state value. For s ∈ Wo, the e-string operator Ls can be used to create e particles at the two endpoints, or to move an e particle from one endpoint to the other.

Identifying m particles is more tricky; the basic insight required is that m particles should correspond to a threading of Z2 flux through “holes” in the planar projection P(G). It is easiest to proceed by defining m-strings, which are defined on cuts t ∈ W¯ . A cut t is defined as follows: (1)

Draw a curve in T 2 that has no intersection with vertices P(v), and whose intersection with each edge P(`) contains at most a ﬁnite number of points, at which the curve is not tangent to P(`).

If the curve is open, we assume its endpoints do not lie in P(G). (2) The cut t is then given by the sequence of edges intersected by the curve. A cut is closed if the curve in (1) is closed, and is simple if the curve has no self-intersections. It is clear that a given curve produces a unique cut, but there are many possible curves that produce the same cut.

¯ m Q x m We deﬁne a m-string operator on a cut t ∈ W by Lt = `∈t σ` . If t is an open cut, then Lt acting on a ground state creates m particles at the two endpoints. The endpoints of the m-string, and thus the m particles it creates, naturally reside at the holes in the planar projection; more precisely, these are the connected components of T 2 − P(G). We denote the set of all holes by H with elements h ∈ H. Not all m excitations can be created as described above, but arbitrary such

m excitations can be created by ﬁrst acting with Lt on a ground state, then acting subsequently with operators localized near the m particles created by the string operator.

Finally, we need to specify the action of symmetry on the spin degrees of freedom themselves.

Letting Ug be the unitary operator representing g ∈ G, we consider

x −1 x x z −1 z z Ugσ` Ug = c` (g)σg`,Ugσ` Ug = c` (g)σg`, (3.24)

x,z −1 assuming symmetries do not swap anyon species. Since Ugσ` Ug are hermitian and unitary x,z simultaneously, we must have c` (g) ∈ {±1}. This satisﬁes a general requirement that space group symmetry should be realized as a product of an on-site operation, with another operation that

µ µ 4 merely moves degrees of freedom (i.e. σ` 7→ σg`). Subject to this requirement, this is the most

4 The origin of these requirements is the fact that these properties holds for hold for all electrically neutral bosonic 42 general action of symmetry with the property that e-strings are taken to e-strings, and m-strings

e −1 e to m-strings; for example, UgLsUg = (±1)Lgs.

Actually we need to impose a further requirement, which is that symmetry must act linearly

(as opposed to projectively) on the spin operators. 4 In particular,

x,z −1 −1 x,z −1 Ug1 Ug2 σ` Ug2 Ug1 = Ug1g2 σ` Ug1g2 . (3.25)

This imposes the restriction

cx,z (g )cx,z(g ) = cx,z(g g ), (3.26) g2` 1 ` 2 ` 1 2 which holds for all ` ∈ E and g1, g2 ∈ G. These conditions do not ﬁx the overall U(1) phase of Ug, which can be adjusted (as a function of g) as desired.

x,z x,z The phase factors c` (g) can be modiﬁed by the unitary “gauge” transformation σ` → x,z x,z x,z γ` σ` , with γ` ∈ {±1}, which sends

x,z x,z x,z x,z c` (g) → γ` γg` c` (g). (3.27)

x,z It is always possible to choose a gauge where c` (T ) = 1, for all ` ∈ E and all translations x,z x,z T ∈ G; this is so because c` (Tx) and c` (Ty) behave under gauge transformation like the x and y components of a ﬂux-free vector potential, residing on the links of a square lattice generated by acting on ` with translation. We shall make this gauge choice without further comment throughout the paper.

x,z If, in addition, it is possible to choose a gauge where c` (g) = 1 for all ` ∈ E and g ∈ G, then by deﬁnition the model is in TC0(G), and we say there is no “spin-orbit coupling.” The reason for this terminology is that, in this case, space group operations have no action on spins beyond moving them from one point in space to another. The case of no spin-orbit coupling is simpler to analyze, and we will discuss it ﬁrst before handling the general case.

It is shown in Appendix B that for L even, it is possible to ﬁnd a ground state |ψ0ei and degrees of freedom (e.g. electron spins, bosonic atoms) that can be microscopic constituents of a condensed matter system. 43 make a choice of phase for Ug so that

Ug|ψ0ei = |ψ0ei (3.28)

e e Lsx |ψ0ei = Lsy |ψ0ei = |ψ0ei, (3.29)

where sx and sy are closed paths chosen as described in Appendix B to wind once around the system in the x and y directions, respectively. For the same phase choice of Ug, combining Eq. (3.28) with

Eq. (3.25) implies Ug1 Ug2 = Ug1g2 . From now on, when we study e particle excitations, we always focus on states that can be constructed by acting on |ψ0ei with e-string operators.

Appendix B also shows that, for L even, there is a ground state |ψ0mi and a phase choice for

Ug, satisfying

Ug|ψ0mi = |ψ0mi (3.30)

m m Ltx |ψ0mi = Lty |ψ0mi = |ψ0mi. (3.31)

Here, the electric strings have been replaced with magnetic strings, with tx and ty appropriately chosen closed cuts winding once around the system in the x and y directions, respectively. When studying m particle excitations, we will always consider states constructed by applying m-string operators to |ψ0mi.

e It should be noted that |ψ0ei and |ψ0mi cannot be the same state, because, for instance, Lsx

m and Lty anticommute. Moreover, the phase choice required to make |ψ0ei symmetry-invariant may not be the same as the corresponding choice for |ψ0mi. These points will not be problematic for us, because we always focus on excited states with either e particles, or m particles, but not both.

Using |ψ0ei to construct e particle states, and similarly |ψ0mi for m particle states, simply provides a convenient means to calculate the e and m fractionalization classes. 44

3.5 Fractionalization and Symmetry Classes

3.5.1 Review of fractionalization and symmetry classes

We now consider in more depth the action of square lattice space group symmetry G in the general class of solvable models introduced in Sec. 3.4, showing how to determine the fractionalization classes of e and m particles, and the corresponding symmetry class. We ﬁrst review the general notions of fractionalization and symmetry classes, before exposing in detail the corresponding structure for the solvable models (Sec. 3.5.2). Readers unfamiliar with this subject may ﬁnd the review rather abstract, so we would like to emphasize that the objects involved appear in concrete and explicit fashion in the discussion of the solvable models.

Each non-trivial anyon (e, m and in the toric code) has a corresponding fractionalization class, that describes the action of symmetry on single anyon excitations of the corresponding type.

(We assume that symmetry does not permute the anyon species.) This structure follows from the fact that the action of symmetry factorizes into an action on individual isolated anyons. Since physical states must contain even numbers of e particles, as an example we consider a state |ψeei with two e particles, labeled 1 and 2. Following the arguments of Ref. [107], we assume that

e e Ug|ψeei = Ug (1)Ug (2)|ψeei, (3.32)

e where Ug (i) gives the action of symmetry on anyon i = 1, 2.

The physics is invariant under a redeﬁnition

e e Ug (i) → λ(g)Ug (i), λ(g) ∈ {±1}, (3.33) which we refer to as a projective transformation. The reason for this terminology is that the

e Ug operators form a projective representation of G, expressed by writing

e e e Ug1 Ug2 = ωe(g1, g2)Ug1g2 , (3.34) where we have suppressed the anyon label i, and ωe(g1, g2) ∈ {±1} is referred to as a Z2 factor set. 45

The factor set satisﬁes the condition

ωe(g1, g2)ωe(g1g2, g3) = ωe(g2, g3)ωe(g1, g2g3), (3.35)

e which follows from the associative multiplication of Ug operators. The factor set is not invariant under projective transformations, but instead transforms as

ωe(g1, g2) → λ(g1)λ(g2)λ(g1g2)ωe(g1, g2). (3.36)

A projective transformation is analogous to a gauge transformation that does not aﬀect the physics, so such transformations should be used to group factor sets into equivalence classes. We denote by

[ωe] the equivalence class containing the factor set ωe. These equivalence classes are the possible fractionalization classes for e particles. It will not be important for the discussion of the present paper, but we mention that the set of fractionalization classes is the second group cohomology

2 H (G, Z2). The discussion proceeds identically for m particles, with ωm the corresponding factor

2 set, and [ωm] ∈ H (G, Z2) the fractionalization class. A complete specification of fractionalization classes defines a symmetry class. It is enough to specify [ωe] and [ωm], because these determine uniquely the fractionalization class.[107] Therefore a symmetry class is specified by the pair

S = h[ωe], [ωm]i. (3.37)

Because all properties of Z2 topological order are invariant under e ↔ m (see Sec. 3.2), symmetry classes related by this relabeling are considered equivalent, that is

h[ωe], [ωm]i ' h[ωm], [ωe]i. (3.38)

Despite the lack of a fundamental distinction between e and m particles, there is a distinction in the solvable toric code models, as is clear from the discussion of these excitations in Sec. 3.4.

While this distinction is only well-deﬁned within the context of the solvable models, it is not just a matter of notation; in general, we do not restrict to planar lattices, so there is not expected to be an exact duality exchanging e ↔ m. Because it is relevant for the construction of solvable models, it 46 will be useful to deﬁne toric code symmetry classes, or TC symmetry classes, that distinguish between e and m particles. A TC symmetry class is simply an ordered pair ([ωe], [ωm]).

To determine fractionalization and symmetry classes, it is convenient to work with the genera-

e tors and their relations [Eqs. (3.17-3.22)]. Focusing on e particles for concreteness, the Ug operators obey the group relations up to possible minus signs, that is

e 2 e (UPx ) = σpx (3.39)

e 2 e (UPxy ) = σpxy (3.40)

e e 2 e (UTx UPx ) = σtxpx (3.41)

e e 4 e (UPx UPxy ) = σpxpxy (3.42)

e e e −1 e −1 e UTx UTy (UTx ) (UTy ) = σtxty (3.43)

e e e e −1 e U U U −1 (U ) = σ , (3.44) Ty Px Ty Px typx

e e e where σpx ∈ {±1}, and similarly for the other σ parameters. The σ ’s are invariant under projective transformations, and moreover uniquely specify the fractionalization class [ωe].[107] In addition, it was shown that each of the 26 = 64 possible choices of the σe’s is mathematically possible; that is, there exists a projective representation for all choices of σe’s.[107] The same considerations lead to six σm parameters characterizing the m fractionalization class. We see that 2080 symmetry classes (4096 TC symmetry classes) are allowed by the classiﬁcation of Ref. [107]. The reader may recall that Ref. [107] found a larger number of symmetry classes by the same type of analysis – the diﬀerence arises because Ref. [107] also considered time reversal symmetry, while here we focus only on space group symmetry.

3.5.2 Fractionalization and symmetry classes in the solvable models

The solvable models are well-suited to the study of fractionalization and symmetry classes

e m because the Ug and Ug operators can be explicitly constructed. We focus ﬁrst on e particles. It is 47 suﬃcient to consider states with only two e particle excitations, of the form

e |ψe(s)i = Ls|ψ0ei, (3.45)

with s an open path, and e particles residing on the endpoints v1(s) and v2(s). The action of symmetry on this state is given by

z Ug|ψe(s)i = cs(g)|ψe(gs)i, (3.46)

z Q z where cs(g) = `∈s cg(`).

e The goal is to construct and study operators Ug that act on single e particles, reproducing the action of Ug on states |ψe(s)i. Consider the pair (g, v) ∈ G × V , where v is the vertex at which an e particle resides, and g ∈ G is the group operation of interest. To each such pair we associate a

e e e e number fg (v) ∈ {±1} and a path sg(v). The path sg(v) has endpoints v and gv. (Note that sg(v) is a cycle or a null path if gv = v.) From this data we form the operator

e e e U (v) = f (v)L e . (3.47) g g sg(v)

By construction, this operator moves an e particle from v to gv, and is thus a reasonable candidate to realize the action of g ∈ G on single e particles. In order to reproduce Eq. (3.46), we require the

e Ug (v) operators to obey the relation

e e Ug|ψe(s)i = Ug [v1(s)]Ug [v2(s)]|ψe(s)i, (3.48)

e which has to hold for all open paths s ∈ Wo and all g ∈ G. We refer to a set of Ug (v) operators satisfying this relation as an e-localization of the symmetry G.

e It should be noted that there is some redundancy in the data used to deﬁne Ug (v). Keeping

e its endpoints ﬁxed, the path sg(v) can be deformed arbitrarily, at the expense of a phase factor.

When acting on states |ψe(s)i as we consider (or even on states with many e particles, but no m particles), this phase factor is independent of the state, and can be absorbed into a redeﬁnition of

e fg (v). 48

At this point, it is important to ask whether it is always possible to ﬁnd an e-localization, and, if it exists, whether the e-localization is in some sense unique. Indeed, in Appendix C we prove that for toric code models as described in Sec. 3.4, it is always possible to ﬁnd an e-localization

e e of G. Moreover, the e-localization is unique up to projective transformations Ug (v) → λ(g)Ug (v), where λ(g) ∈ {±1}. This means that the e-localization is a legitimate tool to study the action of symmetry on e particles in the solvable models.

To determine the e fractionalization class from the e-localization, we consider the product

e e e Ug1 (g2v)Ug2 (v) = F (g1, g2, v)Ug1g2 (v), (3.49) where F (g1, g2, v) ∈ {±1}, and this equation holds acting on all states containing no m particle excitations [including |ψe(w)i]. This relation holds because both sides of the equation are e string operators joining v to g1g2v, and can diﬀer only by a phase factor depending on g1, g2 and v.

We now show that F (g1, g2, v) is independent of v, and forms a Z2 factor set, so that we can write F (g1, g2, v) = ωe(g1, g2). Suppose that for some g1, g2, and some vertices vi, vj, we have

F (g1, g2, vi) 6= F (g1, g2, vj). Then consider the state |ψe(sij)i, where sij is a path joining vi to vj.

We have

Ug1g2 |ψe(sij)i = Ug1 Ug2 |ψe(sij)i (3.50)

e e e e = Ug1 (g2vi)Ug2 (vi)Ug1 (g2vj)Ug2 (vj)|ψe(sij)i

e e = F (g1, g2, vi)F (g1, g2, vj)Ug1g2 (vi)Ug1g2 (vj)|ψe(sij)i

= −Ug1g2 |ψe(sij)i,

a contradiction. This shows F = F (g1, g2), independent of v. The associativity condition required for F (g1, g2) to be a factor set follows from equating the two ways of associating the product in

e e e Ug1 (g2g3v)Ug2 (g3v)Ug3 (v)|ψe(s)i, (3.51) where |ψe(s)i has one e particle at v. Thus we have shown

e e e Ug1 (g2v)Ug2 (v) = ωe(g1, g2)Ug1g2 (v), (3.52) 49 with ωe a Z2 factor set. This operator equation holds acting on all states of the form |ψe(s)i, and more generally on states with any number of e particle excitations created by acting on |ψ0i with e-string operators. The freedom to transform the e-localization via projective transformations induces the usual projective transformation on the factor set, so that only the fractionalization class [ωe] is well deﬁned.

In addition to making explicit the general structure of fractionalization classes in the solvable models, this result also makes it simple to calculate [ωe]. In particular, we may focus on a single e

e particle at any desired location, and determine [ωe] by calculating appropriate products of Ug (v).

In particular, we can calculate the products of generators in Eqs. (3.39-3.44), and determine the σe parameters. There is then no need to check that the resulting σe’s are the same for every possible location of e particle, because we have already established this in general.

The above discussion proceeds in much the same way for m particles, which reside at holes h ∈ H in the planar projection P(G). States with two m particles can be written

m |ψm(t)i = Lt |ψ0mi, (3.53) where t is an open cut. To every pair (g, h) ∈ G × H, where the m particle resides at the hole h,

m m we associate a number fg (h) and a cut tg (h), which joins h to gh. This allows us to write

m m m U (h) = f (h)L m . (3.54) g g tg (h)

From this point, the discussion for e particles goes over to the m particle case, with only trivial

m modiﬁcations. We refer to a set of Ug (h) operators satisfying the m particle analog of Eq. (3.48) as a m-localization. Just as in the case of e-localizations, Appendix C establishes that it is always possible to ﬁnd a m-localization, which is unique up to projective transformations.

3.6 Symmetry Classes Realized by Toric Code Models

Here, we present the main results of this work, on the realization of symmetry classes in toric code models with square lattice symmetry. These results consist of explicit construction of 50 models realizing various symmetry classes, as well as the derivation of general constraints showing that certain symmetry classes are impossible in the family of models under consideration. We have obtained a complete understanding, in the sense that we have found an explicit realization of every symmetry class not ruled out by general constraints.

Below, we present our results in three stages, in order of increasing generality (and decreasing simplicity). First, we exhibit a single model realizing all possible e particle fractionalization classes

[ωe], as the parameters of the Hamiltonian are varied. In this model, the m particles always have trivial fractionalization class. Second, we consider the family of toric code models with no spin-orbit coupling. Finally, we consider toric code models allowing for spin-orbit coupling.

3.6.1 Model realizing all e particle fractionalization classes

Here, we present a model that can realize all possible e-fractionalization classes [ωe], as the parameters of the Hamiltonian are varied. In this model, the m-fractionalization class is always trivial. The model is deﬁned on the lattice shown in Fig. 3.3, and symmetry is chosen to act on the spin degrees of freedom without spin-orbit coupling. The lattice has six types of plaquettes shown in Fig. 3.3, so that only plaquettes of the same type are related by symmetry. Letting Pi ⊂ P be the set of all plaquettes of type i (i = 1,..., 6), the Hamiltonian is

6 e X X m X H = −K Av − Ki Bpi . (3.55) v∈V i=1 pi∈Pi

e m m We choose K = 1, with arbitrary Ki ∈ {±1}, and note that bi ≡ Ki is the ground-state eigenvalue of Bpi . Following the calculation procedure described below, we ﬁnd

e e e σpx = b1, σpxy = b2, σtxpx = b3, (3.56)

e e e σpxpxy = b4, σtxty = b4b5, σtypx = b1b3b4b6, (3.57)

2 from which it is clear that each possible [ωe] ∈ H (G, Z2) is realized in this model for appropriate

m m choice of Ki . In addition we ﬁnd that all the corresponding σ ’s are unity, and thus [ωm] is the trivial fractionalization class. 51

m m K3 K1 m v1 v2 K2 l2 l1 x

(a)

m m m m m m K4 K6 K4 K4 K6 K4

m m m m m m K6 K5 K6 K6 K5 K6

m m m m m m K4 K6 K4 K4 K6 K4

(b) (c)

Figure 3.3: (a) The lattice on which all 26 = 64 e particle fractionalization classes can be realized. There are six types of plaquettes not related by symmetries, and the correponding plaquette terms m are assigned independent coeﬃcients Ki (i = 1, 2, ··· , 6). Nearest-neighbor pairs of vertices are joined by two edges (dark and light; blue and red online), drawn curved to avoid overlapping and to be clear about their movement under space group operations. Plaquetes of type i = 1, 2, 3 are each formed by the two edges joining a nearest-neighbor pair of vertices. Two vertices v1, v2 and e two edges l1, l2 are labeled to illustrate the calculation of σpx discussed in the main text. (b), (c) Subgraphs of the lattice in (a), each containing all the vertices and half the edges. These subgraphs transform into one another under any improper space group operation (i.e. reﬂections). We draw these subgraphs to illustrate the plaquettes of type i = 4, 5, 6. 52

We now illustrate how these results are obtained by working through the determination of

(U e )2 = σe as an example. It follows from the discussion of Sec. 3.5.2 that σe can be obtained Px px px by considering an e particle at any desired vertex v , and then computing (U e )2 acting on this e 1 Px particle. We consider an e particle at vertex v1 as shown in Fig. 3.3a, so that v2 = Pxv1, and the vertices v1 and v2 are joined by edges l1, l2 forming a type i = 1 plaquette. (To be more precise, we should also specify the position of a second e particle at vertex v 6= v1, let s0 be a path joining

e e v1 to v, and consider the state |ψe(s0)i = Ls0 |ψ0ei. However, the result for σpx will be independent of v.)

We are free to choose the e-localization

e z UPx (v1) = σl1 (3.58)

e z UPx (v2) = fσl2 , (3.59) where f = ±1. To determine f, we consider the path s = l1, which has end points v1 and v2. Then we have

z z UPx |ψe(s)i = UPx σl1 |ψ0ei = σl2 |ψ0ei, (3.60) since Pxl1 = l2. But we also have

e e UPx |ψe(s)i = UPx (v1)UPx (v2)|ψe(s)i (3.61)

z z z = (σl1 )(fσl2 )σl1 |ψ0ei (3.62)

z = fσl2 |ψ0ei. (3.63)

Consistency of these two calculations of the action of UPx then requires f = 1.

2 Now that we have ﬁxed the form of the e-localization, we can compute the action of Px on the e particle at v1. We have

e e 2 e e σpx = (UPx ) (v1) = UPx (v2)UPx (v1) (3.64)

z z m = σl2 σl1 = K1 = b1. (3.65)

This should be interpreted as an operator equation that hold acting on any state obtained by acting successively with e-string operators on |ψ0ei. In particular it holds acting on a state of interest, 53

e |ψe(s)i, with one e particle located at v1. The results for the other σ parameters can be obtained by straightforward analogous calculations.

