Interplay between Symmetry and Topological Order in Quantum Spin Systems

Transcription

1 University of Colorado, Boulder CU Scholar Physics Graduate Theses & Dissertations Physics Spring Interplay between Symmetry and Topological Order in Quantum Spin Systems Hao Song University of Colorado Boulder, Follow this and additional works at: Part of the Condensed Matter Physics Commons, and the Quantum Physics Commons Recommended Citation Song, Hao, "Interplay between Symmetry and Topological Order in Quantum Spin Systems" (2015). Physics Graduate Theses & Dissertations This Dissertation is brought to you for free and open access by Physics at CU Scholar. It has been accepted for inclusion in Physics Graduate Theses & Dissertations by an authorized administrator of CU Scholar. For more information, please contact

2 Interplay between Symmetry and Topological Order in Quantum Spin Systems by 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

3 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.

4 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 Z 2 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.

5 To my parents. Dedication

6 v Acknowledgements First and foremost, I am sincerely grateful to my advisor, Prof. Michael Hermele, for his generous 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 comprehensive 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.

7 vi Contents Chapter 1 Introduction Chiral Spin Liquids in Cold Atoms Topological Phases and Symmetries Anomalous Symmetry Fractionalizations and SPT Phases Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms in Three Dimensions Theoretical Model Large-N Ground States Summary of the large-n mean-field results Detailed descriptions of the mean-field ground states Obtaining the mean-field results Relation between bilayer states and square lattice saddle points Discussion Crystal Symmetry Fractionalization Outline and Main Results Review of Z 2 Topological Order Review: toric code model on the square lattice Toric Codes on General Two-dimensional Lattices with Space Group Symmetry Fractionalization and Symmetry Classes

8 vii Review of fractionalization and symmetry classes Fractionalization and symmetry classes in the solvable models Symmetry Classes Realized by Toric Code Models Model realizing all e particle fractionalization classes Toric code models without spin-orbit coupling General toric code models Summary and Beyond Toric Code Models Topological Phases Protected by Reflections and Anomalous Crystalline Symmetry Fractionalization Reflection Symmetry Protected Topological Phyases With a Single Reflection Symmetry With Two Orthogonal Reflection Symmetries Boundary Degrees of Freedom of SPT Phases Dimer model for 1d SPT CZX model for 2d SPT with onsite Z 2 symmetry Model of Fractionalization Anomaly: a Single Reflection (epmp) Spectrum of the model Symmetry behaviour of anyons Model of Fractionalization Anomaly: Two Orthogonal Reflections All Fractionalization Anomalies Associated with the Square Lattice Symmetries Summary and Perspective Summary Perspetive

9 Bibliography 86 viii 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 C.1.1 Toric codes without spin-orbital coupling C.1.2 Toric codes with spin-orbit coupling D Models in T C (G) 119 E A model of etmt 125

10 ix Tables Table 2.1 Ground state saddle-point patterns of χ rr, and the corresponding energies in units of NJ N s for k = 2, 3,..., 10. The different types of large-n ground states are described in the text, and depicted in figures as indicated 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 k = 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 l x,y,z within the given range, the number of times we ran the SCM algorithm with distinct random initial configurations of χ rr 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)

11 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 N s. 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 χ rr is real and spatially constant. (2πn x,y,z /k)-flux. These states have 2πn x /k flux through every plaquette normal to the x-direction, and similarly for y and z, where 0 n x,y,z k 1. Since most of these states break lattice rotation symmetry, the magnitude χ rr is allowed to vary depending on bond orientation, but is fixed to be translation invariant.[3] Notation used in the chapter

12 xi Figures Figure 2.1 Ground-state clusters for k = 2, 3, 4. Shaded bonds are those with χ rr 0. Bonds with different shading (or color in online version) may have different magnitudes χ rr. (a) The k = 2 ground state clusters are dimers and square plaquettes. The square plaquette is pierced by π-flux, and the ratio of χ rr on light (pink online) and dark (blue online) bonds can be chosen arbitrarily. Setting χ rr = 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, χ rr = 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 Φ s -flux through the side plaquettes and Φ t -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

13 xii 2.2 Ground-state saddle point configurations of χ rr for k = 5, 6,..., 10. The right column is a three-dimensional view of each configuration, with larger magnitude χ rr indicated by darker shading. All these saddle points can be viewed as bilayer structures, with χ rr identical on top and bottom layers. The left column thus shows χ rr 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 χ rr can be changed continuously within a single cluster without affecting the energy; only the simplest configurations are shown 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 l x,y edges of the unit cell 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 s x and s y winding periodically around the system are also shown as thick dark bonds (blue online) Depiction of e and m strings in the square lattice toric code. s is an open e-string joining vertices v and v, denoted with thick dark bonds (blue online). t is an open cut joining plaquettes p and p, shown as a dotted line. The cut t contains the thick gray bonds (pink online) intersected by the dotted line

14 xiii 3.3 (a) The lattice on which all 2 6 = 64 e particle fractionalization classes can be realized. There are six types of plaquettes not related by symmetries, and the correponding plaquette terms are assigned independent coefficients K m i (i = 1, 2,, 6). Nearestneighbor 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 v 1, v 2 and two edges l 1, l 2 are labeled to illustrate the calculation of σ e px 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, T C 0 (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 a r is the ground state eigenvalue of A v for v at special points r = o, õ, κ; and b, b 1, b 2 are the ground state eigenvalues of B p for the plaquette p, which in these models is picked to be the smallest cycle made with black edges where b, b 1 or b 2 is written, while b 3 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 origin by ( 1 2, 1 2) results in a transformation (3.74): Px T x P x, σ px σ txpx, σ pxpxy σ pxpxy σ txty. 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

15 xiv 3.5 Two example models in T C (G) that realize TC symmetry classes not possible in T C 0 (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 a r is the ground state eigenvalue of A v for v at special points r = o, õ, κ and b is the ground state eigenvalue of B p for the plaquette p, defined here to be the smallest cycle enclosing the letter b. We write α i = c x l i (P x ), β i = c x l i (P xy ), γ i = c z l i (P x ) and δ i = c z l i (P xy ). (a) A model realizing some TC symmetry classes (and symmetry classes) that cannot be realized without spin-orbit coupling. Here,h 1, h 2 label two positions of a m particle for the calculation of σpx m = α 1 in the main text. (b) A model realizing all 2 6 = 64 possible m particle fractionalization classes [ω m ]. Here, for simplicity, we make the restriction γ i = δ i c i A quantum spin system that respects a single reflection P x : (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) P x [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 Z 2 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

16 xv 4.2 A quantum spin system that respects two orthogonal reflections P x : (x, y, z) ( x, y, z) and P y : (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) P x [U (t)] P y [U (t)] P x P y [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 H 3 (Z 2, U (1)) H 3 (Z 2, U (1)) H 2 (Z 2 Z 2, U (1)) = Z 3 2, 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 The dimer model for one-dimensional SPT phase labelled by ω H 2 (G, U T (1)), where G is the group of on-site symmetries The CZX model for the two-dimensional SPT phase labelled by H 3 (Z 2, U (1)). (a) Each site (circle) contains four spins (dots) and the on-site symmetry of order 2 is generated by U CZX = σ x 1 σx 2 σx 3 σx 4 CZ 12 CZ 23 CZ 34 CZ 41. (b) The local term on each plaquette p in the Hamiltonian is H p = X p P u p P d p P l p P r p, where X p = + acts on the four spins on the plaquette p and P u,d,l,r p = + acts on the up, down, left and right neighbouring pairs of spins around the plaquette p respectively. The ground state is fixed within the bulk; the spins on each plaquette (square) are entangled as p + p 2. But some boundary degrees of freedom are not fixed by the Hamiltonian, like those along the right edge shown here

