PHYSICAL REVIEW X 6, 041068 (2016)

Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface

Meng Cheng,1 Michael Zaletel,1 Maissam Barkeshli,1 Ashvin Vishwanath,2 and Parsa Bonderson1 1Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA 2Department of Physics, University of California, Berkeley, California 94720, USA (Received 15 August 2016; revised manuscript received 26 November 2016; published 29 December 2016) The Lieb-Schultz-Mattis theorem and its higher-dimensional generalizations by Oshikawa and Hastings require that translationally invariant 2D spin systems with a half-integer spin per unit cell must either have a continuum of low energy excitations, spontaneously break some symmetries, or exhibit topological order with anyonic excitations. We establish a connection between these constraints and a remarkably similar set of constraints at the surface of a 3D interacting topological insulator. This, combined with recent work on symmetry-enriched topological phases with on-site unitary symmetries, enables us to develop a framework for understanding the structure of symmetry-enriched topological phases with both translational and on-site unitary symmetries, including the effective theory of symmetry defects. This framework places stringent constraints on the possible types of symmetry fractionalization that can occur in 2D systems whose unit cell contains fractional spin, fractional charge, or a projective representation of the symmetry group. As a concrete application, we determine when a topological phase must possess a “spinon” excitation, even in cases when spin rotational invariance is broken down to a discrete subgroup by the crystal structure. We also describe the phenomena of “anyonic spin-orbit coupling,” which may arise from the interplay of translational and on-site symmetries. These include the possibility of on-site symmetry defect branch lines carrying topological charge per unit length and lattice dislocations inducing degeneracies protected by on- site symmetry.

DOI: 10.1103/PhysRevX.6.041068 Subject Areas: Condensed Matter Physics

I. INTRODUCTION are topologically nontrivial quasiparticles that carry half integer-valued spin [5,6]. In contrast, the local excitations, The celebrated Lieb-Schultz-Mattis (LSM) theorem such as local spin flips, of such systems always carry integer- [1,2], including the higher-dimensional generalizations valued spin. One aim of our work is to put this understanding by Oshikawa [3] and Hastings [4], shows that translation- on more solid formal footing and, thereby, derive the most ally invariant spin systems with an odd number of S ¼ 1=2 general relation between the symmetry properties of the unit moments per unit cell cannot have a symmetric, gapped, cell and the fractional quantum numbers of the emergent and nondegenerate ground state. Either some symmetries of excitations. the system are broken spontaneously, the phase is gapless, Intriguingly similar constraints as those required by the or there is nontrivial topological order with emergent LSM theorem have recently been discovered to arise at the quasiparticle excitations having exotic exchange statistics. surface of symmetry-protected topological (SPT) phases The LSM theorem is remarkable because a microscopic [7–20]. Such phases are short-range entangled states that property, i.e., the spin within a unit cell, constrains the cannot be adiabatically connected to a trivial product state universal long-wavelength physics in a nontrivial way. while preserving the symmetries of the system. An eminent Moreover, it generates powerful practical implications. For example is the 3D topological insulator (TI) protected by example, experimentally ruling out symmetry breaking can time-reversal and charge conservation symmetries, which be sufficient to imply the presence of exotic spin-liquid (at least for weak interactions) has a gapless surface with a physics [5,6]. – Spin-liquid folklore informs us that when a spin- single Dirac cone [21 23]. In general, the surface phases at the boundary of a SPT must also be either symmetry rotationally invariant system with an odd number of S¼1=2 broken, gapless, or gapped with nontrivial “surface topo- moments per unit cell forms a symmetric gapped spin liquid, ” it must also possess deconfined “spinon” excitations, which logical order, mirroring the three options allowed by the LSM theorem. In the case where the 3D SPT phase possesses surface topological order and preserves the symmetries, the 2D Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- surface states are anomalous symmetry-enriched topologi- bution of this work must maintain attribution to the author(s) and cal (SET) phases. Such surface states are anomalous in the the published article’s title, journal citation, and DOI. sense that they cannot be consistently realized in purely 2D

2160-3308=16=6(4)=041068(29) 041068-1 Published by the American Physical Society MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) systems when the symmetry is realized in a local manner. Such anomalies are signaled by an object called the “obstruction class” having nontrivial values (in the corre- sponding cohomology group) [8,18]. The anomalous sur- face SET can provide a nonperturbative characterization of the bulk SPT order, as the symmetry action in the SET must have exactly the right type of “gauge anomaly” required by the nontrivial SPT bulk. In this paper, we develop a new perspective on the LSM theorem by exploiting a relation to the surface of SPT FIG. 1. A 3D system of integer spins can enter a weak SPT states. A crucial role is played by “weak” SPT phases, phase equivalent to stacking together 1D AKLT chains which are SPT phases protected by translational symmetry (as indicated on the left). The weak SPT phase is characterized Ω ∈ [7,22]. We argue that the LSM constraints are, in a precise by a particular element of the cohomology class weak H4 Z3 3 1 sense, a special case of the constraints at the surface of ½ trans ×SOð Þint; Uð Þ. Since each AKLT chain has degen- erate emergent S ¼ 1=2 edge states, the effective Hilbert space of weak SPTs. For example, a 2D spin system with S ¼ 1=2 the 2D surface of the 3D SPT phase behaves like a S ¼ 1=2 per unit cell can be thought of as the boundary of a 3D magnet: it will have a projective (half-integral) representation of Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a cubic SO(3) in each unit cell. Thus, studying surface phases of the 3D lattice with S ¼ 3 per unit cell [24], which is an example of weak SPT is equivalent to studying a 2D S ¼ 1=2 magnet. Given a 3D weak SPT. a proposal for the braiding, statistics, and symmetry fractionali- More specifically, we posit a bulk-boundary “anomaly zation of the 2D magnet, there is a procedure to calculate an O ∈ H4 Z2 3 1 matching” condition when a 2D SET phase is the gapped obstruction class ½ trans ×SOð Þint; Uð Þ, associated and symmetric surface state realized on the boundary with defining an effective theory of the symmetry defects. The of a 3D SPT phase. This bulk-boundary correspondence bulk-boundary correspondence requires O of the boundary and Ω requires that the components of the 2D SET obstruction weak of the bulk to be compatible, constraining the allowed 2D SET orders. class match the corresponding bulk 3D SPT invariants. This anomaly matching allows us to precisely formulate a restriction on what kind of SET order can exist at the when the microscopic degrees of freedom transform as a surface of such a given 3D SPT. A consequence of this nontrivial projective representation of the on-site symmetry bulk-boundary correspondence is that the physical proper- for each unit cell. More significantly, our arguments lead to ties associated with the 3D SPT invariants, namely the a far more general and constraining version of the LSM nature of the emergent boundary modes, imply which SET theorem. It applies to arbitrary on-site unitary symmetries, obstruction classes can be physically realized in purely 2D including discrete symmetries, which cannot be probed by systems in a local and symmetry preserving manner. In the adiabatic flux insertion arguments utilized by Oshikawa particular, the SET obstruction classes permitted in purely and Hastings. Moreover, it also provides sharper restric- 2D systems correspond to SPT phases whose boundary tions on the type of 2D SET order allowed. structure is, at most, a nontrivial projective representation As a concrete application, we address the following of the on-site symmetry per unit cell. This provides an question: under what general conditions must a SET phase understanding of the effective theory of symmetry defects possess a “spinon” excitation? We define a spinon to be an in purely 2D SET phases with translational and on-site excitation which either (a) transforms with the same unitary symmetries, and it requires the SET obstruction projective representation as the unit cell or (b) carries class to match the projective representation characterizing the same fractional U(1) charge as the particle filling the transformation of each unit cell under the on-site ν ¼ p=q. For example, the spinon of a spin liquid in an symmetry. As such, we obtain constraints on the type of S ¼ 1=2 magnet carries S ¼ 1=2, while the Laughlin-type SET order that can be realized in purely 2D systems from quasiparticles (generated by threading a unit quanta of flux) the manner in which the microscopic degrees of freedom of the ν ¼ p=q fractional quantum Hall (FQH) effect carry transform under the symmetry, in particular, from the the fractional charge e ¼ðp=qÞe. This question was projective representation per unit cell. These arguments partially addressed in Ref. [25], which left the question are summarized in Fig. 1. open for certain exotic fractionalization patterns of trans- This point of view immediately provides a simple lation symmetry. We find that a spinon is required if either physical “derivation” of the LSM theorem. Namely, the (1) the on-site symmetry group is continuous and con- bulk-boundary correspondence tells us that if the boundary nected or (2) the anyons are not permuted by the symmetry of a nontrivial 3D SPT phase is gapped and preserves the and there is a point-group symmetry that can relate the two symmetry, then it must also realize nontrivial topological directions of the lattice, such as C3, C4,orC6 lattice order. This further implies that a purely 2D gapped and rotations, or certain reflections. We also examine counter- symmetric phase must have nontrivial topological order examples in which either of the conditions is violated.

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The work is organized as follows. In Sec. II, we outline R of G. R can be either a linear or projective representation the logical structure of the argument. In Sec. III, we review of G. For example, spin S ¼ 1 corresponds to a linear the classification of 3D SPT phases and describe the representation of SO(3), while S ¼ 1=2 corresponds to Künneth formula decomposition of the SPT class in terms a projective representation of SO(3) (a 2π rotation of invariants associated with stacking, packing, and filling results in −1). In general, the possible projective repre- of lower-dimensional SPT phases. In Sec. IV, we review the sentations of a group G are classified by elements of the 2 theory and classification of symmetry fractionalization in cohomology class H ½G; Uð1Þ, where linear representa- symmetric 2D topological phases, explaining the notion of tions correspond to the trivial element ½1. For example, anyons carrying localized fractional charge and projective half-integer spins correspond to the nontrivial element in 2 representations. We discuss the inclusion of translational H ½SOð3Þ; Uð1Þ ¼ Z2. symmetry and the decomposition of the symmetry frac- For a system with U(1) charge conservation (such as tionalization classes in terms of anyonic flux per unit cell, particle number or magnetization), the average charge anyonic spin-orbit coupling, and on-site symmetry frac- ν ¼ðp=qÞ per unit cell defines the filling fraction. In some tionalization. In Sec. V, we review the algebraic theory of cases, the representation R of G is enough to specify the symmetry defects for 2D SET phases with on-site sym- fractional part of the filling with respect to a U(1) subgroup metry and their possible obstructions. We discuss the case of G. For example, in an SO(3) invariant S ¼ 1=2 system, of topological phases with symmetries that do not permute the zero magnetization corresponds to ν ¼ 1=2 filling of anyon types in greater detail. In Sec. VI, we discuss the any U(1) subgroup of SO(3). In other cases, ν may be the bulk-boundary correspondence and anomaly matching in only relevant quantity, such as when G ¼ Uð1Þ [because detail and formulate an understanding of the theory of H2½Uð1Þ; Uð1Þ is trivial]. defects and their obstruction classes for 2D SET phases In order to develop the outline of the argument, we with translational and on-site unitary symmetries. We briefly review SETs. Since anyons are topological objects, consider such 2D SETs in systems with projective repre- they may transform in a fractional manner under G, where sentations per unit cell and fractional charge per unit cell in “fractional” means as compared with the quantum numbers detail. We also describe in detail the phenomena of “any- carried by local operators. Note that even in an S ¼ 1=2 onic spin-orbit coupling,” and a number of resulting magnet, local operators (and, hence, excitations) carry physical consequences. In Sec. VII, we conclude with a integral representations of SO(3). For example, in a discussion of further directions and open questions. quantum spin liquid, all local operators correspond to spin flips and carry integer-valued spin, whereas the topologi- II. OVERVIEW OF THE ARGUMENT cally nontrivial spinon excitations carry half-integer spin. In fractional quantum Hall states, all local operators are Our starting point is a translationally invariant lattice composed of electron operators and therefore carry integer- Hamiltonian in two spatial dimensions, with an internal valued electric charge, while the quasiparticle excitations (on-site) symmetry group G and translational symmetry can carry fractional electric charge. group Z2. We denote the total symmetry group as Thus, in addition to the fusion and braiding statistics G ¼ Z2 × G. It is not only important to know the symmetry specified by the topological order, SET phases are char- group G, but also the particular way in which the micro- acterized by a pattern of “symmetry fractionalization.” We scopic degrees of freedom transform under G. For example, review this structure later, but for now the reader can take a fully spin-rotationally invariant magnet has G ¼ SOð3Þ, for granted the existence of mathematical objects ρ and ½w but it may be composed of integer or half-integer moments describing this pattern. Careful examination [18,26] reveals per unit cell. Likewise, a number conserving system with that not all choices of ρ and ½w allow for a well-defined corresponding symmetry group G ¼ Uð1Þ has a well- SET phase in 2D, as certain choices may be anomalous. defined charge density. These microscopic details can Mathematically, the anomaly is encoded in an obstruction constrain the emergent long wavelength physics in impor- class ½O, which is an element of the cohomology class tant ways. There are two scenarios (exemplified by these H4½G; Uð1Þ. Given the objects ρ and ½w for a proposed two examples) in which the microscopic degrees of free- symmetry fractionalization pattern, there is a well-defined dom in a unit cell transform “fractionally” under the procedure to obtain ½O from the properties of the topo- symmetries: (1) when each unit cell transforms as a logical phase [18,26,27], schematically represented by a projective representation of the on-site symmetry and grinder in Fig. 1. Note that if there are no anyonic (2) when the charge per unit cell of a U(1) symmetry is excitations, the fractionalization pattern is necessarily fractional. We analyze the consequences of having projec- trivial (i.e., ρ ¼ 1 and ½w¼½I), and whenever the tive representations per unit cell and fractional filling fractionalization pattern is trivial, so is the obstruction separately. class ½O¼½1. Given a symmetry G, the microscopic degrees of free- Since 3D SPT phases are also classified by the dom within a unit cell transform under some representation elements ½Ω ∈ H4½G; Uð1Þ, this naturally suggests that