3.6.2 Toric code models without spin-orbit coupling

We now proceed to consider the family of models TC0(G), which includes all toric code models with square lattice space group symmetry as introduced in Sec. 3.4, with the restriction of no spin-orbit coupling. We remind the reader that this means, for any symmetry operation g ∈ G,

µ −1 µ we have Ugσ` Ug = σg`. In words, symmetry acts simply by moving edges and vertices of the lattice, and acts trivially within the Hilbert space of each spin.

In Appendix C.1.1, we obtain a number of constraints on which symmetry classes can occur for models in TC0(G). The main result is the following theorem:

Theorem 1. The TC symmetry classes in A, B, C, M, M1, M2 and M3 are not realizable in

TC0(G), where

e m A = σpxpxy = σpxpxy = −1 ,

e e m m B = σpxpxyσtxty = σpxpxyσtxty = −1 ,

e e m m C = σpxpxyσtypx = σpxpxyσtypx = −1 ,

m m m M = σpx = −1 ∨ σpxy = −1 ∨ σtxpx = −1 ,

m e e M1 = σpxpxy = −1 ∧ σpx = −1 ∨ σpxy = −1 ,

m m e e M2 = σpxpxyσtxty = −1 ∧ σpxy = −1 ∨ σtxpx = −1 ,

m m e e M3 = σpxpxyσtypx = −1 ∧ σpx = −1 ∨ σtxpx = −1 .

Here ∧, ∨ are the logical symbols for “and” and “or” respectively.

This leaves 95 TC symmetry classes not ruled out by the above constraints, corresponding to

82 symmetry classes under e ↔ m relabeling. In addition, all these 95 TC symmetry classes are realized by models in TC0(G).

This theorem is proved in Appendix C.1.1, except for the last statements regarding counting 54

ao ao aΚ b2

b3 b1 b2 b3 b1 b2 b1

b 1 1 b 1 b b 1 1 b 1 b b b 1 b 1 b b 1 (a) 3 1 2 3 (b) 3 1 2 3 (c) 3 1 2 1 1 1 1 ao˜ 1 1 1 1 ao˜ 1 1 1 1 1 1 1 aκ

aΚ ao b aΚ b ao a o ao ao

1 1 1 1 1 b 1 1 1 1 b 1 1 1 1 1 1 1 (d) (e) (f) 1 1 1 ao ao˜ 1 1 1 1 ao 1 aκ 1 1 1 ao ao˜ aκ

Figure 3.4: TC0 (G) models. The shaded square is a unit cell and the origin of our coordinate system is at the center of the square. Below each figure of lattice is the corresponding TC symmetry class in the form (3.72). Here ar is the ground state eigenvalue of Av for v at special points r = o, o,˜ κ; and b, b1, b2 are the ground state eigenvalues of Bp for the plaquette p, which in these models is picked to be the smallest cycle made with black edges where b, b1 or b2 is written, while b3 is for the plaquette made of a pair of black and grey edges (black and pink online). These edges are drawn curved to avoid overlapping and to be clear about their movement under space group operations. The comparison between (a) and (b) gives an explicit example that moving the coordinate system 1 1 origin by 2 , 2 results in a transformation (3.74): Px → TxPx, σpx ↔ σtxpx, σpxpxy ↔ σpxpxyσtxty. The symmetry class differs from (e) by such a transformation can be easily got by moving the coordinate system, so we do not bother drawing a separate lattice for it. 55 and realization of symmetry classes, which are proved here. In fact, we exhibit a model realizing each allowed TC symmetry class. Before proceeding to do this, we would like to give a flavor for how the above constraints are obtained, referring the reader to Appendix C.1.1 for the full details.

m As an illustration, we would like to show that σpx = 1 for any model in TC0(G). (This is part of the fact that TC symmetry classes in M are not realizable in TC0(G).) Consider a m particle located at a hole h ∈ H. If P h = h , then we can choose U m (h ) = 1, and therefore 0 x 0 0 Px 0 σm = (U m )2(h ) = 1. px Px 0

We then consider the case Pxh0 = h1 6= h0. We can always draw a simple cut t joining h0 to h1, so that Pxt = t. We are then free to choose the m-localization

m m UPx (h0) = Lt (3.66)

m m UPx (h1) = fLt , (3.67)

m where f = ±1 needs to be determined. To do this, consider the state |ψm(t)i = Lt |ψ0mi, for which we have

m UPx |ψm(t)i = UPx Lt |ψ0mi = |ψm(t)i, (3.68) where we used the fact that U LmU −1 = Lm = Lm. (Note that here we use the assumption of Px t Px Pxt t no spin-orbit coupling.) But we also have

m m UPx |ψm(t)i = UPx (h0)UPx (h1)|ψm(t)i = f|ψm(t)i. (3.69)

2 Consistency of these two calculations requires f = 1, and we can then calculate Px acting on the m particle located at h0, to obtain

m m 2 σpx = (UPx ) (h0) = (3.70)

m m m 2 = UPx (h1)UPx (h0) = (Lt ) = 1. (3.71)

m We have thus shown that σpx = 1 for any model in TC0(G). Roughly similar reasoning is followed in Appendix C.1.1 to establish the constraints stated in the theorem.

Now we proceed to enumerate and count the TC symmetry classes not ruled out by the constraints of Theorem 1. At the same time, we present the explicit models realizing each class 56

(shown in Figures 3.3 and 3.4). Here, and throughout the paper, we will ﬁnd it convenient to present TC symmetry classes ([ωe] , [ωm]) in the matrix form   σe σe σe σe σe σe σe σe  px pxy txpx pxpxy pxpxy txty pxpxy typx    , (3.72) m m m m m m m m σpx σpxy σtxpx σpxpxy σpxpxyσtxty σpxpxyσtypx or, equivalently,   σe σm  px px     e m   σpxy σpxy       e m   σtxpx σtxpx    . (3.73)    σe σm   pxpxy pxpxy     e e m m   σpxpxyσtxty σpxpxyσtxty     e e m m  σpxpxyσtypx σpxpxyσtypx This form allows for simple comparison to the constraints of Theorem 1. In addition, under the

1 1 change of origin o → 2 , 2 , the entries of the matrix are simply permuted:   σe σe σe σe σe σe  1 2 3 4 5 6    m m m m m m σ1 σ2 σ3 σ4 σ5 σ6   σe σe σe σe σe σe  3 2 1 5 4 6  →   . (3.74) m m m m m m σ3 σ2 σ1 σ5 σ4 σ6 This holds even beyond the setting of solvable toric code models, and can be veriﬁed by replacing

Px as a generator of G by Px → Pfx = TxPx, which corresponds to the desired change of origin. The

σ parameters for the new generators can then be computed in terms of those for the old generators,

a a a a a by noting that U = φ UT UP , where a = e, m and φ ∈ {±1}. Pfx x x The behavior of TC symmetry classes under a change in origin is illustrated in Fig. 3.4a and

Fig. 3.4b. Apart from this example, we do not bother to draw the same lattice twice when the only diﬀerence is a change in origin. So, for example, the model shown in Fig. 3.4e is taken to realized both TC symmetry classes   1 1 1 1 b 1     (3.75) 1 1 1 ao 1 aκ 57 and   1 1 1 b 1 1     , (3.76) 1 1 1 1 ao aκ where the TC symmetry classes (3.75) are realized if we put the origin at the center of the shaded square, and the TC symmetry classes (3.76) are realized if we put the origin at the corner of the shaded square.

Now, we divide the TC symmetry classes not ruled out by Theorem 1 into four collections Di,

m m m i = 0, 1, 2, 3. In Di, there are i of σpxpxy, σtxty and σtypx equal to −1. In D0, we have TC symmetry classes in the form   1        1         1    ,    1         1      1 where the symbol means that the corresponding σ parameter can be chosen to be ±1 indepen-

6 dently of any other parameters. Therefore, |D0| = 2 . These TC symmetry classes are realized in the model discussed in Sec. 3.6.1, and shown in Fig. 3.3.

In D1, we have TC symmetry classes ([ωe] , [ωm]) in the form       1 1 1 1 1                    1 1   1 1   1                     1   1 1   1 1    ,   , or   ,        1 −1   1   1                     1   1 −1   1              1 1 1 −1

3 so |D1| = 3 × 2 . These TC symmetry classes are realized in the models shown in Fig. 3.4(a-c). 58

In D2, we have TC symmetry classes in the form       1 1 1 1 1 1                    1 1   1 1   1 1                     1 1   1 1   1 1    ,   , or   ,        1 −1   1   1 −1                     1 −1   1 −1   1              1 1 −1 1 −1 so |D2| = 3 × 2. These TC symmetry classes are realized in Fig. 3.4(d,e).

In D3, we have only the single TC symmetry class   1 1        1 1         1 1    , (3.77)    1 −1         1 −1      1 −1 which is realized by the model of Fig. 3.4f.

P3 In total, there are thus exactly i=0 |Di| = 95 TC symmetry classes realized by models in TC0 (G). Recalling that the TC symmetry classes ([ωm] , [ωe]) and ([ωe] , [ωm]) correspond to the same symmetry class, it is a straightforward but somewhat tedious exercise to show that 13 symmetry classes are double-counted among the 95 TC symmetry classes. Therefore, the total number of symmetry classes realized by models in TC0(G) is 95 − 13 = 82.

3.6.3 General toric code models

To consider the most general toric code models introduced in Sec. 3.4, we must allow for spin-orbit coupling. As discussed in Sec. 3.4, this means, for any symmetry operation g ∈ G, we

µ −1 µ µ µ have Ugσ` Ug = c` (g)σg`, where c` (g) ∈ {±1}, µ = x, z. The corresponding family of models is referred to as TC(G). Our results on these models are summarized in the following theorem: 59

h1 h2 a l1 b Κ ao l1 l2 l3 ao ao l2

1 1 γ 1 b γ 1 1 1 1 1 1 (a) 2 1 (b) α1 1 1 ao 1 α2 c1 c2 c3 ao ao˜ c1c3aκ

Figure 3.5: Two example models in TC (G) that realize TC symmetry classes not possible in TC0 (G). The shaded square is a unit cell and the origin of our coordinate system is at the center of the square. Below each ﬁgure of lattice is the corresponding TC symmetry class in the form (3.72). Here ar is the ground state eigenvalue of Av for v at special points r = o, o,˜ κ and b is the ground state eigenvalue of Bp for the plaquette p, deﬁned here to be the smallest cycle enclosing the letter “b.” We write α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ). (a) A model i li x i li xy i li x i li xy realizing some TC symmetry classes (and symmetry classes) that cannot be realized without spin- m orbit coupling. Here ,h1, h2 label two positions of a m particle for the calculation of σpx = α1 in 6 the main text. (b) A model realizing all 2 = 64 possible m particle fractionalization classes [ωm]. Here, for simplicity, we make the restriction γi = δi ≡ ci. 60

0 Theorem 2. The TC symmetry classes in P1, P2, P3, A, B and C are not realizable in TC (G), where

e m P1 = σpx = σpx = −1 ,

e m P2 = σpxy = σpxy = −1 ,

e m P3 = σtxpx = σtxpx = −1 ,

e m A = σpxpxy = σpxpxy = −1 ,

e e m m B = σpxpxyσtxty = σpxpxyσtxty = −1 ,

0 e e e e m m C = σpx = σtxpx = σpxpxyσtypx = σpxpxyσtypx = −1 .

This leaves 945 TC symmetry classes not ruled out by the above constraints, corresponding to 487 symmetry classes under e ↔ m relabeling. In addition, all these 945 TC symmetry classes are realized by models in TC(G).

This theorem is proved in Appendices C.1.2 and D. The constraints ruling out some TC symmetry classes are obtained in Appendix C.1.2, while the counting of symmetry classes and the presentation of explicit models is done in Appendix D.

Here, we simply give an illustration how spin-orbit coupling increases the number of allowed symmetry classes. For the model shown in Fig. 3.5a, more TC symmetry classes are

m possible if spin-orbit coupling is included. For example, take the calculation of σpx. Suppose

U σx U = α σx , with α ∈ {±1}. If we choose U m (h ) = σx , then we must have U m (h ) = Px l1 Px 1 l1 1 Px 1 l1 Px 2 α σx to ensure U m (h ) U m (h ) σx |ψ i = U σx |ψ i. Therefore we have σm = (U m )2(h ) = 1 l1 Px 1 Px 2 l1 m0 Px l1 m0 px Px 1 U m (h )U m (h ) = α . Therefore we can have σm = −1, which is impossible without spin-orbit Px 2 Px 1 1 px coupling.

Another particularly interesting example, shown in Fig. 3.5b, is a model realizing all 26 = 64 m particle fractionalization classes. This model is constructed starting with the lattice of Fig. 3.4f and allowing for spin-orbit coupling.

In the next chapter, we will explain why we can only ﬁnd 487 symmetry fractionalization 61 classes here.

3.7 Summary and Beyond Toric Code Models

To summarize, we considered the realization of distinct square lattice space group symmetry fractionalizations in exactly solvable Z2 toric code models. We obtained a complete understanding, in the sense that every symmetry class consistent with the fusion rules is either realized in an explicit model, or is proved rigorously to be unrealizable. In more detail, ﬁrst, we found a single model that realizes all 26 = 64 e particle fractionalization classes as the parameters in its Hamiltonian are varied. Second, we considered a restricted family of models TC0(G) without spin-orbit coupling, but deﬁned on general two-dimensional lattices. We showed that exactly 95 TC symmetry classes

([ωe] , [ωm]), corresponding to 82 symmetry classes h[ωe] , [ωm]i, are realized by models in TC0(G).

This result was established by proving that the other TC symmetry classes cannot be realized by any model in TC0(G), and giving explicit models for those classes not ruled out by such general arguments. Finally, in the most general family of models considered, TC(G), we allowed spin-orbit coupling in the action of symmetry. In this case we found that exactly 945 TC symmetry classes, corresponding to 487 symmetry classes, are realized in TC(G).

These main results are, of course, conﬁned to a special family of exactly solvable models.

Because the symmetry class is a robust characteristic of a SET phase, and thus stable to small perturbations preserving the symmetry,[107] all the symmetry classes that we ﬁnd clearly exist in more generic models. However, there may well be symmetry classes not realized in TC(G) that can occur in more generic models (this is indeed the case, as we see below).

Ideally, we would like to make statements about arbitrary local bosonic models (i.e. those with ﬁnite-range interactions). For example, we can ask the challenging question of which symmetry classes can be realized in the family of all local bosonic models with square lattice space group symmetry. We do not have an answer to this question, but here we provide some partial answers.

First, we show using a parton gauge theory construction that there exist symmetry classes not realizable in TC(G) that can be realized in local bosonic models. Second, we establish a connection 62 between symmetry classes of certain on-site symmetry groups and symmetry classes of the square lattice space group.

Our parton construction allows us to argue that if [ωm] is a m fractionalization class realized

2 for a model in TC0(G), then the symmetry class h[ωe], [ωm]i, where [ωe] ∈ H (G, Z2) is arbitrary, can be realized in a local bosonic model. It is easy to see that some symmetry classes obtained this way cannot be realized in TC(G). For example, the symmetry classes in A are unrealizable in

TC(G) (Theorem 2), but they are possible here.

The starting point for the construction is a Hamiltonian of the form

X e X H = − Kv Av − Bp, (3.78) v∈V p∈P

e where Kv ∈ {±1}. We take the symmetry to act without spin-orbit coupling, so this is a model in

m TC0(G). We have chosen Kp = 1 for all p ∈ P , which implies the e fractionalization class is trivial.

However, Hamiltonians of this form can realize any m fractionalization class allowed in TC0(G), because without spin-orbit coupling the m fractionalization class only depends on the lattice and

e on the Kv coeﬃcients.

We now build a Z2 gauge theory based on the above toric code model. On each vertex v we † introduce a boson ﬁeld created by bvα, where α = 1, . . . , n is an internal index. We also introduce the gauge constraint

† e bvαbvα Av = Kv (−1) , (3.79) with sums over repeated internal indices implied. The gauge theory Hamiltonian is taken to be

X X † X x Hgauge = − Bp + u bvαbvα − h σ` , (3.80) p∈P v∈V `∈E with u > 0. We choose symmetry to act on the boson ﬁeld by

† −1 † UgbvαUg = Dαβ(g)bgv,β, (3.81) where D(g) are unitary matrices giving a n-dimensional projective representation of G. By choosing

D(g), we are choosing a projective symmetry group for the parton ﬁelds.[98] In Ref. [107], it was 63 shown that there exists a ﬁnite-dimensional projective representation for any fractionalization class

2 [ωe] ∈ H (G, Z2), so the bosons can be taken to transform in any desired fractionalization class.

We now discuss two limits of Hgauge. First, we consider the limit h → +∞. In this limit,

x we have σ` = 1, and the only remaining degrees of freedom are the bosons. The gauge constraint becomes   even, Ke = 1 †  v bvαbvα = . (3.82)  e  odd, Kv = −1 In this Hilbert space, all local operators transform linearly under G, and so the model reduces to a legitimate local bosonic model in this limit. Following the usual logic of parton constructions,[96,

98, 105] Hgauge can be viewed as a low-energy eﬀective theory for local bosonic models with the same Hilbert space and symmetry action as in the h → +∞ limit. The expectation is that any phase realized by Hgauge can be realized by some such local bosonic model, although this approach does not tell us how to choose parameters of the local bosonic model to realize the corresponding phase of Hgauge.

Now we consider the exactly solvable limit of Hgauge with h = 0. This limit is deep in the

e † deconﬁned phase of the Z2 gauge theory, and we have Bp = 1, Av = Kv , and bvαbvα = 0 acting on

e ground states. Because Av = Kv , the m particles feel the same pattern of background Z2 charge as in the original TC0(G) toric code model, and their fractionalization class is unchanged. Now, however, the Z2-charged bosons become the e particle excitations, so the e particle fractionalization class [ωe] is determined by the (arbitrarily chosen) projective representation D(g). We have thus obtained a phase with Z2 topological order and symmetry class h[ωe], [ωm]i, as desired. We now present the second result of this section, namely we establish a connection between space group symmetry classes and the symmetry classes of certain on-site symmetries. Suppose that we have a local bosonic model with symmetry G × Go, where G is the space group, and Go is a ﬁnite, unitary on-site symmetry. We do not assume square lattice symmetry here, but allow for a more general space group. We require Go to be isomorphic to some ﬁnite quotient of the space group G. For example, if G is square lattice space group symmetry, we could take Go ' G/T2, 64

2 2 where T2 is the normal subgroup of G generated by translations Tx and Ty . In this case, Go can be nicely described as what remains of the space group when the system is put on a 2 × 2 periodic torus.

Next, we suppose our model has Z2 topological order, and the action of symmetry is described by e and m fractionalization classes [ωe] and [ωm]. Specifying these fractionalization classes in terms of generators and relations, we further assume that the only relations with non-trivial projective phase factors (i.e., σ parameters) are those involving only elements of Go. That is, space group symmetry G acts linearly on e and m particles, and elements g ∈ G commute with go ∈ Go when acting on e and m particles. Basically, we are assuming that we have some non-trivial action of on-site symmetry, where the space group symmetry “comes along for the ride.” As an aside, there are some interesting open questions hidden in our assumptions. For example, is every symmetry class of the on-site Go that can be realized in local bosonic models also compatible with an arbitrary space group symmetry G? Or, are there Go symmetry classes that are only compatible with a given space group G if some elements of Go and G are chosen not to commute acting on e and/or m particles?

With our assumptions speciﬁed, we proceed to break the symmetry down to the subgroup

0 G ⊂ G × Go, deﬁned as the set of all elements of the form (g, φ(g)) ∈ G × Go, where g ∈ G is

0 arbitrary, and φ : G → Go is the quotient map. It is easy to see that G is a subgroup, and that it is isomorphic to G. We thus still have G space group symmetry, but now the space group operations are combined with on-site symmetry operations. Under the new reduced symmetry, it is easy to

0 0 0 see that new [ωe] and [ωm] fractionalization classes for G symmetry are induced by corresponding

Go fractionalization classes before breaking the symmetry. While these remarks remain somewhat abstract at present, this discussion shows that progress in understanding symmetry classes of ﬁnite, unitary on-site symmetry[109, 111, 115, 132] can potentially have direct applications to similar problems for space group symmetry. Chapter 4

Topological Phases Protected by Reﬂections and Anomalous Crystalline Symmetry Fractionalization

In this chapter, we study the bosonic symmetry protected topological (SPT) phases protected by reﬂections. This gives an example of a systematic method for classifying general point group

SPT (pgSPT) phases. Our approach is based on a procedure to reduce pgSPT phases to lower- dimensional SPT phases protected by on-site unitary symmetry. This connection helps us (1) classify pgSPT phases, (2) construct toy models for these phases, (3) analyse their surface degrees of freedom. After understanding these, we further succeed to realize gapped, symmetry preserving surfaces with Z2 topological order on non-trivial pgSPT phases, which produce concrete examples for anomalous crystalline symmetry fractionalization.

In particular, we first study the simple case with only one reflection symmetry. We find a non-trivial pgSPT phase, and show that it can have a gapped, symmetry preserving surface with Z2 topological order (i.e. as in Kitaev’s toric code model), and anomalous symmetry fractionalization.

