Classification of Symmetry Fractionalization in Gapped

Classiﬁcation of symmetry fractionalization in gapped Z2 spin liquids

Yang Qi1, 2 and Meng Cheng3 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 2Institute for Advanced Study, Tsinghua University, Beijing 100084, China∗ 3Station Q, Microsoft Research, Santa Barbara, California 93106-6105, USA† In quantum spin liquids, fractional spinon excitations carry half-integer spins and other fractional quantum numbers of lattice and time-reversal symmetries. Different patterns of symmetry fractionalization distinguish different spin liquid phases. In this work, we derive a general constraint on the symmetry fractionalization of spinons in a gapped spin liquid, realized in a system with an odd 1 number of spin- 2 per unit cell. In particular, when applied to kagome/triangular lattices, we obtain a complete classification of symmetric gapped Z2 spin liquids.

I. INTRODUCTION of central importance to the understanding of SET phases, but also essential to the studies of quantum spin Gapped spin liquids [1, 2] are strongly-correlated quan- liquids. Spin liquids with different patterns of symmetry tum states of spin systems that do not exhibit any fractionalization belong to different phases, and thus are Laudau-type symmetry-breaking orders [3], but instead represented by different symmetric variational wave func- are characterized by the so-called topological order [4– tions [11–15]. Hence, a complete classification exhaust- 7]. The existence of the topological order manifests as ing all possible universality classes of topological spin the emergence of quasiparticle excitations with anyonic liquids greatly facilitates the numerical studies of such braiding and exchange statistics. Although no symme- exotic states. Furthermore, symmetry fractionalization tries are broken, the nontrivial interplay between the may lead to interesting physical signatures [11, 16–19], symmetries and the topological degrees of freedom fur- which can then be detected both numerically and experi- ther enriches the concept of topological orders, leading mentally. Therefore, symmetry fractionalization provides to the notion of symmetry-enriched topological (SET) an important set of features one can use to study the order [8]. In SET phases, global symmetries can have seemingly featureless quantum spin liquid states. nontrivial actions on the quasiparticle excitations, which Viewing symmetric gapped quantum spin liquids as are otherwise not allowed on non-fractionalized local ex- SET states, it has been realized that PSGs of different citations (i.e. magnons). In particular, quasiparticle parton constructions can be unified [20], and together excitations can carry fractionalized symmetry quantum they classify a subset of possible SET states [16]. In ad- numbers, a phenomena known as symmetry fractional- dition to the spinon PSG, the SET states are also charac- ization [9]. terized by fractionalized quantum numbers of the vison An ubiquitous pattern of symmetry fractionalization excitations, which we will briefly refer to as the vison in quantum spin liquids is that there exists a quasipar- PSG, and possibly an additional layer of a symmetry- ticle excitation, dubbed the “spinon” [4, 7], which car- protected topological (SPT) state [21] (in this work, we ries a half-integer spin quantum number [10]. Besides will ignore this further distinction, i.e., we regard two the spin-rotational symmetry, quantum spin liquids of- SET states related by stacking a SPT layer as the same ten have time-reversal symmetry as well as crystalline state). Naively, taking vison PSGs into account would symmetries of the underlying lattice. Correspondingly, significantly enlarge the classification table [16]. How- the spinon can also have fractionalized quantum numbers ever, it turns out that many SETs where both spinons of the space-time symmetries. This was first discussed and visons exhibit nontrivial symmetry fractionalization in the framework of parton constructions of spin liquid are anomalous [22–27]: they can not arise in truly two- state, known as the projective symmetry group (PSG) of dimensional lattice systems, and can only be realized on the parton mean-field ansatz [11]. In this work, we will the surface of a three-dimensional SPT state. For on- use spinon PSG to collectively refer to space-time sym- site unitary symmetries, whether a particular symmetry metry fractionalization on spinons. Together with other fractionalization pattern is anomalous or not can be de- arXiv:1606.04544v3 [cond-mat.str-el] 20 Mar 2018 fractionalized symmetry quantum numbers, PSG classi- termined mathematically [28, 29]. Unfortunately, such a fies symmetric quantum spin liquids that can be realized mathematical framework does not yet exist for space- on a certain type of lattice. time symmetries. Instead, in this work, with the as- Classification of symmetry fractionalization is not only sumption of a background spinon charge per unit cell, we are able to obtain a set of nontrivial constraints on non-anomalous 2D spin liquids.