041068-3 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) a fractionalization pattern given by ρ and ½w with a and, by matching ½O with the weak 3D SPT phase O nontrivial obstruction ½ can occur only at the surface corresponding to packing together type ½νyx 1D chains of a 3D SPT state with ½Ω¼½O. In particular, for a of SPT phases, we derive stringent constraints on the 2D nontrivial SPT phase with ½Ω ≠ ½1, we can only find a SET phase. For instance, we can rule out the double-semion matching nontrivial obstruction class ½O ≠ ½1 if the SET model in a time-reversal invariant S ¼ 1=2 magnet. fractionalization class is also nontrivial ½w ≠ ½I, which Unfortunately, the above argument does not apply to (as previously mentioned) requires there to be nontrivial G ¼ Uð1Þ, which does not have projective representations. anyonic excitations. Thus, a gapped, symmetric surface of a Nevertheless, the constraints are rather simple (and well nontrivial SPT must be topologically ordered. known) for this case, and we can treat them by formalizing Thus far in this overview section, the theory of SET flux-threading type arguments within the SET formalism. phases and their obstructions that we have described Before presenting our results, we introduce the relevant assumes that the microscopic symmetry G is an internal structures on either side of the bulk-boundary mapping: 3D (on-site) symmetry. We need to generalize this theory to SPT phases and 2D SET phases. While much of this include translational symmetries. For symmetry fraction- material is discussed elsewhere [7,26,28–30], our treatment alization, the generalization is straightforward and still of translational invariance is novel. encoded by ρ and ½w. The full extension of the SET theory, which includes symmetry defects and their obstructions, is less clear. For this, we use the bulk-boundary correspon- III. TRANSLATIONALLY INVARIANT dence between 3D SPT phases and 2D SET phases that SPT PHASES exist on their boundaries. Applying the obstruction theory A. Strong SPT phases of 2D SET phases for systems with translational symmetry We first briefly review bosonic SPT phases protected by in the same way as for on-site symmetry, we find that we unitary on-site symmetries (like a spin rotation). It was must have a rather different interpretation of the resulting proposed that, in the absence of topological order, gapped obstruction class. For this, we must first review 3D SPT G-symmetric phases in d spatial dimensions are classified phases with translational symmetry. by an element of the group cohomology Hdþ1½G; Uð1Þ A SPT phase protected by translational symmetry is [7,28,29]. By this, we mean that each distinct phase of this called a “weak” SPT phase. The canonical example is a type in the parameter space of generic G-symmetric weak 3D TI, which arises by vertically stacking 2D TIs. Hamiltonians can be uniquely labeled by some element More generally, a d-dimensional weak SPT phase protected d 1 − of the Abelian group H þ G; U 1 . The identity element by the on-site symmetry group G and the Zd k translational ½ ð Þ of this group is identified with the trivial phase, which can symmetry group can be obtained by “stacking” copies be thought of as the universality class of an unentangled of a k-dimensional SPT phase in a (d − k)-dimensional lattice, where 0 ≤ k ≤ d. As we show, weak SPT phases product state. The physics of this classification is most transparent for can be classified by incorporating translational symmetry in the same manner as internal symmetries, that is, by 1D chains. While a SPT phase has a unique, gapped ground 1 state on a ring, an open segment may have degenerate states Hdþ ½Zd × G; Uð1Þ. associated with its end points. The degeneracy of these The most relevant SPT phases for our discussion are 3D boundary states is protected by their local transformation weak SPT phases that consist of packing together an array properties under g ∈ G. The action of a global symmetry of 1D SPT phases, as illustrated in Fig. 1. At the end of each operation R on a ground state of the open chain can be 1D chain in a SPT phase, there is a degenerate boundary g Ψ state which transforms as a projective representation under localized to the boundaries. For any ground state j i, this takes the form G, characterized by ½νyx. On a surface perpendicular to the direction of the packed 1D chains of SPT phases, we have ð1Þ ð2Þ precisely a 2D system with the projective representation RgjΨi¼Ug Ug jΨi; ð1Þ ½νyx per unit cell. The interpretation is a bit different than can ðjÞ the on-site case, since such a surface be physically where U are unitary operators whose nontrivial action is realized in a strictly 2D system. In particular, one can g localized near the two ends of the chain, labeled j¼1 and 2. simply construct the system from microscopic degrees of ðjÞ freedom in such a way that each unit cell transforms as a The operators Ug do not depend on which of the degenerate ground states they are acting upon. projective representation ½νyx. However, as the obstruction matching condition is still expected to hold, this becomes a It is clear from Eq. (1) that, even if Rg is a linear powerful constraint on the possible SET orders that can representation of G (obeying RgRh ¼ Rgh), the localized ðjÞ occur in such a 2D system. To be more precise, we will later symmetry actions Ug may take the form of projective show that given ρ and ½w for the enlarged symmetry group representations of G. In particular, they only must respect G ¼ Z2 × G, we can determine the obstruction class ½O, group multiplication up to U(1) phases, i.e.,

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ðjÞ ðjÞ ðjÞ ω TIs have three weak indices νi ∈ Z2 for i ¼ x, y, z Ug Uh ¼ jðg; hÞUgh; ð2Þ corresponding to stacking 2D TIs along the iˆ direction as long as the phases ωjðg; hÞ ∈ Uð1Þ obey the constraint [22]. Translational invariance is crucial here; otherwise, neighboring layers can dimerize and leave the state trivial. ω1ðg; hÞω2ðg; hÞ¼1 (assuming the Rg is a linear repre- sentation). Of course, multiplication of these local oper- More generally, there are additional weak invariants. For ðjÞ instance, we can also pack together 1D SPT phases. This ators Ug must be associative, which requires the projective “ ” leads to the following hierarchy of weak invariants in 3D, phases to satisfy the 2-cocycle condition depicted schematically in Fig. 2. (a) νijk, the 0D SPT per unit volume.—A 0D SPT phase ω ðg; hÞω ðgh; kÞ¼ω ðg; hkÞω ðh; kÞ: ð3Þ 1 j j j j is simply a charge, since H ½G; Uð1Þ is the group of U(1) (unitary, Abelian) representations of G. Filling volume with (The projective phases are also known as the “factor set” 0D SPT phases amounts to a charge density given by the characterizing a projective representation.) ν ∈ H1 1 Additionally, one can always trivially redefine the local index ½ xyz ½G; Uð Þ. ν — ~ ðjÞ ðjÞ (b) ij, the 1D SPT per unit area. Packing 1D SPT operators by a phase Ug ¼ ζjðgÞUg , as long as the phases of index νij perpendicular to the ij plane, for phases ζjðgÞ ∈ Uð1Þ satisfy ζ1ðgÞζ2ðgÞ¼1. The corre- i; j ¼ x, y, z, is encoded in three indices, ½νxy; ½νyz; spondingly redefined projective phases are, thus, given by ν ∈ H2 1 ω~ ζ ζ ζ ω ½ zx ½G; Uð Þ. jðg; hÞ¼ð jðgÞ jðhÞ= jðghÞÞ jðg; hÞ. A combination ν — ζ ζ ζ (c) i the 2D SPT per unit length. Stacking 2D SPT of phase factors of the form ð jðgÞ jðhÞ= jðghÞÞ is ˆ phases of index νi along the i direction for i ¼ x, y, z is known as a “2-coboundary.” Hence, one should view 3 encoded in three indices, ½ν ; ½ν ; ½ν ∈ H ½G; Uð1Þ. different choices of projective phases that are related by x y z We find that the weak invariants can be elegantly 2-coboundaries as physically equivalent. incorporated into the cohomology formulation. Assuming Mathematically, taking the quotient of the set of 2- the full symmetry group takes the form G Zd × G for cocycles by the set of 2-coboundaries (i.e., forming ¼ trans int equivalence classes of cocycles related by coboundaries) yields the second cohomology group H2½G; Uð1Þ. (See Appendix A for a brief review of group cohomology.) We denote the equivalence class that contains the cocycle ω as the cohomology class [ω]. In this way, H2½G; Uð1Þ classifies the possible types of projective representations of G, with the trivial cohomology class ½1 ∈ H2½G; Uð1Þ corresponding to linear representations. We note that the constraints relating the phase factors associated with the two end points, i.e., ω1ðg; hÞ¼ −1 −1 ω2ðg; hÞ and ζ1ðgÞ¼ζ2ðgÞ , may be viewed as topo- logical. This is because they are imposed by the topological properties of the system, in this case being that a line segment has precisely two end points. The consequence of these constraints is that there is a correlation between the projective representations localized at the end points of the −1 chain, specifically ½ω1¼½ω2 , and, hence, the possible gapped phases of the system are classified by a single copy of H2½G; Uð1Þ. We emphasize that, when the cohomology class is nontrivial, the corresponding projective represen- tations are necessarily multidimensional, and, hence, there is a protected degeneracy of the boundary states. In higher dimensions, the role of projective representa- tions is played by higher cohomology classes. SPT phases in d spatial dimensions may be characterized by the (d þ 1)th cohomology group Hdþ1½G; Uð1Þ [7].

B. Weak SPT phases FIG. 2. The types of 3D weak SPT phases. (a) Filling 0D SPT A weak SPT phase is a phase protected by translational phases (charge density), νxyz. (b) Packing 1D SPT phases, νxy. symmetry in addition to on-site symmetry. For example, 3D (c) Stacking 2D SPT phases, νz.

041068-5 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) a system with d spatial dimensions, we use the Künneth appearing in the Künneth decomposition. The reader is free formula [7,31] to compute Hdþ1½G; Uð1Þ and find to treat the procedure as a “black box,” but for complete- ness, the key tool is the “slant product,” which we define in Yd Eq. (A7). The slant product induces a homomorphism dþ1 d rþ1 ðdÞ H ½Z × G; Uð1Þ ¼ ðH ½G; Uð1ÞÞ r : ð4Þ that maps n-cohomology classes to (n − 1)-cohomology r¼0 classes: d (See Appendix B for details.) Since ð Þ¼ðd!=r!ðd − rÞ!Þ is −1 r ι ∶ Hn½G; Uð1Þ → Hn ½G; Uð1Þ the number of perpendicular r planes in d space, this formula g ω ↦ ι ω is the obvious generalization of the stacking picture in ½ ½ g : ð6Þ arbitrary dimensions. In 3D, we have More explicitly, for elements p ∈ ZðGÞ in the center of G, H4½Z3 × G;Uð1Þ ¼ H4½G;Uð1Þ × (H3½G;Uð1Þ)3 the slant product acting on 2-, 3-, and 4-cocycles gives (H2 1 )3 H1 1 × ½G;Uð Þ × ½G;Uð Þ: ð5Þ ωðg; pÞ ι ωðgÞ¼ ; ð7Þ p ωðp; gÞ As such, the SPT classes ½Ω ∈ H4½Z3 × G; Uð1Þ naturally Ω ν ν ν ν decompose into octuples as ½ ¼f½ ; ½ i; ½ ij; ½ ijkg, ωðp; g; hÞωðg; h; pÞ 4 ι ω where ν ∈ H G; U 1 is the strong SPT invariant men- p ðg; hÞ¼ ; ð8Þ ½ ½ ð Þ ωðg; p; hÞ tioned in Sec. III A, and ½νi, ½νij, and ½νijk are the weak invariants discussed above. ω ω ι ω ðg; p; h; kÞ ðg; h; k; pÞ In this paper, the most significant weak SPT invariant is p ðg; h; kÞ¼ : ð9Þ ωðp; g; h; kÞωðg; h; p; kÞ the 1D SPT per unit area, characterized by ½νij.As discussed, a 1D SPT phase has degenerate boundary states The slant product can also be applied in succession. that transform under G with a projective representation When p; q ∈ ZðGÞ, one finds the (cohomological) equiv- ν ∈ H2 1 encoded by the index ½ ij ½G; Uð Þ that classifies the alence for applying the slant product in different orders to phase. Consequently, at a boundary surface along the ij −1 be ½ιpιqω¼½ιqιpω . plane, there will be a finite density (one per unit area) of If T is the group element that generates translations low-energy edge states whose fate is determined by the i along the iˆ direction, the 4-tuple of invariants can be surface interactions. Each unit cell of the effective Hilbert recovered by successive application of the slant products ι space at the surface will thus transform under G as a Ti followed by a restriction of the resulting cocycles to the on- projective representation labeled by ½ν . ij site symmetry group G:

1. Example: AKLT spin system ν Ω ðg; h; k; lÞ¼ ðg; h; k; lÞjg;h;k;l∈G; ð10Þ For concreteness, consider a spin system with 3 G ¼ SOð Þ. The second cohomology group (which clas- ν ðg; h; kÞ¼ι Ωðg; h; kÞj ∈ ; ð11Þ 2 i Ti g;h;k G sifies projective representations) is H ½SOð3Þ; Uð1Þ ¼ Z2, the trivial element of which corresponds to the integer spin ν ι ι Ω ijðg; hÞ¼ T T ðg; hÞjg;h∈G; ð12Þ representations and the nontrivial element of which corre- i j sponds to the half-integer spin representations. A well- ν ðgÞ¼ι ι ι ΩðgÞj ∈ : ð13Þ known example of a nontrivial strong 1D SPT phase is ijk Ti Tj Tk g G S 1 the ¼ AKLT chain, which has twofold degenerate For more details on the use of slant product in the Künneth S 1=2 edge states [24]. Packing AKLT chains along the z ¼ decomposition of cocycles, see Appendix C. axis at finite density of one per xy unit cell, we trivially obtain a νxy ¼ −1 weak SPT whose surface theory will behave like an S ¼ 1=2 square-lattice magnet. A more IV. SYMMETRY FRACTIONALIZATION familiar model is the 3D cubic S ¼ 3 AKLT model; in this IN 2D TOPOLOGICAL PHASES ν ν ν −1 case, xy ¼ yz ¼ zx ¼ . However, since we analyze We now review the theory of symmetry fractionalization only one surface at a time, the latter two invariants are in a 2D topologically ordered phase [26,30]. A topological irrelevant. phase supports anyonic quasiparticle excitations whose topologically distinct types are denoted by a; b; c; …, C. Computing the weak invariants which we refer to as anyons or topological charges. We from the slant product assume that the system has total symmetry group 2 Given a 4-cocycle Ωðg1; g2; g3; g4Þ, there is a computa- G ¼ Z × G, where G is the on-site symmetry group tional procedure for finding the invariants fν; νi; νij; νijkg and Z × Z is the translational symmetry group generated

041068-6 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) by T and T , where x and y denote a basis for the Bravais ðjÞ x y symmetry transformations Ug only need to projectively lattice. represent group multiplication:

A. On-site symmetries ðjÞ ðjÞ Ψ η ðjÞ Ψ Ug Uh j fagi¼ aj ðg; hÞUghj fagi; ð15Þ The structure of symmetry fractionalization is organized according to three successively finer distinctions. η ∈ 1 where the projective phase aj ðg; hÞ Uð Þ depends on First, symmetry transformations can permute anyon only the topological properties localized in the neighbor- ν 1 types. For example, consider a bilayer of two ¼ 2 bosonic hood of the jth quasiparticle, which is just the topological quantum Hall states. There are two types of Laughlin charge aj carried by this quasiparticle. Since Rg is a linear quasiparticles, s1, s2, corresponding to the two layers, as representation,Q the projective phases must satisfy the well as their composite f ¼ s1s2. Exchanging the layers is η 1 condition j a ðg; hÞ¼ [34]. g g2 1 j an on-site symmetry (with ¼ ) of the model which The constraints on the projective phases here provide a permutes the two Laughlin quasiparticle types g∶ s1 ↔ s2, richer structure than we observed for 1D SPTs. In particu- while leaving their composite invariant, g∶ f ↔ f. A less lar, this condition is equivalent to the condition that trivial example is Wen’s plaquette model for the Z2 toric η ðg; hÞη ðg; hÞ¼η ðg; hÞ whenever the topological code, where translations by one lattice spacing T for i ¼ x, a b c i charge c is a permissible fusion channel of the topological y exchange electric and magnetic charges, T ∶ e ↔ m i charges a and b (i.e., whenever Nc ≠ 0). It follows that [32,33]. In this paper, we, however, assume that the ab η ðg; hÞ must take the form [26] symmetries G do not permute any of the anyon types, as a this greatly simplifies the computations. η g; h M w ; 16 Second, anyons can carry fractionalized symmetry að Þ¼ a; ðg;hÞ ð Þ quantum numbers. To be more precise, let us prepare a w Ψ where ðg; hÞ is an Abelian anyon and Ma;wðg;hÞ is the state j a1;…;a i with n well-separated quasiparticles, where n mutual braiding statistics between anyons a and wðg; hÞ. the jth quasiparticle carries topological charge aj. (We use the convention where the “vacuum” topological charge is Associativity of the localized operators gives the denoted I and the “topological charge conjugate” of a is condition ¯ denoted a, which is the unique topological charge that can η η η η fuse with a into vacuum.) We assume the overall topo- aðg; hÞ aðgh; kÞ¼ aðg; hkÞ aðh; kÞ; ð17Þ logical charge is trivial I, so that the state can be created from the ground state by applying local operators. (When for all a, which translates into there are non-Abelian anyonic quasiparticles, one must w w w w further specify the fusion channels to specify the state, but ðg; hÞ × ðgh; kÞ¼ ðg; hkÞ × ðh; kÞ: ð18Þ this plays no role in our discussion, so we leave it implicit.) We consider the global symmetry transformation R acting This is precisely the 2-cocycle condition, though not for g elements in U(1), but rather for elements in A, the group on the state jΨ i, where we introduce the shorthand fag fag whose elements are the Abelian anyons, with group for the collection of topological charges a1; …;a carried n multiplication defined by fusion of anyons. by the quasiparticles. Without loss of generality, we shall There is also freedom to trivially redefine the localized assume that the system transforms as a linear representation ðjÞ symmetry transformations by local operators U~ ¼ of G (e.g., if there is a S ¼ 1=2 spin per site, then we g ðjÞ ðjÞ ðjÞ Ψ require the number of sites to be even). Although anyons Zg Ug , whose action on the state is Zg j fagi¼ ζ ðgÞjΨ i, where the phases ζ ðgÞ ∈ Uð1Þ satisfy are nonlocal excitations, the local properties (i.e., local Qaj fag aj ζ 1 density matrix) of regions away from the positions of the j aj ðgÞ¼ . This, similarly, generates a map between anyons remain the same as those of the ground state. ζ these phases and anyons, with the relation aðgÞ¼Ma;zðgÞ, Therefore, the global symmetry transformation Rg should where zðgÞ is an Abelian anyon. This translates the trivial have the following decomposition (when the symmetries do redefinitions of the local operators into the redefinition not permute anyon types): w~ ðg; hÞ¼zðgÞ × zðghÞ × zðhÞ × wðg; hÞ. This is pre- A z Yn cisely redefinition by an 2-coboundary ðgÞ × Ψ ðjÞ Ψ z z Rgj fagi¼ Ug j fagi: ð14Þ ðghÞ × ðhÞ. j¼1 The result is a classification [26] by the Abelian anyon- valued second cohomology group H2½G; A. In particular, ðjÞ A Here, Ug is a local unitary operator whose nontrivial we form equivalence classes of 2-cocycles that are action is localized in the neighborhood of the jth quasi- related to each other by A 2-coboundaries. A given particle. Similar to the localized symmetry actions at end equivalence class ½w ∈ H2½G; A completely specifies points of 1D chains, discussed in Sec. III A, the localized the symmetry fractionalization of the system; i.e., the local