17 xvi 4.5 The dashed line is the axis of the reflection. Applying σ x l (resp. σ y l and σ z l ) is graphically presented by highlighting l in blue (resp. magenta and red). The vertex terms A v at v = v 0, v 1,, v 5 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; A vi = P x ( 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 σ µ l σµ P xl, in general it is not hard to see that P x (A v ) = A Pxv, P x (B p ) = B Pxp and that all vertex terms and plaquette terms commute with each other; [A v, A v ] = [A v, B p ] = [ B p, B p ] = 0, v, v, p, p The red string stands for a product of σ z l on it. The blue string stands for a product of σl x on it. (a) When an m is moved to x = 0, it becomes an excitation with A v = 1 and B p = 1 due to the factor σ y (v x,v y 2) in A v. The excitation is still a 1 boson. (b) When an ε is moved to x = 0, it becomes an excitation with B p = 1. The excitation is still a fermion String operators that create a pair of e s and m s related by reflection. The red string stands for a product of σ z l on it. The blue string stands for a product of σx l on it. (a) A pair of e s. (b) A pair of m s Generalized toric code model defined on coupled layers of square lattice. Spins are associated with the links, whose Pauli operators are generally denoted by σ µ l, µ = x, y, z. Explicitly, we use different notations for spins on vertical and horizontal links; we write σ µ l = τ v µ for spins on vertical links (green online) at v = (v x, v y ) Z Z, and use σ µ l = σµ (l,k) for horizontal links on the layer at z = k with k = 0, 1. In detail the xy-position l of a horizontal link l can be given be the xy-coordinates of its both ends; l = (l 0, l 1 ) with l 0, l 1 Z Z, the xy-coordinates for the ends of l respectively. 78

18 xvii B.1 Graphical argument that L e c ψ 0 = ψ 0, for c = s y Tx 1 s y. The dotted lines show the L L grid of primitive cells, and the paths s y and Tx 1 s y 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 c be the cycle bounding one of the sub-regions, then L e c = L 1 n=0 Le T n x c. In addition, by translation symmetry L e c ψ 0 = L e T yc ψ 0 = ± ψ 0. Since an even number of sub-regions appear in the decomposition of L e c given above, we have L e c ψ 0 = ψ B.2 Graphical illustration of the argument that L e c ψ 0 = ψ 0, for c = s x P x s x. 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 P x. The various symbols are defined in the main text. The vertical dashed line is the P x reflection axis, and the vertex v has been chosen to lie near this axis for convenience. c 1 and c 2 are the boundaries of the left and right shaded regions, respectively. The most important point is that these two regions are related by T L/2 x translation C.1 The calculation of σ m pxpxy. Put an m particle at point h 0, let h j = R j (h 0 ), j = 1, 2, 3, let t 0 W connecting h 0 to h 1 and t j = R j t 0. Then we have UP m x UP m xy (h 0 ) = f0 mlm t 0 ) with f0 m {±1} and UP m x UP m xy (h j ) = L m t 0 t j 1 UP m x UP m xy (h 0 ) R (L m t 0 tj 1 for j = ( ) 4 1, 2, 3. With some calculation, UP m x UP m (h0 xy ) = L m t with t = t 0 t 1 t 2 t 3. Thus, σ e pxpxy = a Vt. If P (v) is enclosed by t with R 2 v v, then v, Rv, R 2 v, R 3 v are four different vertices enclosed by t such as the grey vertices shown here. Since 3 i=0 a R i v = 1, we have σ e pxpxy = a Vt = a P 1 (o) = a Γ(R 2 ). 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 a v = a Pxv = a Pxyv, we have σ m pxpxy = a Γ(Px,P xy)

19 xviii C.2 Illustration of the calculation of σ m typx in Lemmas 7 and 16. Solid squares denote the locations of holes h H, which are chosen so that h 0 is arbitrary (but near the y-axis), and h 1 = P x h 0, h 2 = T y h 0, h 3 = T y h 1, h 4 = P y h 0, h 5 = P y h 1. h 4 and h 5 are not used in Lemma 7. Cuts are represented by solid lines, and h 0 h 1 denotes, for example, a cut joining h 0 to h 1. The cuts t 0 = h 0 h 1, t 1 = h 0 h 2, t 2 = h 1 h 3, and t 3 = h 2 h 3 are labeled, and are chosen to have properties described in the text. The points o = (0, 0) and κ = (0, 1 2 ) are shown

20 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 ɛ, l, ι, respectively. We use ζ and ξ to label diagonal edges. For a diagonal edge, we can associate a unit vector ê running along the direction of the edge, always choosing ê x > 0. Then ζ (ξ) is used to label edges with ê y > 0 (ê y < 0). Panels (a,d). This configuration is only used in Fig. D.2c. The two stacking vertices (blue and red online) together with edge ι 1 connecting them are projected into a point, presented as a ring (blue and red online). Edges ɛ 2, l 3 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

21 xx D.2 T C (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 ι o, ιõ, ι κ, ι κ with the subscript indicating their position, and õ = ( 1 2, ) ( ( 1 2, κ = 0, 1 2), κ = 1 2, 0), in units such that the size of the unit cell is 1 1. For short, we define α i = c x ε i (P x ), β i = c x ε i (P xy ), γ i = c z ε i (P x ) and δ i = c z ε i (P xy ), where ε = l, ɛ, ξ, ζ, ι stands for a generic edge. In addition, e r = a P 1 (r), and b is the eigenvalue of B p for the plaquette (here meaning smallest cycle) p within which b is written. The values of e r and b are well-defined with respect to any local spin frame system satisfying Eqs. (C.15-C.17) D.3 T C (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 ι o, ιõ, ι κ, ι κ with the subscript indicating their position, and õ = ( 1 2, ) ( ( 1 2, κ = 0, 1 2), κ = 1 2, 0), in units such that the size of the unit cell is 1 1. For short, we define α i = c x ε i (P x ), β i = c x ε i (P xy ), γ i = c z ε i (P x ) and δ i = c z ε i (P xy ), where ε = l, ɛ, ξ, ζ, ι stands for a generic edge. In addition, e r = a P 1 (r), and b is the eigenvalue of B p for the plaquette (here meaning smallest cycle) p within which b is written. In panel (l), b is the eigenvalue of B p for the top plaquette. The values of e r and b are well-defined with respect to any local spin frame system satisfying Eqs. (C.15-C.17)

22 xxi D.4 T C (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 ι o, ιõ, ι κ, ι κ with the subscript indicating their position, and õ = ( 1 2, ) ( ( 1 2, κ = 0, 1 2), κ = 1 2, 0), in units such that the size of the unit cell is 1 1. For short, we define α i = c x ε i (P x ), β i = c x ε i (P xy ), γ i = c z ε i (P x ) and δ i = c z ε i (P xy ), where ε = l, ɛ, ξ, ζ, ι stands for a generic edge. In addition, e r = a P 1 (r), and b (or b i ) is the eigenvalue of B p for the plaquette (here meaning smallest cycle) p within which b (or b i ) is written. In panels (w) and (x), b 1 and b are the eigenvalues of B p for the top plaquettes. The values of e r and b (or b i ) 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 require c z l 1 (P xy ) = c z l (P 1 xy ) = c z l 2 (P x ) = 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 l, we use T l 3 to label the black end of l, T l 0 for the other end, and T1 l, T 2 l for the two vertices next to T 3 l. In addition, t is a cut of links connecting xy-plaquettes p and p

23 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 Z 2 spin liquids, which provides a valuable perspective for understanding symmetry protected / enriched topological orders. 1.1 Chiral Spin Liquids in Cold Atoms 1 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, (2013),[4] copyright 2013 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society.

24 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 manybody 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 nuclear spin.[9, 10] This occurs in both the 1 S 0 ground state and a metastable 3 P 0 excited state, where the electronic angular momentum J e = 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 87 Sr, 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

25 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 translational symmetry is preserved. For the simplest AEA Mott insulators (with 1 S 0 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-twodimensionality.

26 4 We now define our model before briefly surveying some related prior work. We consider a fermionic AEA with N spin species, and put m 1 S 0 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 onsite 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, we also consider more generally the case where N m is an integer. Then, the minimum number of spins needed to make a SU(N) singlet is k = N 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 quasilong-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]

27 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 Symmetries 2 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, (2015),[77] copyright 2015 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society.