∗ Current address: Department of Physics, Fudan University, More specifically, we derive constraints on spinon’s Shanghai 200433, China symmetry fractionalization in gapped Z2 spin liquids † Current address: Department of Physics, Yale University, New with a background spinon charge per unit cell, using Haven, CT 06520-8120, USA the triangular/kagome lattice (these two lattices have 2 the same crystalline symmetry group) as the main ex- the space group operations to only change the positions ample. We show that in these systems the spinon PSG of the lattice spins: namely, for g Gs, the symmetry ∈ −1 must match the projective representation of the physi- operation Rg acts on the spin at r as RgSrRg = Sg(r), cal degrees of freedom in a unit cell. Similar constraints without any actions on the internal degrees of freedom. are introduced in Cheng et al. [30] for onsite symme- The concept of symmetry fractionalization refers to the tries. In this work, we show how to generalize this type phenomena that in a 2D topologically ordered state, non- of constraints to include point-group symmetries. To- trivial anyon excitations can carry fractionalized sym- gether with the constraints on vison’s symmetry frac- metry quantum numbers, and more generally, projective tionalization derived in Ref. [26], we have for the first representations of the symmetry group G [40]. On the time obtained a full classification (up to stacking SPT contrary, trivial local excitations must carry “integer” layers) of symmetric gapped Z2 spin liquids on a trian- quantum numbers, or linear representations of the sym- gular/kagome lattice in two dimensions. metry group. Under the assumption of symmetry local- We will focus on possible gapped Z2 spin liquids ization [9], global symmetry transformations acting on 1 in spin- 2 antiferromagnetic(AF) Heisenberg models on a state with multiple well-separated anyons a1, a2,... the triangular and kagome lattices, because they have should factor into unitaries U aj localized near the po- become the most promising candidate systems to re- sitions of the anyons (for space group transformations, alize these exotic phases of quantum spins. Numeri- one also has to move the locations of the anyons). cal simulations using the density matrix renormalization Although the global state must form a linear repre- group (DMRG) method [31–35] have provided evidences sentation of the symmetry group, the local unitaries can that the nearest-neighbor AF Heisenberg model on the obey the group multiplication law projectively: kagome lattice has a gapped spin liquid ground state. Z2 a g a h g h a gh Moreover, recent experiments [36, 37] on the kagome lat- U ( )U ( ) = ωa( , )U ( ), (1) tice material herbertsmithite suggest that there is a spin where each pair of group elements g, h is associated with gap. Although neither the numerical nor the experimen- a U(1) phase ωa(g, h). In addition, the projective phases tal evidences are spotless [38, 39], we still use the kagome must satisfy the fusion rules of anyons: if the anyon type lattice as the main example to demonstrate our method, c appears in the fusion of anyons a and b, then ωaωb = which can be readily generalized to other lattices. ωc. A well-known result from the algebraic theory of The rest of the paper is organized as follows. In Sec. II, anyons [29] says that all such phase factors consistent we briefly review the concept of symmetry fractionaliza- with fusion rules can be expressed using mutual braidings tion and explain its mathematical description, which will with an Abelian anyon: be used throughout this paper. In Sec. III, we derive ω (g, h) = M , w(g, h) . (2) the key result of this work, which is a constraint that re- a a,w(g,h) ∈ A lates the symmetry fractionalization class of the spinon Here, is the group of Abelian anyons, and M is to the projective symmetry representation of the physi- ab the mutualA braiding statistical phase between anyons a cal degrees of freedom in a unit cell. In Sec. IV, we de- and b. Therefore equivalence classes of the phase fac- rive the relation between anyonic flux density created by tors ω (g, h) consistent with both the fusion rule and a symmetry operation, and the fractionalization of com- a the associativity law of are parametrized by the so- mutation relations between this symmetry operation and called fractionalization classesG [w], classified by the sec- translations. This relation is needed in applying the con- ond group cohomology 2[ , ] [28, 29, 41]. straint obtained in Sec. III. In Sec. V, this constraint is Notice that we haveH providedG A two equivalent descrip- applied to the triangular/kagome lattices to obtain a full tions of symmetry fractionalization, in terms of the phase classification of symmetric gapped spin liquid states Z2 factors ω (g, h) or the anyon-valued fractionalization on such lattices. a class w(g, h). We shall use them interchangeably at different places in this paper, depending on convenience. Now we make these abstract definitions more explicit II. SYMMETRY FRACTIONALIZATION IN Z2 for Z2 spin liquids. There are four types of anyons: the SPIN LIQUIDS trivial excitations 1, the bosonic spinon e, the bosonic vison m and the fermionic spinon , with a nontrivial In this section, we briefly review the physical phe- mutual braiding statistics M = M = M = 1. em e m − nomena of symmetry fractionalization in the context of Under fusion, they form a group = Z2 Z2, generated A × gapped 2D Z2 spin liquids [16, 29]. We are interested by e and m. In terms of the fractionalization class [w], 2 in symmetric Z2 spin liquids realized in lattice mod- symmetry fractionalization is classified by [ , ] = 2 2 2 H G A els of spin-1/2 magnets. The largest symmetry group [ , Z2 Z2] = [ , Z2] . In terms of the phase H G × H G of such lattice models is = G Gs with Gs being factors ωa(g, h), the Z2 fusion rules e e = m m = the space group. The “on-site”G part× of the symmetry = 1 constraint them to be 1. Therefore,× for× each T × ± group is G = SO(3) Z2 , where SO(3) denotes the spin- type of anyon a the phase factors ωa are classified by × T 2 rotational symmetry group, and Z denotes the time- [ , Z2]. Furthermore, the fusion rule e m = im- 2 H G × reversal symmetry group. For concreteness, we define plies ω = ωeωm. So the independent phase factors are 3

ωe and ωm, as expected from the group cohomology clas- charge b in each unit cell that carries spin-1/2. Therefore, siﬁcation. [42] b has to be a spinon. The physical meaning of the back- We end this section with two familiar examples of sym- ground anyon charge b is that when an anyon a is adiabat- metry fractionalization. First, consider the time-reversal ically transported around a unit cell, a Berry phase Mab symmetry. In most QSLs, both the bosonic spinon and is accumulated. In other words, the translation symme- 2 (a) (a) (a) (a) the fermionic spinon are Kramers doublets with T = 1 try is fractionalized: Tx Ty = MabTy Tx , or equiv- 2 − 1 (or ωe(T,T ) = 1), while the vison has T = +1 alently, w(Tx,Ty) = b = . For Z2 QSL, without losing − 6 (ωm(T,T ) = 1). According to Eq. (2), the corresponding generality we assume that the e anyon and the anyon anyon-valued fractionalization class is w(T,T ) = m. Our carry a half-integer spin [48]. So we can have b = e, . second example, which is crucial to our discussions, is The discussion in this paper generally applies to both the fractionalization of the translation symmetries gen- cases. erated by T1 and T2. As we will see later, QSLs consid- The existence of a nontrivial background spinon charge ered in this work all have fractional commutation relation puts further constraints on the symmetry fractionaliza- between the two translations for visons: T1T2 = T2T1 tion of the spinon. Ref. [30] shows that for an onsite − (notice that the translation symmetry group is Z Z). In symmetry group G, such constraints relate the fraction- × particular, here we consider spin liquids where both the alization class to the symmetry quantum numbers of the vison m and the fermionic spinon carries T1T2 = T2T1, physical degrees of freedom in the unit cell. Formally, the − and the bosonic spinon e carries T1T2 = T2T1. To for- microscopic degree of freedom in the unit cell can form a mally capture this kind of fractionalized commutation projective representation of the symmetry group, labeled relation, it is convenient to deﬁne the following symbols by [ν] 2[G, U(1)]. The following relation is derived in ∈ H βa(g, h) and b(g, h): Ref. [30]:

ωa(g, h) [ν] = ρ([ωb]) [τ], (4) βa(g, h) = , b(g, h) = w(g, h)w(h, g). (3) · ωa(h, g) where [τ] is a “twist factor” determined by certain sym- This definition only applies to two group elements that metry fractionalization class known as the “anyonic spin- commute with each other (we will see an example of orbit coupling (SOC)”, which describes the fractionaliza- a generalized version of the “commutator” in Eq. (24) tion of commutation relations between translations and between non-commuting symmetry operations). Using other symmetries. The precise form of this additional these notations, the fractionalization of translation sym- factor will be given later in this section, and in Sec. V, metry is described by βe(T1,T2) = +1 and βm(T1,T2) = we shall see that for spin liquids on the kagome lattice, 1, or equivalently by b(T1,T2) = e [see Eq. (2)]. this factor is always trivial. Here, ρ is a natural map − 2 2 from [ , Z2] to [G, U(1)], which is composed of two H G H 2 steps: first, a cocycle in [ , Z2] is mapped to a cocycle III. MICROSCOPIC CONSTRAINTS ON in 2[ , U(1)], by replacingH G the Z2 coefficients with U(1) SPINON’S SYMMETRY FRACTIONALIZATION coefficientsH G +/-1; second, it is further mapped to a cocycle in 2[G, U(1)], by restricting group elements that In this section, we describe constraints on symmetry appear inH a cocycle to be elements of . G fractionalization of spinons in a gapped Z2 spin liquid In this work, we generalize their result to also include with a net spinon charge per unit cell. We first briefly point group symmetry Gpt. The triangular and kagome review the results of Refs. [30 and 43], which relate frac- lattices, which we focus on in this paper, share the same tional quantum numbers of spinons to the symmetry rep- space group p6m, which is a semidirect product of the resentation of the physical degrees of freedom per unit translation symmetry group Gtrans, and the point group cell, for on-site symmetry operations. Then we general- G G /G . As shown in Fig. 4, G is generated pt ' s trans trans ize this relation to point-group symmetry operations. by two translations T1 and T2, and Gpt is generated by We focus on systems with an odd number of spin- two mirror reflections µ and σ. In the following we will 1 ˜ 2 per unit cell, which include most of the candidate refer to G = G Gpt as the extended onsite symmetry spin liquid systems studied numerically and experimen- group. One can always× choose the unit cell to be invariant tally. The Lieb-Schultz-Mattis-Oshikawa-Hastings theo- geometrically under Gpt. Then the microscopic degrees rem [44–47] guarantees that a gapped ground state in of freedom in the unit cell should form a representation such systems preserving translation and spin-rotational of G˜, classified again by [ν] 2[G,˜ U(1)]. We will show symmetries must be topological ordered. Recently, Za- that Eq. (4) should continue∈ to H hold in this more general letel and Vishwanath [43] and Cheng et al. [30] showed setting. that this conclusion can be further refined: symmetry The derivation of Eq. (4) in Ref. [30] is based on the fractionalization in the topological phase is highly con- observation that the 2D lattice system can be viewed strained by symmetry properties of microscopic degrees as surface of a three-dimensional weak SPT state, and of freedom. For example, with SO(3) symmetry and an then the constraint on symmetry fractionalization is ob- odd number of spin-1/2’s per unit cell, it was estab- tained from bulk-boundary correspondence. Unfortu- lished [30, 43] that there must be a background anyon nately, we do not know how to generalize this approach 4 to include point-group symmetries, since the bulk-surface correspondence of 3D SPT states protected by point- group symmetries is not well-understood (however, see Ref. [27] for a recent development). Instead, here we give a purely two-dimensional argument, which not only ~l reproduces the result of Ref. [30] for onsite symmetries, but also generalizes to point-group symmetries. φ = ~b ~l A We consider a (simply-connected) region A that con- · tains n unit cells (thus n background spinons), where n (a) (b) is an odd number. The ground-state wave function can be Schmidt decomposed with respect to the cut ∂A: bx(g) X A Ψ = λ ψ ψ ¯ . (5) | i α| Aiα ⊗ | Aiα y α Lbx(g) Here A¯ is the complement of A. In the Schmidt de- composition, the Hilbert space is decomposed into by(g) by(g) H = ¯, where ¯ are supported on the interior H HA ⊗HA HA/A and the exterior of the cut ∂A, respectively. ψ | Aiα ∈ HA and ψA¯ α A¯ are Schmidt eigenstates in the two sub- x | i ∈ H by(g) spaces. Since ψA α form a complete orthonormal basis L {| i } of , one can define the G˜-representation in the sub- HA bx(g) space A using the basis of Schmidt eigenstates, H (c) X Rg ψA α = Uαβ(g) ψA β, g G.˜ (6) | i | i ∈ FIG. 1. Illustrations of the effect of anyon SOC. β

Here R should be understood as the restriction of the g ˜ global g transformation to the interior of A. The matrices transformations from G can fractionalize. This phenom- U(g) form a projective representation of G˜: ena is dubbed “anyonic spin-orbit coupling (SOC)” in Ref. [30].

U(g)U(h) = ωU (g, h)U(gh), (7) First, we study the symmetry action on the background anyon charges. Intuitively, the entire region A with a U(1) phase ambiguity. Hence, different projective contains a total bn anyon charge. Under the assumption representations are classified by 2[G,˜ U(1)]. of symmetry localization [9], Rg should act projectively Our argument now roughly proceedsH as follows: we will on bn. Denoting the local symmetry transformations by bn first analyze the symmetry action in a region in terms of Ug , they satisfy the topological degrees of freedom, and then match with bn bn n bn the microscopic description of the same symmetry action. U (g)U (h) = [ωbn (g, h)] U (gh). (9) On one hand, we notice that the group G˜ acts on Secondly, the symmetry action can change the Schmidt the Hilbert space A, to which the Schmidt eigenstates H state in a more subtle way: in the presence of anyonic ψA α belong. The action of G˜ on A is determined by |thei symmetry transformation of theH physical degrees of SOC, symmetry action in A can create an “anyonic flux freedom, which is a tensor product of the microscopic density” running across A, which in turn creates anyon constitute on each site. Correspondingly, the factor set charge densities on the boundary of A, but does not affect the local density matrix inside A at all. This anyonic flux ωU is given by density creation contributes additional phase factors to n g h [ωU ] = ([ν]) . (8) ωU ( , ). In general, the anyonic flux density created by a sym- On the other hand, the projective representation U(g) metry operation g can be represented using a “vector” also encodes symmetry actions on the anyonic excita- ~b(g) = bx(g)ˆx + by(g)ˆy. Here, the components of ~b(g) tions. For our purpose, it is of crucial importance to are Abelian anyon charges bx,y(g) . Such an anyon understand the interplay of translation symmetry with flux can be represented by a collection∈ A of anyonic string the extended symmetry group G˜, which naturally come operators, as shown in Fig. 1(a). The total anyonic in two forms: 1) Fractionalization of translation symme- charge carried by these string operators going through tries manifests as background anyon charges (i.e. back- ~l is φ = ~b ~l. In particular, these string operators termi- ground spinons in our case). G˜ can act projectively on nate at the· boundary of the region A, leaving an anyonic the background spinons. 2) The commutation relations charge density, as shown in Fig. 1(b). In Z2 spin liq- between the translation symmetries and other symmetry uids, these background string operators can be thought 5