041068-7 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) projective phases ηaðg; hÞ are given by Eq. (16) for all conditions of a U(1) 2-cocycle. Thus, ηaðg; hÞ anyon types. Thus, we call these cohomology classes the may be viewed as a representative element of the “ ” 2 symmetry fractionalization classes. U(1) 2-cohomology class ½ηa ∈ H ½G; Uð1Þ.As The manifestation of symmetry fractionalization for the previously discussed, H2½G; Uð1Þ classifies projec- anyons may exhibit two characteristic properties: (1) anyons η tive representations of G. Hence, when ½ aj is a can carry a fractional charge (like the Laughlin quasipar- nontrivial element of H2½G; Uð1Þ, the local unitary ticles) and (2) anyons can carry a localized projective operators UðjÞ acting in the neighborhood of the jth representation of the symmetry group G, i.e., they have an g quasiparticle (which has topological charge a ) form internal degeneracy (like spin), which transforms projec- j tively under G. We can determine these symmetry frac- a projective representation of G [36]. In this case, the ðjÞ tionalization properties for an anyon of topological charge Ug must be acting on a multidimensional Hilbert a by examining the corresponding local projective phases space. In other words, there is a symmetry-protected η ðg; hÞ, as we now explain in more detail. local degeneracy associated with anyons of topo- a η (1) An anyon a can carry fractional charge under some logical charge aj. Thus, when ½ a is a nontrivial subgroup H

ðjÞ i2πQ N Ψ aj Ψ B. Incorporating translational symmetry ½U2π=N j fagi¼e j fagi; ð21Þ The analysis of symmetry fractionalization can be in whichQ case the fractional charge can be written as generalized to translational symmetry. In 2D, lattice trans- 2π −1 i Qa N η 2π 2π Z Z e ¼ k¼0 ð =N; k=NÞ [35]. lations form a × group. For the generators Ti of (ii) When the symmetries do not permute anyon types, translation in the i ¼ x and y directions, the decomposition the local projective phases ηaðg; hÞ satisfy the of the action of the symmetry (again assuming the

041068-8 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) symmetries do not permute anyon types) is generalized ηaðg; hÞ ¼ Ma;bðg;hÞ: ð26Þ to [26] ηaðh; gÞ

Yn TiðjÞ A representative 2-cocycle of a fractionalization class in RT jΨ a i¼ U jΨT a i; ð22Þ 2 2 i f g Ti if g H ½Z × G; A can be chosen to take the form j¼1 m n mx y nx y mynx mx wðTx Ty g;Tx Ty hÞ¼½bðTy;TxÞ × ½bðTx; hÞ where Tifag is used here to indicate that the location of the my quasiparticles have all been translated by Ti, and TiðjÞ is × ½bðTy; hÞ × wðg; hÞ; ð27Þ used to indicate that the neighborhood in which the TiðjÞ ∈ G corresponding local operator UT acts nontrivially has for g; h ¼ G. It it straightforward to see that this is i indeed a 2-cocycle, as long as also been translated by Ti [37]. We can then repeat the entire analysis and arrive at essentially the same classi- b b b fication, but now including the translational symmetries. ðTi; gÞ × ðTi; hÞ¼ ðTi; ghÞ; ð28Þ It is very instructive to study the fractionalization classes in H2½Z2 × G; A in greater detail. Using the Künneth for i ¼ x, y. These conditions are just the requirement formula for group cohomology, we have that bðTi; gÞ are group homomorphism from G to A, i.e., 1 that bðTi; gÞ ∈ H ½G; A. H2½Z2 × G; A¼H2½Z2; A × ðH1½G; AÞ2 The terms in the decompositions of Eqs. (23) and (27) can now be understood physically as follows. × H2½G; A; ð23Þ (1) bðTy;TxÞ the “anyonic flux per unit cell.”—The first term, H2½Z2; A¼A, in the Künneth formula 1 1 2 where we have used the fact that H ½G; H ½Z ; A ¼ decomposition of Eq. (23) is characterized by the H1 A2 H1 A 2 H2 Z2 A A ½G; ¼ð ½G; Þ . We note that ½ ; ¼ gauge invariant quantity bðTy;TxÞ ∈ A, which 1 and that H ½G; A is just the (group formed by the) we can interpret as the background topological homomorphisms from G to A. flux through each unit cell (as depicted in Fig. 3) In order to understand the terms in this decomposition, it in the following way. The symmetry operation is useful to consider the symmetry localization of the T−1T−1T T is a sequence of translations corre- −1 −1 y x y x operations of the form Rg Rh RgRh, where the group sponding to a path that encloses one unit cell in a elements g and h commute. In this case, the localized counterclockwise fashion. From Eq. (24), we see operators give that this operation has the corresponding local projective phase factor of Ma;bðT ;T Þ for quasipar- ¯ y x ρ hðjÞ −1ρ−1 ðjÞ −1 ðjÞρ g¯ðjÞρ−1 Ψ ticles of topological charge a. (See also Ref. [25] for h½Ug h ½Uh Ug gUh g j fagi an operational definition.) This phase M b is η ðg; hÞ a; ðTy;TxÞ aj Ψ ¼ j fagi; ð24Þ the mutual braiding statistics associated with an ηa ðh; gÞ j anyon a encircling an anyon bðTy;TxÞ in a counter- clockwise fashion. Thus, we can picture this type of where ρg here is an operator on the physical Hilbert space symmetry fractionalization as being generated by an that acts precisely as the symmetry action does on the Abelian anyon bðTy;TxÞ sitting in each unit cell, as topological state space; i.e., it acts on the topological shown in Fig. 3. This behavior is familiar from quantum numbers of the physical states. (See Ref. [26] “odd” gauge theories [38,39]. ˆ for more details.) For our purposes here, it has the effect of (2) bðTi; gÞ the “anyonic spin-orbit coupling” in the i translating the locations of quasiparticles only when g is a direction for g ∈ G.—The two factors in the second translational symmetry group element. term, ðH1½G; AÞ2, in Eq. (23) are characterized by We define

bðg; hÞ¼bðh; gÞ¼wðg; hÞ × wðh; gÞ; ð25Þ for g; h ∈ G ¼ Z2 × G, which takes values in the set of Abelian anyons A. We note that, when gh ¼ hg, the quantity bðg; hÞ is gauge invariant; i.e., it does not change when wðg; hÞ is modified by a 2-coboundary. η Then, in the gauge where aðg; hÞ¼Ma;wðg;hÞ, we can write the ratio of projective phases as FIG. 3. Anyonic flux per unit cell bðTy;TxÞ.

041068-9 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016)

the two gauge invariant quantities bðTx; gÞ and (3) wðg; hÞ the on-site fractionalization class, for 2 bðTy; gÞ, respectively. These encode the possible g; h ∈ G.—The third term, H ½G; A, in Eq. (23) noncommutativity between the localized operators is just the previously discussed symmetry fraction- for translations and on-site symmetries acting on alization for the on-site symmetries G. anyons. In other words, moving an anyon in the iˆ direction can change its symmetry charge under C. Examples g ∈ G. For this reason, we think of it as a sort of anyonic spin-orbit coupling, which couples the We now illustrate symmetry fractionalization for systems internal quantum numbers of an anyon with its with translational symmetry using three concrete examples. motion. An operational definition of bðTi; gÞ may be 1. Laughlin states given as follows. Consider a state Ψ ¯ with a pair j a;ai The first example is the ν ¼ 1=m Laughlin fractional of well-separated anyons a and a¯ localized at r and 0 quantum Hall states [40] with (magnetic) translational r . We now move the anyon a by one lattice spacing symmetry and U(1) charge conservation. It has been known ei while holding the other anyon fixed, and denote since Laughlin’s seminal work that the fundamental quasi- Ψ0 the resulting state by j a;a¯ i. By translational invari- holes (quasiparticles corresponding to a 2π vortex) in these ance, the local density matrices in a region centered states carry fractional electric charge 1=m (where the ¯ Ψ Ψ0 at a (but far away from a) for j a;a¯ i and j a;a¯ i electron carries charge −1). should be identical, as they are related by a spatial These states support m anyon types, all of which are translation. We then compute the g charge on the Abelian [41]. Let us associate the label ϕ with the anyon two states type of the fundamental quasihole, i.e., a vortex obtained by threading 2π flux. The anyons are then given by 2 −1 Ψ ¯ R Ψ ¯ η A ϕ ϕ … ϕm h a;aj gj a;ai ≈ aðTi; gÞ ¼fI; ; ; ; g, where the corresponding fusion ¼ Ma;bðT ;gÞ: ð29Þ Ψ0 Ψ0 η i ϕj ϕk ϕ½jþkm h a;a¯ jRgj a;a¯ i aðg;TiÞ rules are given by × ¼ . In other words, j A ¼ Zm. The mutual braiding statistics of anyons ϕ ≠ 1 and ϕk (corresponding to taking one around another in a When Ma;bðTi;gÞ , moving the anyon a changes the g charge it carries. Thus, the local operators and counterclockwise fashion) is given by the phase Mϕj;ϕk ¼ processes which move anyons by one lattice spacing ei2πjk=m [42]. are necessarily “charged” under the on-site sym- In the presence of a magnetic field, translations obey a − δ δ l2 ˆ metry. Equivalently, one can view this as there being magnetic algebra: T−δyT−δxTδyTδx ¼ exp½ ið x y= BÞN. an integral symmetry charge (linear representation) It would be interesting to study this more general symmetry per directed unit length of the quasiparticle string group involving continuous translations. However, for our operators (Wilson lines), potentially with a different purposes, we consider only a discrete subset of the trans- value of symmetry charge per directed length in the lations generated by the “unit” translations Tx and Ty, two different directions. A consequence of this which are chosen to define a unit cell with area 2πl2 . This ≠ 1 B property is that, when Ma;bðTi;gÞ , it is not corresponds to one flux quanta enclosed in each unit cell possible to transport a quasiparticle carrying topo- and T T ¼ T T . The corresponding symmetry group is ˆ x y y x logical charge a in the i direction via a process that is thus G ¼ Z2 ×Uð1Þ, as discussed. No anyons are permuted both adiabatic and g symmetric. by symmetry; i.e., ρgðaÞ¼a). The cohomology classes Note that if G is continuous and connected [e.g., decomposing the classification of the possible symmetry G ¼ Uð1Þ or G ¼ SOð3Þ], the spin-orbit fraction- fractionalization [as in Eq. (23)] for this topological order alization H1½G; A is always trivial. This is because 2 2 1 and symmetry group are H ½Z ; A¼Zm, H ½Uð1Þ; A¼ H1 A 2 ½G; consists of all group homomorphisms Z1, and H ½Uð1Þ; A¼Z . A m from G to , so the image of the identity element The symmetry fractionalizations of the Laughlin states A 1inG is the trivial anyon I in . When G is are described by continuous, continuity is also required of the co- n chain C ðG; AÞ. This forces the image of every j k ½jk wðTx;T Þ¼ϕ m ; ð30Þ element in the connected component of the identity y A of G to equal I, since is discrete. Thus, if G is j k 1 wðTy;TxÞ¼I; ð31Þ continuous and connected, H ½G; A¼Z1 and b ðTi; gÞ¼I. On the other hand, if G has multiple w j θ w θ j connected components, e.g., all discrete nontrivial G ðTi ; Þ¼ ð ;Ti Þ¼I; ð32Þ 2 1 ⋊Z or G ¼ Oð Þ¼Uð Þ 2, it can have nontrivial θ θ0− θ θ0 2π w θ θ0 ϕð þ ½ þ 2π Þ= anyonic spin-orbit coupling. ð ; Þ¼ : ð33Þ

