Unification of bosonic and fermionic theories of spin liquids on the kagome lattice
Yuan-Ming Lu,1, 2, 3 Gil Young Cho,1, 4 and Ashvin Vishwanath1, 2 1Department of Physics, University of California, Berkeley, CA 94720 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 3Department of Physics, The Ohio State University, Columbus, OH 43210, USA 4Department of Physics, Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, IL 61801 (Dated: January 1, 2018)
Recent numerical studies have provided strong evidence for a gapped Z2 quantum spin liquid in the kagome lattice spin-1/2 Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of symmetry fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons obeying either bose (Schwinger bosons) or fermi (Abrikosov fermions) statstics, which are then treated within the mean-field theory. A longstanding question has been whether these two approaches are truly distinct, or describe the same phase in complementary ways. Here we show that all 8 Z2 spin liquid phases in Schwinger-boson mean-field (SBMF) construction can also be described in terms of Abrikosov fermions, unifying pairs of theories that seem rather distinct. The key idea is that for Z2 spin liquid states that admit a SBMF description on kagome lattice, the symmetry fractionalization of visions is uniquely fixed. Two promising candidate states for kagome Heisenberg model, Sachdev’s Q1 = Q2 SBMF state and Lu-Ran-Lee’s Z2[0, π]β Abrikosov fermion state, are found to describe the same symmetric spin liquid phase. We expect these results to aid in a complete specification of the numerically observed spin liquid phase. We also discuss a set of Z2 spin liquid phases in fermionic parton approach, where spin rotation and lattice symmetries protect gapless edge states, that do not admit a SBMF description.
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I. INTRODUCTION glement entropy serve as fingerprints of the topological 16 order in Z2 spin liquids. Analogous results have been reported for the frustrated square lattice, although the Z2 spin liquids (SLs) are a class of disordered many- spin states which have a finite energy gap for all bulk correlation lengths in that case are not as small as in the 17–21 excitations. They differ fundamentally from symme- kagome lattice . try breaking ground states such as magnetically ordered Intriguingly, the experimentally studied spin-1/2 phases and valence bond solids, since, in their simplest kagome materials - such as herbertsmithite22 - also re- form, they preserve all the symmetries including spin ro- main quantum disordered down to the lowest temper- tation, time reversal and crystal symmetries. More im- ature scales studied, well below the exchange energy portantly they possess bulk quasiparticles obeying frac- scales. However, in contrast to the numerical studies, 1 tional statistics . For example in the most common Z2 most experimental evidences point to a gapless ground SL of a spin-1/2 system, there are three distinct types of state23,24. It is currently still under debate if the gapless- fractionalized bulk excitations2–7: bosonic spinon b with ness is an intrinsic feature25 or a consequence of impu- half-integer spin, fermionic spinon f with half-integer rities that are known to be present in these materials26. spin, and bosonic vison v (a vortex excitation of Z2 gauge Furthermore the magnetic Hamiltonian of the material theory) with integer spin. They all obey mutual semion may depart from the pure Heisenberg limit. Relating the statistics7: i.e. a bosonic spinon acquires a −1 Berry numerical results to experiments remains an important arXiv:1403.0575v3 [cond-mat.str-el] 29 Dec 2017 phase when it adiabatically encircles a fermionic spinon open question. or a vison. These statistical properties are identical to Since it preserves all symmetries of the system, is a Z2 8 those of excitations in Z2 gauge theory , hence the name SL fully characterized by its topological order? The an- “Z2 spin liquid”. swer is no. In fact, the interplay of symmetry and topo- Recently, interest in Z2 SLs has been recharged by logical order leads to a very rich structure. There are numerical studies on the spin-1/2 Heisenberg model on many different Z2 spin liquids with the same Z2 topo- kagome9–12 lattice, where this state is strongly indicated. logical order and the same symmetry group, but they In particular a topological entanglement entropy13,14 of cannot be continuously connected to each other with- γ = log 2 is observed in the ground state. Just like lo- out breaking the symmetry: they are dubbed “symme- cal order parameters used to describe symmetry breaking try enriched topological (SET)” phases27–35. In a SET phases15, here fractional statistics and topological entan- phase the quasiparticles not only have fractional statis- 2