28 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.[ ] 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, , 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 Z 2 topological order, giving a

29 7 symmetry classification for square lattice space group plus time reversal symmetry, that can easily be generalized to any desired symmetry group. For Z 2 topological order with symmetry group G, a symmetry class is specified by fractionalization classes [ω e ] and [ω m ], for e particle (Z 2 charge) and m particle (Z 2 flux) excitations, respectively. Mathematically, distinct fractionalization classes are elements of the cohomology group H 2 (G, Z 2 ). In more detail, a symmetry class is an un-ordered pair [ω e ], [ω m ] [ω m ], [ω e ], 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 Z 2 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 Z 2 toric code[136] to arbitrary twodimensional lattices with square lattice space group symmetry (a precise definition appears in

30 8 Sec. 3.4). By appropriately choosing the lattice geometry, varying the signs of terms in the Hamiltonian, 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 Z 2 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 Z 2 symmetry. This viewpoint can be extended to give general classifications to pgspt phases. In addition,

31 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 Z 2 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.

32 Chapter 2 Mott Insulators of Ultracold Fermionic Alkaline Earth Atoms in Three Dimensions 1 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 k = 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, (2013),[4] copyright 2013 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society.

33 Theoretical Model The SU(N) Hubbard model H Hubbard = t rr ( ) c α r c r α + h.c. + (U/2) r ( c α r c rα m) 2, (2.1) describes the behavior of fermionic AEAs on an optical lattice.[9] Here c α r and c rα 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 total number of lattice sites is N s. The operator c α r transforms in the fundamental representation of SU(N), while c rα 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 H = J rr (f α r f r α)(f β r f rβ ), (2.2) with the Hilbert space restricted by f α r f rα = m, and J = 2t 2 /U > 0. We now use f α r than c α r rather 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 the large-n mean-field theory is that of a hopping Hamiltonian for the f α r fermions, but it is not correct to interpret this hopping as motion of atoms. Instead, in the large-n mean-field theory, the f α r fermions are spinons, fractional particles that may be either confined or deconfined depending

34 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 H = J rr Ŝ β α(r)ŝα β (r ), (2.3) where Ŝβ α(r) = f β r f rα 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 87 Sr. 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 > N c, where N c 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 > N c. For the square lattice N c = 3,[1] and the argument is easily extended to find N c = 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 ratio k = N 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);

35 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 = DχDχ DλD fdf e S, (2.4) where S = f r α χ rr 2 τ f rα + N τ r τ J rr ( α + χrr f r f r α + h.c. ) +i τ r,r τ ( λ r f α r f rα m ). (2.5) r The field χ rr is a complex Hubbard-Stratonovich field that has been used to decouple the exchange interaction, and λ r is a real Lagrange-multiplier field enforcing the fr α f rα = m constraint. The fermion fields f and f are the usual Grassmann variables. We have introduced J = NJ; J is held fixed in the large-n limit. Finally, τ β 0 i.e. β. dτ. We shall always be interested in zero temperature, 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, χ rr χ rr and λ r iµ r. The saddle-point equations are m = fr α f rα, (2.6) χ rr = J f α N r f rα. (2.7)

36 The above averages are taken in the ground state of the saddle-point (or mean-field) Hamiltonian 14 H MF T = N χ rr 2 + m µ r J rr r + ( ) χ rr fr α f r α + h.c. rr r µ rˆn r, (2.8) where ˆn r f α r f rα. The ground state is determined by finding the global minimum of E MF T ({χ rr }, {µ r }), the ground state energy of H MF T, 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 Large-N Ground States 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 χ rr and µ r. The most important information is contained in χ rr, since typically it is possible for a given χ rr to find µ r 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 χ rr 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 χ rr is physical. The theory has a U(1) gauge redundancy f rα f rα e iφ(r) χ rr χ rr e i(φ(r) φ(r )), (2.9) so the physical information is contained in the following gauge-invariant quantities: (1) magnitude χ rr and (2) flux Φ = a 12 + a 23 + a 34 + a 41 through each plaquette, where 1, 2, 3, 4 indicates the four vertices of a plaquette and a rr is the phase of the χ rr, i.e. χ r r = e ia rr χ rr. (Since χ r r = χ rr, a r r = a rr.)

37 Based on a combination of analytical and numerical techniques described below, we found 15 the ground state configuration of χ rr and µ r 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 χ rr, the lattice is partioned into n-site clusters such that χ rr 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 χ rr is only nonzero for r, r 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 χ, rr lies within either layer; χ rr = J rr connects the two layers. k, (2.10) Moreover, there is a uniform flux Φ = 4π k (2.11) 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 f rα fermion. To fully understand the different ground states, one has to go beyond the N = or meanfield 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 f rα fermions; the

38 k Large-N ground state Sketch of χ rr Energy 2 2/4-site cluster Fig. 2.1a site cluster Fig. 2.1b /8-site cluster Fig. 2.1c site cluster Fig. 2.2a, 2.2b site cluster Fig. 2.2c, 2.2d CSL bilayer Fig. 2.2e, 2.2f site cluster Fig. 2.2g, 2.2h Inhomogeneous bilayer Fig. 2.2i, 2.2j CSL bilayer Fig. 2.2e, 2.2f Table 2.1: Ground state saddle-point patterns of χ rr, and the corresponding energies in units of NJ N s 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 We have not considered the effect of fluctuations in the k = 9 inhomogeneous bilayer ground state 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 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 Φ t through the top and bottom plaquettes (i.e., those lying in the xy-plane), and flux Φ s through the side plaquettes (i.e., those lying in the xz- and yz-planes). Flux passing from the center of the cube

39 to the outside is taken positive. In order to reach the ground state we must have 2Φ t + 4Φ s = ±2π; 17 we choose the positive sign without loss of generality. We let Φ t = 4u and Φ s = π/2 2u; a ground state is obtained if we restrict 0 u π/2. In this situation the magnitude χ rr 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 χ dark = 2 cos u sin u. (2.12) The ground-state patterns of χ rr for 5 k 10 are shown in Fig 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 χ rr 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 χ rr, there is zero flux on the vertical plaquettes connecting the two layers, and the vertical bonds have magnitude χ rr = 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 k = k/2.

40 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 E MF T 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, ±ɛ occuring in the spectrum, and with energy ɛ 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 µ r = 0 and a randomly generated configuration of χ rr. 2: Adjust µ r to satisfy the saddle-point equation f α r f rα = m, for all r. (2.13) µ r is determined by a multidimensional Newton s method.[1, 2, 142] Stop if no solution is found. 3: Generate a new χ rr using the saddle-point equation χ rr = J N f α r f rα. (2.14) 4: Go back to step 2 until χ rr and µ r converge. As long as step 2 is successful, the energy E MF T is guaranteed to decrease with each iteration of the

41 19 Π (a) k = 2 Π Π (b) k = 3 t s s t (c) k = 4 Figure 2.1: Ground-state clusters for k = 2, 3, 4. Shaded bonds are those with χ rr 0. Bonds with different shading (or color in online version) may have different magnitudes χ rr. (a) The k = 2 ground state clusters are dimers and square plaquettes. The square plaquette is pierced by π-flux, and the ratio of χ rr on light (pink online) and dark (blue online) bonds can be chosen arbitrarily. Setting χ rr = 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, χ rr = 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 Φ s -flux through the side plaquettes and Φ t -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.

42 20 (a) k = 5 (b) k = 5 (c) k = 6 (d) k = 6 (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 χ rr for k = 5, 6,..., 10. The right column is a three-dimensional view of each configuration, with larger magnitude χ rr indicated by darker shading. All these saddle points can be viewed as bilayer structures, with χ rr identical on top and bottom layers. The left column thus shows χ rr 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 χ rr can be changed continuously within a single cluster without affecting the energy; only the simplest configurations are shown.

43 21 SCM algorithm.[1, 2] But a random initial configuration of χ rr does not necessarily converge to the ground state, and can instead converge to a local minimum of E MF T. Therefore, in order to find the ground state, we need to try as many independent random initial configurations of χ rr as possible. For those random initial configurations resulting in the lowest energies, we found extremely good convergence in E MF T by the time the SCM procedure is stopped (typically after 300 iterations), and effects of randomness on the reported values of E MF T are thus entirely negligible. To improve the performance of the SCM algorithm, we define χ rr with µ r within some fixed unit cell, which is then repeated periodically to cover a finite-size L x L y L z lattice with periodic boundary conditions. For simplicity, we always choose the unit cell to be a rectangular prism with edge lengths l x,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(l x, L y, L z ) 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 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

44 22 l y l z l y l x (a) l x (c) l y l z 1 l x (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 l x,y edges of the unit cell.