The latter property means the action of mirror reﬂection is anomalous on anyon quasiparticles of the surface state, in the sense that this action cannot be realized in a strictly two-dimensional system. In this case, the reﬂection squares to −1 acting on both the bosonic charge (e) and bosonic

ﬂux (m) quasiparticles.

In addition, we also study the case with two orthogonal reflections. Our classification gives several non-trivial pgSPT phases. One of them is related to the non-trivial one-dimensional SPT protected by onsite unitary Z2×Z2 symmetry. It can provides boundary degrees of freedom carrying 66 projective representation of the reflections. In addition, we find the symmetry fractionalization pattern with the two reflections acting anti-commutatively on both e and m quasiparticles is connected with the projective symmetry transformation of spins.

Finally, our understanding on the connection between the symmetry fractionalization pattern and the symmetry behaviour of spins explains our results in Chapter 3.

4.1 Reﬂection Symmetry Protected Topological Phyases

4.1.1 With a Single Reﬂection Symmetry

For simplicity, we first require our three-dimensional quantum spin system to respect only a single reflection. For concreteness, we are free to choose our coordinate system such that the reflection to be Px :(x, y, z) → (−x, y, z). The plane x = 0 is invariant under the reflection and it divides the system into two halves as illustrated in Fig. 4.1. The class of phases we are considering are SPT phases protected by the reflection. By definition, SPT phases are short-range entangled.

So there is an adiabatic process U (t) tuning one half into a tensor product state, while keeping the other half unchanged. If meanwhile, we also perform another adiabatic process related to the above one by the reﬂection to the other half of the system, then in total we have an adiabatic process

U (t) ⊗ Px [U (t)] that keeps the reﬂection symmetry and makes the system into tensor product state except for the spins on (or near) the grey plane. Thus, the three-dimensional SPT phase protected by a single reﬂection is related to the two-dimensional SPT phase lying on the mirror

1 plane protected by on-site unitary Z2 symmetry. It is generally believed, although not proven rigorously to our knowledge, that there are only two two-dimensional SPT phases protected by Z2

3 unitary on-site symmetry, which are labelled by the two elements of H (Z2,U (1)). This gives us two pgSPT phases.

1 It is pointed to us by Liang Fu that there is another possibility: the above trivialization procedure may lead to some thermal quantum Hall eﬀect type topological orders on the mirror plane. 67

Figure 4.1: A quantum spin system that respects a single reflection Px :(x, y, z) → (−x, y, z). The grey plane x = 0 is invariant under the reflection. It divides the system into two halves. If the system is short-range entangled, then there is an adiabatic process U (t) tuning one half into a tensor product state, while keeping the other half unchanged. If meanwhile, we also perform another adiabatic process related to the above one by the reflection to the other half of the system, then in total we have an adiabatic process U (t) ⊗ Px [U (t)] that keeps the reflection symmetry and makes the system into tensor product state except for the spins on (or near) the grey plane. Thus, the three-dimensional SPT phase protected by a single reflection is related to the two-dimensional SPT phase lying on the mirror plane protected by on-site unitary Z2 symmetry. Further, if the system have some boundary (like the top surface), then on (or near) the line (like the thickened line here) where boundary and the invariant plane of reflection meet, there are some boundary degrees of freedom anomalous under the reflection.

Figure 4.2: A quantum spin system that respects two orthogonal reflections Px :(x, y, z) → (−x, y, z) and Py :(x, y, z) → (x, −y, z). The planes x = 0 (grey) and y = 0 (red) divide the system into four quarters. If the system is short-range entangled, then there is an adiabatic process U (t) tuning one quarter into a tensor product state, while keeping the other quarters unchanged. If meanwhile, we also perform adiabatic processes related by the reflections to the other quarters, then in total we have an adiabatic process U (t) ⊗ Px [U (t)] ⊗ Py [U (t)] ⊗ PxPy [U (t)] that keeps the reflections and makes the system into tensor product state except for the spins on (or near) the planes x = 0 and y = 0. Thus, the pgSPT phases are classified by 3 3 2 3 H (Z2,U (1))×H (Z2,U (1))×H (Z2 × Z2,U (1)) = Z2, with factors corresponding to the on-site SPT phases of the planes x = 0, y = 0 and their intersection line. If the phase corresponds to only the non-trivial element of the last factor, then the symmetry behaviour of the boundary degrees of freedom near the black dot on the top surface is twisted by it. 68

4.1.2 With Two Orthogonal Reﬂection Symmetries

Now suppose the symmetry respect two orthogonal refection symmetries. For concreteness

We are free to choose them to be Px :(x, y, z) → (−x, y, z) and Py :(x, y, z) → (x, −y, z).

The planes x = 0 (grey) and y = 0 (red) divide the system into four quarters. If the system is short-range entangled, then there is an adiabatic process U (t) tuning one quarter into a tensor product state, while keeping the other quarters unchanged. If meanwhile, we also perform adiabatic processes related by the reﬂections to the other quarters, then in total we have an adiabatic process

U (t) ⊗ Px [U (t)] ⊗ Py [U (t)] ⊗ PxPy [U (t)] that keeps the reﬂections and makes the system into tensor product state except for the spins on (or near) the planes x = 0 and y = 0.

As above, on the plane x = 0, the reﬂection Px reduces to a Z2 on-site unitary symmetry and there are two phases protected by it. Let’s denote the non-trivial one by νx, which cannot be adiabatically tuned into tensor product state without breaking Px. Similarly, Py protect two phases related to the on-site SPT phases of the plane y = 0, and the non-trivial one is denoted by

νy.

For convenience, we can reinterpret νx (resp. νy) as inserting a non-trivial two dimensional

SPT protected Z2 on-site unitary symmetry at x = 0 (resp. y = 0). Obviously, nux and νy cannot trivialize each other, because they are spatially separated. Therefore, starting with the tensor product state, the two operations νx, νy generate four pgSPT phases, which thus can be labeled by

3 3 H (Z2,U (1)) × H (Z2,U (1)) = Z2 × Z2. After the four quarters have been put into product states, the remaining parts away from the z-axis (i.e. x = y = 0) can be further trivialized by adding νx, vy if necessary. Eventually we are left with a one-dimensional system along the z-axis possibly non-trivial as a one-dimensional SPT phase, where the reﬂections reduce to Z2 ×Z2 on-site unitary symmetries. According to group cohomology

2 classiﬁcation, the one-dimensional system has two phases labelled by H (Z2 × Z2,U (1)) = Z2.

Let’s denote the non-trivial one by νxy. Then assuming the relevant group cohomology classiﬁcation for on-site SPT phases is complete, our classiﬁcation of pgSPT phases protected by two orthogonal 69

3 3 2 reﬂections is H (Z2,U (1))×H (Z2,U (1))×H (Z2 × Z2,U (1)) = Z2 ×Z2 ×Z2, which is generated by νx, νy and νxy.

4.2 Boundary Degrees of Freedom of SPT Phases

Here we are going to give examples of νx and νxy. In addition, we write down explicit models for their boundary degrees of freedom, which are crucial ingredients in construction of surface topological states to realize anomalous symmetry fractionalizations. But before that, let’s brieﬂy decribe the meaning of boundary degrees of freedom of SPT phases.

Suppose we are given a Hamiltonian, homogeneous in space, describing a non-trivial SPT phase protected by some on-site symmetries. If the system is defined on a closed manifold, then the ground state of the Hamiltonian will be unique. However, if the system is put on a manifold with boundary, it is possible to choose some Hamiltonian such that the ground state will be degenerate and the degeneracy grows exponentially with the length of the boundary. The Hamiltonian fixes the state deep inside the bulk but not at its boundary. So the degrees of freedom that do not affect the energy are localized along the boundary. In addition, the non-triviality of the SPT phase can be reflected in the anomalous behaviour of these boundary degrees of freedom under symmetries.

This explains many astonishing properties of SPT phases and will be the input for constructing our models of anyons with anomalous symmetry fractionalization.

To make sense of the above abstract description, we now present the two models from

Ref. [144] in the following. They are not only simple but also directly related to our models for anomalous crystalline symmetry fractionalization in latter sections.

4.2.1 Dimer model for 1d SPT

2 2 The dimer model gives examples of one-dimensional SPT phases labelled by ω ∈ H (G, UT (1)),

where G is the group of on-site symmetries. Here the degrees of freedom on each site si (i =

2 The subscript T reminds us that G might have anti-unitary symmetries, which act as complex conjugation on U (1). 70

2 Figure 4.3: The dimer model for one-dimensional SPT phase labelled by ω ∈ H (G, UT (1)), where G is the group of on-site symmetries.

0, 1, 2, ··· , n), illustrated by a oval in Fig. 4.3, can be equally divided into right and left parts,

l r −1 denoted si and si , carrying projective representations of G corresponding to ω and ω .So together

l r si = si ⊗ si carries a linear representation of G. The Hamiltonian is chosen to select the ground

r l state with si and si+1 maximally entangled into a singlet (i.e. a state invariant under symmetries).

l The most important property of this state is some boundary degrees of freedom, namely s0 and

r sn, left unﬁxed by the Hamiltonian. They carry projective representations of G. Later we will use the model as means to provide some degrees of freedom carrying projective representations of symmetries.

3 4.2.2 CZX model for 2d SPT with onsite Z2 symmetry

The CZX model gives an example of non-trivial SPT order protected by Z2 on-site unitary symmetry. According the group cohomology classiﬁcation, it corresponds to the non-trivial element

3 of H (Z2,U (1)). The degrees of freedom is speciﬁed in Fig. 4.4, each site is made up with four

1 spin- 2 spins. The symmetry acts on each site as

s x x x x UCZX = σ1 σ2 σ3 σ4 · CZ12 · CZ23 · CZ34 · CZ41,

x where 1, 2, 3, 4 labels the four spins within the site s, σi is the Pauli X operator on the ith spin and CZij is the controlled-Z operator acting on a pair of spins (the ith and jth spins) as

CZij = |↑↑i h↑↑| + |↑↓i h↑↓| + |↓↑i h↓↑| − |↓↓i h↓↓| . (4.1)

3 The model is deﬁned and carefully studied in Ref. [144]. We summarize some key points here for the convenience of the reader. 71

CZ CZ CZ CZ

(a) (b)

3 Figure 4.4: The CZX model for the two-dimensional SPT phase labelled by H (Z2,U (1)). (a) Each site (circle) contains four spins (dots) and the on-site symmetry of order 2 is generated by x x x x UCZX = σ1 σ2 σ3 σ4 · CZ12 · CZ23 · CZ34 · CZ41. (b) The local term on each plaquette p in the u d l r Hamiltonian is Hp = −Xp ⊗ Pp ⊗ Pp ⊗ Pp ⊗ Pp , where Xp = |↑↑↑↑i h↓↓↓↓| + |↓↓↓↓i h↑↑↑↑| acts on u,d,l,r the four spins on the plaquette p and Pp = |↑↑i h↑↑| + |↓↓i h↓↓| acts on the up, down, left and right neighbouring pairs of spins around the plaquette p respectively. The ground state is ﬁxed |↑↑↑↑i +|↓↓↓↓i within the bulk; the spins on each plaquette (square) are entangled as p√ p . But some 2 boundary degrees of freedom are not ﬁxed by the Hamiltonian, like those along the right edge shown here.

The Hamiltonian of the CZX model is

X u d l r H = − Xp ⊗ Pp ⊗ Pp ⊗ Pp ⊗ Pp , (4.2) p where the summation is over all plaquettes,

Xp = |↑↑↑↑i h↓↓↓↓| + |↓↓↓↓i h↑↑↑↑| (4.3) acts on the four spins on the plaquette p and

u,d,l,r Pp = |↑↑i h↑↑| + |↓↓i h↓↓| (4.4) acts on the up, down, left and right neighbouring pairs of spins around the plaquette p respectively.

If we assume periodic boundary conditions for the model, then the ground state is unique, given by

Y |↑↑↑↑ip + |↓↓↓↓ip |Ωi = √ . (4.5) p 2 72

It can be checked directly |Ωi is invariant under the symmetry

Y s UCZX = UCZX . (4.6) s

If we assume open boundary conditions for the model, the Hamiltonian does not completely

fix the boundary degrees of freedom. In particular, any neighbouring pair of spins belonging to two adjacent sites on the boundary (but not the corners) are only constrained to the two-dimensional subspace spanned by |↑↑i and |↓↓i. Effectively, they can be viewed as a spin with two states, E E ˜ ˜ ↑ ≡ |↑↑i and ↓ ≡ |↓↓i. Thus, along the boundary we are left with a series of effective spins, whose Pauli operators we denote as τj with j = 0, 1, 2, ··· labelling positions. Their behaviour under the symmetry UCZX is

z † z UCZX τj UCZX = −τj ; (4.7)

x † z x z UCZX τj UCZX = τj−1τj τj+1; (4.8)

y † z y z UCZX τj UCZX = −τj−1τj τj+1. (4.9)

In the above calculation, we used the ground state property of the spins not on the boundary, which are ﬁxed by the Hamiltonian above. Similar transformations are also written down Ref. [145].

4.3 Model of Fractionalization Anomaly: a Single Reﬂection (ePmP)

In the following, we focus on the symmetry fractionalization pattern of gapped Z2 topological spin liquids with a single reﬂection symmetry. To be concrete, we can choose the reﬂection to be

Px :(x, y) → (−x, y). We are going to show the symmetry fractionalization pattern which we call eP mP is anomalous by the constructing a simple model realizing it on surface of the non-trivial three-dimensional SPT phase protected by the reﬂection.

Explicitly, the reﬂection fractionalization pattern can be characterized by the parity of any string operator creating a pair of anyons related by the reﬂection. The notation eP mP stands for

e e non-triviality of both e amd m particles under the reﬂection. So it means Px (L ) |Ωi = −L |Ωi and

m m e m Px (L ) |Ωi = −L |Ωi, where |Ωi is the ground state and L (resp. L ) is any string operator 73 creating a pair e (resp. m) particles at reﬂection related positions like (x, y), (−x, y). Since e and m have trivial self-statistics, we can alternately say reﬂection acts projectively on e and m;

e 2 m 2 (Px ) = (Px ) = −1 as we did in Chapter 3.

Our surface Z2 topological model is constructed on the square lattice by modifying the toric code to make it compatible with the anomalous action of reﬂection symmetry, which we will specify later. First, let’s clarify the geometry and notations a little bit. The vertices are at

1 1 4 v = (vx, vy) ∈ Z + 2 × Z + 2 . We label plaquettes and links by the coordinates of their 1 1 1 centers p = (px, py) ∈ Z × Z and ` = (`x, `y) ∈ Z + 2 × Z ∪ Z × Z + 2 . We put a spin- 2 µ spin (i.e. qubit) on each link `, whose states are |0i = |↑i and |1i = |↓i, with the Pauli matrices σ` µ µ (µ = x, y, z) generating the operations on it. We also use a shorter notation σj ≡ σ 1 (j ∈ Z) (0,j+ 2 ) for spins lying along the axis of the reﬂection Px, which is illustrated by dashed line in Fig. 4.5.

µ Now let’s specify the anomalous transformation of σ` under Px. Since our two-dimensional model serves as a symmetric gapped surface state of the non-trivial three-dimensional SPT phase protected by the reflection, it includes the boundary degrees of freedom of the SPT phase and keeps their anomalous symmetries. As pointed out in our discussion of pgSPT phases, the three- dimensional SPT phase protected by reflection is related to the SPT phase along the reflection plane protected by the unitary on-site Z2 symmetry. Thus, as illustrated in Fig. 4.1, the degrees of freedom carrying anomalous symmetry in our two-dimensional model lie along the reflection axis, which is the intersection of the surface system and reflection plane. In our surface model, these

µ µ anomalous degrees of freedom are σj ≡ σ 1 . They transform under the reﬂection Px as the (0,j+ 2 )

CZX model boundary under Z2 on-site symmetry (cf. Eqns. (4.7-4.9));

z z σj → −σj , (4.10)

x z x z σj → σj−1σj σj+1, (4.11)

z z y z σj → −σj−1σj σj+1. (4.12)

4 1 3 1 1 3 Z + 2 = ··· , − 2 , − 2 , 2 , 2 , ··· . 74

µ Other spins σ` (`x 6= 0) transform normally;

σµσµ → σµ = σµ , ∀` 6= 0. (4.13) ` (`x,`y) Px` (−`x,`y) x

Finally, we write down the Hamiltonian

X X H = − Av − Bp, (4.14) v p where the plaquette terms are deﬁned the same as in the standard toric code model

Y z Bp = σ` , (4.15) `∈p

1 but the vertex terms at vx = ± 2 need to be modiﬁed due to the above reﬂection symmetry and the form of the vertex terms is summarized as  Q σx, v 6= ± 1 ;  `3v ` x 2   A = x x y x z 1 (4.16) v σ σ 1 1 σ σ σ , vx = − ; (−1,vy) − ,v + − 1 ,v − 1 (0,vy) (0,vy−1) 2  ( 2 y 2 ) ( 2 y 2 )   −σx σx σy σx σz , v = 1 .  (1,vy) 1 1 1 1 (0,vy) (0,vy+1) x 2 ( 2 ,vy+ 2 ) ( 2 ,vy− 2 )

Examples of graphic presentation of Av are given in Fig. 4.5. It can be checked that Px (H) = H under the reﬂection speciﬁed by Eqns. (4.10-4.13).

4.3.1 Spectrum of the model

In the following, we study the spectrum of the model to make sure that it is indeed gapped

Z2 spin liquid.

0 0 Lemma 1. [Av,Av0 ] = [Av,Bp] = Bp,Bp0 = 0, ∀v, v , p, p .

Proof. First, since Bp is the same as in toric code, we have Bp,Bp0 = 0.

Second, if v∈ / p, then each factor in Av commutes with Bp. If v ∈ p, then only two Pauli operator factors in Av anti-commute with with Bp. Hence [Av,Bp] = 0.

Third, we need to be a little more careful to show [Av,Av0 ] = 0 by considering the possible

0 1 cases. (1) Not both of v and v are on x = ± 2 . Then factors in Av,Av0 all commute. (2) Both of v 75

x z x x x x x v v y 0 y 5 z

(a) (d)

x z x x x x x v v y 1 y 4 z

(b) (e)

x z x x x x x v v y 2 y 3 z

x y z Figure 4.5: The dashed line is the axis of the reflection. Applying σ` (resp. σ` and σ` ) is graphically presented by highlighting ` in blue (resp. magenta and red). The vertex terms Av at v = v0, v1, ··· , v5 are presented by the highlighted links which form a t-shape. It can be checked that these six vertex terms commute with each other and that the vertex terms in the two columns are related by the reflection; Avi = Px Av5−i , i = 0, 1, ··· , 5. The highlighted red plaquette in (a) presents a Bp lying on the reflection axis, and it is invariant under the reflection. Because vertex terms and plaquette terms away from the reflection axis are exactly the same as in the toric code model and involve spins transform as σµ → σµ , in general it is not hard to see that ` Px` Px (Av) = APxv, Px (Bp) = BPxp and that all vertex terms and plaquette terms commute with each 0 0 other; [Av,Av0 ] = [Av,Bp] = Bp,Bp0 = 0, ∀v, v , p, p . 76

v v p p

(a) (b)

z Figure 4.6: The red string stands for a product of σ` on it. The blue string stands for a product of x σ` on it. (a) When an m is moved to x = 0, it becomes an excitation with Av = −1 and Bp = −1 y due to the factor σ 1 in Av. The excitation is still a boson. (b) When an ε is moved to (vx,vy− 2 ) x = 0, it becomes an excitation with Bp = −1. The excitation is still a fermion.

0 1 and v are on x = ± 2 . The only non-trivial check involve vertex terms shown in Fig. 4.5; it is not hard to see that [Av,Av0 ] = 0.

Lemma 2. The spectrum of the model is labelled by the eigenvalues of Av and Bp.

Proof. We only need to show how to create all eigenvalue combinations of Av and Bp, and then other properties like topological ground state degeneracy and gappedness can be argued as in standard toric code model.

z An Av excitation can be created and moved as in standard toric code by product of σ` on a path of the lattice. A Bp excitation NOT on x = 0 can be created and moved NOT across x = 0 as in standard toric code as well. We only need to be a little more careful about what happens when an m particle is moved to x = 0. The situation is showed in Fig. 4.6. Then all eigenvalue combinations of Av and Bp are possible. By counting the dimensions of the Hilbert space, we conclude that there are only topological degeneracy left. So the system is gapped, and the spectrum is labelled by eigenvalue combinations of Av and Bp.

In addition, we obviously have

Lemma 3. The ground state of the Hamiltonian (4.14) corresponds to Av = Bp = 1, which is 77

(a) (b) Figure 4.7: String operators that create a pair of e’s and m’s related by reﬂection. The red string z x stands for a product of σ` on it. The blue string stands for a product of σ` on it. (a) A pair of e’s. (b) A pair of m’s.

invariant under the reﬂection transformation deﬁned above.

4.3.2 Symmetry behaviour of anyons

A pair of e’s related by reflection can be created by string operator shown in Fig. 4.7(a). A pair of m’s related by reflection can be created by string operator shown in Fig. 4.7(b). Obviously, the two operators are both odd under reflection. So this system is an example of eP mP .