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θ θ0 ∈ 0 2π Here, ; ½ ; Þ label the elements of U(1), and ½x To see why, consider, for example, the local operation of denotes modulo x. Let us unravel the physics of this two π rotations about the x axis for some anyon a. The fractionalization class. corresponding projective phase is In order to determine the fractional U(1) charge of ϕ, ϕ η 1 consider the localized action of U(1) rotations on . From aðX; XÞ¼Ma;wðX;XÞ ¼ ; ð35Þ i2πQϕ i2π=m Eq. (19), we find that e ¼ Mϕ;ϕ ¼ e , indicating a k 1 −1 fractional charge of Qϕ ¼ð1=mÞ. Similarly, the anyons ϕ where þ corresponds to a carrying trivial spin and corresponds to a carrying spin S ¼ 1=2. Since M ¼ have fractional U(1) charge Qϕk ¼ k=m. e;m −1 1 The anyonic flux per unit cell for this fractionalization Mψ;m ¼ and Mm;m ¼ , and the theory is modular, the class is bðT ;T Þ¼ϕ¯ ¼ ϕm−1. The system is at filling ν ¼ unique choice consistent with our assignment of spin is y x w 1=m electrons per unit cell. Since anyons of topological ðX; XÞ¼m. Similar arguments yield the other terms of ¯ the fractionalization class. charge ϕ carry fractional U(1) charge Qϕ¯ ¼ −1=m,it If we consider any U(1) subgroup of the SO(3), for appears as if the electric charge density of the quantum example, rotations about the x axis, we can determine the Hall liquid arises from the finite density of such anyons ϕ¯ corresponding fractional charge carried by quasiparticles. (one per unit cell). Furthermore, it is known that if a 2π From Eq. (19), we find that ei Qa ¼ M , which indicates quasiparticle of charge Q adiabatically moves along a path a;m that the quasiparticle types have rotational U(1) fractional that encloses a region of area A (containing no quasipar- 0 1 ticles), it will accumulate an Aharanov-Bohm phase of charge values of QI ¼ Qm ¼ and Qe ¼ Qψ ¼ 2. These, −iQA=l2 of course, simply match the corresponding SO(3) projec- γ ¼ e B [42]. Restricting to areas enclosing n flux tive representations (integer or half-integer spin values) quanta, i.e., A n · 2πl2 ,wehaveγ e−i2πQn. Local ¼ B ¼ carried by the quasiparticles, in this case. excitations have charge Q ∈ Z and so accumulate no phase For the translational symmetry, the fact that there is a from encircling n flux quanta. However, a quasiparticle slave particle sitting at each site in the mean-field ansatz ϕk excitation with topological charge will accumulate an leads to the following fractionalization: Aharonov-Bohm phase of γ ¼ e−i2πnk=m when encircling n flux quanta. On the other hand, the mutual statistics bðTy;TxÞ¼e or ψ: ð36Þ b ϕ¯ ϕk −i2πk=m between ðTy;TxÞ¼ and is Mϕ¯ ;ϕk ¼ e . Thus, the Aharonov-Bohm phase γ can equivalently be inter- Which of these two options is actually realized depends on preted as the statistics between quasiparticles and the n whether one uses a Schwinger boson or fermion construc- anyons per unit cell ϕ¯ within the encircled region (con- tion, as well as the chosen mean-field ansatz (0 or π flux per taining n flux quanta). plaquette in the band structure). Note that in the PSG language, bðTy;TxÞ determines the PSG of the anyons via Z η η 1 2. 2 spin liquid aðTy;TxÞ= aðTx;TyÞ¼Ma;bðTy;TxÞ ¼ , for the phase Our second example is translationally invariant Z2 spin factor in Eq. (24). liquids, with G ¼ SOð3Þ. For clarity, we only explicitly Since G is connected, the spin-orbit fractionalization is b write the cocycles for the Z2 × Z2 ¼f1;X;Y;Zg sub- trivial, ðTi; gÞ¼I. group of SO(3), consisting of the three π rotations about the x, y, and z axes. We consider phases with topological order 3. Unconventional symmetry for the toric code of the Z2 toric code type, i.e., Z2 spin liquids. Trial wave An example that realizes a nontrivial anyonic spin-orbit functions of such states can be constructed by Gutzwiller fractionalization class is due to Hermele [48]. We begin projection of a noninteracting Schwinger fermion or boson with Kitaev’s Z2 toric code model [49], where qubits live mean-field ansatz [43–45]. We denote the bosonic spinons on the bonds of a square lattice. The Hamiltonian of the by e, the fermionic spinons by ψ, and the Z2 flux [46,47] model is (also known as a “vison” [38])bym. All anyons are X X A ψ Abelian, defining the group ¼fI;e;m; g with fusion H ¼ − Av − Bp: ð37Þ algebra Z2 × Z2. v p In the familiar Z2 spin liquid, the spinons e and ψ Q Q σx σz carry S ¼ 1=2 under SO(3) while the vison m does not Here, Av ¼ j∈v j and Bp ¼ j∈p j, where the products carry spin. This fixes the fractionalization of the on-site are over the four bonds meeting at vertex v or forming the symmetry G to be the following, in a gauge where plaquette p, respectively. Notice that there are two qubits wð1; gÞ¼I: per unit cell. We now define a global G ¼ Z2 × Z2 ¼f1;X;Z;XZg wðX; XÞ¼wðY;YÞ¼wðZ; ZÞ¼m; symmetry with a peculiar representation that acts only on the vertical (y) bonds. Let the set of all vertical bonds in the bðX; YÞ¼bðY;ZÞ¼bðX; ZÞ¼m: ð34Þ system be denoted Cy and define the operators

041068-11 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) Y Y σx σz b b RX ¼ j ;RZ ¼ j: ð38Þ ðTy;XÞ¼ ðTy;XZÞ¼m; j∈Cy j∈Cy bðTy;ZÞ¼I: ð39Þ 0 It is straightforward to check that ½H; RX¼½H; RZ¼ , Similarly, when we move m along the xˆ direction, the string so these operators are symmetries of the system. However, operator is extended by an extra σx factor for each step, this peculiar choice of operators forms a projective which implies representation of Z2 × Z2, since only the vertical links transform under the symmetry. In particular, we have bðTx;XÞ¼I; R R ¼ð−1ÞLxLy R R , where L and L is the linear X Z Z X x y b b size of the system in the xˆ and yˆ directions, respectively. ðTx;ZÞ¼ ðTx;XZÞ¼e: ð40Þ Moreover, each unit cell locally transforms under the An interesting feature of the model is that the definition of Z2 × Z2 symmetry as a projective representation, since the symmetry breaks the C4 lattice rotation symmetry. We there is one vertical bond in each unit cell. see later that this is unavoidable in such cases. As usual, the e and m excitations correspond to We can interpret the anyonic spin-orbit coupling another violations of the vertex terms (i.e., A ) and plaquette terms Q v way. Recall that an e-string operator applies σz to the (i.e., B ), respectively. The usual definition of the toric- j∈P j p sites contained in a path P running along links between code string operators implies that neither anyon carries a w vertices, and that such an operator creates e quasiparticles projective representation of G,so ðg; hÞ¼I for at the end points of an open path P, while it commutes with g; h ∈ G. Furthermore, our choice of Hamiltonian implies b the Hamiltonian away from the end points (or it commutes trivial anyonic flux per unit cell, so ðTy;TxÞ¼I. everywhere if P is a closed path). Moreover, applying an On the other hand, the interplay of G and translational open e-string operator with one of the path’s end points at symmetries is more interesting and gives nontrivial anyonic the site of an e quasiparticle will move that quasiparticle to spin-orbit coupling. We can move an e anyon by applying a the position of the other end point of P. The global σz string of operators along a path in the lattice. Similarly, symmetry operator R is precisely the product of densely σx Z we can move an m anyon by applying a string of along a packed e-string operators running in the yˆ direction. This is path in the dual lattice. As can be seen from Fig. 4, when we why bðTx;ZÞ¼e; growing the lattice by one unit of Tx move e along the yˆ direction by one lattice spacing, the adds one more of these e-string operators, so ZTx and TxZ extra σz that extends the string operator does not commute differ by the addition of one e string. Similarly, RX can be with RX. Nothing interesting happens, however, when we interpreted as the product of densely packed m strings move m along the yˆ direction. Thus, we find running in the xˆ direction.

V. SYMMETRY DEFECTS AND H4 OBSTRUCTION Not every fractionalization class ½w of a given topo- logical phase and symmetry can be realizable in a strictly 2D system, as some of them may be anomalous. For an on- site, unitary symmetry G, one can determine whether ½w is anomalous by considering the inclusion of symmetry defects, which are extrinsic objects carrying symmetry fluxes. When anyons are transported around the defects they are transformed by the corresponding local group actions. There is an algebraic theory of the fusion and braiding of defects [26] captured by topological data similar to the F symbols and R symbols of anyon models. Importantly, there are consistency conditions that this topological data must satisfy. It was shown mathematically in Ref. [27] that the sufficient and necessary condition for the solvability of the consistency equations is that a certain Z Z FIG. 4. The toric code with an unconventional 2 × 2 object ½O be the trivial cohomology class in H4½G; Uð1Þ. symmetry that exhibits “anyonic spin-orbit coupling.” Dots If this object is nontrivial, it indicates the presence of an represent qubits on each bond; the dark blue ones transform obstruction, meaning that a consistent 2D theory of as projective representations of the global Z2 × Z2 symmetry, while the light dots do not. The string operators that move the e symmetry defects does not exist. This can be interpreted particles vertically (thick black line) and m particles horizontally as an indicator of an anomaly, since symmetry defects can (dashed black line) involve operators on vertical links, so are always be introduced explicitly for any 2D theory realized charged under the symmetry operations. in a lattice model. Given the topological order (i.e., the F

041068-12 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) and R symbols of the anyons), the symmetry group G, We notice that the fusion rules of defects also encode the symmetry action ρ on the topological degrees of the symmetry actions on defect labels by the so-called freedom, and the symmetry fractionalization class ½w, G-crossed braiding. Let us denote the h ∈ G action on the 4 ρ the corresponding obstruction class ½O ∈ H ½G; Uð1Þ is defect charge ag as hðagÞ. The action can be physically uniquely determined and there is a concrete procedure for realized by transporting ag across the branch cut of an h computing it. defect, say Ih, which amounts to a braiding exchange. Since We now briefly review the theory of defects in a the total topological charge in the region containing both topological phase C with symmetry group G that does defects cannot change, and the relative positions of the not permute anyon types [26]. We first define an Abelian defects is interchanged, we must have group A whose elements are the Abelian topological charges of C with multiplication given by the fusion rules. Ih × ag ¼ ρhðagÞ × Ih: ð42Þ Given a fractionalization class ½w ∈ H2½G; A of the symmetry G acting on the topological phase C, we choose This implies a representative w of the class. One can then proceed to −1 ρ w w −1 write down the different types of G defects and their fusion hðagÞ¼½ ðh; gÞ × ðhgh ; hÞ × ahgh : ð43Þ rules. As established in Ref. [26], for each g ∈ G there are jCj topologically distinct types of g defects (i.e., defects that In particular, if gh ¼ hg and wðg; hÞ ≠ wðh; gÞ, then the carry a symmetry flux of g), and they are all related to each two different types of g defects, ag and bðh; gÞ × ag, are other by fusion with anyons. We can label the defect types related to each other by the symmetry action of h. As such, b by arbitrarily picking one Abelian defect type for each g ag and ½ ðh; gÞ × ag must be energetically degenerate. that we label Ig and treat as a reference point within each g Given the topological data of the anyon model (i.e., F sector. It follows that all other g-defect types can be labeled and R symbols), the symmetry fractionalization class ½w, as ag ¼ a × Ig, where a ∈ C are the labels of all the anyon andthefusionrulesinEq.(41), one can then solve the types. The fusion rules then read G-crossed consistency equations for the topological data X (i.e., F, R, U,andη symbols) of the G-crossed defect c w theory. In doing so, one finds that consistent solutions ag × bh ¼ Nab½c × ðg; hÞgh; ð41Þ c∈C exist if and only if the obstruction class ½O¼½1 is trivial, where (a representative of) the obstruction class is found c C where Nab are the fusion coefficients of the anyons in . to be

Fwðg;hÞwðk;lÞwðgh;klÞFwðk;lÞwðh;klÞwðg;hklÞFwðh;kÞwðg;hkÞwðghk;lÞ Oðg; h; k; lÞ¼ Rwðg;hÞ;wðk;lÞ; ð44Þ Fwðg;hÞwðgh;kÞwðghk;lÞFwðk;lÞwðg;hÞwðgh;klÞFwðh;kÞwðhk;lÞwðg;hklÞ which can be shown to be a 4-cocycle. (In this paper, the Generally speaking, if a d-dimensional system exhibits braiding R symbols are written with two superscript gauge anomaly, it can only exist at the boundary of a topological charge labels, in contrast to the global sym- (d þ 1)-dimensional system, where there is an “anomaly metry operators Rg, which are written with a subscript inflow” from the bulk to cancel the gauge anomaly on the group element.) We emphasize that Eq. (44) depends only boundary [50]. This also shows that the bulk cannot be on the symmetry fractionalization class and the F and R adiabatically connected to a trivial product state while the symbols of the Abelian anyons in C. For a derivation of the symmetry is preserved and it is therefore a nontrivial SPT result, we refer the readers to Refs. [18,26]. phase. Recently, it has been shown that in low dimensions In the case of topological orders in which the (non- (d ≤ 3), SPT phases with a unitary symmetry G are vanishing) F symbols involving only Abelian anyons can completely classified by Hdþ1½G; Uð1Þ. In particular, 3D be set to 1 by a choice of gauge (using vertex basis gauge SPT phases are classified by H4½G; Uð1Þ, which is the freedom), the expression for the obstruction significantly same cohomology group to which the obstruction classes simplifies to belong. This is not a coincidence. In fact, the obstruction class precisely determines the bulk SPT phase that is ½Oðg; h; k; lÞ ¼ ½Rwðg;hÞ;wðk;lÞ: ð45Þ needed to cancel the gauge anomaly of the boundary SET phase, a novel kind of bulk-boundary correspondence. This bulk-boundary correspondence has been established In other words, Oðg; h; k; lÞ¼Rwðg;hÞ;wðk;lÞ is a repre- for Abelian topological phases with on-site symmetries that sentative of the obstruction class for such topological do not permute the particle types [18], and is expected to orders. hold more generally, as well.

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VI. INCORPORATING TRANSLATIONAL o ðg; hÞ¼ι ι Oðg; hÞj ∈ : ð50Þ SYMMETRY IN SET PHASES yx Ty Tx g;h G We would like to consider the theory of defects and We can now precisely state the anomaly matching obstructions for 2D SET phases in which there is transla- condition between the 3D bulk SPT phase characterized 4 3 tional symmetry, as well as on-site symmetry, i.e., with by ½Ω¼f½ν; ½νi; ½νij; ½νijkg ∈ H ½Z × G; Uð1Þ and symmetry group G ¼ Z2 × G. Strictly speaking, the theory the 2D boundary SET phase with corresponding obstruc- 4 2 of the symmetry defects described by G-crossed modular tion class ½O¼f½o; ½ox; ½oy; ½oyxg ∈ H ½Z × G; Uð1Þ tensor categories in Ref. [26] is understood to apply to on- to be the requirement that site symmetries G. For translational symmetry, we under- stand how symmetry fractionalization occurs [26],aswe ½ν¼½o; ½νx¼½ox; discuss in Sec. IV B. However, it is not clear, in general, ν ν how to properly define and interpret the algebraic theory of ½ y¼½oy; ½ yx¼½oyx: ð51Þ translational symmetry defects (i.e., dislocations or This assumes the 2D boundary in question is along the xy “movons”). The naive application of the G-crossed con- plane. The remaining terms ν , ν , ν , and ν sistency conditions of Ref. [26] to examples of purely 2D ½ z ½ xz ½ yz ½ xyz systems with projective representations of the on-site characterizing the 3D SPT phase are not involved in symmetry per unit cell, such as certain spin-liquid states, matching the bulk and surface states of boundaries along gives rise to nontrivial obstruction classes ½O. This means the xy plane, though they will be involved in constraints that, in contrast to the case of purely on-site symmetries, the associated with boundaries in the other directions. inclusion of translational symmetry for systems with It is worth considering the different terms in the Künneth projective representations of the on-site symmetry per unit decomposition of the SET obstruction in more detail. The ∈ H4 1 cell requires a refined interpretation of these obstruction term ½o ½G; Uð Þ is simply the obstruction class classes. In the following, we develop such a refined associated with the on-site symmetries G. The terms ∈ H3 1 interpretation. ½ox; ½oy ½G; Uð Þ are associated with the interplay In order to make progress, we develop our understanding between the on-site symmetry fractionalization and the from the following conjecture. anyonic spin-orbit coupling in the xˆ and yˆ directions, ∈ H2 1 Given a 3D SPT with translational and on-site symmetry respectively. The term ½oyx ½G; Uð Þ is associated Z3 trans × Gint and linear representations of Gint per unit cell, with the interplay between the on-site symmetry fraction- if the boundary is gapped and symmetric, it manifests a 2D alization, the anyonic spin-orbit coupling, and the anyonic G Z2 −1 boundary SET with symmetry ¼ trans × Gint whose flux per unit cell. (We note that ½oxy¼½oyx .) defects are described by the G-crossed theory, as described The ½ox and ½oy components of the obstruction class are in Ref. [26], with anomaly matching between the 3D bulk automatically trivial if either the on-site symmetry frac- SPT indices and the 2D boundary SET obstruction class. tionalization class or the anyonic spin-orbit coupling class In order to understand the statement of anomaly match- is trivial [51]. In particular, when G is continuous and ing in this conjecture, we first notice that the cohomology connected, e.g., G ¼ Uð1Þ or SO(3), the anyonic spin-orbit 4 2 group H ½Z × G; Uð1Þ, to which the 2D SET obstruction coupling is necessarily trivial, and, hence, so is ½oi. class ½O belongs, can be decomposed using the Künneth Conversely, nontrivial ½oi can arise only for a noncon- formula to give nected (e.g., discrete) on-site symmetry group G. Examples of such anomalous SET phases occurring at the boundary H4½Z2 × G; Uð1Þ ¼ H4½G; Uð1Þ × (H3½G; Uð1Þ)2 of a 3D system formed by stacking 2D topological insulators has been considered in Ref. [52]. × H2½G; Uð1Þ: ð46Þ Next, we make the following observations regarding the properties of SPT phases. The boundary structure of a The elements of this cohomology group can be similarly nontrivial d-dimensional strong SPT phase cannot be decomposed using the slant product (as we have done physically realized in a local manner as a purely (d − 1)- before). In particular, we can write the 2D SET obstruction dimensional system for d ≥ 2; i.e., it cannot be realized class in this decomposition as ½O¼f½o; ½o ; ½o ; ½o g, x y yx without the corresponding d-dimensional bulk. On the where (for a given representative O of the cohomology other hand, the boundary structure of 0D and 1D strong class) we define SPT phases can be physically realized in a local manner without their corresponding d-dimensional bulk system. oðg; h; k; lÞ¼Oðg; h; k; lÞj ∈ ; ð47Þ g;h;k;l G In particular, the boundary of a 0D SPT phase is trivial, while the boundary of a 1D SPT phase characterized by o ðg; h; kÞ¼ι Oðg; h; kÞj ∈ ; ð48Þ 2 x Tx g;h;k G the index ½ω ∈ H ½G; Uð1Þ gives rise to an emergent projective representation [ω] of the on-site symmetry group o ðg; h; kÞ¼ι Oðg; h; kÞj ∈ ; ð49Þ y Ty g;h;k G G. Projective representations can obviously exist as