45 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 k = 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. There are N s = 2N 2d s saddle point where lattice sites, where N 2d s is the number of sites in a single layer. Consider a χ r1,r 1 = χ r2,r 2 χ rr (2.15) µ r1 = µ r2 µ r (2.16) χ r1,r2 χ v. (2.17) Here, χ v is real and positive, and all other inter-layer χ s are assumed to vanish. We let n label the one-particle eigenstates of a single layer, with energies ɛ 2d n. The full one-particle spectrum is then given by ɛ n,σ = ɛ 2d n + σχ v, (2.18) where σ = ±1. We assume that the energy spectrum and filling are such that only σ = 1 states are occupied by fermions, in which case the two-dimensional spectrum ɛ 2d n (shifted in energy by χ v ) is filled by NN s /k = 2NNs 2d /k fermions. This corresponds to a single-layer problem with twice as many fermions, or filling parameter k = k/2. The saddle point energy is then E MF T = NNs 2d χ 2 v J 2NN2d s k + 2N χ rr 2 + m J r rr χ v (2.19) µ r + E 2d f (k ). Here, m = 2m, and E 2d f (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 k. The first two terms of Eq. (2.19) are minimized with respect to χ v to find χ v = J /k. The last

46 k Cubic lattice Single bilayer k/2 square lattice 5 1 l x,y,z l x,y l x,y l x,y,z l x,y l x,y,z l x,y l x l y l x,y l x,y l z l x,y l x l x l z l y l y l x,y l x,y l z 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 k = 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 l x,y,z within the given range, the number of times we ran the SCM algorithm with distinct random initial configurations of χ rr 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). three terms combine to E 2d MF T (k, J ), the saddle point energy of a single-layer square lattice with filling parameter k and J = J /2. Noting that E 2d MF T (k ) E 2d MF T (k, J ) = 2E 2d MF T (k, J ), (2.20) we obtain the following relation between bilayer and single-layer saddle point energies: E MF T N s N = J 2k E 2d MF T (k/2) N 2d s N. (2.21) This relation allows us to study via SCM the single-layer square lattice with filling parameter k = k/2 as a further check on the cubic lattice results. For integer k, this was already done in Ref. [1]. We carried out SCM calculations for the half-odd integer filling parameters k = 5 2, 7 2, 9 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.

47 k Bilayer (Φ = 2πn/k) k-site cluster Uniform real χ (2πnx,y,z/k)-flux SCM ground state Table 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 N J 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 χrr is real and spatially constant. (2πn x,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 χrr is allowed to vary depending on bond orientation, but is fixed to be translation invariant.[3] 25

48 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

49 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.

50 Chapter 3 Crystal Symmetry Fractionalization 1 We study square lattice space group symmetry fractionalization in a family of exactly solvable models with Z 2 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 Z 2 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 Z 2 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 2 6 = 64 types of symmetry fractionalization allowed for a single anyon species (Z 2 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, (2015),[77] copyright 2015 American Physical Society, and is reproduced here in accord with the copyright policies of the American Physical Society.

51 varied. The paper concludes with a summary and a discussion of two results pertaining to more general bosonic models 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 , and especially Eqs. ( ), 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 Z 2 topological order. We call this family of models T C(G), and it consists of variations of Kitaev s Z 2 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 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 , we examine a subclass of models, T C 0 (G) T C(G), where no spin-orbit coupling is allowed. The main result of Sec is Theorem 1, which describes all symmetry classes that are realized by models in T C 0 (G). Following the statement of the theorem, example models realizing all possible symmetry classes for T C 0 (G) are presented. Finally, in Sec , we treat the general case of T C(G), and state Theorem 2, which describes all symmetry classes that are realized by models in T C(G); the discussion parallels that of Sec The detailed proofs of the theorems are left to the appendices, together with the presentation of models realizing all possible symmetry classes for T C(G). Our results establish that certain symmetry classes are possible in two dimensional models. For symmetry classes that are not realized by models T C(G), a more general understanding

52 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 Z 2 topological order, and Sec. 3.3 gives a review of the simplest Z 2 Kitaev toric code model, on the two-dimensional square lattice. The crucial objects are the e (Z 2 charge) and m (Z 2 flux) excitations of Z 2 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 T C 0 (G), where G is the square lattice space group. We also consider a larger family of models, T C(G), that contains T C 0 (G). In T C(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 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 T C(G) is defined so that permutations of anyons under symmetry never occur. Section 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 T C(G). In our

53 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 ([ω e ], [ω m ]). 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 we describe a single model that realizes all 2 6 = 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 T C 0 (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 T C 0 (G); these models are exhibited in Sec Moving on to the general case of T C(G) where spin-orbit coupling is allowed, Section 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 T C(G). Some examples of such models are described in Sec , with the full catalog of models given in Appendix D. Some of the notation used frequently in the paper is collected in Table Review of Z 2 Topological Order In this paper, we focus on Z 2 topological order in two dimensions, which is in some sense the simplest type of topological order. Z 2 topological order arises in the deconfined phase of Z 2 lattice

54 32 Symbol Table 3.1: Notation used in the chapter. Meaning H Hamiltonian G Symmetry group of H (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 l E Edge l in set of edges E s W Path s in set of paths W C Set of cycles (closed paths) C 0 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 σl x, σz l Pauli matrix spin operators on edge l L e s e-string on path s W L m t m-string on cut t W A v, a v Vertex operator and corresponding eigenvalue B p, b p Plaquette operator and corresponding eigenvalue o = (0, 0) Special points in the plane. õ = ( 1 2, 1 ) 2 (Units of length are chosen such that κ = ( 0, 1 ) 2 the size of the unit cell is 1 1.) κ = ( 1 2, 0) X Size of any finite set X. T C(G) Family of toric code models considered, with spin-orbit coupling allowed T C 0 (G) Family of toric code models considered, no spin-orbit coupling allowed

55 33 gauge theory with gapped bosonic matter carrying the Z 2 gauge charge. 2 There is an energy gap to all excitations, which can carry Z 2 gauge charge and/or Z 2 flux. There is a statistical interaction between charges and fluxes; the wave function acquires a statistical phase factor e iπ 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. Z 2 lattice gauge theory provides a particular concrete realization of Z 2 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 ɛ. In terms of lattice gauge theory, e particles are bosonic gauge charges, m particles are Z 2 -fluxes, and ɛ-particles are e-m bound states. Excitations carrying neither Z 2 charge nor flux are trivial, and are labeled by 1. e, m and ɛ excitations obey non-trivial braiding statistics and are thus referred to as anyons. e and m are bosons, while ɛ 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 e iπ 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 = ɛ ɛ = 1 1 = 1, e 1 = e, m 1 = m, ɛ 1 = ɛ, (3.1) e m = ɛ, e ɛ = m, m ɛ = 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 ɛ (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 ɛ excitations can be created locally. An anyon can be moved from one position to another by acting with a non-local 2 Z 2 lattice gauge theory with fermionic matter also gives rise to Z 2 topological order.