nx as electric and magnetic ﬁeld lines. a 1 : Tx The value of the vector ~b(g) is determined by the fractionalization of commutation relations between g and 3 : g 2 : g translational symmetries. We leave the computation of 4 : T nx ~b(g) to Sec. IV, and examine now how the creation of x (a) an anyonic ﬂux density ~b(g) contributes the extra phase factor τ to Eq. (4). We compute τ for the example of a nx 1 : Tx rectangular region, with nx,y unit cells in x and y direc- a nx tions, respectively, as shown in Fig. 1(c). The symmetry 6 : Tx action creates an n number of anyon string operators of x 4 : P 2 : P the type by(g) along the y direction, and an ny number y ny ny y 5 : Ty 3 : Ty of bx(g) anyon strings along the x direction, respectively. Hence, the action of g not only transforms the anyon within projectively, but also creates anyon string opera- (b) tors. Formally, we write this as

n b x ny y nx FIG. 2. Illustrations of the eﬀect of anyon SOC. U(g) = U (g)[ g ] [ ] . (10) Lbx( ) Lby (g) When combining the actions of g and h, the string operators contribute an additional phase factor from braid- IV. ANYONIC FLUX DENSITY CREATED BY ing: SYMMETRY OPERATIONS

x ny y nx x ny y nx [ g ] [ ] [ h ] [ ] Lbx( ) Lby (g) Lbx( ) Lby (h) In this section, we study the anyonic flux density cre- ny nx y (11) ˜ = M [ x ]ny [ ]nx . ated by a symmetry operation g G. In particular, we g h bx(gh) by (gh) ∈ by ( ),bx( ) L L compute the vector ~b(g) for both onsite and point-group Combining this phase factor with the cocycle factors in symmetry operations, and show that the result can be Eqs. (8) and (9), we obtain related to the fractionalization of commutation relation n n n between g and translational symmetries, known as the ωU (g, h) = [ν(g, h)] = [ωb(g, h)] [M ] , (12) by (g),bx(h) anyonic SOC when g is onsite. The results of this section are not only useful for deriving the extra phase factor for general n = n n , and this in turn implies that x y [τ] in the constraint of Eq. (4), but also provide physical ν(g, h) = ωb(g, h)M . (13) and observable effects of these symmetry fractionaliza- by (g),bx(h) tion quantum numbers. Consequently, the creation of This is the “anomaly-matching” constraint given at the ~b(g) can be used as a way to define and to measure these beginning of this section in Eq. (4), where the twist factor symmetry fractionalization quantum number. [τ] has the following form, In the following we first review the relation between ~b(g) and anyonic SOC, for onsite symmetries, which is τ(g, h) = M g h . (14) by ( ),bx( ) discussed in details in Ref. [30], and then study the gen- We notice that when g and h are onsite symmetries, eralization to point-group symmetries. The general strat- this relation reproduces the twisted anomaly-matching egy is that we design processes that compute the braiding condition, given by Eq. (52) of Ref. [30], up to a cobound- ~ phases Ma,bx,y (g). The flux b(g) is then determined from ary term. In fact, for onsite symmetries, ~b(g) is computed the braidinging phases. in Sec. IV A, and it is given by Eq. (19). Hence, Eq. (13) can be written as A. Onsite symmetries Rb(Tx,g),b(Ty ,h) ν(g, h) =ωb(g, h) Rb(Ty ,g),b(Tx,h) (15) First, we consider an onsite symmetry operation g, Rb(Ty ,gh),b(Tx,gh) , which can be either unitary or anti-unitary (e.g. time- × Rb(Ty ,g),b(Tx,g)Rb(Ty ,h),b(Tx,h) reversal symmetry). With anyonic SOC, g action on the where the last term is a 2-coboundary, and the R symbol ground state effectively creates an anyonic flux density Rab denotes the Berry phase associated with exchanging ~b(g). Such a flux density ~b can be determined by moving two anyons with charge a and b, respectively. Hence, this a test anyon a along a vector ~l = lxxˆ + lyyˆ. The commu- anomaly-matching constraint is equivalent to the follow- tation between the a string and the background string ing: operators results in a phase Ma,φ where φ = ~b ~l. Us- ing this method, we will measure the anyonic flux· density Rb(Tx,g),b(Ty ,h) g h g h creation with the following thought experiment: consider ν( , ) = ωb( , ) b(T ,g),b(T ,h) , (16) R y x creating a pair of anyons a anda ¯ from the vacuum, and which reproduces Eq. (52) of Ref. [30]. translate a by nx unit lengths in the x-direction by apply- 6