4.4 Model of Fractionalization Anomaly: Two Orthogonal Reﬂections

We can choose the coordinate system such that the two reﬂections are Px and Py. The group structure generated by them is G = Z2 ×Z2. Here we are going to give a symmetric Z2 topologically

e e e e m m ordered model realizing symmetry fractionalization patterns with Px Py = −Py Px and Px Py =

m m −Py Px simultaneously. To do this, we need some local degrees of freedom near x = y = 0 carrying

2 a projective representation of G corresponding to non-trivial element of H (Z2 × Z2,U (1)) = Z2. Concretely, we choose the projective representation to be carried by a spin at the vertical link at x = y = 0 as shown in Fig. 4.8.

For the convenience of the model construction, we ﬁrst write down the transformation of the 78 z

Figure 4.8: Generalized toric code model deﬁned on coupled layers of square lattice. Spins are µ associated with the links, whose Pauli operators are generally denoted by σ` , µ = x, y, z. Explicitly, µ µ we use diﬀerent notations for spins on vertical and horizontal links; we write σ` = τv for spins on µ µ vertical links (green online) at v = (vx, vy) ∈ Z × Z, and use σ` = σ(`,k) for horizontal links on the layer at z = k with k = 0, 1. In detail the xy-position ` of a horizontal link ` can be given be the xy-coordinates of its both ends; ` = (`0, `1) with `0, `1 ∈ Z × Z, the xy-coordinates for the ends of ` respectively.

local operators under the two reﬂections  τ x → −τ x ,  v Pxv   Px : τ z → σe τ z , (4.17)  v px Pxv   σµ → σµ ;  (`,0) (Px`,1) and   x x τv → τ ,  Pyv   τ z → −τ z ,  v Pyv Py : (4.18)  σx → σm σx , if P ` = `,  (`,0) py (`,0) y   σµ → σµ , other spin operators;  (`,k) (Py`,k)

e m µ where µ = x, y, z and σpx, σpy = ±1. As shown in Fig. 4.8, τv are Pauli operators for spins associated with vertical links (coloured green), where v = (vx, vy) is the coordinates of either end of the vertical link in the xy-plane. The Pauli matrices for spins on the horizontal links are denoted

µ σ(`,k), where ` labels the position of the link projecting onto the xy-plane and k = 0, 1 labels the µ two layers of square lattices. For convenience, we also use σ` for a generic spin either on a vertical 79 link or on a horizontal link `. In addition, bold v stands for a combination (v, k) ≡ (vx, vy, k), labelling an actual vertex.

It can be checked that Px and Py indeed act projectively on the spin on vertical links x = y =

0, and they act linearly on the other degrees of freedom. Restricted on this spin, the two reﬂections can be represented as

z x e Px = τ(0,0),Py = τ(0,0), if σpx = 1; (4.19)

y x e Px = τ(0,0),Py = τ(0,0), if σpx = −1. (4.20)

2 In both cases, PxPy = −PxPy, which corresponds to non-trivial element of H (Z2 × Z2,U (1)). Now we can write down a toric code type Hamiltonian

X X X H = − A(v,0) − A(v,1) − Bp − B`, (4.21) v∈Z2 p ` where the ﬁrst summation is over all vertices v = (v, k), the second summation is over horizontal plaquettes p on both layers, the third summation is over vertical plaquettes labelled by ` (the position of either constituent horizontal link). The explicit expression for the vertex and plaquette terms are

Y x Av = σ` , (4.22) `3v Y z Bp = σ` , (4.23) `∈p z z z z B` = σ(`,0)σ(`,1)τ`0 τ`1 , (4.24) where the bold letters like v, ` label the actual vertices and links, while ordinary letters v, ` label their position after projecting to the xy-plane, and `0, `1 are the two ends of `. Obviously, all terms in the Hamiltonian commute with one another; it is exactly solvable. The ground state is speciﬁed by

A(v,0) = 1; (4.25)

A(v,1) = −1; (4.26)

Bp = B` = 1. (4.27) 80

It can be checked by direct computation, the Hamiltonian and the corresponding ground state respects the symmetries speciﬁed by Eqns. (4.17, 4.18).

Using the analysis developed in Chapter 3, we can compute

e 2 e (Px ) = σpx; (4.28)

m2 m Py = σpy; (4.29)

e2 m 2 Py = (Px ) = 1. (4.30)

They do not show eP mP type anomaly here. More interestingly, since on the ground state

m m m m A(0,0,0) = −1 and A(0,0,1) = 1, we have Px Py = −Py Px . Meanwhile, to determine the symmetry fractionalization on e, we can start with e(0,0,0) , which is the state with an e at (0, 0, 0), and pick

z Px e(0,0,0) = τ(0,0) e(0,0,0) , (4.31)

Py e(0,0,0) = e(0,0,0) . (4.32)

Then direct computation shows

z PxPy e(0,0,0) = τ(0,0) e(0,0,0) , (4.33)

z z PyPx e(0,0,0) = Py τ(0,0) e(0,0,0) = −τ(0,0) e(0,0,0) , (4.34)

e e e e and hence Px Py = −Py Px . In short, by using a spin at the origin carrying the projective representation of the reﬂections, we have realized all the symmetry fractionalizations with both

e e e e Px Py = −Py Px , (4.35)

m m m m Px Py = −Py Px , (4.36) but without any eP mP anomalies.

Conversely, these symmetry fractionalization patterns necessarily imply the local Hilbert space near the origin must carry a projective representation of the reﬂections corresponding to the

2 non-trivial element of H (Z2 × Z2,U (1)); we can never construct a strictly two-dimensional model 81

e e e e m m m m with spins transforming linearly to realize Px Py = −Py Px and Px Py = −Py Px simultaneously.

The argument will be made by contradiction in the next section.

Further, if we require all spins transform linearly, then the symmetry fractionalization pat-

e e e e m m m m terns with Px Py = −Py Px and Px Py = −Py Px simultaneously can only appear on the surface of non-trivial three-dimensional pgSPT phases. Our discussion of pgSPT phases indicates there is a non-trivial three-dimensional pgSPT phase related to the one-dimensional SPT phase protected by unitary on-site symmetries G = Z2 × Z2, labelled which corresponds to the by the non-trivial

2 element of H (Z2 × Z2,U (1)). Twisted by it, the relevant boundary degrees of freedom lying near x = y = 0 carry a projective representation of G and can be used the desired symmetry fractionalization pattern. The connection between this pgSPT and the surface model is illustrated in

Fig. 4.2.

4.5 All Fractionalization Anomalies Associated with the Square Lattice Symmetries

In Section 4.3, we ﬁrst realized the eP mP fractionalization pattern on the surface of the non- trivial pgSPT phase protected by a single reﬂection. This actually reveals the anomalous nature of the eP mP fractionalization pattern; namely it can never appear in strictly two-dimensional systems.

To better understand this, roughly, we can make an argument as follows. Suppose otherwise we find a strictly two-dimensional quantum spin system realizing the eP mP fractionalization pattern. Then we stick this two-dimensional system with our surface model of eP mP . Before tuning on interactions between them, the two together form a system with 16 anyon types, which are denoted by (a, b) with a labelling anyons in the two-dimensional system and b labelling anyons in the surface model. Now we notice both (e, e) and (m, m) will be trivial under the reflection (i.e. the reflection squares to 1 on both sectors). In addition, they have trivial self and mutual statistics.

So we can imagine a procedure to condense both of them without breaking the reﬂection symmetry.

Then we check that other anyons, like (e, 1), will be conﬁned during this procedure because of their 82 non-trivial mutual statistics with either (e, e) or (m, m). Eventually, we obtained a symmetric short-range entangled state on the surface of a non-trivial pgSPT phase. By the connect between on-site SPT and pgSPT phases, this also means that we have gapped the edge of the non-trivial two-dimensional on-site SPT phase without breaking the symmetry. This contradicts the theory of on-site SPT phases. Therefore, our assumption is false; we can never ﬁnd a strictly two-dimensional quantum spin system realizing the eP mP fractionalization pattern.

Similarly, our model in Section 4.4 reveals that we can neither ﬁnd a strictly two-dimensional

e e e e m m m m quantum spin system realizing Px Py = −Py Px and Px Py = −Py Px simultaneously if the spins transform linearly under the two reﬂections. Interestingly, these two types of anomaly explains all symmetry fractionalization patterns we missed in Chapter 3.

Theorem 2 in Chapter 3 summarize the missed fractionalization patterns. Actually, the fractionalization patterns in C0 do not add more to the list because of the freedom of relabelling e ↔ m, as pointed out by the counting in AppendixD. So we only need to look at P1, P2, P3,

A and B. Obviously, the reason that we missed the fractionalization patterns in P1, P2 and

e,m e,m 4 e,m P3 is the eP mP anomaly. Further, to explaining A, we notice that (Px Pxy ) = σpxpxy implies

e,m e,m e,m e,m e,m e,m e,m e,m e,m Px Py = σpxpxyPy Px , where Py = Pxy Px Pxy . However, among all the models deﬁned in Chapter 3, it can be checked all spins transform linearly under Px and Py. Therefore, the fractionalization patterns in A cannot be realized in those models. Finally, for B, the reasoning is

1 the same by considering the reﬂections about the edges of the center unit cell, namely about x = 2 1 and y = 2 .

There is no obvious way to write an exactly solvable model with Z2 topological order and with spins transform projectively under the D4 symmetries generated by Px and Pxy. Our parton construction at the end of Chapter 3 can overcome this limitation. Chapter 5

Summary and Perspective

5.1 Summary

In this thesis, we study the topological phases of quantum spin systems, with particular attention to the interplay between symmetry and topological order.

In spite of their remarkable theoretical properties, non-trivial topological order are not easy to be realized and measured experimentally in spin systems. Earlier studies showed the analogue of fractional quantum Hall eﬀect in spin system, chiral spin liquids (CSLs), might be found in two- dimensional (d = 2) Mott insulators of ultracold fermionic alkaline earth atoms (AEAs) loaded in a optical lattice, which was described by the anti-ferromagnetic SU(N) Heisenberg model. Thanks to the quenched hyperﬁne structure, the scattering between AEAs shows SU(N) spin rotation symmetry with variable N going up to 10. This enhanced spin symmetry makes CSLs energetically favourable as Mott insulator ground states in two spatial dimensions when N is large. Chapter 2, promoting this investigation to d = 3, was a numerical study of the anti-ferromagnetic SU(N)

Heisenberg model on the cubic lattice, where a rich phase diagram was mapped as N varied. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional CSL state.[4] Recently, this SU(N) symmetry has been conﬁrmed in a cold atom experiment and we hope the study of

SU(N) magnetism will be carried out further experimentally in the future.

The intrinsic topological phases diﬀer from one another due to distinct patterns of quantum entanglement, which are stable even when all symmetries are explicitly broken. Among them, the 84 trivial one is called short-range entangled (SRE) meaning that it can be adiabatically tuned into the tensor product state. The non-trivial topological phases are called topologically ordered or long-range entangled (LRE). Usually in two-dimensional, topologically ordered system has anyonic excitations. Further, if we require the system to respect some symmetries, then the parameter space for the corresponding Hamiltonians is restricted. As a result, one original topological phase may break into several pieces even without the mechanism of spontaneous symmetry breaking. If this happens to the SRE phase, we will get several symmetry protected topological (SPT) phases.

If the original phase is topologically ordered, the resulting pieces are called symmetry enriched topological (SET) phases.

Chapter 3 studies the phenomenon of symmetry fractionalization on anyons in two-dimensional

SET phases with Z2 topological order and crystalline symmetries. If the system have the symmetries of the square lattice, then there 2080 possible symmetry fractionalization patterns. We try our best to construct model to realize as many of them as possible and get 487 in the end.[77]

Later in Chapter 4, we succeed to prove these are all we can find in strictly two-dimensional systems. The other symmetry fractionalization patterns are anomalous, meaning they can only appear in the surface state of some three-dimensional system. In order to achieve this, we obtain two important results. First, we established the connection between SPT phases protected by reflections with SPT phases protected by on-site symmetries in lower dimensions. This connection gives two three-dimensional SPT phases protected a single reflection, eight three-dimensional SPT phases protected two orthogonal reflections, and it can be used to classify point group SPT phases in general. Second, we construct toy models of Z2 topologically ordered surface state carrying anomalous symmetry fractionalization.

5.2 Perspetive

The following questions naturally follows our study of symmetric topological phases.

• Besides Z2 topological order, what are the other possibilities of topologically ordered surface 85

states that respect the crystalline symmetries? To answer this, maybe we need to establish

a general theory for crystalline symmetry fractionalization anomaly.

• Can we prove the eT mT anomaly by studying the boundary degrees of freedom? We have

got an interesting toric code type model (see Appendix E) following this logic.

• Can we establish a parallel story for fermionic systems?

• How to study SET phases in three dimensions?

• Are there any signature measurements to determine SET/SPT phases? Bibliography

[1] M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. Lett. 103, 135301 (2009).

[2] M. Hermele and V. Gurarie, Phys. Rev. B 84, 174441 (2011).

[3] The calculation procedure is to set up special initial data of χrr0 for iteration under the SCM procedure. For those χrr0 6= 0 in a given state, we choose |χrr0 | = 0.1, with the required pattern of fluxes. After finding the lowest energy, we checked whether the flux pattern is preserved. For (2πnx,y,z/k)-flux states, when k ≥ 7 the computation time becomes large, so we restricted the range of flux to 0 ≤ nx ≤ 1, and 0 ≤ ny,z ≤ k − 1. [4] H. Song and M. Hermele, Phys. Rev. B 87, 144423 (2013).

[5] D. Jaksch and P. Zoller, Annals of Physics 315, 52 (2005), special Issue.

[6] M. Lewenstein, A. Sanpera, V. Ahuﬁnger, B. Damski, A. Sen(De), and U. Sen, Advances in Physics 56, 243 (2007).

[7] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).

[8] We do not consider bosonic AEAs, because all stable bosonic atoms with similar electronic structure have zero nuclear spin.

[9] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. M. Rey, Nat. Phys. 6, 289 (2010).

[10] M. A. Cazalilla, A. F. Ho, and M. Ueda, New J. Phys. 11, 103033 (2009).

[11] M. Foss-Feig, M. Hermele, and A. M. Rey, Phys. Rev. A 81, 051603 (2010).

[12] M. Foss-Feig, M. Hermele, V. Gurarie, and A. M. Rey, Phys. Rev. A 82, 053624 (2010).

[13] C. Xu, Phys. Rev. B 81, 144431 (2010).

[14] T. A. T´oth,A. M. L¨auchli, F. Mila, and K. Penc, Phys. Rev. Lett. 105, 265301 (2010).

[15] P. Corboz, A. M. L¨auchli, K. Penc, M. Troyer, and F. Mila, Phys. Rev. Lett. 107, 215301 (2011).

[16] S. R. Manmana, K. R. A. Hazzard, G. Chen, A. E. Feiguin, and A. M. Rey, Phys. Rev. A 84, 043601 (2011). 87

[17] A. Rapp and A. Rosch, Phys. Rev. A 83, 053605 (2011).

[18] G. Szirmai, E. Szirmai, A. Zamora, and M. Lewenstein, Phys. Rev. A 84, 011611 (2011).

[19] B. Bauer, P. Corboz, A. M. L¨auchli, L. Messio, K. Penc, M. Troyer, and F. Mila, Phys. Rev. B 85, 125116 (2012).

[20] L. Bonnes, K. R. A. Hazzard, S. R. Manmana, A. M. Rey, and S. Wessel, Phys. Rev. Lett. 109, 205305 (2012).

[21] L. Messio and F. Mila, Phys. Rev. Lett. 109, 205306 (2012).

[22] A. Tokuno and T. Giamarchi, Phys. Rev. A 86, 053614 (2012).

[23] K. R. A. Hazzard, V. Gurarie, M. Hermele, and A. M. Rey, Phys. Rev. A 85, 041604 (2012).

[24] Z. Cai, H.-h. Hung, L. Wang, D. Zheng, and C. Wu, Phys. Rev. Lett. 110, 220401 (2013).

[25] Z. Cai, H.-H. Hung, L. Wang, and C. Wu, Phys. Rev. B 88, 125108 (2013).

[26] P. Corboz, K. Penc, F. Mila, and A. M. L¨auchli, Phys. Rev. B 86, 041106 (2012).

[27] P. Corboz, M. Lajk´o,A. M. L¨auchli, K. Penc, and F. Mila, Phys. Rev. X 2, 041013 (2012).

[28] T. Fukuhara, Y. Takasu, M. Kumakura, and Y. Takahashi, Phys. Rev. Lett. 98, 030401 (2007).

[29] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76, 051604 (2007).

[30] T. Fukuhara, S. Sugawa, M. Sugimoto, S. Taie, and Y. Takahashi, Phys. Rev. A 79, 041604 (2009).

[31] Y. N. Martinez de Escobar, P. G. Mickelson, M. Yan, B. J. DeSalvo, S. B. Nagel, and T. C. Killian, Phys. Rev. Lett. 103, 200402 (2009).

[32] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck, Phys. Rev. Lett. 103, 200401 (2009).

[33] B. J. DeSalvo, M. Yan, P. G. Mickelson, Y. N. Martinez de Escobar, and T. C. Killian, Phys. Rev. Lett. 105, 030402 (2010).

[34] S. Taie, Y. Takasu, S. Sugawa, R. Yamazaki, T. Tsujimoto, R. Murakami, and Y. Takahashi, Phys. Rev. Lett. 105, 190401 (2010).

[35] M. K. Tey, S. Stellmer, R. Grimm, and F. Schreck, Phys. Rev. A 82, 011608 (2010).

[36] S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A 82, 041602 (2010).

[37] S. Sugawa, K. Inaba, S. Taie, R. Yamazaki, M. Yamashita, and Y. Takahashi, Nat. Phys. 7, 642 (2011).

[38] S. Blatt, T. L. Nicholson, B. J. Bloom, J. R. Williams, J. W. Thomsen, P. S. Julienne, and J. Ye, Phys. Rev. Lett. 107, 073202 (2011). 88

[39] M. Bishof, Y. Lin, M. D. Swallows, A. V. Gorshkov, J. Ye, and A. M. Rey, Phys. Rev. Lett. 106, 250801 (2011).

[40] M. Bishof, M. J. Martin, M. D. Swallows, C. Benko, Y. Lin, G. Qu´em´ener,A. M. Rey, and J. Ye, Phys. Rev. A 84, 052716 (2011).

[41] N. D. Lemke, J. von Stecher, J. A. Sherman, A. M. Rey, C. W. Oates, and A. D. Ludlow, Phys. Rev. Lett. 107, 103902 (2011).

[42] S. Stellmer, R. Grimm, and F. Schreck, Phys. Rev. A 84, 043611 (2011).

[43] S. Stellmer, R. Grimm, and F. Schreck, Phys. Rev. A 87, 013611 (2013).

[44] I. Aﬄeck and J. B. Marston, phys. Rev. B 37, 3774 (1988).

[45] J. B. Marston and I. Aﬄeck, Phys. Rev. B 39, 11538 (1989).

[46] N. Read and S. Sachdev, Nucl. Phys. B 316, 609 (1989).

[47] V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987).

[48] V. Kalmeyer and R. B. Laughlin, Phys. Rev. B 39, 11879 (1989).

[49] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989).

[50] D. V. Khveshchenko and P. B. Wiegmann, Mod. Phys. Lett. B 3, 1383 (1989).

[51] D. V. Khveshchenko and P. B. Wiegmann, Mod. Phys. Lett. B 4, 17 (1990).

[52] H. Yao and S. A. Kivelson, Phys. Rev. Lett. 99, 247203 (2007).

[53] D. F. Schroeter, E. Kapit, R. Thomale, and M. Greiter, Phys. Rev. Lett. 99, 097202 (2007).

[54] M. Greiter and R. Thomale, Phys. Rev. Lett. 102, 207203 (2009).

[55] R. Thomale, E. Kapit, D. F. Schroeter, and M. Greiter, Phys. Rev. B 80, 104406 (2009).

[56] B. Scharfenberger, R. Thomale, and M. Greiter, Phys. Rev. B 84, 140404 (2011).

[57] B. Sutherland, Phys. Rev. B 12, 3795 (1975).

[58] I. Aﬄeck, Nucl. Phys. B 305, 582 (1988).

[59] C. Honerkamp and W. Hofstetter, Phys. Rev. Lett. 92, 170403 (2004).

[60] F. F. Assaad, Phys. Rev. B 71, 075103 (2005).

[61] V. L. Pokrovskii and G. V. Uimin, Sov. Phys. JETP 34, 457 (1972).

[62] Y. Q. Li, M. Ma, D. N. Shi, and F. C. Zhang, Phys. Rev. Lett. 81, 3527 (1998).

[63] M. van den Bossche, F. C. Zhang, and F. Mila, Eur. Phys. J. B 17, 367 (2000).

[64] A. L¨auchli, F. Mila, and K. Penc, Phys. Rev. Lett. 97, 087205 (2006).

[65] D. P. Arovas, Phys. Rev. B 77, 104404 (2008). 89

[66] F. Wang and A. Vishwanath, Phys. Rev. B 80, 064413 (2009).

