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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, USAy 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 fraction- alization 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 y 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 2 −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 ( ) = !a( ; )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 !a(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 !a!b = which can be readily generalized to other lattices. !c. 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 ! (g; h) = M ; w(g; h) : (2) the key result of this work, which is a constraint that re- a a;w(g;h) 2 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 ! (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 ! (g; h) or the anyon-valued fractionalization on such lattices. a class w(g; h). We shall use them interchangeably at dif- ferent places in this paper, depending on convenience. Now we make these abstract definitions more explicit II.
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