x ing a string operator a. This can be thought as applying In Fig. 2(b) this sequence is 1 2 3. The second nx L → x → Tx to a. We then apply the g symmetry transformation sequence is applying Py and then La, which maps to first ny to the state. The sequence is illustrated in Fig. 2(a) as applying Py, then applying Ty to move the anyon back 1 2. We can also switch the order of translating a to the beginning of the string operator, and finally ap- → nx and the g action, corresponding to 3 4 in Fig. 2(a). ply Tx . This is illustrated as 4 5 6 in Fig. 2(b). → → → ny Via the assumption of symmetry localization, it is easy Here, additional translations in the y direction (Ty ) are x to see that the Berry phase in this process is given by the applied to ensure that the string operator a appears at nx L commutator of Tx and g, and by definition is equal to the same location in the two sequences. The Berry phase nx [βa(Tx, g)] . Using Eq. (2), this phase can be expressed can be computed as the following, as nx ny nxny ny nx Tx Ty Pya = [ωa(Tx,Ty)] Ty Tx Pya nx nx [βa(Tx, g)] = [Ma,b(T ,g)] . (17) nxny nx ny nx x = [ωa(Tx,Ty)] [ωa(Tx,Py)] Ty PyTx a. (20) Since the braiding phase Ma,c is also the commutator of two crossing string operators moving a and c respectively, The two interpretations of the same Berry phase should one can re-interpret Eq. (17) in the following manner: be equated: first, the g action creates a string y along y per unit Lby (g) ny length in the x-direction. The phase in Eq. (17) then Ma,by (Py ) = [ωa(Tx,Ty)] ωa(Tx,Py). (21) x results from the nontrivial commutator between a and (the x string have to cross n a strings):L Formally we should have Lbx(g) La x Ly n x y nx nx y nx x b b y [ ] = [Ma,b (g)] [ ] . (18) by(Py) = (Tx,Py)[ (Tx,Ty)] . (22) La Lbx(g) x Lbx(g) La When b(T ,T ) = 1, this equation only gives consistent This is illustrated in Fig. 2(a). Comparing the Berry x y 6 results if ny has a fixed parity. phases in Eqs. (17) and (18), we conclude that by(g) = b(T , g) [49]. The other component b (g) can be com- Let us go back to Eq. (21). To determine by(Py), we x x need at least two test anyons, which we will choose to ~ g puted similarly. Finally, we get the relation between b( ) be the background spinon b = b(T ,T ) and the vison and the anyonic SOC, x y v. For the background spinon, ωb(Tx,Ty) = 1 so the choice of ny does not matter. If the test anyon is a vison, by(g) = b(Tx, g), bx(g) = b(Ty, g). (19) we now argue that the parity of ny only depends on the geometric property of the mirror axis. For example, consider the site-centered reflection Py. To see why, consider B. Point-group symmetries − ny −nx ny nx moving a vison by Ty Tx Ty Tx , so that the path is symmetric under P . n n unit cells are enclosed by Next, we show that point-group symmetries have a y x y the path and the Berry phase is ( 1)nxny . On the other similar effect of creating anyonic flux densities associated hand, since the area enclosed by− the path is symmetric with the fractionalization of their commutation relations under P , it is obvious that the number of physical spins with translations. Here, we only discuss mirror reflec- y inside the area must be n mod 2. We thus conclude that tions, since other point-group symmetry operations can x ( 1)nxny = ( 1)nx , which implies n is odd. Combining be reduced to them (as shown in Fig. 4, the point group y these− arguments,− we can unambiguously conclude that is generated by two mirror reflections). Without losing generality, we describe our results in the context of mir- by(Py) = b(Tx,Py)b(Tx,Ty). (23) ror reflections on a square lattice. For simplicity, for the rest of this section we restrict ourselves to the case of a We also notice that site-centered Py is needed for our Z2 (toric code) topological order. general proof to work, since there must be an odd number We first consider a mirror symmetry Py, which maps y of spin-1/2’s in the region A. to y. The anyonic flux created by P in the y direction, − y Similarly, by(Px) is related to the symmetry fraction- which is perpendicular to the mirror axis, can also be re- alization associated with Tx and Px. Because Px and Tx lated to the fractionalization of the commutation relation do not commute, but instead satisfy a twisted commu- between T and P . To detect this flux density, we draw −1 x y tation relation TxPx = PxTx , we can not naively apply a test string operator along the mirror axis. The Berry the previous definition of commutation relation fraction- x phase of exchanging this test string operator a with the alization. As we show below, by(Px) is determined by a y n L anyonic flux [ ] x , can be interpreted as the Berry twisted version of commutation relation fractionalization, Lby (Py ) phase of exchanging two sequences of symmetry actions x −1 −1 shown in Fig. 2(b): The first sequence is applying La by(Px) = w(Tx,Px)w(Tx,Tx )w(Px,Tx ). (24) and then Py, which maps to translating an a anyon by nx ny Tx followed by applying Py, and finally applying Ty to Similar conclusions hold for Py,Ty as well as the inversion move the anyon back to the end of the string operator. I with Tx,y. 7

µ σ µ σ Pxa T2 x T2 P x P a La a x nx Pxa Tx Pxa σ′ σ′ a Is Is

T1 P x T1

2 (a)Kagome lattice (b)Triangular lattice Px a x nx 2 La PxTx Pxa Px a

nx 2 FIG. 4. Generators of the space group of the (Tx Px) a kagome/triangular lattice, Gs = p6m.

FIG. 3. Illustration of the physical process determining the anyonic flux creation by(Px). itive picture for the physics. The effect of anyonic flux density creation can also be exploited in numerical diagonsis of the corresponding The anyonic flux density b (P ) can be computed from y x symmetry fractionalization classes. The fractionalization the Berry phase associated with applying a test string of commutation relations between one translation and operator Lx and P in opposite orders. In particular, a x another (unitary) symmetry operation g can be measured the Berry phase M is obtained through the fol- a,by (Px) as the difference in eigenvalues of g operators on ground lowing operations: ( x)−1P −1 xP = xP xP . This La x La x La xLa x states in different topological sectors [18, 19], with appro- Berry phase can be related to symmetry fractionaliza- priate lattice geometry. This result can be readily repro- tion, by viewing these operations as symmetry actions duced using anyonic flux density creation. For simplicity, on test anyon charges. As shown in Fig. 3, we first cre- we consider the example of an onsite unitary symmetry ate two test charges of type a from the vacuum, to the g. The symmetry operation g creates anyonic flux along right of the mirror axis, and denote the initial state by the y direction, which can be represented as string oper- y a a. The operation Px reflects both anyons to the left, ators [ ]nx . On a minimally entangled state (MES) and× we denote the result by P a P a. Next, the test Lby (g) x x on a torus, with an anyonic flux a going through along string x is created by translating× one of the anyon to a the x direction, these string operators contributes a non- the right,L and the state becomes P a T P a. Next, x x x trivial phase factor [M ]nx = [β (T , g)]nx . Thus, if P maps them to P T P a P 2a. Finally,× applying the a,by (g) a x x x x x x n is odd, the g eigenvalues of different MESs encode the test string operator moves the× anyon on the left back to x 2 commutation relation fractionalization βa(Tx, g). the right, and we get TxPxTxPxa Px a. Comparing this to the initial state, we conclude× that the Berry phase a a accumulated is ω (TxPx,TxPx)ω (Px,Px) = Ma,by (Px). Hence, we conclude that V. APPLICATION TO GAPPED SPIN LIQUIDS ON THE KAGOME/TRIANGULAR LATTICE