[67] N. Fukushima, “Vanishing neel ordering of SU(n) heisenberg model in three dimensions,” (2005), arXiv:cond-mat/0502484.

[68] S. Pankov, R. Moessner, and S. L. Sondhi, Phys. Rev. B 76, 104436 (2007).

[69] C. Xu and C. Wu, Phys. Rev. B 77, 134449 (2008).

[70] C. Wu, J.-P. Hu, and S.-C. Zhang, Phys. Rev. Lett. 91, 186402 (2003).

[71] S. Chen, C. Wu, S.-C. Zhang, and Y. Wang, Phys. Rev. B 72, 214428 (2005).

[72] P. Lecheminant, E. Boulat, and P. Azaria, Phys. Rev. Lett. 95, 240402 (2005).

[73] C. Wu, Phys. Rev. Lett. 95, 266404 (2005).

[74] C. Wu, Mod. Phys. Lett. B 20, 1707 (2006).

[75] C. Wu, Physics 3, 92 (2010).

[76] E. Szirmai and M. Lewenstein, EPL (Europhysics Letters) 93, 66005 (2011).

[77] H. Song and M. Hermele, Phys. Rev. B 91, 014405 (2015).

[78] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).

[79] M. Z. Hasan and J. E. Moore, Annual Review of Condensed Matter Physics 2, 55 (2011).

[80] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).

[81] X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011).

[82] L. Fidkowski and A. Kitaev, Phys. Rev. B 83, 075103 (2011).

[83] N. Schuch, D. P´erez-Garc´ıa, and I. Cirac, Phys. Rev. B 84, 165139 (2011).

[84] A. M. Turner, F. Pollmann, and E. Berg, Phys. Rev. B 83, 075102 (2011).

[85] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 87, 155114 (2013).

[86] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).

[87] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

[88] R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, Nature 389, 162 (1997).

[89] L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne, Phys. Rev. Lett. 79, 2526 (1997), arXiv:cond-mat/9706307 .

[90] J. Martin, S. Ilani, B. Verdene, J. Smet, V. Umansky, D. Mahalu, D. Schuh, G. Abstreiter, and A. Yacoby, Science 305, 980 (2004).

[91] X. G. Wen, F. Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989). 90

[92] X. G. Wen, Phys. Rev. B 44, 2664 (1991).

[93] N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991).

[94] S. Sachdev, Phys. Rev. B 45, 12377 (1992).

[95] L. Balents, M. P. A. Fisher, and C. Nayak, Phys. Rev. B 60, 1654 (1999), arXiv:cond- mat/9811236 .

[96] T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850 (2000), arXiv:cond-mat/9910224 .

[97] R. Moessner, S. L. Sondhi, and E. Fradkin, Phys. Rev. B 65, 024504 (2001), arXiv:cond- mat/0103396 .

[98] X.-G. Wen, Phys. Rev. B 65, 165113 (2002), arXiv:cond-mat/0107071 .

[99] X.-G. Wen, Phys. Rev. D 68, 065003 (2003).

[100] A. Kitaev, Annals of Physics 321, 2 (2006).

[101] F. Wang and A. Vishwanath, Phys. Rev. B 74, 174423 (2006).

[102] S.-P. Kou and X.-G. Wen, Phys. Rev. B 80, 224406 (2009).

[103] Y. Huh, M. Punk, and S. Sachdev, Phys. Rev. B 84, 094419 (2011).

[104] G. Y. Cho, Y.-M. Lu, and J. E. Moore, Phys. Rev. B 86, 125101 (2012).

[105] G. Chen, A. Essin, and M. Hermele, Phys. Rev. B 85, 094418 (2012).

[106] M. Levin and A. Stern, Phys. Rev. B 86, 115131 (2012).

[107] A. M. Essin and M. Hermele, Phys. Rev. B 87, 104406 (2013), arXiv:1212.0593 .

[108] A. Mesaros and Y. Ran, Phys. Rev. B 87, 155115 (2013).

[109] L.-Y. Hung and X.-G. Wen, Phys. Rev. B 87, 165107 (2013).

[110] L.-Y. Hung and Y. Wan, Phys. Rev. B 87, 195103 (2013).

[111] Y.-M. Lu and A. Vishwanath, ArXiv e-prints (2013), arXiv:1302.2634 [cond-mat.str-el] .

[112] C. Xu, Phys. Rev. B 88, 205137 (2013).

[113] C. Wang and T. Senthil, Phys. Rev. B 87, 235122 (2013).

[114] A. M. Essin and M. Hermele, Phys. Rev. B 90, 121102 (2014).

[115] X. Chen, F. J. Burnell, A. Vishwanath, and L. Fidkowski, Phys. Rev. X 5, 041013 (2015).

[116] Y. Gu, L.-Y. Hung, and Y. Wan, Phys. Rev. B 90, 245125 (2014).

[117] Y.-M. Lu, G. Y. Cho, and A. Vishwanath, ArXiv e-prints (2014), arXiv:1403.0575 [cond- mat.str-el] .

[118] C.-Y. Huang, X. Chen, and F. Pollmann, Phys. Rev. B 90, 045142 (2014). 91

[119] M. Hermele, Phys. Rev. B 90, 184418 (2014). [120] J. Reuther, S.-P. Lee, and J. Alicea, Phys. Rev. B 90, 174417 (2014). [121] T. Neupert, C. Chamon, C. Mudry, and R. Thomale, Phys. Rev. B 90, 205101 (2014). [122] A. M. Turner, Y. Zhang, R. S. K. Mong, and A. Vishwanath, Phys. Rev. B 85, 165120 (2012). [123] A. M. Turner, Y. Zhang, and A. Vishwanath, Phys. Rev. B 82, 241102 (2010). [124] L. Fu, Phys. Rev. Lett. 106, 106802 (2011). [125] T. L. Hughes, E. Prodan, and B. A. Bernevig, Phys. Rev. B 83, 245132 (2011). [126] J. C. Y. Teo and T. L. Hughes, Phys. Rev. Lett. 111, 047006 (2013). [127] C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 86, 115112 (2012). [128] Z. Wang, X.-L. Qi, and S.-C. Zhang, Phys. Rev. B 85, 165126 (2012). [129] C.-K. Chiu, H. Yao, and S. Ryu, Phys. Rev. B 88, 075142 (2013). [130] F. Zhang, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 111, 056403 (2013). [131] R.-J. Slager, A. Mesaros, V. Juriˇcić, and J. Zaanen, Nature Physics 9, 98 (2013). [132] L. Fidkowski, N. H. Lindner, and A. Kitaev, to appear. [133] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, ArXiv e-prints (2014), arXiv:1410.4540 . [134] A. Vishwanath and T. Senthil, Phys. Rev. X 3, 011016 (2013). [135] M. A. Metlitski, C. L. Kane, and M. P. A. Fisher, Phys. Rev. B 88, 035131 (2013). [136] A. Yu. Kitaev, Ann. Phys. 303, 2 (2003), arXiv:quant-ph/9707021 . [137] Y. Qi and L. Fu, ArXiv e-prints (2015), arXiv:1505.06201 [cond-mat.str-el] . [138] M. Hermele and X. Chen, ArXiv e-prints (2015), arXiv:1508.00573 [cond-mat.str-el] . [139] R. Moessner and J. T. Chalker, Phys. Rev. Lett. 80, 2929 (1998). [140] For k = 5, 6, where we find a continuous ground state degeneracy within each cluster, these statements hold for a particular choice of ground state in each cluster. [141] D. S. Rokhsar, Phys. Rev. B 42, 2526 (1990). [142] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge Univer- sity Press, 1992). [143] We also considered some more general parallelepiped unit cells, but less systematically than the rectangular prism case. [144] X. Chen, Z.-X. Liu, and X.-G. Wen, Phys. Rev. B 84, 235141 (2011). [145] M. Levin and Z.-C. Gu, Phys. Rev. B 86, 115109 (2012) Appendix A

Complete set of commuting observables1

e e We show here that the operators {Av|v ∈ V }, {Bp|p ∈ P }, Lsx , Lsy , as deﬁned in Sec. 3.4, form a complete set of commuting observables for any model in the family TC(G). The approach is to construct a basis that is completely labeled by the simultaneous eigenvalues of these operators.

We recall that plaquettes P together with sx, sy, form an elementary set of cycles, so that for

e e any c ∈ C, Lc can be decomposed into a product of Lp’s, with the product possibly also including

e e Lsx and/or Lsy . However, the plaquettes are in general not independent, in the sense that there may be non-trivial relations of the form Bp1 ··· Bpn = 1, for some p1, . . . , pn ∈ P . For the present purpose, it will be convenient to construct an elementary and independent set of cycles.

Let T be a spanning tree of the graph G. By deﬁnition, T is a subgraph of G containing all vertices of G (T spans G), so that T is connected and has no cycles (T is a tree). Any tree with n vertices has n − 1 edges, so T has |V | − 1 edges. We denote the edge set of T by ET , and let

0 0 E = E − ET . For every ` ∈ E , there is a unique cycle c(`) ∈ C containing only ` and edges in

0 ET . We claim {c(`)|` ∈ E } is an elementary, independent set of cycles.

To show the c(`) cycles are elementary, suppose c is a cycle. Without loss of generality, we

0 assume c has no repeated edges. Viewing c as a subset of E, let c ∩ E = {`1, . . . , `n}. Then we claim the desired result, namely

Le = Le ···Le . (A.1) c c(`1) c(`n)

e e e Q z 0 To show this, consider the product L L ···L = 0 σ . c lies entirely in E , and must c c(`1) c(`n) `∈c ` T be empty or a union of disjoint cycles. These two facts are only consistent if c0 is empty, and so

LeLe ···Le = 1, equivalent to Eq. (A.1). c c(`1) c(`n) e The c(`) cycles are also independent: we can choose the eigenvalues of Lc(`) independently for

0 z all ` ∈ E . To see this, consider a reference state |{Φ`}i, deﬁned as the eigenstate of σ` satisfying   |{Φ }i, ` ∈ E z  ` T σ` |{Φ`}i = (A.2) 0  Φ`|{Φ`}i, ` ∈ E ,

|E|−|V |+1 where Φ` ∈ {±1}. There are clearly 2 such reference states, which form an orthonormal set, because E0 contains |E| − |V | + 1 edges. Also, we clearly have

e Lc(`)|{Φ`}i = Φ`|{Φ`}i. (A.3)

e From the above discussion, it is clear that for every set of Lc(`) eigenvalues {Φ`} there is a

e e corresponding distinct consistent choice of {Bp}, Lsx and Lsy eigenvalues, and vice versa. For the purpose of constructing a complete set of commuting observables, we can therefore replace {Bp},

e e e 0 Lsx and Lsy by {Lc(`)|` ∈ E }. We will complete the discussion by exhibiting an orthonormal basis, where the basis states

e are simultaneous eigenvalues of {Av} and {Lc(`)}. We construct the basis states starting from the Q reference states |{Φ`}i. Let av ∈ {±1}, subject to the constraint v av = 1, then we consider the state 1 Y 1 |{av}, {Φ`}i = √ √ (1 + avAv)|{Φ`}i. (A.4) 2 2 v∈V These states are normalized, and satisfy

Av|{av}, {Φ`}i = av|{av}, {Φ`}i (A.5)

e Lc(`)|{av}, {Φ`}i = Φ`|{av}, {Φ`}i, (A.6)

|V |−1 thus forming an orthonormal set. Moreover, since there are 2 possible choices of {av}, the

|V |−1 |E|−|V |+1 |E| number of states |{av}, {Φ`}i is 2 · 2 = 2 . This is the dimension of the Hilbert

e space, so we exhibited a basis completely labeled by the eigenvalues of {Av} and Lc(`). Appendix B

Symmetry-invariant ground states1

For an even by even lattice (i.e. L even), it is always possible to choose Ug and ﬁnd a ground state |ψ0ei satisfying

Ug|ψ0ei = |ψ0ei (B.1)

e e Lsx |ψ0ei = Lsy |ψ0ei = |ψ0ei, (B.2)

where sx and sy are closed paths that wind around the system once in the x and y directions,

respectively. From this it also follows that Ug1 Ug2 = Ug1g2 ; this equation holds acting on |ψ0ei, so the linear action of symmetry on local operators [Eq. (3.25)] implies it holds on all states.

In fact, it is also possible to ﬁnd a ground state |ψ0mi satisfying similar properties but for m-string operators:

Ug|ψ0mi = |ψ0mi (B.3)

m m Ltx |ψ0mi = Lty |ψ0mi = |ψ0mi. (B.4)

Here, tx and ty are closed cuts winding once around the system in x and y directions, respectively.

e m Because, for instance, Lsx and Lty must anti-commute, |ψ0ei and |ψ0mi cannot be the same state.

We now show the existence of |ψ0ei; the argument for |ψ0mi is essentially identical, apart from one subtlety that we address at the end of this Appendix. We deﬁne sx by ﬁrst drawing a

0 0 0 path sx joining the an arbitrary v ∈ V to Txv. The path sx is then formed by joining sx, Txsx,

2 0 Tx sx, and so on, to obtain

0 0 L−1 0 sx = (sx)(Txsx) ··· (Tx sx), (B.5) a closed path winding once around the system in the x-direction. We then choose sy = Pxysx.

With these paths speciﬁed, we specify a unique state in the four-dimensional ground state manifold by requiring

e e Lsx |ψ0i = Lsy |ψ0i = |ψ0i. (B.6)

By symmetry, Ug|ψ0i must also lie in the ground state manifold for all g ∈ G. We will show that

e Lsµ Ug|ψ0i = Ug|ψ0i, (B.7)

iφg iφg for µ = x, y, which implies Ug|ψ0i = e |ψ0i, for some phase factors e . It is enough to show this for the generators g = Tx,Px,Pxy. Once this is established, we can make trivial phase redeﬁnitions

−iφTx UTx → e UTx , and similarly for the other generators, thus setting φg = 0 to obtain the desired result.

Before proceeding to show Eq. (B.7) for each generator in turn, we obtain an equivalent simpler condition. We have

e −1 e z e L U |ψ i = U U L U |ψ i = c −1 (s )U L −1 |ψ i, (B.8) sµ g 0 g g sµ g 0 g µ g g sµ 0

L/2 for µ = x, y. Now, it is clear we can break sµ into two paths, sµ = sµ1sµ2, so that sµ2 = Tµ sµ1.

Then we have

z z z L/2 cg−1 (sµ) = cg−1 (sµ1)cg−1 (Tµ sµ1). (B.9)

z L/2 z Using Lemma 11 of Appendix C.1.2, cg−1 (Tµ sµ1) = cg−1 (sµ1), so that

z cg−1 (sµ) = 1. (B.10)

Therefore we have shown

e e L U |ψ i = U L −1 |ψ i. (B.11) sµ g 0 g g sµ 0 96

L T-1s sy x y 2 c¢

L 2

e −1 Figure B.1: Graphical argument that Lc|ψ0i = |ψ0i, for c = sy ∪ Tx sy. The dotted lines show the −1 L × L grid of primitive cells, and the paths sy and Tx sy are shown. c encloses a region of area L, which can be broken (dashed lines) into L smaller sub-regions each of unit area. Let c0 be the cycle e QL−1 e bounding one of the sub-regions, then L = L n 0 . In addition, by translation symmetry c n=0 Tx c e e L 0 |ψ i = L 0 |ψ i = ±|ψ i. Since an even number of sub-regions appear in the decomposition of c 0 Tyc 0 0 e e Lc given above, we have Lc|ψ0i = |ψ0i.

This implies Eq. (B.7) will hold if, for each generator g,

e L −1 |ψ i = |ψ i. (B.12) g sµ 0 0

−1 Now we consider g = Tx. Since Tx sx = sx, Eq. (B.12) is satisifed for µ = x. For µ = y, we have

e e e e L −1 |ψ0i = L L |ψ0i = L |ψ0i = |ψ0i, (B.13) Tx sy c sy c

−1 where c = sy ∪Tx sy, and the last equality follows from a graphical argument in Fig. B.1. Here and in the following, for the union ∪ operation to make sense, we can view paths as multisets of edges.

e z And the meaning of Lc is obvious; it is a product of σl with multiplicities taken into account.

Next we consider g = Pxy, and Eq. (B.12) becomes

e LPxysµ |ψ0i = |ψ0i. (B.14)

2 This clearly holds, because Pxysy = Pxysx = sx, and Pxysx = sy. 97

Finally, we consider g = Px. For µ = x, we have

e e e e LPxsx |ψ0i = LcLsx |ψ0i = Lc|ψ0i = |ψ0i, (B.15)

where c = sx ∪ Pxsx, and the last equality follows from an argument we now provide. We ﬁrst cut

L/2 sx into two equal-length pieces sx1 and T sx1, which meet at a vertex v. We then have

L/2 −L/2 Pxsx = (Pxsx1)(PxTx sx1) = (Pxsx1)(Tx Pxsx1)

L/2 = (Pxsx1)(Tx Pxsx1), (B.16)

L L/2 where the last equality holds since Tx = 1. We have thus decomposed c = (sx1)(Tx sx1) ∪

L/2 0 e e e (Pxsx1)(Tx Pxsx1). Now we draw a path s joining v to Pxv, and we decompose Lc = Lc1 Lc2 , introducing the cycles

0 L/2 L/2 0 c1 = (sx1)(s )(Tx Pxsx1)(Tx s ) (B.17)

L/2 L/2 0 0 c2 = (Tx sx1)(Tx s )(Pxsx1)(s ). (B.18)

For g = Px and µ = y, we have

e e e e LPxsy |ψ0i = LcLsy |ψ0i = Lc|ψ0i = |ψ0i, (B.19)

where c = sy ∪ Pxsy, and the last equality follows from an argument given below, which is similar to that already given in the case µ = x. We ﬁrst break sy into two equal-length paths related by

L/2 Ty translation, that is

L/2 sy = (sy1)(Ty sy1), (B.20)

L/2 and let v be a vertex where sy1 and Ty sy1 meet. Since Px and Ty commute, we have

L/2 Pxsy = (Pxsy1)(Ty Pxsy1). (B.21)

We can then proceed following the discussion for g = Px, µ = x to obtain the desired result. 98

P v x

c1 c2 v

e Figure B.2: Graphical illustration of the argument that Lc|ψ0i = |ψ0i, for c = sx ∪ Pxsx. It is important to note that, in the interest of clarity, this figure is schematic in the sense that it accurately shows the connectivity of the paths involved, and their properties under translation symmetry, but not their properties under Px. The various symbols are defined in the main text. The vertical dashed line is the Px reflection axis, and the vertex v has been chosen to lie near this axis for convenience. c1 and c2 are the boundaries of the left and right shaded regions, respectively. L/2 The most important point is that these two regions are related by Tx translation.

The argument for the existence of |ψ0mi is essentially identical. However, there is one subtlety that should be addressed. In establishing symmetry-invariance of |ψ0ei, we had to choose the phase of Ug appropriately. The same step arises in the corresponding discussion for |ψ0mi, and the two phase choices may not be compatible. Fortunately, this is not an issue for our purposes, because we never need to work with |ψ0ei and |ψ0mi at the same time. We simply make (possibly) diﬀerent phase choices for Ug depending on the ground state we are working with in a given calculation. Appendix C

General construction of e and m localizations in toric code models1

e Here, we show by explicit construction that an e-localization Ug (v) always exists for the toric code models, and also that this e-localization is unique up to projective transformations

e e Ug (v) → λ(g)Ug (v), with λ(g) ∈ {±1}. The explicit form for the e-localization we obtain is useful for obtaining general constraints on symmetry classes in Appendix C.1. The corresponding results and explicit form also hold for m-localizations. We focus ﬁrst on e particles and e-localizations, postponing discussion of m particles to the end of this Appendix.

We ﬁx g ∈ G, and arbitrarily single out a vertex v0. v0 may depend on g, but we do not

e e e e write this explicitly. We then choose U (v0) = f (v0)L e , where f (v0) ∈ {±1} is arbitrary, and g g sg(v0) g e sg(v0) is arbitrary so long as it joins v0 to gv0. In addition, for each v 6= v0, we choose a path sv joining v0 to v. We will now show that

e e e e Ug (v) = Lsv Ug (v0)g(Lsv ) (C.1)

−1 gives an e-localization. Here, we have introduced the notation g(O) = UgOUg for any operator

e e e e O. It is clear that U (v) can be put into the form U (v) = f (v)L e . g g g sg(v) To proceed, we need to show that

e e Ug|ψe(s)i = Ug [v1(s)]Ug [v2(s)]|ψe(s)i (C.2)

for all open paths s. The endpoints of s are denoted v1(s), v2(s). We ﬁrst show that Eq. (C.2)

1 This appendix has been published as a portion of Hao Song and M. Hermele, Phys. Rev. B 91, 014405 (2015),[77] copyright 2015 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society. 100 holds for all pairs of e particle positions (i.e. all pairs of endpoints v1(s), v2(s)), using a speciﬁc choice of paths. Then we proceed to show Eq (C.2) it holds for any open path s.

If s = sv, the endpoints of s are v0 and v, and an easy calculation shows Eq. (C.2) holds.