two distinct Z2 SLs in f SR can have the same PSG for tics, but can also carry projective representation of the fermionic spinons while only one of them has symmetry- symmetry group. This phenomena is dubbed “symmetry protected gapless edge modes. However, they differ in fractionalization”30,34,35, a well-known example being the the topology of spinon band structures, which provides fractional charge carried by the quasiholes (or quasielec- an interesting link between symmetry implementation trons) in the fractional quantum Hall effect36. Differ- and topological edge states. By arguing the absence of ent SET phases are characterized by different patterns of symmetry-protected gapless edge states in any SBMF symmetry fractionalization, mathematically classified by state, we show that all Z2 spin liquids in SBMF con- 30,34,35 2 the 2nd group cohomology H (Gs, A), where Gs is struction must have a trivial PSG (or symmetry fraction- the symmetry group of the system and A is the (fusion) alization pattern) for vison v. The knowledge of bosonic group of Abelian anyons in the topological order. In the spinon PSG and vison PSG in a SBMF state leads to its case of Z2 spin liquid, A = Z2 ×Z2 according the Abelian fermionic spinon PSG, with the help of proper twist fac- fusion rules summarized in (8). tors, therefore establishing the correspondence between a In the literature Z2 SLs have been constructed SBMF state and an Abrikosov-fermion Z2 SL. Applying in various slave-particle (or parton) frameworks: the these general principles to Z2 SLs on kagome lattice, we most predominant two approaches fractionalize physi- show that all 8 different Schwinger-boson (bSR) states cal spin-1/2’s into bosonic spinons2,37–39 and fermionic have their partners in the Abrikosov-fermion (f SR) rep- 3,5,27,40–43 38 spinons respectively. Both approaches yield resentation. In particular Q1 = Q2 state in Schwinger variational wavefunctions with good energetics44–46 for boson representation belongs to the same SET phase 48 the kagome lattice model. It was proposed that sym- as Z2[0, π]β state in Abrikosov fermion representation. metric Z2 SLs are classified by the projective symmetry This correspondence allows us to identify the possible 27 groups (PSGs) of bosonic/fermionic spinons. However symmetry-breaking phases in proximity to Z2 SLs on it has been a long-time puzzle to understand the relation kagome lattice. In fact all 8 SBMF states have their between different PSGs in bosonic-spinon representation Abrikosov fermion counterparts, as summarized in Table (bSR) and fermionic-spinon representations (f SR)47. To II). Part of these correspondences (for 4 SBMF states be specific in the kagome lattice Heisenberg model, in with p2 + p3 = 1 in TABLEII) has been obtained previ- bSR (Schwinger-boson approach) there are 8 different Z2 ously in Ref. 51, by explicitly identifying their projected 39 38 SLs among which the so-called Q1 = Q2 state is con- wavefunctions. These results serve as a useful guide in sidered a promising candidate according to variational future studies of Z2 SLs. 44 48 calculations . Meanwhile there are 20 distinct Z2 SLs This article is organized as follows. After a brief re- in f SR (Abrikosov-fermion approach), including the so- view on symmetry fractionalization, PSG and their re- 48 called Z2[0, π]β state which is in the neighborhood of lations in Z2 SLs in sectionIIA-IIB, we first establish energetically favorable U(1) Dirac SL45. Are these two the twist factors between PSGs of different anyons in candidate states actually two different descriptions of the a Z2 spin liquid in sectionIIC. In sectionIII we show same gapped phase? If not, what are their counterparts that absence of protected edge states and defect bound in the other representation? states in a Z2 spin liquid can determine the vison PSG In this paper we establish the general connection be- (see TABLEI). These results allow us to compute the vison and fermion PSG in any SBMF state, which estab- tween different Z2 SLs in bSR and f SR. We show that Z2 SLs constructed by projecting parton mean-field states lishes to the correspondence between Schwinger-boson in bSR (Schwinger-boson representation) cannot host and Abrikosov-fermion mean-field states of Z2 spin liq- symmetry-protected gapless edge states. This impor- uids, as studied in sectionIV and summarized in TABLE tant observation allows us to determine how visons trans- II. In sectionV we analyze and argue the most promising Z spin liquid candidate for spin-1/2 kagome Heisenberg form under symmetry in Schwinger-boson Z2 SLs, and to 2 further relate a Schwinger-boson state