by(Px) = w(TxPx,TxPx)w(Px,Px). (25) In this section, we apply the constraint Eq. (4) w g h w gh k 1 Using the cocycle equation ( , ) ( , ) = to gapped spin liquids in spin- 2 models on the w(h, k)w(g, hk), one can rewrite this result in the kagome/triangular lattice. This constraint greatly re- form of Eq. (24). duces the number of possible symmetry fractionalization We notice that to determine the anyonic flux density classes on spinons. Together with previously obtained created by a mirror symmetry operation, it is more conve- constraints on symmetry fractionalization of visons [26], nient to decompose the flux density ~b into components in we obtain a complete classification of gapped symmetric directions parallel and perpendicular to the mirror axis, Z2 spin liquids. as in our discussion above, because in such a setup, the To begin, we recall that the space group of the tri- test string is invariant under the mirror symmetry. In the angular/kagome lattice, Gs = p6m, is an extension of contrary, the flux density creation associated with onsite the translational symmetry group Gtrans, generated by and inversion symmetries can be determined using de- T1 and T2, by the point group Gpt = C6v, generated by compositions in any directions. two mirror reflections µ and σ, as shown in Fig. 4. It is Besides determining the twist factor [τ] in Eq. (14), the also convenient to define a site-centered inversion Is, as effect of anyonic flux density creation can also be used in shown in Fig. 4, and the mirror reflection with the mirror 0 defining and detecting the (twisted) commutation rela- axis along the T1 direction σ = Isµ. Here, we choose two tion fractionalization between translations and elements mirror symmetries µ and σ0, whose axes intersect at a lat- of G˜. The Berry phase computations we have presented tice site (consequently, the product of mirror reflections can serve as physical definitions of these fractionalization gives site-centered rotations), such that a region symmet- classes, and the anyonic flux creation provides an intu- ric under two mirror reflections contain an even number 8 of unit cells. We notice that the relation between σ0 (and The first four invariants already appear in Eq. (26) also Is) and the generators of p6m is different for the two [i.e. they are actually U(1) invariants], and the last 0 different lattices, although in both cases we have defined one ωµ,σ = β(µ, σ0) = 1 represents the commuta- 0 ± 0 Is = µσ . In terms of notations introduced in Sec. II and tion relation fractionalization between µ and σ . For T Sec. III, the total symmetry group is = SO3 Z2 Gs = spin-1/2 magnets on the triangular/kagome lattices, it T G × × SO3 Z2 p6m, and the extended onsite symmetry group is straightforward to check that the invariants are given ˜× × T T 1 is G = SO3 Z2 Gpt = SO3 Z2 C6v. by [ωphys] = ( 2 , 1, 1, 1, +1). To classify× symmetry× fractionalizations× × we first com- − − − 2 ˜ 2 ˜ The natural map ρ between [G, Z2] and [G, U(1)] 2 8 H H pute the second group cohomology [ , Z2] = Z2. The can be understood intuitively. We already mentioned 8 H G 2 = 256 different cohomology classes are labeled by eight that ωspin, ωT , ωµT , ωσT are identified between the two, 0 Z2 invariants: and ωµ,σ = ωIs ωµωσ. [This is because the symmetry fractionalization of I2 = (µσ0)2 can be factorized as the spin 12 µ σ Is T µT σT s (ω , ω , ω , ω , ω , ω , ω , ω ), (26) product of µ2, σ02, and the commutation relation between µ and σ0.] ωµ and ωIs are the only true “ ” invariants in where ωspin = 0 or 1 denotes whether the anyon car- Z2 2 2[G,˜ ]. In summary, using the notations in Eqs. (26) ries an integer or a half-integer spin, respectively; ω12 = Z2 andH (28), the map ρ has the form β(T ,T ) = 1; the other six variables are all given in 1 2 ± the form of ωX = ω(X,X) = 1, denoting the frac- ρ([ω]) = (ωspin, ωT , ωµT , ωσT , ωIs ωµωσ). (29) tional quantum number X2 = ±1. As we have already We now apply the constraint in Eq. (4) to the frac- discussed in Sec. II, we need to± specify [ω ] and [ω ]. e m tionalization of the background spinon b. First, we can The fact that ωe = 1/2 and b(T ,T ) = e completely spin x y show that the twist factor τ in Eq. (4) is always trivial. determines the symmetry fractionalization of the vison We observe that, among the invariants listed in Eq. (26) in a gapped Z2 spin liquid [26]. First, the background 2 that label different fractionalization classes in [ , Z2], spinon charge density implies that the T1 and T2 anti- H G none of them involves translations and G˜ at the same commute when acting on a vison, i.e. βm(T1,T2) = 1. Intuitively, a vison acquires a Berry phase of π when− it time, i.e. no anyonic SOC appears in the list. In fact, is adiabatically transported around a unit cell, because as shown by an explicit computation in Appendix A, all of the braiding with the background spinon charge. All anyonic SOC fractionalization classes b(T1,2, g) [includ- the other fractional quantum numbers can be constrained ing the twisted version in Eq. (24)], and hence the any- 1 using the method of flux-fusion anomaly test [25, 26]. To onic flux densities bx,y(g), are either or the background ˜ spinon b = b(T1,T2) for all g G. Since in a Z2 spin liq- summarize, the symmetry fractionalization of visons is ∈ fixed: uid, we always have M1,b = Mb,b = +1, the twist factor [τ] evaluates to 1 identically. [ωm] = (0, 1, +1, +1, 1, +1, +1, +1). (27) With [τ] out of the way, the constraint in Eq. (4) states − − that the symmetry fractionalization of the background We notice that, for the convenience of applying the con- spinon must match the projective representation of the straint in Eq. (4), we consider a site-centered inversion Is, physical degrees of freedom in each unit cell. In particu- instead of a plaquette-centered inversion Ip as in Ref. [26]. lar, according to Eq. (29), it means that among the eight 2 spin µT As a result, the vison carries Is = 1. As discussed in T σT − invariants, ωb , ωb , ωb , ωb are directly fixed by the Ref. [26], this is equivalent to I2 = +1 [50], and the vi- σ p symmetry properties of the unit cell, and ωb is not in- son PSG specified in Eq. (27) is the same as the odd dependent. In summary, we establish the following con- Ising gauge theory on the kagome/triangular lattice [51– straints: 54]. We also notice that, although the fractional quan- 0 spin spin σ µ,σ Is µ tum numbers in Eq. (27) appear to be the same for vison ωb = ωphys, ωb = ωphysωb ωb , (30) PSGs on the kagome and triangular lattices, the two vi- X X ωb = ωphys,X = T, µT, σT. son PSGs are indeed different [55]. This is because these 12 µ quantum numbers are defined in terms of symmetry op- The remaining unconstrained ones are just ωb , ωb Is 3 erations Is, which has different forms on the two types and ωb , so we have 2 = 8 classes of spinon PSGs. of lattices, in terms of the generators of the p6m space Since all the vison PSGs are completely fixed, we con- group. clude that there are at most eight distinct types of Now we turn to the symmetry fractionalization of symmetric Z2 spin liquid with a background spinon spinons. To apply Eq. (4) we first study the symme- charge. There are actually no further constraints since try representations of physical degrees of freedom in a all eight possible states have been constructed previ- unit cell. As discussed in Sec. III, they are classified by ously using bosonic [13] or fermionic parton construc- 2 ˜ 5 the [G, U(1)] = Z2, where the extended onsite symme- tions [14, 20, 56, 57], and tensor product states [15]. H ˜ T try group G = SO3 Z C6v. These comology classes Therefore, we have obtained a complete classification of × 2 × are labeled by the following invariants: gapped symmetric Z2 spin liquids in translation-invariant spin- 1 kagome/triangular lattice models. The results are 0 2 (ωspin, ωT , ωµT , ωσT , ωµ,σ ). (28) summarized in Table I. 9