041068-14 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) independent objects (i.e., without the support of a corre- cell, such as a S ¼ 1=2 magnet, by imposing the constraint sponding 1D bulk system) within systems of any that ½oyx must be equal to the projective representation per dimensionality. unit cell for purely 2D SET phases. For this, we consider Z3 In the context of a 3D SPT with symmetry trans × Gint, the properties of ½oyx in more detail and examine several these observations imply the following properties. (1) The special cases. boundary structure corresponding to nontrivial [ν] requires In the following, we focus on topological orders in the 3D bulk to exist, so it cannot be physically realized in a which the F symbols involving only Abelian anyons can be purely 2D system in a local manner that preserves sym- set to 1 by an appropriate gauge choice, and symmetry metry. (2) The boundary structure corresponding to non- G ¼ Z2 × G that does not permute the anyon types. In this ν trivial ½ i requires the array of 2D SPT phases (stacked in case, the obstruction class is given by Eq. (45), and we, the 3D bulk, as explained in Sec. III B) to exist, so it cannot thus, have be physically realized in a purely 2D system in a local b b manner that preserves symmetry. (3) The boundary struc- R ðTx;gÞ; ðTy;hÞ w b ture corresponding to nontrivial ½ν can be physically oyxðg; hÞ¼M ðg;hÞ; ðTy;TxÞ b b ; ð52Þ yx R ðTy;gÞ; ðTx;hÞ realized in a purely 2D system. Specifically, this class manifests emergent degrees of freedom on the boundary for g; h ∈ G. of the 3D system, which transform under G such that each For anyon models with nontrivial F symbols, Eq. (52) 2D unit cell on the surface transforms as the projective acquires an additional contribution o~ yxfFg, which is a ν representation ½ yx of G. This structure can instead be combination of several F symbols (see Appendix D for physically realized in a purely 2D system by constructing it details). However, we have verified that if either (a) G is from physical degrees of freedom that transform under G in Abelian, (b) G is connected and continuous, or (c) there is a ~ such a way that each unit cell transforms as the projective point-group symmetry relating Tx to Ty, then ½oyxfFg¼½1. ν representation ½ yx of G. In other words, for these cases, the modification of oyx is Finally, we make the observation that any purely 2D SET simply a 2-coboundary, and we can still use Eq. (52) as a phase (i.e., one that can be physically realized in a strictly representative of the cohomology class ½oyx. 2D system in a local manner) can be realized on the It is sometimes useful to consider quantities constructed boundary of a 3D SPT. Combining these observations with from the cocycles, such as the conjectured bulk-boundary correspondence leads to the following understanding of the 2D SET obstruction class oyxðg; hÞ MbðTx;gÞ;bðTy;hÞ Mb b ; 53 when translational symmetry is included. ¼ ðg;hÞ; ðTy;TxÞ ð Þ oyxðh; gÞ MbðTy;gÞ;bðTx;hÞ The obstruction class ½O¼f½o; ½ox; ½oy; ½oyxg of a G Z2 purely 2D SET phase with symmetry ¼ trans × Gint which is gauge invariant when gh ¼ hg, and thus may 1 must have trivial components ½o¼½ox¼½oy¼½ , readily indicate when the cohomology class is nontrivial. while the component ½oyx must equal the projective representation of G per unit cell of the 2D system. If each 1. Trivial topological order unit cell of the system transforms locally as a linear representation under the on-site symmetries G, then we A gapped topological phase with trivial topological must have ½oyx¼½1. order, for example, a featureless paramagnet or insulator, If these conditions on ½O of the 2D SET phase are not will necessarily have a vanishing obstruction class O 1 . Consequently, the above interpretation of the satisfied, i.e., if [o], ½ox,or½oy are nontrivial or if ½oyx is ½ ¼½ not equal to the projective representation of G per unit cell, obstruction class implies that it is forbidden for such phases then the SET cannot be physically realized as a strictly 2D to occur in a model with a nontrivial projective represen- system. We emphasize that this gives an interpretation of tation of G per unit cell. This is the result of the (higher- the obstruction class that differs from the interpretation dimensional) LSM theorem, but now generalized to the when there is only on-site symmetry. In particular, a case of arbitrary on-site unitary symmetries G, including discrete symmetries. nontrivial obstruction class component ½oyx does not rule out physical realization of the system in purely 2D, but rather requires that each 2D unit cell transforms as the 2. Trivial contribution to ½oyx from anyonic projective representation ½oyx. This result provides con- spin-orbit coupling straints on the allowed SET order of a given 2D system. We now consider the case when the anyonic spin-orbit coupling symmetry fractionalization class gives trivial A. Projective representation per unit cell contribution to ½oyx, so that Eq. (52) reduces to We are finally prepared to analyze LSM-type constraints ½o ðg; hÞ ¼ ½Mw b ¼½ηb ðg; hÞ: ð54Þ for a 2D system with a projective representation per unit yx ðg;hÞ; ðTy;TxÞ ðTy;TxÞ

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There are a number of scenarios in which this may occur, form a projective representation of G, with multiplication including (a) the anyonic spin-orbit coupling class is trivial, given by i.e., bðTi; gÞ¼I for i ¼ x, y (as previously mentioned, this is automatically the case when the on-site symmetry group UgUh ¼ oyxðg; hÞUgh: ð57Þ G is continuous and connected) and (b) the anyonic spin- orbit coupling classes are equal in the two directions, i.e., In other words, the successive applications of this pro- bðTx; gÞ¼bðTy; gÞ for all g (this will occur when the cedure with Ig and Ih defects is related to an application of system preserves a point-group symmetry that relates the xˆ the procedure with an Igh by the phase oyxðg; hÞ. and yˆ directions, for example, C3, C4, C6 lattice rotations, On the other hand, we can arrange the processes involved or reflections that interchange x and y). in successively applying the local actions for g and h in a Recalling the discussion from Sec. IV B, we know that slightly different order. First, create both the Ig and Ig pair ηb g; h is the projective representation of G carried ½ ðTy;TxÞð Þ from vacuum and also the Ih and Ih pair from vacuum. b w by the anyon ðTy;TxÞ, which is the anyonic flux per unit Next, fuse Ig and Ih, which results in ½ ðg; hÞgh ¼ w w cell. In other words, Eq. (54) implies that, for purely 2D ðg; hÞ × Igh. Then, adiabatically transport ½ ðg; hÞgh SET phases with such anyonic spin-orbit coupling, the around a unit cell in a counterclockwise fashion. Finally, projective representation of G carried by the anyonic flux annihilate all the defects to vacuum. Transporting w per unit cell must be precisely equivalent to the projective ½ ðg; hÞgh around the unit cell acquires an extra phase representation describing the local symmetry action of each Mwðg;hÞ;bðTy;TxÞ relative to transporting Igh around the unit unit cell (i.e., of the microscopic degrees of freedom). In cell, because of the extra topological charge w g; h that is b ð Þ this sense, the anyonic flux per unit cell, ðTy;TxÞ,isa being transported around the anyonic flux per unit cell spinon which screens the projective representation per unit bðT ;T Þ. Comparing the two results produces Eq. (54). 1 2 y x cell. Indeed, this is precisely how the S ¼ = parton In summary, we have derived the following sufficient construction works, as there is a density of one parton conditions for the existence of a spinon. 1 2 (spinon) per S ¼ = moment. If (i) anyons are not permuted by the symmetries and We may also be interested in quasiparticles that play a (ii) either G is continuous and connected or the system has Z2 role similar to that of the vison in a spin liquid. For this, a point-group symmetry such as C3, C4, C6 or a reflection we consider that relates x to y, then quasiparticles that carry the topological charge b T ;T are spinons, as they carry η ð y xÞ oyxðg; hÞ bðg;hÞðTy;TxÞ ¼ Mb b ¼ ; ð55Þ the same projective representation ½oyx of G that is carried ðg;hÞ; ðTy;TxÞ η oyxðh; gÞ bðg;hÞðTx;TyÞ by each unit cell. which is gauge invariant when gh ¼ hg. When this 3. Nontrivial contribution to ½oyx from anyonic quantity is nontrivial (i.e., not equal to 1), the anyon spin-orbit coupling bðg; hÞ has nontrivial mutual statistics with the anyonic We now examine a situation where the system need flux per unit cell, i.e., the spinon, and, hence, transforms −1 −1 not have a spinon, even though it has a nontrivial projective projectively under Ty Tx TyTx. In this sense, bðg; hÞ is Z representation per unit cell. Let us consider the case similar to a vison excitation. In particular, for the 2 spin where the anyonic spin-orbit coupling is nontrivial, i.e., liquid, one finds bðTi; gÞ ≠ I, and the relevant point-group symmetry is broken explicitly or spontaneously (e.g., by a nematic o X; Z yxð Þ −1 order parameter). In this case, it is possible to satisfy the ¼ MbðX;ZÞ;bðTy;TxÞ ¼ Mm;e=ψ ¼ ; ð56Þ oyxðZ; XÞ condition for a purely 2D SET phase that oyx in Eq. (52) matches the projective representation per unit cell, even b where ðX; ZÞ¼m is the vison. when the on-site symmetry fractionalization class is trivial, We can understand Eq. (54) when the anyonic spin-orbit i.e., ½w¼½I, and the anyonic flux per unit cell is trivial, coupling is trivial through the following heuristic i.e., bðTy;TxÞ¼I. In such a situation, none of the anyons flux-insertion type argument. First, create a g-defect and carry a nontrivial projective representation of G and there is ¯ g-defect pair from vacuum, say, carrying labels Ig and Ig, no background anyonic flux. This yields respectively. Next, adiabatically transport the Ig defect b b along a path enclosing one unit cell in a counterclockwise R ðTx;gÞ; ðTy;hÞ oyxðg; hÞ¼ b b ; ð58Þ fashion. Finally, annihilate the pair of defects into vacuum, R ðTy;gÞ; ðTx;hÞ returning the system to the ground state. This process can be thought of as applying the g symmetry transformation so ½oyx can still match a nontrivial projective representation locally to a unit cell. We schematically denote the g action per unit cell, but it requires a rather unconventional way of on the local degrees of freedom by Ug. Such local actions doing so.

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The toric code with unconventional symmetry model creating a bðTi; gÞ − bðTi; gÞ pair, leaving bðTi; gÞ on described in Sec. IV C 3 realizes this scenario. In particular, the new segment of defect branch line, and compounding we see that the bðTi; gÞ topological charge with the defect to change its topological charge value. An additional implication o ðX; ZÞ Mb T ;X ;b T ;Z yx ð x Þ ð y Þ is then that we cannot adiabatically transport a g defect ¼ ¼ Mm;e ¼ −1; ð59Þ o ðZ; XÞ Mb b ˆ yx ðTx;ZÞ; ðTy;XÞ in the i direction while respecting h symmetry when ≠ 1 MbðTi;gÞ;bðTi;hÞ . which indicates that ½oyx is the nontrivial element in 2 2 One way to verify this interpretation is to braid a H ½Z2; Uð1Þ ¼ Z2. This model breaks C4 symmetry, as quasiparticle of topological charge c0 around a unit seg- the symmetry was defined to act nontrivially only on the ment of the defect branch line in a counterclockwise vertical bonds. fashion, as shown in Fig. 5(a). Since crossing the defect branch line applies the symmetry action to the object B. Physics of the anyonic spin-orbit coupling crossing the branch, this has the same effect as the local −1 −1 action of R Rg RT Rg on the anyon c0, which is to say it While the physics of the anyonic flux per unit cell Ti i generates a phase equal to ðη ðT ; gÞ=η ðg;T ÞÞ ¼ bðTy;TxÞ is straightforward and has been understood quite c0 i c0 i M b . As this holds for arbitrary topological charge well in the context of quantum Hall states and topological c0; ðTi;gÞ ˆ spin liquids, the anyonic spin-orbit coupling has received c0, it implies that a unit length in the i direction of g-defect much less attention. In Sec. IV B, we discuss the physics of branch line carries topological charge bðTi; gÞ. anyonic spin-orbit coupling fractionalization classes with Note that the vertical height of the loop in Fig. 5(a) respect to quasiparticles, which yield the interpretation that should be taken to be much larger than the width of the g- quasiparticle string operators (Wilson lines) carry sym- defect branch line, which is set by the correlation length of metry charge per directed unit length. In this section, we the system. In some cases, the g-defect branch line is not examine in more detail the properties of symmetry defects localized, e.g., if the g defect results from a lattice when there is anyonic spin-orbit coupling symmetry dislocation [54,55] or if the G symmetry results from fractionalization. spontaneously breaking a larger continuous symmetry. In We know that the application of the translational this case, one can instead compare the result of braiding c0 symmetry action to an on-site symmetry g defect (sym- along two paths enclosing the defect, one of which is ˆ metry flux) carrying topological charge ag transforms the translated along the i direction relative to the other, as ρ b topological charge value to Ti ðagÞ¼½ ðTi; gÞ × ag.In shown in Fig. 5(b). other words, the translational symmetry action changes the Another way to verify this is to adiabatically transport a topological charge of a g defect by bðTi; gÞ. Naively, it defect carrying topological charge ag around a dislocation, might seem that the implication of such a translational i.e., a translational symmetry defect (movon). As shown in symmetry action would be that g defects cannot be Fig. 6,ifag-defect branch line carries topological charge ˆ b ˆ adiabatically transported in the i direction, since the ðTi; gÞ per unit length in the i direction, then the total topological charge value carried by the defect must change when it moves. Indeed, it is not possible to adiabatically −1 c0 Tx transport the position of the defect in a manner where the c0 Hamiltonian of the defect is simply translated, as this would −1 either involve a level crossing or violate conservation of g g g topological charge. However, it is possible to adiabatically g transport a g defect by suitably changing the form of the Tx defect Hamiltonian to favor a different value of topological Tx charge localized at the end point of the defect branch line. (a) (b) In particular, for a g defect carrying the energetically favored topological charge of ag, adiabatically transporting FIG. 5. (a) Braiding an anyon c0 around a unit length of the defect by one unit length in the iˆ direction involves g-defect branch line is equivalent to the local action of R−1R−1R R on the anyon (shown here for i ¼ x). This implies extending the defect branch line by one unit length and Ti g Ti g b ending with a Hamiltonian that energetically favors topo- that g-defect branch lines carry topological charge ðTi; gÞ per ˆ b unit length in the i direction. (b) The same result is obtained by logical charge ½ ðTi; gÞ × ag at the new end point of the comparing the results of braiding c0 around two loops, each of branch line [53]. This can be interpreted to mean that which encloses the g defect, and where one is translated relative g-defect branch lines carry topological charge bðTi; gÞ per ˆ ˆ to the other in the i direction. This alternative derivation allows a unit length in the i direction. Thus, adiabatically trans- generalization to the case where the g-defect branch line is not porting a g defect by a unit length in the iˆ direction involves localized.