56 34 p v s y s x 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 s x and s y 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 Z 2 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 Z 2 topological order. We consider a L L square lattice with periodic boundary conditions (forming a torus), and we label vertices by v, edges by l, and square plaquettes by p. The degrees of freedom are spin-1/2 spins, residing on the edges. Local operators are then built from Pauli matrices σ µ l (µ = x, y, z) acting on the spin at l. We introduce operators associated with vertices and plaquettes, A v = σl x (3.2) l star(v) B p = l p σ z l, (3.3) where p contains the four edges in the perimeter of a square plaquette, and star(v) is the set of

57 35 s t 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 v, denoted with thick dark bonds (blue online). t is an open cut joining plaquettes p and p, 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 H = K e v A v K m p B p, (3.4) with K e, K m > 0. It is easy to see that [A v, A v ] = [ B p, B p ] = [Av, B p ] = 0, (3.5) rendering the Hamiltonian exactly solvable. The energy eigenstates can be chosen to satisfy A v ψ = a v ψ (3.6) B p ψ = b p ψ, (3.7) where a v, b p {±1}. The Hilbert space has dimension 2 2L2, so we need 2L 2 independent Hermitian operators with eigenvalues ±1 to form a complete set of commuting observables (CSCO), whose eigenvalues uniquely label a basis of states. Due to the periodic boundary conditions, v A v = p B p = 1, and the A v and B p only give 2L 2 2 independent operators. To obtain a CSCO, we need two additional operators, and one choice is given by L e x = σl z, Le y = σl z, (3.8) l s x l s y

58 36 with eigenvalues lx,y e {±1}, where s x, s y are non-contractible loops winding around the system in the x and y directions, respectively, as shown in Fig The eigenvalues {a v, b p, lx, e ly} e uniquely label a basis of energy eigenstates. In particular, there are four ground states with a v = b p = 1, a sign of Z 2 topological order. Excitations above the ground state reside at vertices with a v = 1, and plaquettes with b p = 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 a v = 1 vertices as e particles, and b p = 1 plaquettes as m particles. ɛ excitations are e-m pairs. Acting on a ground state with σ z l creates a pair of e particles, at the two vertices touching l. Similarly, acting with σ x l creates two m particles in the two plaquettes touching l. 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 l forming a connected path, which may be either closed or open (see Fig. 3.2). Then we define L e s = σl z. (3.9) l s Suppose s is an open path with endpoints v 1 and v 2. If L e s acts on a ground state, it creates e particles at v 1 and v 2. Alternatively, acting on a state with an e particle at v 1 and none at v 2, L e s moves the e particle from v 1 to v 2. On the other hand, if s is a closed path and is contractible (i.e. does not wind around the torus), and if ψ 0 is a ground state, then L e s ψ 0 = ψ 0. 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 Alternatively, t can be viewed as a set of edges in the dual lattice forming a connected path. The m-string operator is then L m t = σl x. (3.10) l t Just as with e-strings, if t is an open cut, with endpoints in two plaquettes p 1, p 2, L m t can be used

59 to create a pair of m particles or to move a single m particle from one plaquette to another. If t is 37 a closed, contractible cut, L m t gives unity acting on a ground state. If the path s and the cut t cross n c (s, t) times, then L e sl m t = ( 1) nc(s,t) L m t L e s. (3.11) This can be used to verify that the e, m and ɛ excitations indeed obey the braiding statistics of Z 2 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 T C(G), where in this paper G is always the square lattice space group. We will also introduce a smaller family of models T C 0 (G) T C(G). These two families are distinguished in that spin-orbit coupling (as defined below) is allowed for models in T C(G), but is absent in T C 0 (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. Spin-1/2 degrees of freedom reside on edges, and we again denote work with Pauli matrices σ µ l (µ = x, y, z) acting on the spin at edge l E. To proceed, it is helpful to introduce some notation and terminology. A path is a sequence of edges s = l 1 l 2 l n joining successive vertices; that is, l i and l i+1 are incident on a common vertex. Paths are considered unoriented, so that l 1 l 2 l n = l n l 2 l 1. The set of all paths is denoted W. A path may either be open with distinct endpoints v 1, v 2 V, or closed. Two open paths s and s sharing an endpoint can be composed into the path ss. Since an edge may appear in s more than once, more precisely the definition (3.9) of e-string operator should be understood

60 38 as L e s = l s σ z l = σz l 1 σ z l 2 σ z l n, (3.12) for for s = l 1 l 2 l n. Since operators in the product commute, there is no harm to interpret s as multiset of edges as well. In this paper, we use the product notation l X for all three cases: X is a set, a multiset or a sequence of edges. The set of open paths is denoted W o W, while closed paths are called cycles, and the set of cycles is C W. e-string operators are defined on paths s W by L e s = l s σz l. 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, A v = l star(v) σ x l (3.13) B p = l p σ z l, (3.14) where p P, and star(v) is again the set of edges touching v. It is again easy to see that [A v, A v ] = [ B p, B p ] = [Av, B p ] = 0. (3.15) The Hamiltonian is H = v V K e va v p P K m p B p, (3.16) where now the coefficients K e v, K m p are allowed to depend on the vertex or plaquette. Only the signs of the coefficients will be important, so for convenience of notation we take K e v, K m p {±1}. Energy eigenstates can again be labeled by a v, b p {±1}, the eigenvalues of A v and B p, respectively. Any ground state will satisfy a v = Kv e and b p = Kp m, 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 state with a v = Kv, e b p = Kp m. 3 3 This is the case provided Kv e and Kp m are compatible with constraints obeyed by A v and B p operators. More

61 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 gv, g gl. G is generated by translation x x + 1 (T x ), reflection x x (P x ), and reflection x y (P xy ). Translation by y y + 1 is given in terms of the generators by T y = P xy T x P xy. The group G can be defined in terms of the generators by requiring them to obey the relations, P 2 x = 1, (3.17) P 2 xy = 1, (3.18) (T x P x ) 2 = 1, (3.19) (P x P xy ) 4 = 1, (3.20) T x T y Tx 1 Ty 1 = 1, (3.21) T y P x Ty 1 Px 1 = 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 v = (T x ) nx (T y ) ny v if and only if n x, n y = 0 mod L, with the same statement holding for all l 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 precisely, we have v Av = 1, which implies Ke v must satisfy v Ke v = 1. In addition, suppose P is a subset of P for which p P B p = 1, then we must have p P K m p = 1.

62 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(v 1 ) = P(v 2 ) for v 1 v 2. It is always possible to choose P(l) 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 C 0 C be the set of contractible cycles. The assumption that contractible cycles can be decomposed into plaquettes means that, given s C 0, there exists {p 1,..., p n } P so that L e s = n i=1 B p i. 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 s x and s y that wind around the torus in the x and y directions, respectively. The operators {A v }, {B p }, L e s x and L e s y form a complete set of commuting observables (Appendix A). Denoting eigenvalues of L e s x,s y by l e x,y {±1}, it is then easy to see that H has a four-fold degenerate ground state, corresponding to the four choices of l e x,y with the other eigenvalues fixed to a v = K e v and b p = K m p. Just as for the square lattice toric code, e particles lie at vertices where a v = K e v; that is,

63 41 where a v differs from its ground state value. For s W o, the e-string operator L e s 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 Z 2 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(l) contains at most a finite number of points, at which the curve is not tangent to P(l). 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. We define a m-string operator on a cut t W by L m t = l t σx l. If t is an open cut, then Lm t 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 excitations can be created by first acting with L m t 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 U g be the unitary operator representing g G, we consider U g σ x l U 1 g = c x l (g)σx gl, U gσ z l U 1 g = c z l (g)σz gl, (3.24) assuming symmetries do not swap anyon species. Since U g σ x,z l Ug 1 are hermitian and unitary simultaneously, we must have c x,z l (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 merely moves degrees of freedom (i.e. σ µ l σµ gl ).4 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

64 general action of symmetry with the property that e-strings are taken to e-strings, and m-strings to m-strings; for example, U g L e su 1 g = (±1)L e gs. 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, 42 U g1 U g2 σ x,z l Ug 1 2 Ug 1 1 = U g1 g 2 σ x,z l Ug 1 1 g 2. (3.25) This imposes the restriction c x,z g 2 l (g 1)c x,z l (g 2 ) = c x,z l (g 1 g 2 ), (3.26) which holds for all l E and g 1, g 2 G. These conditions do not fix the overall U(1) phase of U g, which can be adjusted (as a function of g) as desired. γ x,z l The phase factors c x,z l (g) can be modified by the unitary gauge transformation σ x,z l σ x,z l, with γ x,z l {±1}, which sends c x,z l (g) γ x,z l γ x,z gl cx,z l (g). (3.27) It is always possible to choose a gauge where c x,z l (T ) = 1, for all l E and all translations T G; this is so because c x,z l (T x ) and c x,z l (T y ) 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 l with translation. We shall make this gauge choice without further comment throughout the paper. If, in addition, it is possible to choose a gauge where c x,z l (g) = 1 for all l E and g G, then by definition the model is in T C 0 (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 ψ 0e and degrees of freedom (e.g. electron spins, bosonic atoms) that can be microscopic constituents of a condensed matter system.