[ω] Relations [ωe][ωm] bulk-surface correspondence can be generalized to point- spin iπSx,y,z 2 1 group symmetries as well, and for gapped symmetry- ω (e ) = ±1 2 0 ω12 T T = ±T T ±1 −1 preserving surface states Eq. (4) can be viewed as the 1 2 2 1 precise form of such correspondence [30]. On the other ωσ σ2 = ±1 ±1 +1 hand, gapless spin liquids violating Eq. (30) found in µ 2 σ I Z2 ω µ = ±1 ωe ωe +1 Schwinger/Abrikosov fermion constructions should also Is 2 0 2 ω Is = (µσ ) = ±1 ±1 −1 satisfy such a bulk-surface relation, i.e. they exhibit the ωT T 2 = ±1 −1 +1 correct anomaly. It will be interesting to investigate the ωσT (σT )2 = ±1 −1 +1 relation further, which we will leave for future studies. ωµT (µT )2 = ±1 −1 +1 We notice that our constraints on the symmetry fractionalization of spinons can be easily adapted to TABLE I. Fractional quantum numbers in symmetric gapped spin models with only U(1)Sz spin-rotational symme- Z2 spin liquids on the kagome lattice. try [e.g., the SO(3) symmetry is broken by easy-plane anisotropies], as long as there is still a background “spinon” charge per unit cell. Here, a spinon should VI. CONCLUSION be understood as an anyon carrying 1/2 charge under