to an Abrikosov- model, i.e. Q1 = Q2 SBMF state which is equivalent fermion one. Since a bosonic spinon and vison fuse to to the Abrikosov-fermion Z2[0, π]β state. Finally in sec- a fermionic spinon as shown in (8), their correspond- tionVI we discuss possible Z2 spin liquid s with mirror- ing PSG coefficients naively should follow a product rule. symmetry protected edge states in the Abrikosov fermion However crucially, in some cases such as the PSG coeffi- representation. cients involving inversion symmetry30, extra twist fac- tors enter, modifying the naive fusion rule. Here we identify two additional instances where such nontrivial II. SYMMETRY FRACTIONALIZATION IN A PSG fusion rules occur, as explained in SectionIIC. Z2 SPIN LIQUID Related results can also be established using different techniques49,50. A. A brief review on symmetry fractionalization Next, we demonstrate that knowledge of just the bosonic (or just the fermionic) spinon PSG, with no fur- Symmetry fractionalization30,34,35 is a mathematical ther information such as the existence of a SBMF ansatz, framework that characterizes and classifies different SET is not enough to fully characterize a Z2 SL. For example, phases in two spatial dimensions (2d). The key point is 3
3rd one can be obtained by fusing the other two anyons. that a global (or crystalline) symmetry can act projec- These 2 “elementary” anyons can be chosen as any two tively on the anyons in a gapped 2d topological order. types out of all three, such as b and v. Since all anyons a More precisely, the action Ug of symmetry element g on in (8) satisfy a Z2 fusion rule a × a = 1, this leads to an anyon a satisfies the following condition in a gapped 2d Abelian fusion group of A = Z2 × Z2, where the two Z2 87 topological order : factors are associated with f and v separately. a a a Now that phase factors {ωa(g, h)} must be compatible Ug · Uh = ωa(g, h) Ugh (1) with the A = Z2 × Z2 fusion rules according to (5), since where ω1(g, h) ≡ 1 for an arbitrary local excitation 1 we must have ωa(g, h) = ±1 in a Z2 spin liquid. Therefore the projective action of symmetry group G on the anyons in ωa(g, h) = hω(g, h), ai ∈ U(1) (2) s a Z2 spin liquid is fully determined by is the U(1)-valued mutual braiding phase between a and an Abelian anyon ω(g, h). This can be understood by {ωf (g, h) = ±1|g, h ∈ Gs} × {ωv(g, h) = ±1|g, h ∈ Gs} 2 considering defect τg of symmetry element g, which must ∈ H (Gs,Z2 × Z2) (9) satisfy the following fusion rule up to gauge redundancy. This completely characterizes τg × τh = τgh × ω(g, h) (3) the symmetry fractionalization pattern in a symmetric Z2 spin liquid. where ω(g, h) can be an Abelian anyon in the topological There is one more issue to emphasize: relation (5) from order. These phase factors must satisfy the associativity fusion rules only apply to global (“onsite”) symmetries. condition When elements g or h are crystalline symmetries, there 30,49 c can be an extra twist factors Ωa,b(g, h) ∈ U(1) when ωa(f, g)ωa(fg, h) = ωa(f, gh)ωa(g, h) (4) we consider the implication of fusion rules on {ωa(g, h)} and be compatible with the fusion rules of anyons c a × b = c =⇒ ωa(g, h)ωb(g, h) = Ωa,b(g, h)ωc(g, h) (10)
a × b = c =⇒ ωa(g, h)ωb(g, h) = ωc(g, h), In the case of Z2 spin liquid, as will be discussed if g, h are global (onsite) symmetries. (5) soon in sectionIIC, the Z2-valued twist factors c Ωa,b(g, h) = ±1 can be nontrivial for cases involving Notice that there is a gauge redundancy for phase factors crystalline rotation30, mirror reflection and time rever- {ωa(g, h)}: we can always redefine the symmetry opera- 49 a sal symmetries . tion Ug by adding an extra braiding phase ha, αgi ∈ U(1), where αg is an arbitrary Abelian anyon. This gauge transformation modifies ωa(g, h) by B. Projective symmetry group (PSG) and its relation to symmetry fractionalization ha, α i ω (g, h) −→ ω (g, h) gh (6) a a ha, α i · ha, α i g h Here we briefly review the concept of projective sym- As a result, the gauge-inequivalent phase factors metry group (PSG) in the slave-particle construction of spin-1/2 quantum spin liquids27. A more detailed discus- {ωa(g, h)} are classified by the 2nd group cohomology sion will be given later in sectionIV. (Gs denotes the symmetry group) In the slave-particle construction, each spin-1/2 Sr on 2 {ωa(g, h)} ∈ H (Gs, A) (7) lattice site r is represented by a Kramers pair of slave particles {χr,α|α =↑, ↓} as with a discrete coefficient belonging to an Abelian group 1 X A, i.e. the (fusion) group of Abelian anyons in the topo- S = χ† ~σ χ (11) r 2 r,α α,β r,β logical order. α,β=↑,↓ Take the Z2 spin liquid for example, it features the following Abelian fusion rules7: where ~σ represents three Pauli matrices. The