0 0 0 Now we consider vertices v, v 6= v0 and v 6= v , which are joined by the path svv0 = svsv, and we choose s = svv0 . We have

e e e |ψ (s 0 )i = L |ψ i = L L |ψ i. (C.3) e vv svv0 0 sv sv0 0

Then, for the left-hand side of Eq. (C.2),

e e e U |ψ (s 0 )i = g(L )|ψ i = g(L )g(L )|ψ i. (C.4) g e vv svv0 0 sv sv0 0

The right-hand side of Eq. (C.2) can easily be veriﬁed after observing that

e e e e e e Ug (sv)Ug (sv0 ) = Ug (sv)Ug (v0)Ug (sv0 )Ug (v0), (C.5)

e 2 since [Ug (v0)] = 1.

e 0 0 Now, consider |ψ (s)i, where s has endpoints v, v , with v, v 6= v0. We have

e e |ψ (s)i = cL |ψ (s 0 )i = |ψ (s 0 )i, (C.6) ssvv0 e vv e vv where we used the fact that |ψe(svv0 )i is an eigenstate of any closed e-string operator, and where

e c = ±1 is the eigenvalue of L acting on |ψ (s 0 )i. The corresponding result holds when s has ssvv0 e vv endpoints v0, v. Therefore, Eq. (C.2) holds independent of the choice of s.

e To consider uniqueness of the symmetry localization, it is convenient to use the form Ug (v) =

e e e f (v)L e . The endpoints of s (v) are fixed, but the path is otherwise arbitrary. However, we are g sg(v) g e always free to deform the paths sg(v) to some fixed set of reference paths, since this only affects the

e e e overall phase factor fg (v). Therefore it is enough to consider the redeﬁnition Ug (v) → λ(g, v)Ug (v).

We now show that Eq. (C.2) requires λ(g, v) = λ(g), independent of v, which is precisely the general form of projective transformations. Suppose there exist vertices v1, v2 with λ(g, v1) 6= λ(g, v2). Such a transformation changes the right-hand side of Eq. (C.2) by a minus sign for a state with e particles at v1 and v2, and is not consistent. Therefore the most general redeﬁnition of the e-localization is

e e the projective transformation Ug (v) → λ(g)Ug (v). 101

The obvious parallel discussion establishes the corresponding results for m-localizations. The corresponding explicit form for the m-localization is

U m(h) = LmU m(h )g(Lm). (C.7) g th g 0 th

This is obtained following the above discussion upon replacing vertices by holes (v0 → h0, v → h), and paths by cuts (sv → th).

C.1 General constraints on symmetry classes in toric code models

C.1.1 Toric codes without spin-orbital coupling

Here, we consider models in the family TC0(G), and prove Theorem 1 stated in Section 3.6.2.

We first introduce some additional notation to be used below. Recalling that av is the Q ground state eigenvalue of Av, we define aX = v∈X av for any finite subset X ⊂ V . If t is a simple closed cut (see Sec. 3.4 for a definition), we define Vt = {v ∈ V |P (v) is enclosed by t}. In addition, we define Γ (g1, . . . , gn) to be the set of vertices that are fixed under each of g1, . . . , gn.

That is, Γ(g1, . . . , gn) = {v ∈ V |giv = v, i = 1, ··· , n}. To shorten various expressions, we write

−1 g (O) = UgOUg for the transformation of any local operator O under g ∈ G, and deﬁne R = PxPxy

(π/2 counterclockwise rotation) and Py = PxyPxPxy (reﬂection y → −y).

In calculations below, we will use the e and m symmetry localizations given in Eqs. (C.1)

m m and (C.7), and discussed in Appendix C. In addition, we will often write equations like σpxpxy = Lt .

Such equations hold when acting on |ψ0mi, or more generally on states created by acting on |ψ0mi with m-string operators, and should be interpreted in this way.

m −1 Lemma 4. For any model in TC0(G), we have σpxpxy = aP (o) = aΓ(Px,Pxy), where o = (0, 0).

Proof. As shown in Fig. C.1, consider an m particle located at an arbitrary hole h0 and let hj =

j j R h0, j = 1, 2, 3. Then draw a cut t ∈ W¯ connecting h0 and h1. Let tj = R t0, j = 1, 2, 3 and

e m t = t0t1t2t3. The cuts are chosen so that t is simple. Then we choose UR (h0) = Lt0 , and using the 102

h1 t0

t1 h0 o

h2 t3 t2 h3

m j Figure C.1: The calculation of σpxpxy. Put an m particle at point h0, let hj = R (h0) , j = 1, 2, 3, let t ∈ W¯ connecting h to h and t = Rjt . Then we have U m U m (h ) = f mLm with 0 0 1 j 0 Px Pxy 0 0 t0 f m ∈ {±1} and U m U m (h ) = Lm U m U m (h ) R Lm for j = 1, 2, 3. With some 0 Px Pxy j t0···tj−1 Px Pxy 0 t0···tj−1 4 calculation, U m U m (h ) = Lm with t = t t t t . Thus, σe = a . If (v) is enclosed by Px Pxy 0 t 0 1 2 3 pxpxy Vt P t with R2v 6= v, then v, Rv, R2v, R3v are four diﬀerent vertices enclosed by t such as the grey Q3 e vertices shown here. Since i=0 aRiv = 1, we have σpxpxy = aVt = aP−1(o) = aΓ(R2). The above statements are also true in the cases with spin-orbital coupling using the gauge choice described in Appendix C.1.2. In the case without spin-orbital coupling, since av = aPxv = aPxyv, we have m σpxpxy = aΓ(Px,Pxy).

t3 h2 h3

h4 h5

t1 o t2

h0 h1 t0

m Figure C.2: Illustration of the calculation of σtypx in Lemmas 7 and 16. Solid squares denote the locations of holes h ∈ H, which are chosen so that h0 is arbitrary (but near the y-axis), and h1 = Pxh0, h2 = Tyh0, h3 = Tyh1, h4 = Pyh0, h5 = Pyh1. h4 and h5 are not used in Lemma 7. Cuts are represented by solid lines, and h]0h1 denotes, for example, a cut joining h0 to h1. The cuts t0 = h]0h1, t1 = h]0h2, t2 = h]1h3, and t3 = h]2h3 are labeled, and are chosen to have properties 1 described in the text. The points o = (0, 0) and κ = (0, 2 ) are shown. 103 results of Appendix C,

m m m m UR (h1) = Lt0 Lt0 R Lt0 , (C.8)

m m m m UR (h2) = Lt0t1 Lt0 R Lt0t1 , (C.9)

m m m m UR (h3) = Lt0t1t2 Lt0 R Lt0t1t2 , (C.10)

m 4 m m m m m (UR ) (h0) = Lt0 Lt2 R Lt0 Lt2 = Lt . (C.11)

m 2 2 3 Therefore, σpxpxy = aVt . For v ∈ Vt, if P (v) 6= o or R v 6= v, then v, Rv, R v, R v are four Q3 diﬀerent vertices in Vt, with av = aRv = ··· . Then i=0 aRiv = 1, and these vertices do not

m contribute to aVt . We have thus shown σpxpxy = aVt = aP−1(o) = aΓ(R2), part of the desired result.

2 2 2 For v ∈ Γ R , we have Pxv, Pxyv, Rv ∈ Γ R since R commutes with Px, Pxy. Let Go denote the subgroup generated by Px, Pxy, which is the same as the subgroup ﬁxing the origin o.

2 Let Gov be the orbit of v under Go and Gv = {g ∈ G|gv = v}. Then Gov ⊆ Γ R and Gv is a

2 subgroup of Go. Because |Go| = 8 and R v = v, we have |Gov| = |Go/Gv| = 1, 2, 4. Now with

the assumption that there is no spin-orbital coupling, then av = aPxv = aPxyv = aRv and hence

m 2 aGov = 1 unless |Gov| = 1. Therefore, σpxpxy = aΓ(R ) = aΓ(Px,Pxy).

4 a a a a Lemma 5. Let Pfx = TxPx and σ = U UP for a = e, m. Then we have σ = pxpxyf Pfx xy pxpxyf a a σpxpxyσtxty, for a = e, m.

4 a a a a a a a 4 Proof. Since U = ±UT UP , we have U UP = UT UP UPxy . It is then straightforward Pfx x x Pfx xy x x to bring the U a operators to the left side of this product, using the relations of Eqs. (3.39-3.44), Tx and the result follows.

Remark. This lemma is valid even with spin-orbital coupling allowed.

m m Lemma 6. For any model in TC (G), we have σ σ = a −1 = a , where o = 0 pxpxy txty P (oe) Γ(Pxy,TxPx) e 1 1 2 , 2 .

1 1 m Proof. Repeat the proof to Lemma 4, replacing Px → Pfx = TxPx, o → oe = 2 , 2 , and σpxpxy → m m σ , obtaining the result σ = a −1(o) = a . The desired result then follows imme- pxpxyf pxpxyf P e Γ(Pfx,Pxy) diately from Lemma 5. 104

m m Lemma 7. For any model in TC0(G), we have σpxpxyσtypx = a −1 1 = aΓ(Px,TyPy). P ((0, 2 ))

Proof. As shown in Fig. C.2, pick a hole h0 near the y-axis, let h1 = Pxh0, h2 = Tyh0, h3 = Tyh1.

Draw a cut t0 connecting h0, h1 and a cut t1 joining h0 and h2. Let t2 = Pxt1, t3 = Tyt0 and t = t0t1t3t2. The cuts, some of which may contain no edges, are chosen so that t is simple. We choose h0 and t such that all vertices enclosed by t are located on the y-axis and no vertex located

1 above 0, 2 is enclosed by t. We choose U m (h ) = Lm, U m (h ) = Lm, then following Appendix C we can choose Px 0 t0 Ty 0 t1

m m m m UTy (h1) = Lt0 Lt1 Ty(Lt0 ),

m m m m UPx (h2) = Lt1 Lt0 Px(Lt1 ). −1 These results can be used to evaluate the product U m U m U m U m −1 acting on a m particle Ty Px Ty Px initially located at h3. We obtain

m m m m m σtypx = Lt0 Lt1 Ty Lt0 Px Lt1 . (C.12)

So far we have not assumed the absence of spin-orbit coupling. Now making this assumption,

m m m m m m we have Px Lt1 = Lt2 and Ty Lt0 = Lt3 . Thus, σtypx = Lt = aVt . If v ∈ Vt and P (v) 6= 1 o, 0, 2 , then v, Pyv are two diﬀerent vertices in Vt by construction. Since av = aPyv, their product

m m does not contribute to aVt . So σtypx = aP−1(o)a −1 1 . Lemma 4 says σpxpxy = aP−1(o). Thus, P ((0, 2 )) m m σpxpxyσtypx = a −1 1 , part of the result to be shown. P ((0, 2 )) 1 −1 Let κ = 0, 2 , Gκ = {g ∈ G|gκ = κ} and Gκv the orbit of v under Gκ. Then Gκv ⊆ P (κ) if P (v) = κ, and Gκ is generated by Px, TyPy. In addition, |Gκ| = 4 and hence |Gκv| = 1, 2, 4.

0 m m 0 Since av = av for v ∈ Gκv, we have aGκv = 1 unless |Gκv| = 1. Thus, σpxpxyσtypx = aP−1(κ) = aΓ(Px,TyPy).

m m m Lemma 8. For any model in TC0(G), we have σpx = σpxy = σtxpx = 1.

Proof. Given a hole h0, let h1 = gh0, where g = Px,Pxy, or TxPx. We can always draw a simple

m m cut joining h0 to h1 so that gt = t. We choose Ug (h0) = Lt , and by Appendix C we can choose

m m m m m Ug (h1) = Lt Lt g (Lt ) = Lt , 105

m m where we used the assumption of no spin-orbit coupling. Then Ug (h1) Ug (h0) = 1. We place a m

m 2 m 2 m m particle at h0, and compute (Ug ) acting on this m particle, ﬁnding (Ug ) (h0) = Ug (h1)Ug (h0) =

m m m 1. Thus, σpx = σpxy = σtxpx = 1.

2 e2 Lemma 9. Suppose g ∈ G such that g = 1 and there is v ∈ V such that gv = v. Then Ug = 1.

e Proof. Because gv = v, we have Ug (v) = 1 or −1. So

e2 e 2 Ug (v) = Ug (v) = 1.

e2 Therefore, Ug = 1.

Remark. This lemma is valid even with spin-orbital coupling allowed.

Theorem 1. The TC symmetry classes in A, B, C, M, M1, M2 and M3 are not realizable in

TC0(G), where

e m A = σpxpxy = σpxpxy = −1 ,

e e m m B = σpxpxyσtxty = σpxpxyσtxty = −1 ,

e e m m C = σpxpxyσtypx = σpxpxyσtypx = −1 ,

m m m M = σpx = −1 ∨ σpxy = −1 ∨ σtxpx = −1 ,

m e e M1 = σpxpxy = −1 ∧ σpx = −1 ∨ σpxy = −1 ,

m m e e M2 = σpxpxyσtxty = −1 ∧ σpxy = −1 ∨ σtxpx = −1 ,

m m e e M3 = σpxpxyσtypx = −1 ∧ σpx = −1 ∨ σtxpx = −1 .

Here ∧, ∨ are the logical symbols for “and” and “or” respectively.

This leaves 95 TC symmetry classes not ruled out by the above constraints, corresponding to

82 symmetry classes under e ↔ m relabeling. In addition, all these 95 TC symmetry classes are realized by models in TC0(G).

Proof. The unrealizability of M is a restatement of Lemma 8. 106

m To prove the unrealizability of A and M1, suppose σpxpxy = −1, then Lemma 4 implies that

e e e there is v ∈ V ﬁxed under Px, Pxy and hence ﬁxed under R. So σpxpxy = σpx = σpxy = 1 by

Lemma 9 and hence the TC symmetry classes in A and M1 are unrealizable. The unrealizability of

B and M2 follows from an almost identical argument, using Lemma 6 and Lemma 9.

m m To prove the unrealizability of C and M3, suppose σpxpxyσtypx = −1. Then Lemma 7 implies there exists v0 ∈ V ﬁxed under Px, TyPy. It follows that Pxyv0 is a vertex ﬁxed under TxPx.

e e Therefore, σpx = σtxpx = 1 by Lemma 9. So the TC symmetry classes in M3 are not realizable in

j TC0(G). Further, for the unrealizability of C, let vj = R v0 for j = 1, 2, 3, pick s0 ∈ W joining v0 to v , and let s = Rjs for j = 1, 2, 3. We choose U e (v ) = 1, U e (v ) = Le and U e (v ) = Le . 1 j 0 Px 0 R 0 s0 Ty 2 s0s1 Using Appendix C, we have U e (v ) = Le for j = 1, 2, 3, and U e (v ) = Le . Thus, R j sj Px 2 s0s1Px(s1s0)

e e 4 e σpxpxy = (UR) (v0) = Ls0s1s2s3 , −1 σe = U e U e U e U e −1 (v ) = Le . typx Ty Px Ty Px 2 s0s1Px(s1s0)

Therefore,

σe σe = Le Le typx pxpxy s0s1Px(s1s0) s0s1s2s3

e e = Ls2Pxs1 Ls3Pxs0

e e = Ls2Pxs1 R Ls2Pxs1 = 1,

where the last equality holds because s2Pxs1 (and hence also s3Pxs0) is a closed path. In short,

m m e e σtypxσpxpxy = −1 implies σtypxσpxpxy = 1. So the TC symmetry classes in C are not realizable in

TC0(G).

The statements about counting and realization of symmetry classes are proved in Sec. 3.6.2.

C.1.2 Toric codes with spin-orbit coupling

We now allow for spin-orbit coupling and consider models in the family TC(G). We prove the following Theorem, which was also stated in Sec. 3.6.3: 107

0 Theorem 2. The TC symmetry classes in P1, P2, P3, A, B and C are not realizable in TC (G), where

e m P1 = σpx = σpx = −1 ,

e m P2 = σpxy = σpxy = −1 ,

e m P3 = σtxpx = σtxpx = −1 ,

e m A = σpxpxy = σpxpxy = −1 ,

e e m m B = σpxpxyσtxty = σpxpxyσtxty = −1 ,

0 e e e e m m C = σpx = σtxpx = σpxpxyσtypx = σpxpxyσtypx = −1 .

The unrealizability in TC(G) of the above TC symmetry classes is proved below in this

Appendix. Appendix D describes the counting of TC symmetry classes not ruled out by the

Theorem, and gives explicit examples of models for these classes.

2 µ µ Lemma 10. If g = 1, then c` (g) = cg` (g) , ∀` ∈ E.

Proof. By Eq. (3.26), if g2 = 1, then for all ` ∈ E

µ µ µ 2 µ c` (g) cg` (g) = c` g = c` (1) = 1.

µ µ Hence c` (g) = cg` (g).

µ As discussed in Sec. 3.4, we can redeﬁne the local axes for each spin such that c` (T ) = 1, for T ∈ G any translation. We always work in such a gauge.

µ Lemma 11. For any translations T,T1,T2 ∈ G, and for all ` ∈ E, g ∈ G, we have cT ` (T1gT2) = x,z cµ (g). 108

Proof. We have

µ µ µ µ cT ` (T1gT2) = c` (T ) cT ` (T1gT2) = c` (T1gT2T )

µ 0 µ µ 0 µ = c` T g = c` (g) cg` T = c` (g).

0 0 Here, we have used the fact that there is a translation T ∈ G such that T1gT2T = T g, which

−1 follows from the fact that translations are a normal subgroup of G (so, in particular, g T1g is a translation).

z,x To proceed, we need to consider further gauge ﬁxing of c` (g) by choosing the local frame of spins. By Lemma 11, it is suﬃcient to restrict to g ∈ Go, where Go = {g ∈ G|go = o} with o = (0, 0) the origin. Let ` be the orbit of some ` ∈ E under translations. By Lemma 11, we can

µ µ µ write c` (g) = c` (g), for all g ∈ Go. Gauge transformations γ` that are constant on translation µ orbits ` do not aﬀect the choice c` (T ) = 1 for translations T . Therefore, it is natural to think µ µ of the allowed gauge transformations as functions of `, and write γ` instead of γ` . The gauge transformation Eq. (3.27) then becomes

µ µ µ µ c` (g) → γ` γg`c` (g). (C.13)

Now, consider some ﬁxed translation orbit `0. Let GoR be the rotation subgroup of Go.

Denote the orbit of `0 under Go by Go`0, and the orbit of `0 under rotations by GoR`0. Then

|GoR`0| = 4, 2, 1 and |Go`0| = 8, 4, 2, 1. We have the following possibilities:

(1) |Go`0| = 8. In this case, elements ` ∈ Go`0 are in one-to-one correspondence with group

elements g ∈ Go. That is, for each ` ∈ Go`0, we can write uniquely ` = g`0 for some

g ∈ Go. We make a gauge transformation by choosing

γµ = γµ = cµ (g). (C.14) ` g`0 `0

Then, in the transformed gauge, cµ (g) → 1 for all g ∈ G , by construction. We now `0 o µ consider c` (g) for arbitrary ` ∈ Go`0, g ∈ Go, in the transformed gauge. We can write 109

` = g1`0 for some unique g1 ∈ Go, and

cµ(g) = cµ (g) = cµ (g)cµ (g ) ` g1`0 g1`0 `0 1

= cµ (gg ) = 1. `0 1

µ Therefore, we are free to choose a gauge where c` (g) = 1. In particular, we have shown µ µ cR2`(g) = c` (g).

(2) |Go`0| = 4 and |GoR`0| = 4. In this case, elements ` ∈ Go`0 = GoR`0 are in one-to-one

correspondence with gR ∈ GoR. Therefore, the same argument given in the previous case

z implies we can choose a gauge so that c`(gR) = 1 for all ` ∈ Go`0 and all gR ∈ GoR. Now,

for arbitrary g ∈ Go, we consider

µ µ µ 2 µ 2 cR2`(g) = cR2`(g)c` (R ) = c` (gR )

µ 2 µ µ 2 µ = c` (R g) = c` (g)cg`(R ) = c` (g).

µ µ Therefore, we have also chosen a gauge in this case where cR2`(g) = c` (g).

2 µ (3) |GoR`0| < 4. In this case, R ` = ` for all ` ∈ L. Therefore it holds trivially that cR2`(g) = µ c` (g).

We have thus shown the following fact, which will be useful in later calculations:

µ µ Lemma 12. It is possible to choose a local spin frame so that cR2` (g) = c` (g), for all ` ∈ E and g ∈ G, with µ = x, z.

To be more concrete, below, we always work in a local spin frame such that ∀µ = x, z,

µ c` (T ) = 1, for any translation T, (C.15)

µ c` (g) = 1, ∀g ∈ Go, if |Go`| = 8, (C.16)

µ c` (g) = 1, ∀g ∈ GoR, if |GoR`| = 4, (C.17) and hence Lemma 12 can be applied. 110

Proposition 1. No TC symmetry classes in P1, P2 or P3 are realizable in TC(G).

Proof. We deﬁne

px E0 = {` ∈ E|Px` = ` with ends of ` ﬁxed} ,

px E1 = {` ∈ E|Px` = ` with ends of ` interchanged} .

px px px Then E − (E0 ∪ E1 ) can be partitioned into pairs {`, Px`}. Let E2 be a set formed by selecting one edge from each such pair. Now, we put a m particle at h0 and draw a cut t ∈ W¯ joining h0 with h = P h such that P t = t. Then we choose U m (h ) = Lm, and by Appendix C we may 1 x 0 x Px 0 t further choose

e m m m m UPx (h1) = Lt Lt Px (Lt ) = Px (Lt ).