65 43 make a choice of phase for U g so that U g ψ 0e = ψ 0e (3.28) L e s x ψ 0e = L e s y ψ 0e = ψ 0e, (3.29) where s x and s y 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 U g, combining Eq. (3.28) with Eq. (3.25) implies U g1 U g2 = U g1 g 2. From now on, when we study e particle excitations, we always focus on states that can be constructed by acting on ψ 0e with e-string operators. Appendix B also shows that, for L even, there is a ground state ψ 0m and a phase choice for U g, satisfying U g ψ 0m = ψ 0m (3.30) L m t x ψ 0m = L m t y ψ 0m = ψ 0m. (3.31) Here, the electric strings have been replaced with magnetic strings, with t x and t y 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 ψ 0m. It should be noted that ψ 0e and ψ 0m cannot be the same state, because, for instance, L e s x and L m t y anticommute. Moreover, the phase choice required to make ψ 0e symmetry-invariant may not be the same as the corresponding choice for ψ 0m. 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 ψ 0e to construct e particle states, and similarly ψ 0m for m particle states, simply provides a convenient means to calculate the e and m fractionalization classes.

66 Fractionalization and Symmetry Classes 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 ). 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 ɛ 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 ψ ee with two e particles, labeled 1 and 2. Following the arguments of Ref. [107], we assume that U g ψ ee = U e g (1)U e g (2) ψ ee, (3.32) where U e g (i) gives the action of symmetry on anyon i = 1, 2. The physics is invariant under a redefinition U e g (i) λ(g)u e g (i), λ(g) {±1}, (3.33) which we refer to as a projective transformation. The reason for this terminology is that the U e g operators form a projective representation of G, expressed by writing U e g 1 U e g 2 = ω e (g 1, g 2 )U e g 1 g 2, (3.34) where we have suppressed the anyon label i, and ω e (g 1, g 2 ) {±1} is referred to as a Z 2 factor set.

67 45 The factor set satisfies the condition ω e (g 1, g 2 )ω e (g 1 g 2, g 3 ) = ω e (g 2, g 3 )ω e (g 1, g 2 g 3 ), (3.35) which follows from the associative multiplication of U e g operators. The factor set is not invariant under projective transformations, but instead transforms as ω e (g 1, g 2 ) λ(g 1 )λ(g 2 )λ(g 1 g 2 )ω e (g 1, g 2 ). (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 [ω e ] the equivalence class containing the factor set ω e. 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 H 2 (G, Z 2 ). The discussion proceeds identically for m particles, with ω m the corresponding factor set, and [ω m ] H 2 (G, Z 2 ) the fractionalization class. A complete specification of fractionalization classes defines a symmetry class. It is enough to specify [ω e ] and [ω m ], because these determine uniquely the ɛ fractionalization class.[107] Therefore a symmetry class is specified by the pair S = [ω e ], [ω m ]. (3.37) Because all properties of Z 2 topological order are invariant under e m (see Sec. 3.2), symmetry classes related by this relabeling are considered equivalent, that is [ω e ], [ω m ] [ω m ], [ω e ]. (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 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

68 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 ([ω e ], [ω m ]). To determine fractionalization and symmetry classes, it is convenient to work with the generators and their relations [Eqs. ( )]. Focusing on e particles for concreteness, the Ug e operators obey the group relations up to possible minus signs, that is (U e P x ) 2 = σ e px (3.39) (U e P xy ) 2 = σ e pxy (3.40) (U e T x U e P x ) 2 = σ e txpx (3.41) (U e P x U e P xy ) 4 = σ e pxpxy (3.42) U e T x U e T y (U e T x ) 1 (U e T y ) 1 = σ e txty (3.43) UT e y UP e x U e (U e Ty 1 P x ) 1 = σtypx, e (3.44) where σpx e {±1}, and similarly for the other σ e parameters. The σ e s are invariant under projective transformations, and moreover uniquely specify the fractionalization class [ω e ].[107] In addition, it was shown that each of the 2 6 = 64 possible choices of the σ e s is mathematically possible; that is, there exists a projective representation for all choices of σ e s.[107] The same considerations lead to six σ m 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 Fractionalization and symmetry classes in the solvable models The solvable models are well-suited to the study of fractionalization and symmetry classes because the U e g and U m g operators can be explicitly constructed. We focus first on e particles. It is

69 47 sufficient to consider states with only two e particle excitations, of the form ψ e (s) = L e s ψ 0e, (3.45) with s an open path, and e particles residing on the endpoints v 1 (s) and v 2 (s). The action of symmetry on this state is given by U g ψ e (s) = c z s(g) ψ e (gs), (3.46) where c z s(g) = l s cz g(l). The goal is to construct and study operators Ug e that act on single e particles, reproducing the action of U g on states ψ e (s). 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 number fg e (v) {±1} and a path s e g(v). The path s e g(v) has endpoints v and gv. (Note that s e g(v) is a cycle or a null path if gv = v.) From this data we form the operator U e g (v) = f e g (v)l e s e g(v). (3.47) 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 U e g (v) operators to obey the relation U g ψ e (s) = U e g [v 1 (s)]u e g [v 2 (s)] ψ e (s), (3.48) which has to hold for all open paths s W o and all g G. We refer to a set of Ug e (v) operators satisfying this relation as an e-localization of the symmetry G. It should be noted that there is some redundancy in the data used to define Ug e (v). Keeping its endpoints fixed, the path s e g(v) can be deformed arbitrarily, at the expense of a phase factor. When acting on states ψ e (s) 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 fg e (v).

70 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 of G. Moreover, the e-localization is unique up to projective transformations Ug e (v) λ(g)ug e (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 U e g 1 (g 2 v)u e g 2 (v) = F (g 1, g 2, v)u e g 1 g 2 (v), (3.49) where F (g 1, g 2, v) {±1}, and this equation holds acting on all states containing no m particle excitations [including ψ e (w) ]. This relation holds because both sides of the equation are e string operators joining v to g 1 g 2 v, and can differ only by a phase factor depending on g 1, g 2 and v. We now show that F (g 1, g 2, v) is independent of v, and forms a Z 2 factor set, so that we can write F (g 1, g 2, v) = ω e (g 1, g 2 ). Suppose that for some g 1, g 2, and some vertices v i, v j, we have F (g 1, g 2, v i ) F (g 1, g 2, v j ). Then consider the state ψ e (s ij ), where s ij is a path joining v i to v j. We have U g1 g 2 ψ e (s ij ) = U g1 U g2 ψ e (s ij ) (3.50) = Ug e 1 (g 2 v i )Ug e 2 (v i )Ug e 1 (g 2 v j )Ug e 2 (v j ) ψ e (s ij ) = F (g 1, g 2, v i )F (g 1, g 2, v j )Ug e 1 g 2 (v i )Ug e 1 g 2 (v j ) ψ e (s ij ) = U g1 g 2 ψ e (s ij ), a contradiction. This shows F = F (g 1, g 2 ), independent of v. The associativity condition required for F (g 1, g 2 ) to be a factor set follows from equating the two ways of associating the product in U e g 1 (g 2 g 3 v)u e g 2 (g 3 v)u e g 3 (v) ψ e (s), (3.51) where ψ e (s) has one e particle at v. Thus we have shown U e g 1 (g 2 v)u e g 2 (v) = ω e (g 1, g 2 )U e g 1 g 2 (v), (3.52)

71 49 with ω e a Z 2 factor set. This operator equation holds acting on all states of the form ψ e (s), and more generally on states with any number of e particle excitations created by acting on ψ 0 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 [ω e ] 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 [ω e ]. In particular, we may focus on a single e particle at any desired location, and determine [ω e ] by calculating appropriate products of Ug e (v). In particular, we can calculate the products of generators in Eqs. ( ), and determine the σ e parameters. There is then no need to check that the resulting σ e 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 ψ m (t) = L m t ψ 0m, (3.53) where t is an open cut. To every pair (g, h) G H, where the m particle resides at the hole h, we associate a number f m g (h) and a cut t m g (h), which joins h to gh. This allows us to write U m g (h) = f m g (h)l m t m g (h). (3.54) From this point, the discussion for e particles goes over to the m particle case, with only trivial modifications. We refer to a set of Ug m (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

72 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 [ω e ], 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 Model realizing all e particle fractionalization classes Here, we present a model that can realize all possible e-fractionalization classes [ω e ], 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 P i P be the set of all plaquettes of type i (i = 1,..., 6), the Hamiltonian is H = K e v V A v 6 i=1 K m i p i P i B pi. (3.55) We choose K e = 1, with arbitrary K m i {±1}, and note that b i K m i is the ground-state eigenvalue of B pi. Following the calculation procedure described below, we find σ e px = b 1, σ e pxy = b 2, σ e txpx = b 3, (3.56) σ e pxpxy = b 4, σ e txty = b 4 b 5, σ e typx = b 1 b 3 b 4 b 6, (3.57) from which it is clear that each possible [ω e ] H 2 (G, Z 2 ) is realized in this model for appropriate choice of K m i. In addition we find that all the corresponding σ m s are unity, and thus [ω m ] is the trivial fractionalization class.