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along the links in the xˆ direction). For vertex terms −Av that σz straddle the branch line, we conjugate by k for the site k on the vertical bond on the right-hand side of the branch line, which gives the replacement −Av ↦ þAv. In other words, we modify the Hamiltonian to energetically favor an e quasiparticle at each vertex v along the Z-defect branch line running in the xˆ direction, as shown in Fig. 7(a). This is consistent with the interpretation of the Z-defect branch b ˆ line carrying topological charge ðTx;ZÞ¼e per unit FIG. 6. Braiding a g defect around an i dislocation (Ti defect) length in the xˆ direction. If we apply a translation R leaves a g-defect branch line loop encircling the dislocation. If g- Tx to the system, the positions of the flipped vertex terms are defect branch lines carry bðT ; gÞ per unit length, the branch line i simply moved by one lattice spacing in the xˆ direction, as loop carries a total topological charge of bðT ; gÞ. The topological i shown in Fig. 7(b). Locally, the form of the Hamiltonian at charge of the g defect will change by bðTi; gÞ, which matches the ρ b the translated defect (branch line end point) does not translational symmetry action T ðagÞ¼½ ðTi; gÞ × ag. Here, i change; i.e., it energetically favors the same value of we display a Tx defect with a g-defect loop encircling it, topological charge and the local density matrix around explicitly labeling the bðTx; gÞ topological charge per directed unit length of the g-defect branch line [bðTy; gÞ labels are left the defect is related to the previous one by translation. implicit]. However, we cannot reach such a configuration from the initial state by adiabatically transporting the defect, since it would violate topological charge conservation (e anyons topological charge of the g-defect branch line loop left must be created or annihilated in pairs). b encircling the Ti defect will equal ðTi; gÞ. [This g-defect In order to adiabatically transport the defect by one branch line loop can be shrunk onto the Ti defect and lattice spacing in the xˆ direction, we adiabatically change removed, leaving behind an extra topological charge of −Av ↦ þAv for the vertex immediately to the right of the b ðTi; gÞ on the Ti defect.] By conservation of topological defect, but must also modify the Hamiltonian in the vicinity charge, the topological charge of the g defect must change of the defect (branch line end point) in a way that conserves b from ag to ½ ðTi; gÞ × ag as a result of the braiding topological charge. For example, if we wish to move a process. This is consistent with the fact that transporting defect with topological charge IZ by one lattice spacing in an object around a defect effects the symmetry action upon the xˆ direction, we can conserve topological charge by that object, i.e., transporting ag around a Ti defect yields adiabatically changing −Av ↦ þAv for the two vertices to ρ b Ti ðagÞ¼½ ðTi; gÞ × ag. Note that this is also consistent the right of the defect, as shown in Fig. 7(c). This creates ρ two new e anyons: the first e anyon is interpreted as being with the g symmetry action on Ti defects; i.e., gðaTi Þ¼ the unit length extension of the Z-defect branch line in the xˆ ½bðTi; gÞ × a . Ti direction, and the second e anyon is interpreted as being It is worth considering the on-site symmetry defects associated with the Z-defect itself, modifying its topologi- of the toric code with unconventional Z2 × Z2 symmetry cal charge from I to e . If we subsequently adiabatically (introduced in Sec. IV C 3) in detail. The defect branch Z Z transport the defect a second time in the xˆ direction, we lines for this example can be drawn along the links of the lattice in the xˆ direction and across the middle of could do so by simply leaving the Hamiltonian unchanged plaquettes in the yˆ direction [56]. Following the prescrip- and subsequently interpreting the second e anyon as being tion described in Ref. [26], g defects are included by the second unit length extension of the Z-defect branch line in the xˆ direction, as shown in Fig. 7(d). If, instead, we had modifying the Hamiltonian in the following way: for each ˆ term in the Hamiltonian that straddles the defect branch subsequently adiabatically transported the defect in the y ˆ line, denote the sites on the right-hand side of the directed direction, the second e anyon would be transported in the y direction along with the defect branch line end point (while branch line as Cr, and conjugate theQ corresponding term in ðkÞ the first e anyon remains fixed at its location), as shown in the Hamiltonian by the operator ∈ Rg , i.e., apply the k Cr Fig. 7(e). symmetry transformation to only the sites on one side of the Returning to the general case, we can use the interpre- branch line. The Hamiltonian at the end points is ambigu- tation of the defect branch line carrying topological charge ous under this prescription, and may possibly be chosen to per directed unit length to reexamine the heuristic argument favor localization of different topological charge values. in Sec. VI A 2 explaining the physical meaning of the ˆ Let us focus on a Z-defect branch line along the x obstruction matching condition for the case when there is direction in this example and apply this defect construction. anyonic spin-orbit coupling. In particular, we consider the ðkÞ σz The on-site symmetry operator corresponds to RZ ¼ k. local g action, denoted Ug, on a unit cell by pair creating a − σz ¯ Conjugating plaquette terms Bp by k obviously leaves it g-g defect pair, adiabatically transporting the g defect invariant (which is why we are free to draw the branch line around a path enclosing one unit cell in a counterclockwise

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fashion, and then pair annihilating the defects. To be more concrete (without any loss of generality), let us choose the path to be r → r − yˆ → r − yˆ þ xˆ → r þ xˆ → r. Keeping track of the topological charge creation and annihilation due to the adiabatic transportation of the g defect creating a (a) loop of defect branch line, we have (1) r → r − yˆ creates

topological charge bðTy; gÞ on the left segment of defect branch line, (2) r − yˆ → r − yˆ þ xˆ creates topological charge bðTx; gÞ on the lower segment of defect branch line, (3) r − yˆ þ xˆ → r þ xˆ creates topological charge bðTy; gÞ on the right segment of defect branch line, and ˆ → b (b) (4) r þ x r creates topological charge ðTx; gÞ on the upper segment of defect branch line. The corresponding configuration of topological charges for this g-defect branch loop is illustrated in Fig. 8(a). One can think of these topological charge values as anyons connected by anyonic string operators (Wilson lines) of the correspond- ing types, which can be deformed to the configuration of (c)

(d) (a) (b)

(e) (c) (d) FIG. 7. (a) In the toric code with unconventional symmetry, Z defects exist at the end points of a defect branch line (indicated FIG. 8. Physical interpretation of the contribution to the by the dashed line), which is produced by conjugating the spins obstruction from the anyonic spin-orbit coupling from topological on the vertical links on one side of the branch line by σz for terms charge per directed unit length of defect branch lines. (a) Pair in the Hamiltonian that straddle the branch line. This changes creating a g-g¯ defect pair, adiabatically transporting the g defect −Av ↦ þAv for vertex terms straddling the branch line, ener- counterclockwise along a path enclosing a unit cell, and then getically favoring an e quasiparticle (indicated by solid black annihilating the defects generates a loop of g-defect branch line dots) at those vertices. (b) The configuration obtained from that in encircling the unit cell (dashed blue line). This applies local g action U on the unit cell. Each edge of the unit cell can then be (a) by acting on the system with the translation operator RTx. g (c) The configuration obtained from (a) by adiabatically trans- associated with a topological charge, as indicated in the figure. porting the defect in the xˆ direction by one lattice spacing. In this (b) The topological charges associated with each edge of the unit case, the topological charge of the defect must change from IZ to cell can be obtained by applying Wilson lines in the horizontal eZ. (d) The configuration obtained from that in (c) by adiabati- and vertical directions, as shown. The precise manner in which cally transporting the defect in the xˆ direction by one lattice the crossing point is resolved does not affect the final result. spacing. In this case, the configuration of the physical system is (c) The configuration of anyon strings associated to the process unchanged, but is interpreted differently. (e) The configuration UgUh. Solid lines correspond to the Ug process and dashed lines obtained from that in (c) by adiabatically transporting the defect to the Uh process. (d) The configuration of anyon strings ˆ in the y direction by one lattice spacing. associated to the process UhUg.

041068-19 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) anyonic string operators shown in Fig. 8(b). [The inter- is one vertex site with three edges. The vertex term at this x x x section of anyonic string operators W and Wb T ;g in site σ1σ2σ3 breaks the Z symmetry and, therefore, must bðTx;gÞ ð y Þ Fig. 8(b) must be resolved to provide a proper definition, have its coefficient set to zero. This results in a degeneracy σxσxσx 1 but the details of how they are resolved does not affect the at the dislocation labeled by the stabilizer 1 2 3 ¼ , quantities of interest to us, so we simply leave it drawn as which can be interpreted as the two degenerate topological an intersection.] charge values of the translational symmetry defect, differ- b We can now compare the corresponding anyonic string ing from each other by ðTx;ZÞ¼e. operator configurations that arise from successive applica- tions of g and h symmetry actions and vice versa, i.e., C. Fractional U(1) charge per unit cell UgUh and UhUg, which are shown in Figs. 8(c) and 8(d), The higher-dimensional LSM theorem also holds for respectively. Using the relation for Abelian anyonic string U(1)-symmetric translationally invariant systems with frac- operators tional charge density ν ¼ p=q per unit cell [57]. Since the U(1) group has no projective representations, there are no 3D weak SPT phases with nontrivial νxy invariants, nor 2D SET ð60Þ phases with nontrivial oyx, and our previous argument does not apply. Proceeding more heuristically than in the projec- tive case, our goal is to establish the following well-known claims [3,4,58], within the language of the SET formalism. we find that these two configurations are equivalent up to (1) Adiabatically threading 2π flux results in an anyonic appropriate phase factors obtained from passing the string excitation called the “vison” v. The mutual statistics between v and an excitation carrying topological charge operators for Ug through those of Uh. Specifically, we find a determines the fractional U(1) charge Qa of a through the i2πQ Mb b relation e a M . −1 −1 ðTx;gÞ; ðTy;hÞ oyxðg; hÞ ¼ v;a UgUhUg Uh ¼ ¼ : ð61Þ (2) There is an anyon b T ;T per unit cell whose Mb b o ðh; gÞ ð y xÞ ðTy;gÞ; ðTx;hÞ yx ν fractional U(1) charge is equal to the filling QbðTy;TxÞ ¼ This is the desired obstruction matching condition. (mod integers). Quasiparticles that carry topological charge b “ ” Finally, let us mention another interesting manifestation ðTy;TxÞ are called spinons. i2πσH of anyonic spin-orbit coupling. Consider applying an on- (3) The Hall conductance satisfies e ¼ Mv;v. site symmetry action to a translational symmetry defect Consider a translationally invariant system with a non- (i.e., a dislocation), which gives ρ a b T ; g × a , integer filling fraction ν of some U(1) charge, so gð Ti Þ¼½ ð i Þ Ti changing the topological charge of a translational sym- G ¼ Z2 ×Uð1Þ. According to the results in Sec. IV B, we can completely specify the symmetry fractionalization metry defect (movon) by bðTi; gÞ. Since the two types of translational symmetry defect topological charges are class by the following data: related by symmetry transformations, they must be ener- b w w getically degenerate. We can see this explicitly in the ðTy;TxÞ¼ ðTy;TxÞ × ðTx;TyÞ; ð62Þ example of the toric code model with unconventional 1 2π θ θ0− θ θ0 symmetry. In Fig. 9, we show an xˆ dislocation (associated wðθ; θ0Þ¼vð = Þð þ ½ þ 2π Þ; ð63Þ with the pentagonal plaquette), and on the dislocation there where these are all elements in the set of Abelian anyons A. Here, we label the U(1) group elements by an angle θ ∈ ½0; 2πÞ. The second line serves as the definition of 2 3 1 the vison excitation v ∈ A. (As described in Sec. IV C 1, the corresponding excitation in a fractional quantum Hall state is the Laughlin-type fundamental quasihole ϕ.) Physically, one can argue that v is the anyon nucleated by adiabatically threading 2π U(1) flux as follows. Threading flux θ into the system (e.g., into a plaquette) results in a defect Iθ. If we nucleate another defect Iθ0 nearby and then fuse them, we obtain FIG. 9. An xˆ dislocation in the toric code model. When the unconventional Z2 × Z2 symmetry is imposed, the coefficient of 0 0 x x x Iθ × Iθ0 ¼½wðθ; θ Þ θ θ0 ¼ wðθ; θ Þ × I θ θ0 : ð64Þ the vertex term σ1σ2σ3 must be set to zero to respect the Z ½ þ 2π ½ þ 2π symmetry. This results in a degeneracy between topological 0 charge values of the translational symmetry defect that differ by If we let θ ¼ 2π − θ, this gives Iθ × I2π−θ ¼ v. Similarly, if bðTx;ZÞ¼e. we did this repeatedly with θ ¼ 2π=N, we would find