slave par- ticles, or simply “partons”, can obey either fermi or bose b × f = v, b × v = f, f × v = b, statistics: they correspond to the Abrikosov-fermion40,43 b × b = f × f = v × v = 1. (8) (f SR) or Schwinger-boson52 (bSR) representation respec- tively. Since the Hilbert space of partons are gener- where b stands for the spin-1/2 bosonic spinon, v for the ally larger than the physical Hilbert space of spin-1/2, spinless vison and f for the spin-1/2 fermionic spinon. a “single-occupancy” constraint must be applied to the Here 1 stands for local excitations carrying integer spins, parton Hilbert space obeying the trivial bose statistics. From the above fusion X † rules, it is clear that all 3 types of Abelian anyons can be nˆr ≡ χr,αχr,α = 1, ∀ r. (12) generated by 2 types of anyons among them, while the α=↑,↓ 4
{Ug|g ∈ Gs} is coined a “projective symmetry group” As a result, the physical spin-1/2 wavefunction must be (PSG)27, i.e. an extension of symmetry group G satis- ˆ s obtained by a Gutzwiller projection Pnˆr=1 on the parton fying wavefunction PSG/IGG = G (21) Y ˆ s |Ψspini = Pnˆr=1|Ψpartoni. (13) r It’s straightforward to show that IGG elements {Λ(g, h)} also satisfy the following associativity condition or in other words Y Λ(f, gh)Λ(g, h) = Λ(f, g)Λ(fg, h) (22) hα1, α2, ··· , αN |Ψspini = h0| χr,αr |Ψpartoni (14) r In the slave-particle formalism, typically a mean-field Hamiltonian Hˆ φ of partons will be constructed, giving Physically, a symmetric spin-1/2 ground state must re- MF rise to a parton ground state |Ψ i (for details see sec- main invariant under all symmetry operations {g ∈ G } parton s tionIV). In this case, IGG is the group of gauge rotations of the symmetry group G . On the other hand, is the s that keep the parton mean-field Hamiltonian invariant. parton state |Ψ i also invariant under all symme- parton In the case of Z spin liquids, there are both parton try operations? This is not necessarily true, because af- 2 hopping and pairing terms in the mean-field Hamiltonian, ter the Gutzwiller projection (13), any unitary rotations and hence only the parton number parity is conserved in (“gauge rotations”) in the unphysical Hilbert space vi- ˆ φ olating single-occupancy constraint (12) will not affect HMF and |Ψpartoni. This leads to a Z2 group structure the physical spin-1/2 state |Ψ i. As a result, the (un- of IGG = {χr,α → ±χr,α}' Z2, generated by gauge spin P nˆ rotation W = (−1) r r . In such a Z spin liquid, PSG projected) parton state |Ψpartoni only needs to remain 0 2 invariant up to a gauge rotation. Therefore the symme- elements {Ug} acts on partons in a projective fashion try action Ug on the partons can be decomposed into the −1 −1 −1 UgUhχr,αU Ug = η(g, h)Ughχr,αU , product of physical symmetry operation Og and gauge h gh rotations Gg η(g, h) = ±1. (23) P nˆr Ug = Og · Gg, ∀ g ∈ Gs (15) with IGG element Λ(f, g) = η(g, h) r . A set of gauge invariant Z2-valued phases {η(g, h) = ±1|g, h ∈ Note that gauge rotations {Gg} must all preserve the Gs} therefore fully labels the PSG. single-occupancy constraint (12) and the Gutzwiller- From associativity relation (4) and (22), gauge redun- projected (physical) Hilbert space. dancy (6) and (19), and their definitions (1) and (20), While the physical symmetry operation Og must form the similarity between symmetry fractionalization pat- a linear representation of symmetry group Gs with tern {ωa(g, h)} and PSG pattern {η(g, h)} is obvious. In fact, Z2 PSGs are nothing but physical manifestions of Og · Oh = Ogh (16) abstract symmetry fractionalizations in gapped Z2 spin 30 the gauge rotations generally form a projective represen- liquids , where the anyon a is determined by the statis- tics of partons {χ }. In other words, {η(g, h) = ±1}' tation of Gs as r,α {ωb(g, h) = ±1} in the Schwinger boson representation Gg · Gh = Λ(g, h) Ggh (17) where χ ∼ b, and {η(g, h) = ±1}'{ωf (g, h) = ±1} in the Abrikosov fermion representation where χ ∼ f. where the parton state |Ψpartoni must remain invariant In the framework of symmetry fractionalization, re- under pure gauge rotation Λ(g, h) lation (10) determined from fusion rules (8) allows us to relate symmetry fractionalization patterns (or PSGs Λ(g, h)|Ψpartoni = |Ψpartoni, Λ(g, h) ∈ IGG. (18) patterns in parton constructions) of all 3 anyon species a = f, v, b. Specifically, (10) allows us to determine the We define IGG (invariant gauge group)27 as the set of fermion PSG {ω (g, h)} from boson PSG {ω (g, h)} and all gauge rotations that keep parton state |Ψ i in- f b parton vison PSG {ω (g, h)}. This fact is crucial for the unifi- variant. Clearly one can redefine the gauge rotation as v cation of Schwinger-boson and Abrikosov-fermion repre- G → G · W by an extra IGG element W ∈ IGG, g g g g sentations for gapped symmetric Z spin liquids, as will leading to the following gauge redundancy on Λ(g, h) 2 become clear soon. −1 Λ(g, h) → Λ(g, h)Wg · Wh · Wgh , {Wg} ∈ IGG. (19) Due to associativity relations (16) and (17), the symme- C. Twist factors for symmetry fractionalization in a Z2 spin liquid try actions {Ug} on partons {χr,α} also form a projective representation (or extension) of symmetry group Gs: In this section we establish the nontrivial twist fac- Ug · Uh = Λ(g, h) Ugh, Λ(g, h) ∈ IGG. (20) tors for symmetry fractionalization patterns of different 5