Therefore,

m 2 m m m m UPx (h0) = UPx (h1) UPx (h0) = Px (Lt ) Lt

Y x |`∩t| m = [c` (Px)] = σpx. `∈E

Since |Px` ∩ t| = |` ∩ Pxt| = |` ∩ t|, we have

m Y x |`∩t| Y x x |`∩t| σpx = [c` (Px)] c` (Px)cPx`(Px) px px px `∈E0 ∪E1 `∈E2 Y x |`∩t| = [c` (Px)] , px `∈E0 where we used the fact that |` ∩ t| is even for ` ∈ Epx, and also cx(P )cx (P ) = cx(P 2) = cx(1) = 1. 1 ` x Px` x ` x `

m So σpx = −1 implies that there is ` ∈ E such that Px` = ` with its ends ﬁxed, and therefore

e there is a vertex v with Pxv = v. Hence σpx = 1 by Lemma 9.

m e In short, σpx = −1 implies σpx = 1 and hence P1 is not realizable in TC(G). The same arguement applies to P2 and P3.

Lemma 13. In the chosen gauge, for any v ∈ V , avaRvaR2vaR3v = 1 and avaPxvaPyvaR2v = 1. 111

Proof. First we show that avaR2v = aRvaR3v. We have

h Y x ih Y x i R (AvAR2v) = c` (R) c` (R) ARvAR3v `3v `3R2v h Y x ih Y x i = c` (R) cR2` (R) ARvAR3v `3v `3v

= ARvAR3v,

where the last equality follows from Lemma 12. Because R is a symmetry, this implies avaR2v = aRvaR3v, and hence avaRvaR2vaR3v = 1.

Similarly,

h Y x ih Y x i Px (AvAR2v) = c` (Px) c` (Px) APxvAPyv `3v `3R2v h Y x ih Y x i = c` (Px) cR2` (Px) APxvAPyv `3v `3v

= APxvAPyv.

Therefore, avaR2v = aPxvaPyv, and hence avaPxvaPyvaR2v = 1.

m Lemma 14. For any model in TC (G), we have σpxpxy = aP−1(o) = aΓ(R2), where o = (0, 0).

Proof. We repeat the ﬁrst paragraph of the proof to Lemma 4. The last equality in Eq. C.11

e e e e e e is no longer obvious; it still holds because R L L = R L L 2 = L L , by Lemma 12. t0 t2 t0 R t0 t1 t3 Q3 In addition, the argument given in the proof of Lemma 4 that i=0 aRiv = 1 for v ∈ Vt (with

P (v) 6= o or R2v 6= v) is no longer correct. Instead, this fact follows directly from Lemma 13.

Proposition 2. No TC symmetry classes in A are realizable in TC(G).

m 2 Proof. Suppose that σpxpxy = −1. Then Lemma 14 tells us that there exists v such that R v = v.

e e e 2 But then we can choose UR2 (v) = 1, implying σpxpxy = UR2 (v) = 1.

m m Lemma 15. For any model in TC (G), we have σ σ = a −1 = a 2 , where o = pxpxy txty P (oe) Γ(TxTyR ) e 1 1 2 , 2 . 112

Proof. We repeat the proof of Lemma 14, replacing Px → Pfx and using Lemma 5. We also use

2 2 the fact that (PfxPxy) = TxTyR It should be noted that Lemma 12 still holds upon replacing

2 2 R → TxTyR , by Lemma 11.

Proposition 3. No TC symmetry classes in B are realizable in TC (G).

Proof. This follows by the same argument used to prove Prop. 2, using Lemma 15 in place of

Lemma 14.

e e m m −1 2 Lemma 16. For any model in TC (G), if σpx = σtxpx = −1, then σpxpxyσtypx = aP (κ) = aΓ(TyR ),

1 where κ = 0, 2 .

Proof. As shown in Fig. C.2, choose h0 ∈ H near the y-axis, let h1 = Pxh0, h2 = Tyh0, h3 = Tyh1, h4 = Pyh0, h5 = Pyh1. Draw a simple cut t ∈ W¯ joining h0, h1, h5, h3, h2, h4, h0 in turn.

We denote the part of t joining two successive holes by, for example, h]0h1, and that joining three successive holes by, for example, h^0h1h5 = h]0h1h]1h5. We let t0 = h]0h1, t1 = h^0h4h2, t2h^1h5h3 and t3 = h]2h3. We choose h0 and t such that any vertices enclosed by t are located on the y-axis, and no vertex with y-coordinate greater than 1/2 is enclosed by t. Moreover, t is constructed so that

Tyt0 = t3, Pxt1 = t2, Pyh]0h4 = h]0h4, TyPyh]2h4 = h]2h4. Then we have

m m m m m σtypx = Lt0 Lt1 Ty Lt0 Px Lt1 , (C.18) using the same argument leading to Eq. C.12 in the proof of Lemma 7.

m m m m m In our chosen gauge, Ty Lt0 = Lt3 . We now prove Px Lt1 = Lt2 by showing Px L = h]0h4 m m m L and Px L = L . h]1h5 h]4h2 h]5h3 m m First, to show Px L = L , let h]0h4 h]1h5

py E0 = {` ∈ E|Py` = ` with ends of ` ﬁxed} ,

py E1 = {` ∈ E|Py` = ` with ends of ` interchanged} .

py py py Then E − (E0 ∪ E1 ) can be divided into pairs {`, Py`}. Let E2 be a set formed by picking one 113

edge from each such pair. Since Py` ∩ h]0h4 = ` ∩ Pyh]0h4 = ` ∩ h]0h4 , we have

`∩h]0h4 m Y x `∩h]0h4 Y x x L = (σ` ) σ` σPy` . h]0h4 py py py `∈E0 ∪E1 `∈E2

e We notice σpx = −1 implies there is no v such that Pxv = v, by Lemma 9. Hence there is no v

py such that Pyv = v; otherwise Pxyv is ﬁxed under Px. Thus, E0 is empty. In addition, ` ∩ h]0h4 py is even for ` ∈ E1 . Finally,

x x x x c (P ) c (P ) = c (P ) c 2 (P ) ` x Py` x ` x R Py` x

x x x 2 x = c` (Px) cPx` (Px) = c` Px = c` (1) = 1, (C.19) so we have

m m Px L = L . h]0h4 h]1h5

m m Similarly, to show Px L = L , let h]4h2 h]5h3

typy E0 = {` ∈ E|TyPy` = ` with ends of ` ﬁxed} ,

typy E1 = {` ∈ E|Py` = ` with ends of ` interchanged} .

typy typy typy Then E − E0 ∪ E1 can be divided into pairs {`, TyPy`}. Let E2 be a set formed by typy typy typy choosing one edge from each such pair, and let E01 = E0 ∪ E1 . Then,

`∩h]4h2 m Y x `∩h]4h2 Y x x L = (σ` ) σ` σTyPy` . h]4h2 typy typy `∈E01 `∈E2

e We notice σtxpx = −1 implies there is no v such that TxPxv = v, by Lemma 9. Hence there is no

typy v such that TyPyv = v; otherwise Pxyv is ﬁxed under TxPx. Thus, E0 is empty. In addition,

typy l ∩ h]4h2 is even for l ∈ E1 . Finally,

x x x x c` (Px)cTyPy`(Px) = c` (Px)cPy`(Px) = 1, where the last equality was shown in Eq. (C.19). Therefore, we have

m m Px L = L . h]4h2 h]5h3 114

m m m m Therefore, Px Lt1 = Lt2 , and hence σtypx = Lt = aVt . For v ∈ Vt, if P (v) 6= o, κ, then v,

e Pxv, Pyv, PxPyv are four diﬀerent vertices in Vt. This holds because σpx = −1 requires v 6= Pxv and

m Pyv 6= PxPyv. Since avaPxvaPyvaPxPyv = 1 by Lemma 13, we have σtypx = aVt = aP−1(o)aP−1(κ).

m m Hence, using Lemma 14, σpxpxyσtypx = aP−1(κ).

−1 2 Further, if v ∈ P (κ) but TyR v 6= v, then v, Pxv, TyPyv and PxTyPyv are distinct vertices

−1 e e in P (κ); σpx = −1 requires that v 6= Pxv and σtxpx = −1 requires that v 6= TyPyv. Using

Lemma 13, and the fact that aT v = av for any translation T , we have

avaPxvaTyPyvaPxTyPyv = avaPxvaPyvaPxPyv = 1.

−1 2 This implies that only those vertices v ∈ P (κ) satisfying v = TyR v give non-trivial contributions

m m to σpxpxyσtypx, and we have shown

m m −1 2 σpxpxyσtypx = aP (κ) = aΓ(TyR ).

e e m m Lemma 17. If σpx = σtxpx = σpxpxyσtypx = −1, then there exists v ∈ V and s = l1l2 ··· lq ∈ W

0 2 2 connecting v, v = Pxv such that TyR v = v and TyR l = l with ends ﬁxed for each edge l in s.

m m 2 e Proof. By Lemma 16, σpxpxyσtypx = −1 implies Γ TyR is non-empty. In addition, σpx = −1

2 implies that there is no v ∈ V such that Pxv = v. Let J = v ∈ Γ TyR |avaPxv = −1 . Then

0 0 0 0 J = {v1, v1, v2, v2, ··· , vn, vn} with vi = Pxvi for i = 1, 2, ··· , n. Here n must be odd, since

m m n 0 0 2 −1 = σpxpxyσtypx = aΓ(TyR ) = aJ = (−1) . In addition, vi = TyPyvi, because vi = Pxvi =

2 PxTyR vi = TyPyvi.

We consider the graph G0 = (Y,E0), where

2 x E0 = ` ∈ E|TyR ` = ` with ends ﬁxed, c` (Px) = −1 ,

2 Y = Γ TyR .

v Let E0 = {` ∈ E0|` 3 v}. Now we show that in G0, the degree of each vertex v ∈ J is odd, while

v v the degree of v ∈ Y − J is even. That is, |E0 | is odd for v ∈ J and |E0 | is even for v ∈ Y − J. To 115 show this, we consider v ∈ Y and notice the following partition

h 2 i 2 star (v) = ∪j `j,TyR `j ∪ starTyR (v) ,

2 where star (v) = {` ∈ E|` 3 v}, j labels all distinct pairs `j,TyR `j for `j ∈ star(v) with `j 6=

2 2 2 TyR `j, and starTyR (v) = ` ∈ star (v) |TyR ` = ` . Then we have   Y x Px (Av) = Px  σ`  `∈star(v) Y x Y x x = c (P ) c (P ) c 2 (P ) A ` x `j x TyR `j x Pxv `∈star (v) j TyR2   Y =  cx (P ) A  ` x  Pxv `∈star (v) TyR2 v |E0 | = (−1) APxv,

v x x x |E0 | where we used the fact that c 2 (P ) = c 2 (P ) = c (P ). It follows that a a = (−1) . So TyR ` x R ` x ` x v Pxv v v |E0 | is odd for v ∈ J and |E0 | is even for v ∈ Y − J.

Now we claim that there exists v ∈ J and a path s = `1`2 ··· `q in G0 connecting v with Pxv, which is a more detailed version of the result to be shown. We prove this claim by contradiction, and assume there is no v ∈ J such that v and Pxv are in the same connected component of G0. Then,

0 without loss of generality, we relabel pairs vi ↔ vi, so that each component has empty intersection

0 0 with at least one of the sets {v1, . . . , vn} or {v1, . . . , vn}. Since n is odd, there must then be at least one component of G0 containing an odd number of vertices in J. This is a contradiction, since the number of vertices of odd degree is even in any graph.

Proposition 4. No TC symmetry classes in C0 are realizable in TC(G).

e e m m Proof. Assume σpx = σtxpx = σpxpxyσtypx = −1. Lemma 17 tells us that there exists v ∈ V and

0 2 2 sκ ∈ W connecting v, v = Pxv such that TyR v = v and TyR sκ = sκ, where the subscript κ

0 1 indicates that sκ connects vertices with P (v) = P (v ) = κ ≡ 0, 2 . Choose s ∈ W joining Pxv

j 0 j to Rv. Let vj = R v, vj = R Pxv for j = 1, 2, 3. 116

e In order to compute σpxpxy, we follow Appendix C to choose

e e UR (v) = Lsκs,

e e e e UR (v1) = LsκsLsκsR Lsκs , U e (v ) = Le Le R Le , R 2 sκsR(sκs) sκs sκsR(sκs)

e e e e U (v ) = L 2 L R L 2 . R 3 sκsR(sκs)R (sκs) sκs sκsR(sκs)R (sκs)

We have

e e 4 e e σ = (U ) (v) = L 2 R(L 2 ). pxpxy R sκsR (sκs) sκsR (sκs)

e e e e Noticing R L L 2 = L L 3 by Lemma 12, we then have sκs R (sκs) R(sκs) R (sκs)

e e σ = L 2 3 . pxpxy sκsR(sκs)R (sκs)R (sκs)

e To calculate σtypx, we choose

e e e U (v2) = L −1 = L 2 , Px Ty sκ R sκ

U e (v ) = Le . Ty 2 sκsR(sκs)

Following Appendix C, we further choose

e e e e U (v) = L L 2 P L , Px sκsR(sκs) R sκ x sκsR(sκs)

e 0 e e e U v = L 2 L T L 2 Ty 2 R sκ sκsR(sκs) y R sκ

e e e L 2 L L . R sκ sκsR(sκs) sκ

Thus,

−1 e e e e e −1 0 σtypx = UTy UPx UTy UPx v −1 e 0 e e e −1 = UTy v2 UPx (v2) UTy (v2) UPx (v)

e e e e = L L 2 P L L . sκsR(sκs) R sκ x sκsR(sκs) sκ

Finally, we have

e e e e e σ σ = L 2 3 L P L . (C.20) pxpxy typx R (s)R (sκs) sκ x sκsR(sκs) 117

This can be simpliﬁed, ﬁrst noting that U e (v0) = Le U e (v)P (Le ), and therefore Px sκ Px x sκ

e e e 0 e e − 1 = σpx = UPx (v)UPx (v ) = Lsκ Px Lsκ . (C.21)

0 In addition, we have TxPxv3 = v3, so we choose

e e U (v ) = L 3 TxPx 3 R sκ

e e e e U (v ) = L 3 U (v )(T P )(L 3 ) TxPx 3 R sκ TxPx 3 x x R sκ

e e e = L 3 U (v )P (L ). R sκ TxPx 3 x Rsκ

Therefore,

e e e 0 e e − 1 = σ = U (v )U (v ) = L 3 P (L ). (C.22) txpx TxPx 3 TxPx 3 R sκ x Rsκ

Substituting Eqs. (C.21) and C.22) into Eq. (C.20), we have

e e σpxpxyσtypx

z z e e e e = cs (Px) cRs (Px) LR2sLR3sLPxsLPxRs

z z z −1 e −1 e = c (P ) c (P ) c 3 R L 3 R L 3 s x Rs x PxsR s PxsR s PxsR s

z z z −1 = c (P ) c (P ) c 3 R , s x Rs x PxsR s

3 z z z where in the last line we used the fact that PxsR s is a closed path. Since cs1s2 (g) = cs1 (g)cs2 (g), we have

e e z z z −1 σ σ = c (P ) c (P ) c 3 R pxpxy typx s x Rs x PxsR s

z z −1 z z −1 = cs (Px) cPxs(R )cRs (Px) cR3s R .

To simplify this further, we make repeated use of Eq. (3.26). First, we note that

z z −1 z cs (Px) cPxs(R ) = cs(Pxy), and so

e e z z z −1 σpxpxyσtypx = cs(Pxy)cRs (Px) cR3s R . (C.23) 118

Next, cz (P )cz (P ) = cz (P 2) = 1, and so Pxys x Rs x Pxys x

z z cRs(Px) = cPxys(Px).. (C.24)

z z −1 z −1 Moreover, cR2s(R)cR3s(R ) = cR3s(R R) = 1, so

z −1 z cR3s(R ) = cR2s(R). (C.25)

Substituting Eqs. (C.24, C.25) into Eq. (C.23), we have

e e z z z σpxpxyσtypx = cs(Pxy)cPxys(Px)cR2s(R)

z z = cs(R)cR2s(R) = 1, where the last equality follows from Lemma 12.

e e m m e e In short, σpx = σtxpx = σpxpxyσtypx = −1 implies σpxpxyσtypx = 1. Therefore, no TC symmetry classes in C0 are realizable. Appendix D

Models in TC (G)1

To complete the proof of Theorem 2, we need to give explicit models in TC (G) for the TC symmetry classes that are not excluded by the theorem. These models are summarized in Fig. D.2,

Fig. D.3 and Fig. D.4. In some of the models, we use lattices with stacking of vertices and/or edges; that is, there can be distinct edges or vertices with the same image under P. We use single solid lines and points to present edges and vertices that do not stack, while the meaning of other line and point types used is illustrated in Fig. D.1. We use diﬀerent letters l, , ι, ξ, ζ to label edges with diﬀerent direction, as illustrated in Fig. D.1. In particular, l labels vertical edges, and

horizontal edges, with ξ and ζ indicating diagonal edges. The symbol ι is reserved for edges that project to a single point under P.

Following this discussion, it is easy but tedious to verify that all TC symmetry classes not excluded by Theorem 2 are realized by the models in Fig. D.2, Fig. D.3 and Fig. D.4.

Finally, let’s compute the total number of realizable symmetry classes. Let D = P1 ∪P2 ∪P3 ∪

A ∪ B be a subset of unrealizable TC symmetry classes, and let T be the set of all TC symmetry classes. From the form Eq. (3.72) and the deﬁnition of D, it is apparent that

|T − D| = 35 × 4 = 972.

The TC symmetry classes in C0 − C0 ∩ D are of the form     σe σm −1 1  px px         e m     σpxy σpxy             e m     σtxpx σtxpx   −1 1    =   ,      σe σm     pxpxy pxpxy         e e m m     σtxtyσpxpxy σtxtyσpxpxy         e e m m    σpxpxyσtypx σpxpxyσtypx −1 −1

0 0 3 with classes in P2, A, B excluded, so |C − C ∩ D| = 3 = 27. Thus, the total number of TC symmetry classes realizable in TC (G) is |(T − D) − (C0 − C0 ∩ D)| = 972 − 27 = 945.