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N ðI2π=NÞ ¼ v (where N is chosen to be a multiple of jAj). VII. DISCUSSION Since U(1) is continuous (and we can take the limit A. Symmetries that permute the anyon types N → ∞), these give the same result as adiabatically In this paper, we focus on SET phases in which no anyon creating a defect Iθ and increasing θ from 0 to 2π. Thus, the vison v is the result of 2π-flux insertion. Note types are permuted by the symmetries. Here, we briefly that H2½Uð1Þ; A¼A, which is consistent with this physi- discuss the case when there are permutations. cal interpretation. If G is continuous and connected (so that G cannot Next, let us suppose that a vison v is adiabatically permute anyon types), it was argued in Ref. [25] that there transported around a path that encloses an anyon a in a must be a spinon regardless of whether translations permute counterclockwise fashion. On one hand, it will accumulate anyon types. This is not surprising, as the Oshikawa flux- the mutual statistics phase M due to v encircling the threading arguments should apply. Presumably, this result v;a can be derived from the obstruction theory, but the anyon a. On the other hand, this process is equivalent to a O 2π flux encircling the anyon a, and the Aharonov-Bohm calculation of ½ seems to be so involved that we do 2π not attempt doing so in this paper. effect indicates that it should acquire the phase ei Qa due to If G is discrete, we first consider the case where the flux 2π encircling a charge Q . Equating the two gives a translations permute the anyon types (while G does not). the relation In this case, the LSM constraint can be satisfied without spinons, if the on-site symmetry group G is not connected i2πQa e ¼ Mv;a: ð65Þ and continuous. One example is Wen’s plaquette model on a square lattice, as defined in Ref. [32] and discussed in this We also consider letting v be transported along a path context in Ref. [25]. In this model, the e and m anyons are that encloses a unit cell that is in the vacuum state. Similar violations of single plaquette terms of the Hamiltonian on to above, transporting a 2π flux around a unit cell measures the two sublattices, so translations by one lattice spacing the total charge enclosed by the path, which is precisely the permute the two types of anyons: ρT ∶ e ↔ m for i ¼ x, y. ν i fractional U(1) charge per unit cell, giving the accumu- We also define a global Z2 × Z2 symmetry generated by 2πν Q Q lated phase ei . We can also view this process as the the operators R ¼ σx and R ¼ σz. Apparently, b X r r Z r r braiding between the vison v and the anyon ðTy;TxÞ per there is a projective representation of the Z2 × Z2 sym- unit cell, which gives the phase Mv;bðTy;TxÞ. Equating the metry on each site. Therefore, we would expect that the two phases, we have the relation symmetry fractionalization class should give the obstruc- tion class consistent with the nontrivial weak SPT order in 2π i2πν i QbðT ;T Þ 3D. With anyon permutation, the classification of sym- e ¼ M b ¼ e y x : ð66Þ v; ðTy;TxÞ metry fractionalization is now given by the second coho- 2 mology group HρðG; Z2 × Z2Þ. The implication is twofold. (1) bðT ;T Þ should carry a y x We can determine the fractionalization class in this fractional U(1) charge ν (mod 1). (2) The vison v has example. It is easy to see that the on-site symmetry does b i2πν nontrivial braiding statistics with ðTy;TxÞ, given by e . not fractionalize at all, so there are no spinons. However, (We note that, with this definition of filling, the Laughlin the fractionalization class corresponding to noncommuta- ν −1 states have filling ¼ =m, corresponding to a density of tivity between the on-site symmetries G and the translations 1 −1 =m charge electrons per unit cell.) gives a nontrivial anyonic spin-orbit coupling. In particular, Finally, the Hall conductance, which is equal to the U(1) a ψ fermion can be created by a string of σy, which 2π charge moved into a region by threading a flux through anticommutes with both σx and σz. Thus, we have it, is simply equal to the charge Qv of the vison, since v is the anyon that results from threading the 2π flux. Thus, we ηψ ðTi;XÞ ηψ ðTi;ZÞ have the relation ¼ ¼ −1; ð68Þ ηψ ðX; TiÞ ηψ ðZ; TiÞ 2πσ 2π ei H ¼ ei Qv ¼ M : ð67Þ v;v for i ¼ x, y. This gives bðTi;XÞ¼bðTi;ZÞ¼e or m (these two values are gauge equivalent). We again emphasize that these relations can be viewed as On the other hand, one may consider the case when only imposing sharp constraints on the SET orders allowed in a the on-site symmetries in G can permute the anyon types. system. For example, a fractional Hall conductance When the anyonic spin-orbit fractionalization is trivial, one requires the existence of anyons with matching statistics, can apply the heuristic argument given in Sec. VI A, since i.e., there must be an Abelian anyon v with topological the adiabatic processes of moving on-site symmetry defects iπσH twist θv ¼e , which characterizes the U(1) symmetry (which may be non-Abelian) remain well defined as long fractionalization associated with winding the phase by 2π as the on-site symmetry defects are not permuted under and is the anyon created by threading a 2π flux. translations. The heuristic argument strongly suggests the

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Ly existence of spinon per unit cell. Again, this can presum- when an object crosses the defect line, it is acted on by Ty , ably be resolved by calculating the obstruction in this most transporting it forward, which is equivalent to periodic general case. boundary conditions. When a model with fractional symmetry charge per B. Constraints from additional symmetries unit cell is placed on a cylinder with circumference Ly odd (or, more generally, a circumference incommensurate with the Our results can be further strengthened when combined fractional filling), the ground states break translational with other symmetries. For example, we have shown that, symmetry in a subtle fashion. This is most familiar in the for G ¼ SOð3Þ, the anyonic flux per unit cell bðT ;T Þ y x thin-torus limit of the FQH effect, where the orbital occupa- must be a spinon. Now consider some other symmetry tions become charge-density wave patterns 0101 , element h which commutes with all elements of G. Since breaking T translational symmetry. Note that according to the anyonic flux per unit cell bðT ;T Þ is physically well x y x Eq. (43),whenT acts on the defect it transforms as defined and gauge invariant, it must be invariant under h: x ρ (b ) b h ðTy;TxÞ ¼ ðTy;TxÞ.Ifh is an antiunitary (i.e., time- ρ b Ly T ða Ly Þ¼ ðTx;TyÞ × a Ly : ð69Þ reversing) or spatial parity reflecting symmetry, the topo- x Ty Ty logical twist is complex conjugated under the action of the b Ly ≠ When ðTx;TyÞ I, this implies that the MES jaT Ly i symmetry; i.e., θρ ¼ θ [26]. Thus, anyons that are left y hðaÞ a breaks translational symmetry, in which case it is actually invariant under h must have θ ¼1. In particular, this a permuted into another MES. This is precisely the increase in would require that bðT ;T Þ is a boson or a fermion. x y the unit cell expected of a phase at fractional filling. This fact excludes the double-semion topological order It is also clear why a nontrivial anyonic spin-orbit from possessing a spinon. Recall that the double-semion 0 0 coupling fractionalization class can result in an exception model has four types of anyons, fI; s; s ;b¼ s × s g, to the cylinder argument for the existence of a spinon. where s and s0 are, respectively, semions of opposite θ θ − The cylinder argument requires that some MES of an odd- chirality (i.e., s ¼ i and s0 ¼ i). The only nontrivial circumference cylinder is invariant under the on-site boson or fermion is b, the boson formed by fusing (or 0 symmetry group G. However, the defect type transforms forming a bound state) of s and s . Time reversal inter- under an on-site symmetry as changes s and s0, while leaving b invariant. As such, s and 0 s must have conjugate representations of G (i.e., they must Ly ρ ða Ly Þ¼bðg;T Þ × a Ly : ð70Þ map to each other under time reversal), so the bound state b g Ty y Ty must carry a trivial representation of G, and, hence, cannot When bðg;T ÞLy ≠ I, the MESs are permuted by the be a spinon. This is the “no-go” theorem proved in y symmetry, and it is impossible to define the symmetry Ref. [25]. properties of the entanglement spectrum as was required in the cylinder argument. C. Relation to cylinder arguments We can also understand the argument why a spinon must We can make connections with the arguments in exist when a rotation relates Tx and Ty, even in the presence Ref. [25] that demonstrated the existence of a spinon by of a nontrivial anyonic spin-orbit coupling fractionalization considering the action of symmetries on the degenerate class. In this case, one can change the basis of the Bravais ˆ ˆ0 ˆ − ˆ b ground states of an “infinite cylinder.” When a topological lattice to x and y ¼ y x. We then have ðg;Ty0 Þ¼I,soif phase is put on an infinitely long cylinder, the ground state we compactify the lattice along the yˆ0 direction into a degeneracy is the same as that of the torus, which is one cylinder (while still being extended in the xˆ direction), the ground state per topologically distinct anyon type a. In the MESs are now invariant under all g and we can apply the “minimally entangled state” (MES) basis for the ground argument to prove the existence of a spinon per unit cell. state manifold, each basis state can thus be labeled by an anyon types as jai. More generally, we can thread sym- D. Possible generalizations metry flux g through the cylinder by twisting the boundary Finally, we speculate on some future directions. condition. It is natural to assume that the ground states are now labeled by the defect sectors jagi. In either case, the 1. Time-reversal symmetry labeling of the MES basis is presumably ambiguous up to In the presence of spin-orbit coupling, the symmetry of a fusion with an Abelian anyon, just like the defect sectors; A spin system is broken down to time reversal. In a Mott i.e., the MES basis is a torsor over . insulator, each unit cell contains a Kramers doublet with To incorporate translational symmetry into this picture, 2 T ¼ −1. Thus, there is a projective representation of T in we assert that a cylinder of circumference L corresponds y each unit cell, and an extension of our result should hold. to defect sector a Ly . After all, a cylinder can be viewed as Ty Indeed, it was argued in Ref. [25] that when translations do Ly an infinite plane with a Ty defect running along the x axis; not permute anyons, there is a spinon excitation in each unit

041068-22 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) cell carrying T 2 ¼ −1. However, at present we cannot 4. Gapless phases and the FL phase address the exotic cases, i.e., when anyons are permuted by When a system with fractional symmetry charge per unit translations or there is anyonic spin-orbit coupling between cell is gapless, there must also be constraints on the T and Ti. The issue is that, while the fractionalization of T ’ 2 resulting phase. The most obvious example is Luttinger s can easily be incorporated into the Hρ classification, we do theorem [63]: at filling ν, the volume of the Fermi sea is not know how to define a defect theory or the correct proportional to ν. Another example is 1D spin chains with conditions for computing an obstruction O. spin-1=2 per unit cell, which can be viewed as the edge of a 2D AKLT state. The 2D AKLT state is a weak SPT, and the 2. Nonsymmorphic symmetries bulk-boundary correspondence now implies that the edge and magnetic translation algebras spin chain cannot be gapped without breaking the spin A series of works have shown that translations can be rotation symmetry or the 1D translational symmetry. (Note supplemented by nonsymmorphic glide symmetries in that the option of topological order is not available in 1D.) order to get tighter LSM-type filling bounds [59–61].It This is, of course, the content of the original LSM theorem. would be interesting to understand the corresponding Analogously, one can further examine the symmetry action generalization of this work [62]. However, it is not in the gapless phase. It was argued that symmetries are completely known how glides can be incorporated into implemented in an anomalous fashion in the resulting the SET formalism. In the least, glides will quite generi- conformal field theory [64] of the spin chain, which is the cally permute anyon types as they conjugate topological 1D (gapless) analog of our constraint. In 2D, there can be a spin. combination of topological and gapless degrees of freedom. Another interesting case is the magnetic algebra Presumably the constraints can be accommodated in a −1 −1 2πϕ ˆ shared fashion between the topological and gapless degrees T T T T ¼ ei N, particularly in lattice-type models y x y x of freedom, suggesting a generalized version of the FL used to generate Chern bands. For ϕ ¼ a=b, we can apply scenario [65]. our results at the cost of considering a magnetic unit cell (as we do for the example of FQH states), but there may be stronger constraints (perhaps “b times” more powerful) that ACKNOWLEDGMENTS arise from considering the full algebra. We thank B. Bauer, M. Hermele, S. Parameswaran, C. Wang, and Z. Wang for helpful conversations. In 3. Itinerant systems particular, we thank M. Hermele and C. Wang for bringing An S ¼ 1=2 magnet is ultimately composed of spinful the model of Sec. IV C 3 to our attention. A. V. acknowl- fermions, which have some residual charge fluctuations. edges support from the Templeton Foundation and a Thus, the true on-site symmetry group is not a projective Simons Investigator award. representation of SO(3), but is instead a linear representa- tion of U(2) [a central extension of SO(3) by U(1)]. More generally, we can consider models in which some U(1) APPENDIX A: REVIEW OF GROUP conserved particles at integer filling ν carry a projective COHOMOLOGY representation [ω]. Intuitively, it seems that we should be Here, we provide a brief review of group cohomology for able to take a limit where charge fluctuations freeze out and finite groups. Given a finite group G, let M be an Abelian consider an effective model with representation ½ων per group equipped with a G action ρ∶ G × M → M, which is unit cell. In 1D, it was shown that a LSM-type result of this compatible with group multiplication. In particular, for any form indeed holds (though there are important differences g ∈ G and a; b ∈ M,wehave in the itinerant case when LSM is further extended to include point-group symmetries) [61]. ρ ab ρ a ρ b : A1 We can speculate about what version of our story we gð Þ¼ gð Þ gð Þ ð Þ expect to hold in 2D for itinerant systems. All local (We leave the group multiplication symbols implicit.) excitations carry charge Q ¼ 1 and a projective represen- Such an Abelian group M with G action ρ is called a G tation [ω]. The relevant form of symmetry fractionalization is representation-charge separation. In other words, there module. ω g1; …; g ∈ M n may be anyons that carry projective representations [ω], but Let ð nÞ be a function of group elements g ∈ G for j 1; …;n. Such a function is called no charge Q ¼ 0, or anyons that carry charge Q ¼ 1,but j ¼ an n-cochain and the set of all n-cochains is denoted only linear representations [1]. Such anyons may differ n only by a local excitation. These are the “spinons” and as C ðG; MÞ. They naturally form a group under holons; presumably they play the same role as the spinon in multiplication, our previous discussions. We leave the full structure of the ω ω0 … ω … ω0 … itinerant case to future work. ð · Þðg1; ; gnÞ¼ ðg1; ; gnÞ ðg1; ; gnÞ; ðA2Þ

041068-23 MENG CHENG et al. PHYS. REV. X 6, 041068 (2016) and the identity element is the trivial cochain ιgωðg1; …; gn−1Þ ω … 1 ðg1; ; gnÞ¼ . Yn−1 n −1 We now define the “coboundary” map d∶ C ðG; MÞ → ð−1Þn þj ¼ ωðg1; …; gj; g; gj 1; …; gn−1Þ : ðA7Þ nþ1 þ C ðG; MÞ acting on cochains to be j¼0

ω … It can be shown that dðι ωÞ¼ι ðdωÞ. Therefore, ι is, in d ðg1; ; gnþ1Þ g g g ρ ω … fact, a group homomorphism: ¼ g1 ½ ðg2; ; gnþ1Þ Yn ι ∶ HnðG; MÞ → Hn−1ðG; MÞ: ðA8Þ ωð−1Þj … … g × ðg1; ; gj−1; gjgjþ1; gjþ2; ; gnþ1Þ j¼1 ð−1Þnþ1 ω g1; …; g : × ð nÞ ðA3Þ APPENDIX B: KÜNNETH FORMULA