73 51 y K 3 m K 1 m v 1 v 2 K 2 m l 2 l 1 x (a) K 4 m K 6 m K 4 m K 4 m K 6 m K 4 m K 6 m K 5 m K 6 m K 6 m K 5 m K 6 m K 4 m K 6 m K 4 m K 4 m K 6 m K 4 m (b) (c) Figure 3.3: (a) The lattice on which all 2 6 = 64 e particle fractionalization classes can be realized. There are six types of plaquettes not related by symmetries, and the correponding plaquette terms are assigned independent coefficients Ki m (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 v 1, v 2 and two edges l 1, l 2 are labeled to illustrate the calculation of σpx e 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.

74 52 We now illustrate how these results are obtained by working through the determination of (UP e x ) 2 = σpx e as an example. It follows from the discussion of Sec that σpx e can be obtained by considering an e particle at any desired vertex v 1, and then computing (UP e x ) 2 acting on this e particle. We consider an e particle at vertex v 1 as shown in Fig. 3.3a, so that v 2 = P x v 1, and the vertices v 1 and v 2 are joined by edges l 1, l 2 forming a type i = 1 plaquette. (To be more precise, we should also specify the position of a second e particle at vertex v v 1, let s 0 be a path joining v 1 to v, and consider the state ψ e (s 0 ) = L e s 0 ψ 0e. However, the result for σpx e will be independent of v.) We are free to choose the e-localization UP e x (v 1 ) = σl z 1 (3.58) UP e x (v 2 ) = fσl z 2, (3.59) where f = ±1. To determine f, we consider the path s = l 1, which has end points v 1 and v 2. Then we have U Px ψ e (s) = U Px σl z 1 ψ 0e = σl z 2 ψ 0e, (3.60) since P x l 1 = l 2. But we also have U Px ψ e (s) = UP e x (v 1 )UP e x (v 2 ) ψ e (s) (3.61) = (σl z 1 )(fσl z 2 )σl z 1 ψ 0e (3.62) = fσl z 2 ψ 0e. (3.63) Consistency of these two calculations of the action of U Px then requires f = 1. Now that we have fixed the form of the e-localization, we can compute the action of Px 2 on the e particle at v 1. We have σpx e = (UP e x ) 2 (v 1 ) = UP e x (v 2 )UP e x (v 1 ) (3.64) = σl z 2 σl z 1 = K1 m = b 1. (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 ψ 0e. In particular it holds acting on a state of interest,

75 ψ e (s), with one e particle located at v 1. The results for the other σ e parameters can be obtained by straightforward analogous calculations Toric code models without spin-orbit coupling We now proceed to consider the family of models T C 0 (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, we have U g σ µ l U 1 g = σ µ gl. 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 T C 0 (G). The main result is the following theorem: Theorem 1. The TC symmetry classes in A, B, C, M, M 1, M 2 and M 3 are not realizable in T C 0 (G), where A = { σpxpxy e = σpxpxy m = 1 }, B = { σpxpxyσ e txty e = σpxpxyσ m txty m = 1 }, C = { σpxpxyσ e typx e = σpxpxyσ m typx m = 1 }, M = { σpx m = 1 σpxy m = 1 σtxpx m = 1 }, M 1 = { σpxpxy m = 1 ( σpx e = 1 σpxy e = 1 )}, M 2 = { σpxpxyσ m txty m = 1 ( σpxy e = 1 σtxpx e = 1 )}, M 3 = { σpxpxyσ m typx m = 1 ( σpx e = 1 σtxpx e = 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 T C 0 (G). This theorem is proved in Appendix C.1.1, except for the last statements regarding counting

76 54 a o a o a Κ b 2 b 3 b 1 b 2 b 3 b 1 b 2 b 1 b 3 (a) ( b3 1 1 b 1 1 b 2 b aõ 1 ) (b) ( 1 1 b3 1 b 1 b 2 b aõ 1 1 ) ( 1 b3 1 b (c) 1 b a κ ) b a o a o a Κ a o b a o a Κ a o (d) ( b a o aõ 1 ) ( b 1 (e) a o 1 a κ ) ( (f) a o aõ a κ ) Figure 3.4: T C 0 (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 a r is the ground state eigenvalue of A v for v at special points r = o, õ, κ; and b, b 1, b 2 are the ground state eigenvalues of B p for the plaquette p, which in these models is picked to be the smallest cycle made with black edges where b, b 1 or b 2 is written, while b 3 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 origin by ( 1 2, 2) 1 results in a transformation (3.74): Px T x P x, σ px σ txpx, σ pxpxy σ pxpxy σ txty. 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.

77 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. As an illustration, we would like to show that σ m px = 1 for any model in T C 0 (G). (This is part of the fact that TC symmetry classes in M are not realizable in T C 0 (G).) Consider a m particle located at a hole h 0 H. If P x h 0 = h 0, then we can choose UP m x (h 0 ) = 1, and therefore σpx m = (UP m x ) 2 (h 0 ) = 1. We then consider the case P x h 0 = h 1 h 0. We can always draw a simple cut t joining h 0 to h 1, so that P x t = t. We are then free to choose the m-localization U m P x (h 0 ) = L m t (3.66) U m P x (h 1 ) = fl m t, (3.67) where f = ±1 needs to be determined. To do this, consider the state ψ m (t) = L m t ψ 0m, for which we have U Px ψ m (t) = U Px L m t ψ 0m = ψ m (t), (3.68) where we used the fact that U Px L m t U 1 P x = L m P xt = Lm t. (Note that here we use the assumption of no spin-orbit coupling.) But we also have U Px ψ m (t) = U m P x (h 0 )U m P x (h 1 ) ψ m (t) = f ψ m (t). (3.69) Consistency of these two calculations requires f = 1, and we can then calculate P 2 x acting on the m particle located at h 0, to obtain σ m px = (U m P x ) 2 (h 0 ) = (3.70) = U m P x (h 1 )U m P x (h 0 ) = (L m t ) 2 = 1. (3.71) We have thus shown that σpx m = 1 for any model in T C 0 (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

78 56 (shown in Figures 3.3 and 3.4). Here, and throughout the paper, we will find it convenient to present TC symmetry classes ([ω e ], [ω m ]) in the matrix form σe px σpxy e σtxpx e σpxpxy e σpx m σpxy m σtxpx m σpxpxy m ( σ e pxpxy σtxty e ) ( σ e pxpxy σtypx) e ( σ m pxpxy σtxty m ) ( σ m pxpxy σtypx) m, (3.72) or, equivalently, σpx e σpxy e σtxpx e σpxpxy e σpxpxyσ e txty e σpx m σpxy m σtxpx m σpxpxy m σpxpxyσ m txty m. (3.73) σ e pxpxyσ e typx σ m pxpxyσ m typx This form allows for simple comparison to the constraints of Theorem 1. In addition, under the change of origin o ( 1 2, 1 2), the entries of the matrix are simply permuted: σe 1 σ2 e σ3 e σ4 e σ5 e σ6 e σ1 m σ2 m σ3 m σ4 m σ5 m σ6 m σe 3 σ2 e σ1 e σ5 e σ4 e σ6 e σ3 m σ2 m σ1 m σ5 m σ4 m σ6 m. (3.74) This holds even beyond the setting of solvable toric code models, and can be verified by replacing P x as a generator of G by P x P x = T x P x, 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, by noting that U ã P x = φ a U a T x U a P x, where a = e, m and φ a {±1}. 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 b a o 1 a κ (3.75)

79 57 and b a o a κ, (3.76) 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 D i, i = 0, 1, 2, 3. In D i, there are i of σpxpxy, m σtxty m and σtypx m equal to 1. In D 0, we have TC symmetry classes in the form , where the symbol means that the corresponding σ parameter can be chosen to be ±1 independently of any other parameters. Therefore, D 0 = 2 6. These TC symmetry classes are realized in the model discussed in Sec , and shown in Fig In D 1, we have TC symmetry classes ([ω e ], [ω m ]) in the form ,, or , so D 1 = These TC symmetry classes are realized in the models shown in Fig. 3.4(a-c).