the U(1)Sz symmetry. However, the constraints on the symmetry fractionalization of visons are much less strin- In this work, we study the classification of symmetry T µ gent: the flux-fusion anomaly test only fixes ωm, ωm and fractionalization carried by spinons in gapped 2D Z2 spin σ liquids. For gapped spin liquids realized in spin systems ωm [25], leaving other quantum numbers of visons un- 1 determined. We leave a systematic classification of non- with an odd number of spin- 2 degrees of freedom per unit cell, we generalize the constraint of Cheng et al. [30] anomalous gapped spin liquids with U(1)Sz symmetry to point-group symmetries. The generalized constraint for future works. We also notice that our constraints establishes a relation between the symmetry fractional- can be straightforwardly generalized to other 2D lattices, ization of the spinon and the symmetry representation like the square lattice. We will also leave this for future of the physical degrees of freedom in one unit cell. Ap- works. plying this constraint to triangular/kagome lattices, we obtain a full classification (up to stacking SPT layers) of ACKNOWLEDGMENTS gapped Z2 spin liquids. In particular, we show that the eight spin liquid states constructed previously exhaust all possibilities [13–15, 20, 56, 57]. M.C. would like to thank M. Zaletel for many enlight- We notice that the Schwinger/Abrikosov fermion con- ening conservations on related topics. Y.Q. and M.C. struction actually yields 20 different PSGs for Z2 spin are grateful for Michael Hermele for very constructive liquid states on the kagome lattice [14, 20]. Eight of comments on the manuscript. Y.Q. is supported by the them correspond to symmetric gapped spin liquids in Ta- Ministry of Science and Technology of China under Grant ble I [20], while the rest, if naively translating the PSG No. 2015CB921700. This research was supported in part to fractionalization class of the fermionic spinons, vio- by Perimeter Institute for Theoretical Physics. Research late the constraints in Eq. (30). However, the fermionic at Perimeter Institute is supported by the Government mean-field states are always gapless [58] with these 12 of Canada through Industry Canada and by the Province classes of PSG, i.e. they are gapless Z2 spin liquids. of Ontario through the Ministry of Research and Innova- Moreover, on other lattices, spin liquids where all anyons tion. carry integer spins and/or T 2 = +1 [and thus violating Note added. Recently, we were informed of a related the constraint in Eq. (4)] can also be realized as gapless work [61]. spin liquids [11, 59, 60]. It was known that the gaplessness of fermionic spinons are protected by the PSG at the level of mean-field states, and our derivation of the Appendix A: Anyonic spin-orbit coupling on the constraints in Eq. (30) provides a non-perturbative un- kagome lattice derstanding of the symmetry-protected gaplessness. As we mentioned earlier, Ref. [30] argued that 2D sys- In this appendix, we derive the symmetry fractional- tems with a spin-1/2 per unit cell can be viewed as the ization classes b(T1,2, g), for symmetry operations g G˜ surface of a 3D weak SPT state protected by SO(3) and on the kagome lattice. Because of the six-fold rotational∈ 2 translation symmetries, if only physical degrees of free- symmetry, the second group cohomology [G, Z2] fac- dom carrying linear representations of SO(3) are allowed. torizes as the following: H The surface of such a 3D SPT state must be anomalous 2 2 2 ˜ in a precise manner that can be canceled by “anomaly in- [ , Z2] = [Gtrans, Z2] [G, Z2], (A1) flow” to the bulk, and a symmetric surface state can ei- H G H × H 2 ther be gapless or gapped with an intrinsic topological or- where [Gtrans, Z2] = Z2 is labeled by the quantum H 12 der satisfying the constraint in Eq. (4). We believe such a number ω = β(T1,T2). Therefore, there are no inde- 10 pendent quantum numbers describing the anyon SOC, Table II. The last root class with ω12 = 1 can be re- which must then be completely fixed by the value of ω12. alized using a 16-dimensional projective representation.− We first show that an onsite symmetry operation g Here, we use e(i, j), i, j = 0,..., 3 to denote the 16 basis must have a trivial anyonic SOC, i.e. β(T1,2, g) = 1. vectors, and the projective representation is specified as 2 Using the cocycle condition of [ , Z2], one can show follows that H G φ(T1)e(i, j) = e(i + 1, j) i β(gh, k) = β(g, k)β(h, k), (A2) φ(T2)e(i, j) = ( 1) e(i, j + 1) − (A4) φ(µ)e(i, j) = ( 1)j(j+1)/2e( i + j, j) if both g and h commute with k. Since T1 and T2 are − − φ(σ)e(i, j) = ( 1)ije(j, i). related by the C6 rotational symmetry, the commutation- − relation fractionalization between g and T must be 1,2 In these equations, i and j are defined mod 4, i. e. the same: β(T , g) = β(T , g). Formally, this can be 1 2 e(i, j) = e(i+4, j) = e(i, j +4). This projective represen- proved using Eq. (A2) and the relation C2T C−2 = T . 6 1 6 2 tation is obtained by rewriting the (p , p , p ) = (1, 0, 0) Then, using Eq. (A2) one more time, we get β(T T , g) = 1 2 3 1 2 state in Ref. [13] as a matrix representation. β(T , g)β(T , g) = +1, i. e. g must commute with T T . 1 2 1 2 Using this projective representation, one can explicitly However, the translational symmetry operation of T T 1 2 compute the fractionalization of commutation relations is also related to both T and T by the C symmetry. 1 2 6 associated with the mirror symmetries µ and σ, for the Therefore, we obtain β(T , g) = β(T , g) = 1. In partic- 1 2 cohomology class ω12 = 1. First, we calculate the com- ular, the anyonic SOC associated with the time-reversal mutation relations between− mirror reflections and trans- symmetry, β(T ,T ), must always be trivial. 1,2 lations along the mirror axes: Next, we study the symmetry fractionalization classes β(T1,2, g) when g is a point-group symmetry operation. 0 2 β(T1T2, σ) = β(T1, σ ) = β(T1T2 , µ) = 1. (A5) Instead of deriving them formally using the algebraic re- − lations, here we compute them using an alternative ap- Since in the other root cohomology classes with ω12 = proach, by explicitly constructing matrix representations +1 all these phase factors are equal to +1, we conclude 4 2 0 2 of all 2 = 16 cohomology classes in [Gs, Z2]. A ma- that, β(T T , σ) = β(T , σ ) = β(T T , µ) = β(T ,T ) H 1 2 1 1 2 1 2 trix representation is a map φ : G GL(V ) where V is a holds for all cohomology classes. This implies the follow- → finite-dimensional complex vector space. Each projective ing results in 2[ , ]: representation gives a cocycle ω, as the group multipli- H G A 0 2 cation is only realized projectively, b(T1T2, σ) = b(T1, σ ) = b(T1T2 , µ) = b(T1,T2). (A6)

φ(g)φ(h) = ω(g, h)φ(gh), ω(g, h) = 1, (A3) Using Eq. (23), we see that no nontrivial anyonic flux ± densities are created by these mirror symmetries along and ω satisfies the cocycle condition automatically, be- the mirror axis. cause matrix multiplications are associative. Therefore, Next, we consider the twisted commutation relation each projective representation belongs to a certain class between mirror reflections and translations perpendicular 2 to the mirror axes. Motivated by Eq. (24), we define the of ω [G, Z2]. If we tensor product two projective ∈ H following quantum number, representations φ = φ1 φ2, the factor set of φ is the product of those of φ and⊗ φ . Therefore, we only need 1 2 β˜(g, h) = ω(g, h)ω(g, h−1gh)/ω(h, h−1gh). (A7) to construct projective representations realizing each of the four root classes in 2[G , ] = 4, which has one s Z2 Z2 It is easy to check that β˜(g, h) is a coboundary- of the four invariants ω12H, ωµ, ωσ and ωI being 1 and − independent invariant of a cohomology class [ω] all the other three being +1. 2 ∈ [ , Z2] for any two group elements. Using the ex- plicitH G form of the projective representation in Eq. (A4), ωµ ωσ ωI φ(µ) φ(σ) we can show that all the twisted commutation relations −1 +1 +1 iτ 2 τ 3 have trivial fractionalization, +1 −1 +1 τ 3 iτ 2 β˜(T −1T , σ) = β˜(T T 2, σ0) = β˜(T , µ) = +1. (A8) +1 +1 −1 τ 3 τ 1 1 2 1 2 1 Hence, for all cohomology classes in 2[ , ], the results TABLE II. Projective representations realizing root comology H G A 12 i of Eq. (24) are trivial. In other words, the mirror sym- classes with ω = 1. Here, τ denotes the Pauli matrices. metries do not create any anyonic flux density in the direction perpendicular to the mirror axes. Therefore, the As discussed in Sec. V, the cohomology classes with mirror reflections µ, σ and σ0 do not create any nontriv- 12 ω = 1 can be realized by a representation with φ(T1) = ial anyonic flux density. As a result, we can safely ignore φ(T2) = 1. In particular, the three root classes can be the [τ] factor, when applying the constraint in Eq. (4) to realized using projective representations summarized in the triangular/kagome lattices in Sec. V. 11

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