Take the first algebraic identity in the left column of TA- BLEII for example
−1 −1 T2 T1 T2T1 = e (28) Its associated gauge-invariant phase factor is
−1 −1 ωa(T2 T1 T2T1) ≡ −1 −1 ωa(T2,T1) = ωa(T ,T ) · ωa(T2,T1) = (29) 2 1 ωa(T1,T2) As a result, the gauge-invariant twist factors are also as- sociated with such algebraic identities
b Ωf,v(g1 ··· gN ) ≡ b b b QN−1 ωf,v(g1, g2) · ωf,v(g1g2, g3) ··· ωf,v( i=1 gi, gN ) (30) FIG. 1: Crystal symmetries of kagome lattice with 4 gener- In the following we discuss three different algebraic ators {T1,2,Rπ/3,Ry}. Translations (T1, T2) along the direc- identities, each leading to one nontrivial twist factor in tion 1 and 2 are drawn as the directed arrow. Rπ/3 stands for 60 degree rotation about a hexagon center. The mirror a Z2 spin liquid. We’ll first present a general physical picture based on toric code model7 of Z topological reflection 1 is denoted by Ry, while reflection 2 corresponds 2 order, and then demonstrate them in projected parton to Rπ/3Ry. wavefunctions for a Z2 spin liquid. We notice that aside b from the twist factor Ωf,v(I,I) = −1 associated with 2 6 algebraic identity I = (Rπ/3) = e, the other two anyons a = f, b, v in a Z2 spin liquid. As discussed earlier nontrivial twist factors related to reflection symmetry on fusion rules (8), only two types of anyons are “elemen- Rx,y and time reversal symmetry T are missed in tary” in the sense that the 3rd type can be obtained by previous studies30. fusing these two types. Without loss of generality, here we choose fermionic spinon f and vison v as the two elementary anyons. Meanwhile the bosonic spinon b is simply a bound state of f and v, according to fusion rule b 1. Inversion I: Ωf,v(I,I) = −1 b = f × v. Focusing on 2d kagome lattice (space group P 6mm, see FIG.1) with symmetry group On the kagome lattice, hexagon-centered inversion T 3 Gs = P 6mm × SO(3) × Z2 (24) symmetry operation I = (Rπ/3) is the triple action of π/3 rotation Rπ/3 (see FIG.1 and 8th row in TABLE where T refers to time reversal symmetry, we will reveal I). Clearly when inversion acts twice, all particles rotate the nontrivial twist factors counterclockwise around the hexagon center by a full cir- b ωf (g, h)ωv(g, h) cle i.e. Ωf,v(g, h) = (25) ωb(g, h) 2 6 I = (Rπ/3) = e. (31) These twist factors generally apply to an arbitrary 2d lattice. Being a bound state of a vison v and a fermionic spinon Note that ωa(g, h) for certain symmetry elements g, h f, a bosonic spinon b would collect an extra −1 phase will depend on gauge choices due to redundancy (6). factor30, because the fermionic spinon encircles the vison Therefore we need to focus on the gauge-invariant quan- once in this process. tities, which sometimes can be a product of multi- To be more precise, let’s introduce toric code model7 ple {ωa(g, h)} factors. Note that each PSG coefficient as a concrete demonstration of Z2 topological orders (or ωa(g, h) is associated with algebraic identity g × h = gh Z2 spin liquids). In the toric code model, ends of various for group elements g, h and gh. Quite generally, the open strings represent fractionalized excitations such as gauge-invariant coefficients can always be obtained from spinons and visons. There are three different types of the following algebraic identity (summarized in the left strings, corresponding to three different anyons (bosonic column of TABLEI) spinon b, fermionic spinon f and vison v) in a Z2 spin liquid. In the figures we use solid lines to represent g · g ··· g = e, e ≡ identity element. (26) 1 2 N fermionic spinon (solid red circle) strings, and dashed as lines for vison (blue cross) strings. Anyons of different types obey mutual semion statistics, which means each ωa(g1 ··· gN ) ≡ crossing of two different types of strings will yield a −1 QN−1 ωa(g1, g2) · ωa(g1g2, g3) ··· ωa( i=1 gi, gN ) (27) phase factor. 6
As the dashed blue string Sˆv crosses with the solid black string Oˆf , an extra −1 sign will appear as we commute the I2 symmetry operator for vison and the string operator for fermionic spinon. As a result the PSGs ωa(I,I) associated with two inversion operations have a b nontrivial twist factor Ωf,v(I,I) = −1. This conclusion remains true no matter the inversion symmetry is plaquette-centered or site-centered.