80 58 In D 2, we have TC symmetry classes in the form ,, or , so D 2 = 3 2. These TC symmetry classes are realized in Fig. 3.4(d,e). In D 3, we have only the single TC symmetry class which is realized by the model of Fig. 3.4f , (3.77) In total, there are thus exactly 3 i=0 D i = 95 TC symmetry classes realized by models in T C 0 (G). Recalling that the TC symmetry classes ([ω m ], [ω e ]) and ([ω e ], [ω m ]) 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 T C 0 (G) is = 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 have U g σ µ l U 1 g = c µ l (g)σµ gl, where cµ l (g) {±1}, µ = x, z. The corresponding family of models is referred to as T C(G). Our results on these models are summarized in the following theorem:

81 59 h 1 h 2 l 1 b a Κ a o l 1 l 2 l 3 a o l 2 a o ( 1 1 γ2 1 b γ (a) 1 α a o 1 α 2 ) ( (b) c 1 c 2 c 3 a o aõ c 1 c 3 a κ ) Figure 3.5: Two example models in T C (G) that realize TC symmetry classes not possible in T C 0 (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 a r is the ground state eigenvalue of A v for v at special points r = o, õ, κ and b is the ground state eigenvalue of B p for the plaquette p, defined here to be the smallest cycle enclosing the letter b. We write α i = c x l i (P x ), β i = c x l i (P xy ), γ i = c z l i (P x ) and δ i = c z l i (P xy ). (a) A model realizing some TC symmetry classes (and symmetry classes) that cannot be realized without spinorbit coupling. Here,h 1, h 2 label two positions of a m particle for the calculation of σ m px = α 1 in the main text. (b) A model realizing all 2 6 = 64 possible m particle fractionalization classes [ω m ]. Here, for simplicity, we make the restriction γ i = δ i c i.

82 Theorem 2. The TC symmetry classes in P 1, P 2, P 3, A, B and C are not realizable in T C (G), where 60 P 1 = { σpx e = σpx m = 1 }, P 2 = { σpxy e = σpxy m = 1 }, P 3 = { σtxpx e = σtxpx m = 1 }, A = { σpxpxy e = σpxpxy m = 1 }, B = { σpxpxyσ e txty e = σpxpxyσ m txty m = 1 }, C = { σpx e = σtxpx e = σpxpxyσ e typx e = σpxpxyσ m typx m = 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 T C(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 possible if spin-orbit coupling is included. For example, take the calculation of σpx. m Suppose U Px σl x 1 U Px = α 1 σl x 1, with α 1 {±1}. If we choose UP m x (h 1 ) = σl x 1, then we must have UP m x (h 2 ) = α 1 σl x 1 to ensure UP m x (h 1 ) UP m x (h 2 ) σl x 1 ψ m0 = U Px σl x 1 ψ m0. Therefore we have σpx m = (UP m x ) 2 (h 1 ) = UP m x (h 2 )UP m x (h 1 ) = α 1. Therefore we can have σpx m = 1, which is impossible without spin-orbit coupling. Another particularly interesting example, shown in Fig. 3.5b, is a model realizing all 2 6 = 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

83 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 Z 2 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 2 6 = 64 e particle fractionalization classes as the parameters in its Hamiltonian are varied. Second, we considered a restricted family of models T C 0 (G) without spin-orbit coupling, but defined on general two-dimensional lattices. We showed that exactly 95 TC symmetry classes ([ω e ], [ω m ]), corresponding to 82 symmetry classes [ω e ], [ω m ], are realized by models in T C 0 (G). This result was established by proving that the other TC symmetry classes cannot be realized by any model in T C 0 (G), and giving explicit models for those classes not ruled out by such general arguments. Finally, in the most general family of models considered, T C(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 T C(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 T C(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 T C(G) that can be realized in local bosonic models. Second, we establish a connection

84 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 [ω m ] is a m fractionalization class realized for a model in T C 0 (G), then the symmetry class [ω e ], [ω m ], where [ω e ] H 2 (G, Z 2 ) 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 T C(G). For example, the symmetry classes in A are unrealizable in T C(G) (Theorem 2), but they are possible here. The starting point for the construction is a Hamiltonian of the form H = v V K e va v p P B p, (3.78) where K e v {±1}. We take the symmetry to act without spin-orbit coupling, so this is a model in T C 0 (G). We have chosen K m p = 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 T C 0 (G), because without spin-orbit coupling the m fractionalization class only depends on the lattice and on the Kv e coefficients. We now build a Z 2 gauge theory based on the above toric code model. On each vertex v we introduce a boson field created by b vα, where α = 1,..., n is an internal index. We also introduce the gauge constraint A v = Kv( 1) e b vαb vα, (3.79) with sums over repeated internal indices implied. The gauge theory Hamiltonian is taken to be H gauge = p P B p + u v V b vαb vα h l E σ x l, (3.80) with u > 0. We choose symmetry to act on the boson field by U g b vαu 1 g = D αβ (g)b gv,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

85 63 shown that there exists a finite-dimensional projective representation for any fractionalization class [ω e ] H 2 (G, Z 2 ), so the bosons can be taken to transform in any desired fractionalization class. We now discuss two limits of H gauge. First, we consider the limit h +. In this limit, we have σ x l = 1, and the only remaining degrees of freedom are the bosons. The gauge constraint becomes even, K b v e = 1 vαb vα = odd, Kv e = 1. (3.82) 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] H gauge 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 H gauge 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 H gauge. Now we consider the exactly solvable limit of H gauge with h = 0. This limit is deep in the deconfined phase of the Z 2 gauge theory, and we have B p = 1, A v = Kv, e and b vαb vα = 0 acting on ground states. Because A v = Kv, e the m particles feel the same pattern of background Z 2 charge as in the original T C 0 (G) toric code model, and their fractionalization class is unchanged. Now, however, the Z 2 -charged bosons become the e particle excitations, so the e particle fractionalization class [ω e ] is determined by the (arbitrarily chosen) projective representation D(g). We have thus obtained a phase with Z 2 topological order and symmetry class [ω e ], [ω m ], 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 G o, where G is the space group, and G o 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 G o 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 G o G/T 2,

86 64 where T 2 is the normal subgroup of G generated by translations Tx 2 and Ty 2. In this case, G o 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 Z 2 topological order, and the action of symmetry is described by e and m fractionalization classes [ω e ] and [ω m ]. 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 G o. That is, space group symmetry G acts linearly on e and m particles, and elements g G commute with g o G o 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 G o that can be realized in local bosonic models also compatible with an arbitrary space group symmetry G? Or, are there G o symmetry classes that are only compatible with a given space group G if some elements of G o 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 G G G o, defined as the set of all elements of the form (g, φ(g)) G G o, where g G is arbitrary, and φ : G G o 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 see that new [ω e ] and [ω m ] fractionalization classes for G symmetry are induced by corresponding G o 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.

87 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 lowerdimensional 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 Z 2 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 Z 2 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 Z 2 Z 2 symmetry. It can provides boundary degrees of freedom carrying

88 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 Reflection Symmetry Protected Topological Phyases 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 P x : (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 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) P x [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 Z 2 symmetry. 1 It is generally believed, although not proven rigorously to our knowledge, that there are only two two-dimensional SPT phases protected by Z 2 unitary on-site symmetry, which are labelled by the two elements of H 3 (Z 2, 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.

89 67 Figure 4.1: A quantum spin system that respects a single reflection P x : (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) P x [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 Z 2 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 P x : (x, y, z) ( x, y, z) and P y : (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) P x [U (t)] P y [U (t)] P x P y [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 H 3 (Z 2, U (1)) H 3 (Z 2, U (1)) H 2 (Z 2 Z 2, U (1)) = Z 3 2, 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.