One can directly verify that ddω ¼ 1 for any The Künneth formula for the group cohomology of the Hd 0 ω ∈ CnðG; MÞ, where 1 is the trivial cochain in direct product of two groups ðG × G ;MÞ reads: Cnþ2 G; M d “ ð Þ. This is why is considered a boundary d operator.” Hd½G × G0;M¼⨁ Hk½G; Hd−k½G0;M: ðB1Þ With the coboundary map, we next define ω ∈ k¼0 CnðG; MÞ to be an n-cocycle if it satisfies the condition Z dω ¼ 1. We denote the set of all n-cocycles by Here, M is Abelian and finitely generated (e.g., M ¼ or a finite Abelian group), and we only consider M being a 0 n n trivial G × G module. This formula has been derived in ZρðG; MÞ¼fω ∈ C ðG; MÞjdω ¼ 1g: ðA4Þ Ref. [31] for finite G and G0. We shall briefly review the relevant mathematical results and show that the formula n We also define ω ∈ C ðG; MÞ to be an n-coboundary generally holds for discrete groups and Lie groups. if it satisfies the condition ω ¼ dμ for some (n − 1)- We start from the Künneth formula for topological −1 cochain μ ∈ Cn ðG; MÞ. We denote the set of all n- cohomology. Let X and X0 be two topological manifolds, coboundaries by we have [31]

d n n n−1 BρðG; MÞ¼fω ∈ C ðG; MÞj∃μ ∈ C ðG; MÞ∶ω ¼ dμg: Hd½X × X0;M¼⨁ Hk½X; Hd−k½X0;M: ðB2Þ k 0 ðA5Þ ¼ Notice that Hd½X; M is the topological cohomology of n n n n Clearly, BρðG; MÞ ⊂ ZρðG; MÞ ⊂ C ðG; MÞ. In fact, C , chain complexes X with coefficient in M. Now we set Zn, and Bn are all groups and the coboundary maps are X ¼ BG and X0 ¼ BG0, i.e., the classifying spaces of n 0 homomorphisms. It is easy to see that BρðG; MÞ is a normal principle G and G bundles. In order to derive Eq. (B1) n d Hd subgroup of ZρðG; MÞ. Since d is a boundary map, we from Eq. (B2), we need H ½BG; M¼ ½G; M.Itiswell think of the n-coboundaries as being trivial n-cocycles, and known that this is true for discrete G by construction. it is natural to consider the quotient group For a compact (continuous) Lie group G,atheoremof Wigner [66,67] shows that, if we define Hd½G; M as the n Borel cohomology (which is exactly what we need in the ZρðG; MÞ Hn ρðG; MÞ¼ n ; ðA6Þ physical applications), the relation also holds. Therefore, BρðG; MÞ Eq. (B1) holds generally for discrete groups and compact Lie groups. which is called the nth group cohomology. In other words, We also need to establish the formula for U(1) coef- n HρðG; MÞ collects the equivalence classes of n-cocycles ficients. This is achieved by making use of the following that only differ by n-coboundaries. relation: The algebraic definition we give for group cohomology is most convenient for discrete groups. For continuous Hd½G; Uð1Þ ¼ Hdþ1½G; Z: ðB3Þ group, formally the same definition applies, but one has to impose proper continuity conditions on the cocycle Again, when G is continuous, we should use Borel functions. cohomology. Let us now consider M being a G module with trivial We now apply the Künneth formula to compute action, and let g ∈ G be an arbitrary element. We define the the decomposition of Hn½Zd × G; Uð1Þ. First, we can n n−1 slant product ιg∶ C ðG; MÞ → C ðG; MÞ: show that

041068-24 TRANSLATIONAL SYMMETRY AND MICROSCOPIC … PHYS. REV. X 6, 041068 (2016) ðdÞ Z n for 0 ≤ n ≤ d Hn½Zd; Z¼ ðB4Þ Z1 for d

To see this result, we notice that BZd ¼ T d, the d-dimensional torus, and so the right-hand side of Eq. (B4) is the well- known (topological) cohomology of the torus. We then have

Hn½Zd × G; Uð1Þ ¼ Hnþ1½Zd × G; Z Ynþ1 ¼ Hk½G; Hnþ1−k½Zd; Z k¼0 Ynþ1 k ð d Þ ¼ H ½G; Z n−kþ1 k¼maxf1;n−dþ1g Yn k ð d Þ ¼ H ½G; Uð1Þ n−k : ðB5Þ k¼maxf0;n−dg

APPENDIX C: EXPLICIT REPRESENTATIONS OF KÜNNETH DECOMPOSITION We now give explicit representative n-cocycles for each equivalence class in Hn½G × H;Uð1Þ, in terms of lower- dimensional cohomology classes. We write an element of G × H as ðg; hÞ. A U(1) n-cohomology class [ω] can be represented by an n-cocycle of the form

Yn ω( … ) ν … ; … ðg1; h1Þ; ; ðgn; hnÞ ¼ kðh1; h2; ; hk gkþ1; ; gnÞ; ðC1Þ k¼0 where each νk represents a normalized n-cocycle (i.e., dνk ¼ 1). Explicitly, they must satisfy the condition

Yk n 1 p ν kþ1; nþ1 νð−1Þ þ k ; n νð−1Þ … … ; nþ1 kðfhigi¼2 fgjgj¼kþ2Þ k ðfhigi¼1 fgjgj¼kþ1Þ k ðh1; ; hp−1; hphpþ1; hpþ2; ; hkþ1 fgjgj¼kþ2Þ p¼1 Yn q νð−1Þ k ; … … 1 · k ðfhigi¼1 gkþ1; ; gq−1; gqgqþ1; ; gnþ1Þ¼ : ðC2Þ q¼kþ1 1 In this equation, if we set gkþ1 ¼ , we get

Yk p k 1 ν kþ1; nþ1 νð−1Þ … … ; nþ1 νð−1Þ þ k ; nþ1 1 kðfhigi¼2 fgjgj¼kþ2Þ k ðh1; ; hp−1; hphpþ1; hpþ2; ; hkþ1 fgjgj¼kþ2Þ · k ðfhigi¼1 fgjgj¼kþ2Þ¼ ; p¼1 ðC3Þ

which is exactly the k-cocycle condition on H.In Now let us substitute Eq. (C3) back into Eq. (C2). We see other words, for fixed values of gj, the quantity νkðh1; h2; ν … ; … that, when hi are fixed, kðh1; h2; ; hk gkþ1; ; gnÞ is a … ; … k 1 ; hk gkþ1; ; gnÞ is a k-cocycle in Z ½H; Uð Þ.Ifwe (n − k)-cocycle in Zn−k½G; Uð1Þ. Combined with the modify ν by a k-coboundary of H (still fixing all the g ), it k j previous results, it follows that ν corresponds to a is not difficult to see that ν as a n-cocycle of G × H is then k k − Hn−k Hk 1 modified by an n-coboundary, which is cohomologically (n k)-cocycle in ½G; ½H; Uð Þ. We also notice that this parametrization provides a trivial. Therefore, we can say that νk corresponds to a cohomology class in Hk½H; Uð1Þ. More abstractly, we can concrete recipe to write down a representative n-cocycle Hn 1 view ½νk as a function of (n − k) elements of G for the cohomology classes in ½G × H; Uð Þ, given a to Hk½H; Uð1Þ. cohomology class in Hn−k½G; Hk½H; Uð1Þ.

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Furthermore, given the decomposition of Eq. (C1), one This follows from a straightforward application of the can extract νk by applying slant products: definition of the slant product, so we omit the details.

APPENDIX D: SIMPLIFYING ½oyx WHEN THE F SYMBOLS ARE NONTRIVIAL ω( 1 … 1 ) i 1;h i 1;h −1 i 1;h1 ðgk 1; Þ; ; ðgn; Þ ð kÞ Yð k Þ ð Þ þ We simplify the expression of the obstruction class ν … ; … σðpÞ ¼ ðhp1 ; ; hpk gkþ1; ; gnÞ : ðC4Þ involving nontrivial F symbols. First let us display the p full expression of the obstruction:

Fwðg;hÞwðk;lÞwðgh;klÞFwðk;lÞwðh;klÞwðg;hklÞFwðh;kÞwðg;hkÞwðghk;lÞ Oðg; h; k; lÞ¼ Rwðg;hÞ;wðk;lÞ: ðD1Þ Fwðg;hÞwðgh;kÞwðghk;lÞFwðk;lÞwðg;hÞwðgh;klÞFwðh;kÞwðhk;lÞwðg;hklÞ

For a derivation of the result, we refer the readers to neglected in the main text where we assume all F symbols Refs. [18,26]. can be made trivial. We do not bother to write down the We first need to have an explicit general expression of general expression since it is too complicated and not very the F symbols. The Abelian subcategoryQA can always be enlightening. However, before we proceed to calculate A Z ~ decomposed into cyclic groups as ¼ j Nj , i.e., each oðg; hÞ, we first discuss how it is affected by the gauge a ∈ A can be uniquely represented as a tuple transformations on F and R symbols. Recall that F symbols … ∈ Z have the following gauge degrees of freedom: a ¼ða1;a2; Þ, where aj Nj . For Abelian theories, the possible F symbols that satisfy the pentagon consis- tency equation are classified by H3½A; Uð1Þ. These are uða; bÞuða × b; cÞ Fabc → Fabc: ðD6Þ further constrained by the hexagon consistency equations, uða; b × cÞuðb; cÞ which must be satisfied by a braided theory of anyons. The allowed F symbols can then be brought to the form Here, uða; bÞ is an arbitrary U(1)-valued function. X p Mathematically, the gauge transformations modify the F Fabc exp i2π jk a b 3(A 1 ) ¼ ½ jNj ð½ k symbols by a coboundary in B ; Uð Þ . Under such a ≤ NjNk j k gauge transformation, one finds that − þ½ck ½bk þ ckÞN ; ðD2Þ k uðx ;y Þuðx ;y Þ o~ðg; hÞ → o~ðg; hÞ g h h g ð2 − coboundaryÞ: uðyg;xhÞuðyh;xgÞ where pjk ∈ Z. In the nomenclature of Ref. [68], this only involves 3-cocycles of type I and type II, as the type III ðD7Þ cocycles are excluded by the hexagon equations. To shorten the expressions, we define When combining o~ðg; hÞ with the definition of oyx, the extra piece ðuðx ;y Þuðx ;y Þ=uðy ;x Þuðy ;x ÞÞ, which X p g h h g g h h g f a; b exp i2π jk a b : D3 is not generally a coboundary, will be canceled by a similar ð Þ¼ ½ jNj ½ kNk ð Þ j≤k NjNk factor coming from the R symbols, so that oðg; hÞ indeed acquires just a coboundary. So the F symbols can be written as Now let us compute o~ using the gauge choice of Eq. (D4):

abc fða; bÞfða; cÞ F ¼ : ðD4Þ β β β fða; b × cÞ fð ;xghyghÞ fðxg; ygÞfðxh; yhÞ o~ðg; hÞ¼ fðβ;xgygÞfðβ;xhyhÞ fðxg; βyghÞfðxh; βyghÞ We also define for convenience fðy ;x Þfðy ;x Þ fðx ;y Þfðx ;y Þ × g gh h gh g h h g : fðyg;xgÞfðyh;xhÞ fðyg;xhÞfðyh;xgÞ β ≡ bðTx;TyÞ;xg ¼ bðg;TxÞ;yg ¼ bðg;TyÞ: ðD5Þ ðD8Þ

1 We now define o~ðg; hÞ formally as the contribution to Note that xgh ¼ xhg since bð·;TxÞ ∈ H ½G; A. It follows the slant product oyx from the F symbols, which is that

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o~ðg; hÞ that o~ðg; hÞ=o~ðg; hÞ does not depend on the gauge choice ¼ 1: ðD9Þ o~ðh; gÞ of F and R symbols. We now assume lattice rotational symmetry so For Abelian G, Eq. (D9) is the necessary and sufficient xg ¼ yg. Note that if G is continuous, xg ¼ yg ¼ I, and condition to show o~ðg; hÞ is a coboundary. One may the analysis also holds. Again, we can show o~ðg; hÞ is a wonder whether this statement depends on gauge choices coboundary: for F and R symbols. But from Eq. (D7), we can be assured

fðβ;x2 Þ f x ; βx f x ; βx f x ;x f x ; βx f x ;x f x ;x ~ gh ð g gÞ ð h hÞ ð gh ghÞ ð gh ghÞ ð g ghÞ ð h ghÞ oðg; hÞ¼ β 2 β 2 β β β : ðD10Þ fð ;xgÞfð ;xhÞ fðxgh; xghÞ fðxg;xgÞfðxh;xhÞ fðxg; xghÞfðxh; xghÞ fðxgh;xghÞ

The first three factors are exactly in the form of coboundaries. We now write out the last two more explicitly:

β X fðxgh; xghÞ fðxg;xghÞfðxh;xghÞ pjk j j j k k k ¼ exp i2π ð½x − ½xg − ½x Þ ð½β þ x − ½x Þ β β gh h Nj gh gh Nk fðxg; xghÞfðxh; xghÞ fðxgh;xghÞ ≤ NjNk j k X p exp i2π jk xj − xj − xj βk xk − xk − βk ¼ ð½ gh ½ g ½ hÞNj ð½ þ gh ½ gh ½ ÞNk ≤ NjNk j k X p × exp i2π jk xj − xj − xj βk ð½ gh ½ g ½ hÞNj ½ Nk ≤ NjNk j k X p exp i2π jk xj − xj − xj βk ; D11 ¼ ð½ gh ½ g ½ hÞNj ½ Nk ð Þ j≤k NjNk which is also a coboundary. From Eq. (D7), we see that [8] A. Vishwanath and T. Senthil, Physics of Three- when xg ¼ yg, under the gauge transformation of F and R Dimensional Bosonic Topological Insulators: Surface- symbols, o~ðg; hÞ is modified only by a coboundary, so Deconfined Criticality and Quantized Magnetoelectric although we have chosen a particular gauge Eq. (D2) for Effect, Phys. Rev. X 3, 011016 (2013). our computation, our conclusion holds generally. [9] C. Wang and T. Senthil, Boson Topological Insulators: A Window into Highly Entangled Quantum Phases, Phys. In summary, we shown that if either G is Abelian, G is ~ 1 Rev. B 87, 235122 (2013). continuous, or there is a rotation symmetry, then ½o¼½ [10] F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath, and the analysis of the main text holds. Exactly Soluble Model of a Three-Dimensional Symmetry- Protected Topological Phase of Bosons with Surface Topological Order, Phys. Rev. B 90, 245122 (2014). [11] L. Fidkowski, X. Chen, and A. Vishwanath, Non-Abelian [1] E. Lieb, T. Schultz, and D. Mattis, Two Soluble Models of an Topological Order on the Surface of a 3D Topological Antiferromagnetic Chain, Ann. Phys. (Berlin) 16, 407 Superconductor from an Exactly Solved Model, Phys. Rev. (1961). X 3, 041016 (2013). [2] I. Affleck and E. H. Lieb, A Proof of Part of Haldane’s [12] C. Wang, A. C. Potter, and T. Senthil, Gapped Symmetry Conjecture on Spin Chains, Lett. Math. Phys. 12, 57 (1986). Preserving Surface State for the Electron Topological [3] M. Oshikawa, Topological Approach to Luttinger’s Theo- Insulator, Phys. Rev. B 88, 115137 (2013). rem and the Fermi Surface of a Kondo Lattice, Phys. Rev. [13] P. Bonderson, C. Nayak, and X.-L. Qi, A Time-Reversal Lett. 84, 3370 (2000). Invariant Topological Phase at the Surface of a 3D [4] M. B. Hastings, Lieb-Schultz-Mattis in Higher Dimensions, Topological Insulator, J. Stat. Mech. (2013) P09016. Phys. Rev. B 69, 104431 (2004). [14] C. Wang, A. C. Potter, and T. Senthil, Classification of [5] P. A. Lee, An End to the Drought of Quantum Spin Liquids, Interacting Electronic Topological Insulators in Three Science 321, 1306 (2008). Dimensions, Science 343, 629 (2014). [6] L. Balents, Spin Liquids in Frustrated Magnets, Nature [15] X. Chen, L. Fidkowski, and A. Vishwanath, Symmetry (London) 464, 199 (2010). Enforced Non-Abelian Topological Order at the Surface of [7] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry a Topological Insulator, Phys. Rev. B 89, 165132 (2014). Protected Topological Orders and the Group Cohomology [16] M. A. Metlitski, C. L. Kane, and M. P. A. Fisher, of Their Symmetry Group, Phys. Rev. B 87, 155114 (2013). Symmetry-Respecting Topologically Ordered Surface

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