Now we confirm this intuitive picture using the pro- jected parton wavefunction in the Abrikosov fermion rep- resentation. The same calculation also goes through in the Schwinger boson representation. Consider an excited state with a pair of fermionic spinons f1,2 related by in- version symmetry Iˆ
b −1 FIG. 2: (Color online) Nontrivial twist factor Ωf,v(I,I) = −1 |f1,2i ≡ f1f2|Ψ0i, f2 = UI f1UI . (35) 6 2 associated with inversion square operation (Rπ/3) = I = e, as discussed in sectionIIC1. The symmetry operator associ- where |Ψ0i represents the parton ground state. By defi- 6 2 ated with (Rπ/3) = I is illustrated by the solid (dashed) nition of symmetry fractionalization and PSGs we have closed hexagon string with red (blue) color for fermionic spinons f (vison v). The phase factor acquired by a bosonic 2 −2 UI f1,2UI = ωf (I,I)f1,2, ωf (I,I) = ±1. (36) spinon b = f × v in this process is ωb(I,I) = −ωf (I,I) · ωv(I,I), where the extra −1 sign comes from the crossing of fermion string (black solid line) and vison string (blue dashed It’s straightforward to check that line). hf1,2|Iˆ|f1,2i = −ωf (I,I) (37) hΨ0|Iˆ|Ψ0i As illustrated in FIG.2, we consider a fermionic spinon where the extra −1 sign shows up because two fermionic f on the end of a (black) solid string, and another vison spinon operators f and f are exchanged under inversion v on the end of a (black) dashed string. The bound state 1 2 operation Iˆ. Similarly for excited state of these two object is a bosonic spinon b = f × v. When ˆ ˆ the solid (Sf ) and dashed (Sv) black string operators act −1 |v1,2i ≡ v1v2|Ψ0i,UI v1UI = v2. (38) on the ground state (vacuum) |Ψ0i, such an excited state
|f × vi = Sˆ Sˆ |Ψ i (32) with a pair of visons v1,2 on top of mean-field ground f v 0 state, we have is created. As π/3 rotation acts for six times or ˆ equivalently inversion symmetry operation acts twice hv1,2|I|v1,2i 2 −2 6 2 = ωv(I,I),UI v1,2UI = ωv(I,I)v1,2(39) i.e. (Rπ/3) = I , the phase factor acquired in this pro- hΨ0|Iˆ|P si0i cess is given by vacuum expectation value of the I2 sym- metry operator And the excited state with a pair of bosonic spinons bi = fi × vi is created by 6 ha|(Rπ/3) |ai 6 = ωa(I,I), a = b, f, v. (33) hΨ0|(Rπ/3) |Ψ0i |b1,2i = f1f2v1v2|Gi. (40) which is the solid (dashed) closed hexagon string Clearly we have 30 operator Oˆf (Oˆv) with red (blue) color for fermionic spinon f (vison v). More concretely we have hb |Iˆ|b i 1,2 1,2 = ω (I,I) = −ω (I,I) · ω (I,I) (41) ˆ b f v 6 ˆ −6 ˆ ˆ hΨ0|I|Ψ0i (Rπ/3) Sf (Rπ/3) = Of · Sf , 6 ˆ −6 ˆ ˆ (Rπ/3) Sv(Rπ/3) = Of · Sv, And this proves the nontrivial twist factor 6 hf×v|(Rπ/3) |f×vi −1 −1 6 = hΨ0|S S Of Sf OvSv|Ψ0i hΨ0|(R ) |Ψ0i v f ω (I,I) · ω (I,I) π/3 Ωb (I,I) ≡ f v = −1 (42) −1 −1 f,v ω (I,I) = (SvOf Sv Of )hΨ0|Of |Ψ0ihΨ0|Ov|Ψ0i b hf|(R )6|fi hv|(R )6|vi π/3 π/3 2 6 = − 6 6 . (34) hΨ0|(Rπ/3) |Ψ0i hΨ0|(Rπ/3) |Ψ0i associated with algebraic identity I = (Rπ/3) = e. 7
b 2. Mirror reflection R: Ωf,v(R,R) = −1
In this case we consider a pair of fermionic spinons f1,2 connected by a solid black string, and another pair of visons v1,2 connected by a dashed black string as de- picted in FIG.3. We assume all these anyons lie on the reflection axis so that they are symmetric under reflec- tion operation R. We’ll reveal the nontrivial twist factor b Ωf,v(R,R) = −1 associated with two reflection opera- tions R2 = e, by studying the reflection quantum number carried by these anyons. A key ingredient of our discus- sions is that the pair of fermionic spinons f1,2 (and visons v1,2) must be related by translation Ta, as shown in FIG. 3. This guarantees two anyons of the same species share the same symmetry quantum number. As illustrated in FIG.3, the total reflection ( R) quan- tum number for the pair of fermionic spinons f1,2 (vi- sons v1,2) is the ground state expectation value of solid (dashed) closed string operator Oˆf (Oˆv) with red (blue) color. Without loss of generality, we can assume this excited state with a pair of bosonic spinons is created by first applying the (solid black) fermionic spinon open string Sˆf on the ground state and then the (dashed black) vison open string Sˆv:
|b1b2i = Sˆv · Sˆf |Ψ0i, (43) ˆ ˆ |f1f2i = Sf |Ψ0i, |v1v2i = Sv|Ψ0i, b FIG. 3: (Color online) Nontrivial twist factor Ωf,v(R,R) = −1 2 URaiUR = ai, a = b, f, v; i = 1, 2. −1 associated with reflection square operation R = e, as discussed in section IIC2. Reflection axis of R is denoted by More concretely, in a Z2 spin liquid we have (see FIG.3) the dotted green line. The symmetry operator associated with reflection R is illustrated by the solid (dashed) closed hexagon ˆ −1 ˆ ˆ RSaR = Oa · Sa, a = f, v. (44) string Of (Ov) with red (blue) color for fermionic spinons f1,2 (vison v1,2), in the bottom of the figure. The phase factor 2 2 as (Sˆa) = (Oˆa) = 1 in a toric code. As a result we have acquired by each bosonic spinon bi = fi × vi, i = 1, 2 in 2 the process of R = e is ωb(R,R) = −ωf (R,R) · ωv(R,R), where the extra −1 = S O S−1O−1 sign comes from the ha1a2|R|a1a2i ˆ f v f v = hΨ0|Oa|Ψ0i, a = f, v; (45) crossing of open fermionic spinon string S (black solid line) hΨ0|R|Ψ0i f and closed vison string Ov (blue dashed line). Note that the and pair of fermionic spinons f1,2 (and visons v1,2) are related by translation Ta (parallel to reflection axis), to guarantee that hb1b2|R|b1b2i = hΨ0|Sf Sv (Ov Sv )(Of Sf )R|Ψ0i they share the same reflection quantum number. hΨ0|R|Ψ0i hΨ0|R|Ψ0i ˆ ˆ −1 −1 = hΨ0|Of |Ψ0ihΨ0|Ov|Ψ0i · (Sf OvSf Ov ) spinon and vison. As a result when reflection R acts = − hf1f2|R|f1f2i hv1v2|R|v1v2i . (46) hΨ0|R|Ψ0i hΨ0|R|Ψ0i twice, the phase factor acquired by each bosonic spinon b = f × v , i = 1, 2 in this process is given by Since the dashed blue (closed) string anti-commute i i i 2 with black solid string, the reflection quantum number of ωb(R,R) = (±i) · ωf (R,R) · ωv(R,R) bosonic spinon pair in FIG.3 equals the reflection quan- = −ω (R,R) · ω (R,R) (47) tum number of fermionic spinon pair multiplying that f v of vison pair, with an extra −1 sign. Since the pair of Therefore the twist factor associated with two reflection anyons are related by translation symmetry, this −1 sign operations R2 = e is indeed nontrival should be evenly split into two halves: i.e. upon reflec- tion operation R, each bosonic spinon acquires an ex- b ωf (R,R) · ωv(R,R) Ωf,v(R,R) ≡ = −1. (48) tra ±i phase in addition to the product of the fermionic ωb(R,R) 8
time reversal symmetry T . The Kramers doublets obey To validate this conclusion for reflection square 2 a Z2 fusion rule, in the sense that the bound state of two R = e in the Abrikosov fermion representation, we can Kramers doublets becomes one Kramers singlet. This follow exactly the same calculations as in the preceding b 2 2 implies a trivial twist factor Ωf,v(T,T ) = 1 for T = e section for inversion square I = e, by simply replacing associated withe time reversal symmetry. inversion I with reflection R. Now we consider an excitation d located on the reflec- tion plane, hence invariant under mirror reflection. Now that TR is also an anti-unitary Z2 symmetry just like T , for similar reasons there are also “non-Kramers doublets” 3. Time reversal T and mirror reflection R: with ω (TR,TR) = −1, which features two-fold degen- Ωb (R,T ) = −Ωb (T,R) d f,v f,v eracy protected by symmetry TR. From the viewpoint of this excitation d on reflection plane, TR symmetry can Unlike other global (on-site) unitary symmetries, time be treated in exactly the same fashion as time reversal reversal is an anti-unitary symmetry operation involving T . Therefore non-Kramers doublets of anti-unitary Z2 a complex conjugation. Below we show a nontrivial twist symmetry TR should also obey a Z2 fusion rule, i.e. two factor non=Kramers doublets fuse into a non-Kramers singlet Ωb (R,T ) s with ωs(TR,TR) = 1. If we consider one fermonic b −1 −1 f,v spinon f and a vison v, both on the reflection plane, it Ωf,v(R T RT ) = b = −1 (49) Ωf,v(T,R) is straightforward to verify the trivial twist factor
−1 −1 associated with algebraic identity R T RT = e. b ωf (TR,TR)ωv(TR,TR) Let’s first consider the combination T · R of time re- Ωf,v(TR,TR) = = 1 (53) ωb(TR,TR) versal T and mirror reflection R, which is an anti-unitary spatial symmetry. When this symmetry acts twice, each based on the Z2 fusion rules of non-Kramers doublets. anyon a acquires a phase factor ωa(TR,TR) = ±1. In this case, is there a nontrivial twist factor To summarized, we have established the nontriv- ial twist factor (52) associated with algebraic identity b ωf (TR,TR)ωv(TR,TR) R−1T −1RT = e using two different arguments. Notice Ωf,v(TR,TR) = (50) ωb(TR,TR) that inversion symmetry I is a combination of two re- flection symmetries with perpendicular reflection planes when fusing different anyons or not? The answer is no. i.e. I = RxRy, as shown in FIG.1. Therefore the twist In the same setup as in FIG.3, each bosonic spinon gets factor associated with I−1T −1IT is trivial an extra ±i phase (in addition to the phase factors ac- quired by fermionic spinon and vison) under reflection Ωb (I,T ) Ωb (I−1T −1IT ) ≡ f,v = +1 (54) R. However when time reversal acts, it takes complex f,v Ωb (T,I) conjugation and hence we have (±i)∗(±i) = 1. This ex- f,v tra phase hence cancels out upon symmetry operation Before closing of this section, we want to mention that 2 (TR) = e. Therefore twist factor (50) associated with all arguments used here can be made more rigorous by 2 (TR) = e is trivial and equals unity. Meanwhile, as an considering a thin cylinder geometry, which relates the algebraic identity we have phase factors {ωa(g, h)} to one-dimensional invariants of 53 2 −1 −1 2 2 symmetry protected topological (SPT) phases . This (TR) = (R T RT ) · (T ) · (R ) = e. (51) dimensional reduction approach is discussed in Ref. 49. and therefore from associativity we have