To count realizable symmetry classes (as opposed to TC symmetry classes), we ﬁrst count the number of symmetry classes obtained from T − D. Under e ↔ m relabeling, every TC symmetry class in T − D either goes into itself, or goes into another TC symmetry class in T − D. It is easy to see that only 2 TC symmetry classes in T − D are invariant under e ↔ m relabeling, so the number

1 of distinct symmetry classes obtained from T − D is 2 (972 − 2) + 2 = 487. Now consider a TC symmetry class in C0 − C0 ∩ D. Under e ↔ m, we obtain a TC symmetry class not contained in C0 − C0 ∩ D, but which is contained in T − D. Therefore the resulting TC symmetry class is realizable. This means that removing C0 − C0 ∩ D from T − D does not reduce the number of symmetry classes, even though the number of TC symmetry classes is reduced. The total number of realizable symmetry classes is thus 487. This completes the proof of Theorem 2. 121

y l3 x (a) (b) (c)

y l3 x (d) (e) (f)

Figure D.1: Depiction of the graphical notation used to represent stacking of vertices and edges. The first row shows the connectivity of vertices and edges, and the second row gives the corresponding two-dimensional presentation. It is convenient to imagine the graph of the lattice as first being embedded in three-dimensional space, and then projected into the two-dimensional plane. When these structures are present, we always assume top edges (blue online) are transformed to bottom edges (red online) under improper space group operations (i.e. reflections), while translations do not swap edges with different colors. Edges parallel to the x-axis, y-axis, z-axis are labeled by symbols , l, ι, respectively. We use ζ and ξ to label diagonal edges. For a diagonal edge, we can associate a unit vectore ˆ running along the direction of the edge, always choosinge ˆx > 0. Then ζ (ξ) is used to label edges withe ˆy > 0 (êy < 0). Panels (a,d). This configuration is only used in Fig. D.2c. The two stacking vertices (blue and red online) together with edge ι1 connecting them are projected into a point, presented as a ring (blue and red online). Edges 2, l3 pass through the ring but do not end on it. The triple-stacking edges are presented as double lines. Panels (b,e). A configuration with double-stacking vertices and no stacking edges. We use a darker point (blue online) to represent the upper vertex, and a lighter ring (red online) to represent the lower vertex. The edges linked to the upper vertex are darker (blue online) and the edges linked to the lower vertex are lighter (red online). Panels (c,f). A situation with double-stacking vertices and edges. The vertices are represented as in (b,e). The lower edge is represented by a lighter double line (red online), and the upper edge is a single darker line (blue online) drawn in the center of the double line. 122

l1 Ε4 Ε1 Ε4 l3 l2 Ζ2 l3 Ζ2 l3 Ε5 l1 Ε5

1 1 1 1 1 γ γ 1 1 1 1 γ 1 δ 1 1 1 γ γ (a) 1 (b) o 3 (c) o 1 3 α1 β2 α3 eo eo˜ α4α5 1 α2 α3 eo eo˜ α1 α1 1 α2 eo eo˜ α4α5eκ

Ε7 Ε1 b l2 l Ζ4 Ζ 1 Ζ3 l5 1 l2 Ε6

1 1 1 γ 1 γ γ δ 1 1 1 γ γ 1 γ 1 1 bγ γ (d) 1 2 (e) o o 2 (f) o o˜ o o˜ α1 β3 α5 1 eo˜ α6α7 1 1 α2 eo eo˜ α1 1 β1 1 eo eo˜ 1

Ε2 l2 Ζ4 Ζ l1 Ζ Ζ1 1 3 l3 Ε5

γ 1 γ 1 1 γ γ γ 1 1 δ 1 γ 1 1 γ γ 1 γ (g) o o˜ o o˜ (h) 2 1 3 (i) 5 1 2 1 β1 1 eo eo˜ eκ 1 β1 α3 1 eo˜ α2 α1 β3 1 1 eo˜ α5

Figure D.2: TC (G) models (Part I). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin o at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (3.72). The edges are labeled by different letters according to their directions as described in the text and in Fig. D.1. Edges that map to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ 1 1 1 1 with the subscript indicating their position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the unit cell is 1 × 1. For short, we define α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ), i εi x i εi xy i εi x i εi xy where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, er = aP−1(r), and b is the eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b is written. The values of er and b are well-defined with respect to any local spin frame system satisfying Eqs. (C.15-C.17). 123

l2 l2 b Ε Ε 3 3 Ζ5 l l1 l4 1 Ζ4 l6 Ε5 Ε7

1 δ 1 γ 1 γ 1 1 1 γ δ γ γ δ γ 1 1 bγ γ (j) o˜ 1 2 (k) 1 5 2 (l) o o o˜ o o˜ α1 1 α4 1 eo˜ α3α5 α1 β4 α6 1 1 α3α7 1 1 1 eo eo˜ 1

Ι Ι Κ o Ε1 l2 Ε1 Ιo ΙΚ b l2 l3

γ δ γ 1 1 γ γ γ δ 1 b 1 γ γ δ 1 1 γ γ (m) o o o˜ o o˜ (n) 1 o˜ 2 (o) o o 2 3 1 1 1 eo eo˜ eκ 1 1 α2 1 eo˜ α1 1 1 α3 eo 1 α1

b Ε Ε1 1 Ζ3 Ζ2 Ζ1 Ζ2 l4

γ 1 γ δ 1 bγ γ γ 1 γ δ 1 γ γ γ 1 1 δ δ γ (p) 1 o˜ 2 1 o˜ (q) κ o˜ 1 κ o˜ (r) 1 2 3 4 1 β2 1 1 eo˜ 1 1 β1 1 1 eo˜ eκ 1 β2 α4 1 1 α1

Figure D.3: TC (G) models (Part II). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin o at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (3.72). The edges are labeled by different letters according to their directions as described in the text and in Fig. D.1. Edges that map to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ 1 1 1 1 with the subscript indicating their position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the unit cell is 1 × 1. For short, we define α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ), i εi x i εi xy i εi x i εi xy where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, er = aP−1(r), and b is the eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b is written. In panel (l), b is the eigenvalue of Bp for the top plaquette. The values of er and b are well-defined with respect to any local spin frame system satisfying Eqs. (C.15-C.17). 124

l2 Ε4 l3 l6 l b b Ε4 Ε7 5 Ζ3 Ξ3 Ε2 Ξ5 l1 l1 Ζ1 Ε2 Ε8 Ε4

1 γ 1 γ γ γ 1 γ γ γ b γ γ 1 γ δ δ bγ γ (s) 5 1 6 3 (t) 3 4 1 5 (u) 2 4 1 3 2 4 α1 1 α6 1 1 α4α8 α1 1 1 1 1 α4 1 β1 1 1 1 1

ΙΚ Ι l1' l1 b1 Κ b Ζ2 Ι Ε Ιo Ι Ζ1 o 2 Κ ΙΚ Ε2' b2

γ 1 γ δ δ γ γ γ δ γ b 1 b γ b γ δ γ 1 b γ γ (v) κ κ˜ 1 2 κ κ˜ (w) o o o 2 1 1 2 (x) o o κ˜ o κ˜ 1 β1 1 1 1 eκ 1 1 1 eo 1 1 1 1 1 eo 1 eκ

ΙΚ b2 Ξ1 b1 ΙΚ

γ δ γ b b γ γ (y) κ 1 κ˜ 1 2 κ κ˜ 1 1 1 1 1 eκ

Figure D.4: TC (G) models (Part III). The shaded square is a unit cell and the TC symmetry classes are calculated with the origin o at the center of the shaded square. Below each lattice is the corresponding TC symmetry class in the form (3.72). The edges are labeled by different letters according to their directions as described in the text and in Fig. D.1. Edges that map to a single point under P are labeled by ιo, ιo˜, ικ, ικ˜ 1 1 1 1 with the subscript indicating their position, ando ˜ = 2 , 2 , κ = 0, 2 ,κ ˜ = 2 , 0 , in units such that the size of the unit cell is 1 × 1. For short, we define α = cx (P ), β = cx (P ), γ = cz (P ) and δ = cz (P ), i εi x i εi xy i εi x i εi xy where ε = l, , ξ, ζ, ι stands for a generic edge. In addition, er = aP−1(r), and b (or bi) is the eigenvalue of Bp for the plaquette (here meaning smallest cycle) p within which b (or bi) is written. In panels (w) and (x), b1 and b are the eigenvalues of Bp for the top plaquettes. The values of er and b (or bi) are well-defined with respect to any local spin frame system satisfying Eqs. (C.15-C.17) except in (w), where a further gauge z z z fixing is needed and we require c (Pxy) = c 0 (Pxy) = c (Px) = 1. l1 l1 l2 Appendix E

A model of eTmT

In this appendix, we construct a quantum spin model with Z2 topological order and time reversal symmetry T showing symmetry fractionalization pattern (T e)2 = (T m)2 = −1, which is also called eT mT for short. In this situation, each e and m excitation carries a Kramers degeneracy.

The geometry of the system is given by the lattice shown in Fig. 4.8, which is obtained by

1 connecting vertices of two square lattices with vertical links. We put a spin- 2 spin on each link `,

Figure E.1: The image of the lattice in Fig. 4.8 projected into the xy-plane. The resulting vertices, links and plaquettes are called xy-vertices, xy-links and xy-plaquettes respectively. Auxiliary dashed lines are added to triangularize the plane. In our eT mT model, vertices are colored diﬀerently ` according to the form of vertex term on them. For each xy-link `, we use T3 to label the black end ` ` ` ` of `, T0 for the other end, and T1 , T2 for the two vertices next to T3 . In addition, t is a cut of links connecting xy-plaquettes p and p0. 126

µ whose Pauli matrices are generically denoted by σ` , µ = x, y, z. We choose the coordinate system such that the vertices are at v = (vx, vy, k) ∈ Z × Z × Z2 with k labelling the layer. In order to set up convenient notations for more local operators, we need first to label it position in the extended two dimensions. So we project our system into the xy-plane and get the square lattice as shown in Fig. E.1. Here each vertex is the projection image of a vertical link. They are colored differently according to the their symmetry behaviours and the vertex terms defined on them, which will be explained latter. To be distinguished from the vertex in Fig. 4.8, we called them xy-vertices, and they are labelled by their xy-coordinates v = (vx, vy). There are two vertices, v = (v, k) = (vx, vy, k) with k = 0, 1, projected to the same xy-vertex v. Similarly, we also the terms xy-links and xy-plaquettes here. Each xy-link, labelled by `, is the projection image of two horizontal links, labelled by ` = (`, k) with k = 0, 1 for layers. Each xy-plaquette, labelled by p, is the projection image of two horizontal plaquettes, labelled by p = (p, k) with k = 0, 1 for layers. More concretely, ` is specified by (v, ρˆ) withρ ˆ = ±x,ˆ ±yˆ for the direction of ` from v. Or alternatively, ` is specified by both of its ends (v, v0).

With the above notations for geometric elements of the lattice, the spin operators are expressed more concretely as   σµ = σµ , ` ⊥ zˆ; µ  `,k v,ρ,kˆ σ` = (E.1)  µ τv , ` k zˆ; where ` = (`, k) = (v, ρ,ˆ k) for horizontal link (i.e. ` ⊥ zˆ), v is the projection image of ` for vertical link (i.e. ` k zˆ) and µ = x, y, z. We use a diﬀerent Greek letter τ for the spins on vertical links to remind us that they are surface degrees of freedom with anomalous symmetry.

It is conjectured that eT mT is anomalous, which means that it can only appear on the surface of some three-dimensional system with non-trivial SPT order protected by time reversal symmetry. The cohomology models are constructed to give examples of non-trivial SPT phases. We investigate surface degrees of freedom of the cohomology model associated with the non-trivial ele-

4 T ment of H Z2 ,U (1) for onsite time reversal symmetry. By studying their symmetry behaviour, 127 we propose the following transformation of the local degrees of freedom under the time reversal symmetry T

x,y x,y σ`,0 ↔ σ`,1 , (E.2)

z z σ`,0 ↔ −σ`,1, (E.3)

x,y x,y τv → uvτv , (E.4)

z z τv → −τv , (E.5) where  z z z z z z z z  1−τv+ˆxτv+ˆy 1−τv+ˆyτv−xˆ 1−τv−xˆτv−yˆ 1−τv−yˆτv+ˆx  2 2 2 2 i i i i , vx + vy is odd;   z z z z z z z z  1−τv+ˆxτv+ˆx+ˆy 1−τv+ˆx+ˆyτv+ˆy 1−τv+ˆyτv−xˆ+ˆy 1−τv−xˆ+ˆyτv−xˆ uv = i 2 i 2 i 2 i 2 (E.6)   1−τz τz 1−τz τz 1−τz τz 1−τz τz  v−xˆ v−xˆ−yˆ v−xˆ−yˆ v−yˆ v−yˆ v+ˆx−yˆ v+ˆx−yˆ v+ˆx i 2 i 2 i 2 i 2 , vx + vy is even.

In Fig. E.1, the xy-vertices v = (vx, vy) with vx + vy odd are colored black, while the xy-vertices

µ with vx + vy even are colored blue or red. Obviously, the time reversal symmetry of τv look strange; here we imagine they are associated with surface degrees of freedom of some non-trivial three-dimensional SPT phase.

If we deﬁne   {v +x, ˆ v +y, ˆ v − x,ˆ v − yˆ} , vx + vy is odd; N (v) =  {v +x, ˆ v +x ˆ +y, ˆ v +y, ˆ v − xˆ +y, ˆ v − x,ˆ v − xˆ − y,ˆ v − y,ˆ v +x ˆ − yˆ} , vx + vy is even; (E.7) the set of neighbor vertices around v according to the triangularization shown in Fig. E.1, then

1 #dw[N(v)] uv = (−1) 2 , where #dw [N (v)] counts the number of domain walls among the spins along

N (v). It is easy to check that

2 uv = 1, (E.8)   z z −τ τ , v ∈ N (v0) , x,y 2  N−(v0,v) N+(v0,v) (uv0 τv ) = (E.9)  0, v∈ / N (v0) , 128 where N− (v0, v) and N+ (v0, v) are the vertices before and after v in N (v0) (in counterclockwise order).

Modifying the toric code model to make it compatible with the above time reversal symmetry, we get our Hamiltonian X X H = − Av − Bp, (E.10) v p where the ﬁrst summation is over all vertices and the second is over all plaquettes either horizontal or vertical. The plaquette term is kept exactly the same as in the toric code model;

Y z Bp = σ` . (E.11) `∈p The vertex terms are given by  Q x z x  σ σ τ , if vx + vy is odd (v black);  `3v `,0 `,1 v    Q σx τ xw  `3v `,0 v (v,1)    Q 1+B(p,1) 1−B(p,1) z · p3v 2 + 2 τv.p , if vx, vy are both even (v red); A(v,0) =  Q x x  `3v σ`,0 τv w(v,1)    1+B(v,v+ˆρ) 1−B(v,v+ˆρ) · Q + σz σz  ρˆ=±xˆ 2 2 v+ˆρ,y,ˆ 1 v+ˆρ,−y,ˆ 1    Q 1−B(v,v+ˆρ) 1+B(v,v+ˆρ) z z · ρˆ=±yˆ 2 + 2 σv+ˆρ,x,ˆ 1σv+ˆρ,−x,ˆ 1 , if vx, vy are both odd (v blue); (E.12) and  Q x z x  σ σ τ uv, if vx + vy is odd (v black);  `3v `,1 `,0 v    Q σx τ xu w  `3v `,1 v v (v,0)    Q 1+B(p,0) 1−B(p,0) z · p3v 2 + 2 τv.p , if vx, vy are both even (v red); A(v,1) =  Q x x  `3v σ`,1 τv uvw(v,0)    1+B(v,v+ˆρ) 1−B(v,v+ˆρ) · Q + σz σz  ρˆ=±xˆ 2 2 v+ˆρ,y,ˆ 0 v+ˆρ,−y,ˆ 0    Q 1−B(v,v+ˆρ) 1+B(v,v+ˆρ) z z · ρˆ=±yˆ 2 + 2 σv+ˆρ,x,ˆ 0σv+ˆρ,−x,ˆ 0 , if vx, vy are both odd (v blue); (E.13) 129 where v B p is the diagonal corner of p with respect to v, B(v,v+ˆρ) is the vertical plaquette containing ` = (v, ρ,ˆ k) with k = 0, 1, and

1 1 w = 1 + σz + τ z − σz τ z · 1 + σz + τ z − σz τ z (v,1) 2 v,x,ˆ 1 v+ˆx−yˆ v,x,ˆ 1 v+ˆx−yˆ 2 v,x,ˆ 1 v+ˆx+ˆy v,x,ˆ 1 v+ˆx+ˆy 1 1 1 + σz + τ z − σz τ z · 1 + σz + τ z − σz τ z 2 v,y,ˆ 1 v+ˆx+ˆy v,y,ˆ 1 v+ˆx+ˆy 2 v,y,ˆ 1 v−xˆ+ˆy v,y,ˆ 1 v−xˆ+ˆy 1 1 1 + σz + τ z − σz τ z · 1 + σz + τ z − σz τ z 2 v,−x,ˆ 1 v−xˆ−yˆ v,−x,ˆ 1 v−xˆ−yˆ 2 v,−x,ˆ 1 v−xˆ+ˆy v,−x,ˆ 1 v−xˆ+ˆy 1 1 1 + σz + τ z − σz τ z · 1 + σz + τ z − σz τ z , 2 v,−y,ˆ 1 v+ˆx−yˆ v,−y,ˆ 1 v+ˆx−yˆ 2 v,−y,ˆ 1 v−xˆ−yˆ v,−y,ˆ 1 v−xˆ−yˆ (E.14)

w(v,0) = T w(v,1) 1 1 = 1 − σz − τ z − σz τ z · 1 − σz − τ z − σz τ z 2 v,x,ˆ 0 v+ˆx−yˆ v,x,ˆ 0 v+ˆx−yˆ 2 v,x,ˆ 0 v+ˆx+ˆy v,x,ˆ 0 v+ˆx+ˆy 1 1 1 − σz − τ z − σz τ z · 1 − σz − τ z − σz τ z 2 v,y,ˆ 0 v+ˆx+ˆy v,y,ˆ 0 v+ˆx+ˆy 2 v,y,ˆ 0 v−xˆ+ˆy v,y,ˆ 0 v−xˆ+ˆy 1 1 1 − σz − τ z − σz τ z · 1 − σz − τ z − σz τ z 2 v,−x,ˆ 0 v−xˆ−yˆ v,−x,ˆ 0 v−xˆ−yˆ 2 v,−x,ˆ 0 v−xˆ+ˆy v,−x,ˆ 0 v−xˆ+ˆy 1 1 1 − σz − τ z − σz τ z · 1 − σz − τ z − σz τ z . 2 v,−y,ˆ 0 v+ˆx−yˆ v,−y,ˆ 0 v+ˆx−yˆ 2 v,−y,ˆ 0 v−xˆ−yˆ v,−y,ˆ 0 v−xˆ−yˆ (E.15)

In Fig. E.1, vertices with diﬀerent forms of vertex term are colored diﬀerently. Here A(v,0) and A(v,1) are constructed such that A(v,1) = T A(v,0) . Hence the Hamiltonian is invariant under the

2 2 time reversion; T (H) = H. Further, we can check that Av = Bp = 1 and [Av,Av0 ] = [Av,Bp] = Bp,Bp0 = 0.

The system is Z2 topologically ordered and its ground state |Ωi is speciﬁed by Av = Bp = 1. This can be justiﬁed by writing down the string operators for e and m excitations. Here an e

0 excitation at v means Av = −1. Obviously, a pair of e excitations at v, v can be created by

e Q z 0 Ls = `∈s σ` for any path s connecting v and v . If we denote the e excitation at v by |evi, then

e |evev0 i = Ls |Ωi (up to a phase factor).

Now let’s take a look at how to create a pair of m excitations. Given a cut of links like the 130 one shown in Fig. E.1, we can deﬁne an m-string operator on it as

m Y m Lt = L` , (E.16) `∈t where the elementary pieces are

1−τz τz 1−τz τz 1−τz 1−τz  T ` T ` T ` T ` T ` T ` y y 0 1 0 2 1 2 ` σ σ i 2 i 2 i 2 i 2 ,T blue;  (`,0) (`,1) 0   1−B 1+B 1−B 1+B m  (T `,T `) (T `,T `) (T `,T `) (T `,T `) L = σy σy τ z 1 3 + 1 3 τ z 2 3 + 2 3 τ z , ` k xˆ & T ` red; ` (`,0) (`,1) T ` 2 2 T ` 2 2 T ` 0  0 1 2   1+B ` ` 1−B ` ` 1+B ` ` 1−B ` `  y y z (T1 ,T3 ) (T1 ,T3 ) z (T2 ,T3 ) (T2 ,T3 ) z ` σ σ τ ` 2 + 2 τ ` 2 + 2 τ ` , ` k yˆ & T0 red. (`,0) (`,1) T0 T1 T2 (E.17)

` ` ` As illustrated in Fig. E.1, T0 is the blue (resp. red) end of ` and T3 is the black end of `, while T2

` ` 1 and T3 are the two red (resp. blue) vertices next to T3 . In addition, the xy-plaquette containing

` ` ` ` m both ` and T1 (resp. T2 ) is denoted by p1 (resp. p2). It can be checked that (1) L` anticommutes

` ` m with A ` , A ` , B ` , B ` for k = 0, 1, if T and T are red; (2) L anticommutes with (T1 ,0) (T2 ,0) (p1,k) (p2,k) 1 2 ` ` m B ` , B ` for k = 0, 1, if T is red; (3) L commutes with other Av and Bp not mentioned (p1,k) (p2,k) 0 ` m above. Then Lt deﬁned by Eq. (E.16) only anticommutes with A(v,0), A(v0,0) at red vertices and

B(p,0), B(p,1), B(p0,0), B(p0,1) on plaquettes located at the ends of t, as shown in Fig. E.1. Therefore,

m m Lt acting on the ground state |Ωi leads to a state Lt |Ωi with local excitations at the ends of t.

Let’s denote this type of local excitation at p by |mpi corresponding to A(v,0) = B(p,0) = B(p,1) = −1,

m where v is the red vertex on p. Then mpmp0 = Lt |Ωi (up to a phase factor).

In order to justify the notation, we need to determine the statistics of |mpi. It is created and

m m m moved by string operators of the form Lt . Since Lt always commutes with Lt0 independent of the

0 relative positions of t and t , the excitation |mpi has trivial self-statistics. In addition, it is easy to

m e e m see that Lt Ls = −LsLt if t crosses s, and hence |mpi has π mutual statistics with e. Therefore,

|mpi is indeed an m excitation and the system is Z2 topologically ordered. Now let’s determine the symmetry fractionalization pattern here. We can check

m |t| z z m T (Lt ) |Ωi = (−1) τv τv0 Lt |Ωi , (E.18)

1 ` ` ` To ﬁx the order convention between T2 and T3 , we choose the x-component (resp. y-component) of T2 is smaller ` than T3 , if ` is parallel toy ˆ (resp.x ˆ), denoted ` k yˆ (resp. ` k xˆ). 131 where |t| is the length of the cut t (i.e. the number of times that t goes across xy-links). In the example shown in Fig. E.1, |t| = 4. Therefore,

iθ z T |mpi = e τv |mpi , (E.19)

z iθ where τv is at the red corner of p and e is some phase factor unimportant for latter calculation.

This localization of time reversal symmetry can be double-checked by verifying that the excitation in Eq. (E.19) corresponds to T A(v,0) = T B(p,0) = T B(p,1) = −1 (i.e. A(v,1) = B(p,1) =

B(p,0) = −1). So we have

In short, this means (T m)2 = −1.

z Similarly, we have T e(v,0) = τv e(v,0) up to some unimportant phase factor and hence

2 e 2 T e(v,0) = − e(v,0) . So (T ) = −1 as well. To summarize, the model has time reversal symmetry with symmetry fractionalization pattern (T e)2 = (T m)2 = −1.