b III. IMPLICATIONS ON VISON PSGS FROM b −1 −1 Ωf,v (R,T ) Ωf,v(R T RT ) ≡ b ABSENCE OF EDGE STATES Ωf,v (T,R) b Ωf,v (T R,T R) = b b = −1 (52) Ωf,v (T,T )Ωf,v (R,R) In previous sections, we introduced symmetry frac- tionalization patterns {ωa(g, h)|g, h ∈ Gs; a = f, v} to Note that for global time reversal symmetry, the twist characterize a gapped symmetric Z2 spin liquid. In par- b factor is trivial i.e. Ωf,v(T,T ) = +1. ticular, the PSG coefficients {ωf (g, h)} ({ωb(g, h)}) in b The above twist factor Ωf,v(TR,TR) = 1 can also be the Abrikosov fermion (Schwinger boson) representation argued in an alternative way. It is well-known that any are nothing but symmetry fractionalization patterns for excitation (labeled as a here) in a time reversal invariant fermionic (bosonic) spinons f (b). In other words, in each system can be categorized into Kramers doublets with parton construction, the parton PSG only specifies the ωa(T,T ) = −1, and Kramers singlets with ωa(T,T ) = 1. symmetry fractionalization pattern for one anyon species In particular for each excitation a with ωa(T,T ), there is (f or b). Meanwhile by establishing the nontrivial twist a two-fold degeneracy (Kramers “doublet”) protected by factors (25) in sectionIIC, we can relate fermionic spinon 9
Algebraic Identities bosonic spinon bσ fermionic spinon fσ vison v Phase factors (27) Twist factors (30) −1 −1 p1 T2 T1 T2T1 = e (-1) η12 -1 ωa(T2,T1)/ωa(T1,T2) 1 −1 −1 T1 Rπ/3T2Rπ/3 = e 1 1 1 +1 by gauge choice 1 −1 −1 T1 T2Rπ/3T1Rπ/3 = e 1 1 1 +1 by gauge choice 1 −1 T1Rx T1Rx = e 1 1 1 ωa(T1Rx,T1Rx)/ωa(Rx,Rx) 1 −1 TyRy TyRy = e 1 1 1 ωa(TyRy,TyRy)/ωa(Ry,Ry) 1 2 2 p2+p3 Rx ≡ (Rπ/3Ry) = e (-1) ησ 1 ωa(Rx,Rx) -1 2 p2 (Ry) = e (-1) ησησC6 1 ωa(Ry,Ry) -1 6 2 p1+p3 (Rπ/3) = I = e (-1) ηC6 1 ωa(I,I) -1 −1 −1 T1 T T1T = e 1 1 1 ωa(T1,T )/ωa(T,T1) 1 −1 −1 T2 T T2T = e 1 1 1 ωa(T2,T )/ωa(T,T2) 1 −1 −1 p2 Ry T RyT = e (-1) ησT ηC6T 1 ωa(Ry,T )/ωa(T,Ry) -1 −1 −1 p2+p3 Rx T RxT = e (-1) ησT 1 ωa(Rx,T )/ωa(T,Rx) -1 2 T = e -1 -1 1 ωa(T,T ) 1
TABLE I: Algebraic identities g1 ··· gN = e, their associated phase factors (or PSG coefficients) ωa(g1 ··· gN ) in (27), and twist b factors Ωf,v(g1 ··· gN ) in (30). Considering spin-1/2 Z2 spin liquids on the kagome lattice, we list ±1-valued PSG coefficients 39 48 54 (phase factors) {ωa(g1 ··· gN )} for bosonic spinon with a = b, fermionic spinon with a = f and vison with a = v. 3 3 Mirror reflection Rx is defined as Rx ≡ (Rπ/3) Ry and inversion I ≡ (Rπ/3) , see FIG.1. We also introduce the translation −1 2 39 Ty ≡ T1 T2 alongy ˆ-axis. Here bosonic spinon (bσ) PSGs are labeled by three Z2 integers pi = 0, 1 (i = 1, 2, 3), while 48 fermionic spinon (fσ) PSGs are labeled by six signs (η12, ησ, ησC6 , ηC6 , ησT , ηC6T ) where η = ±1. Choosing a proper gauge −1 −1 −1 −1 we can always fix ωa(T1 Rπ/3T2Rπ/3) = ωa(T1 T2Rπ/3T1Rπ/3) ≡ 1 for all anyons {a = b, f, v}. Note that the vison PSG in any SBMF Z2 SL (i.e. in bSR) is completely fixed as summarized above. If one Z2 spin liquid state in bSR and one in f SR belong to the same SET phase, according to fusion rule condition (10), their PSG coefficients must satisfy relation (64).
(velocity is set to unity): PSG {ωf (g, h)} and bosonic spinon PSG {ωb(g, h)} via the vison PSG {ω (g, h)}. In other words, knowing the P † † v L0 = α iψR,α(∂t − ∂x)ψR,α − iψL,α(∂t + ∂x)ψL,α vison PSG {ω (g, h)} in a Schwinger boson mean-field v (55) (SBMF) state will allow us to also compute the fermionic spinon PSG {ωf (g, h)} in this state. By comparing it where α denotes different branches of left/right movers. with the Abrikosov fermion Z2 spin liquid states, we can One can always add backscattering terms to gap out the establish the correspondence between this SBMF state edge modes {ψR/L,α} and another possible Abrikosov fermion state. X † L1 = ψ Mα,βψL,β + ψR,α∆α,βψL,β + h.c. (56) In this section, we answer the following question: what R,α α,b are the physical manifestations of the vison PSG in a Z2 spin liquid? As will become clear later in section if they are not forbidden by symmetry. In a different IV, understanding this question will allow us to fix the language, the above “mass” terms correspond to Bose vison PSGs in all SBMF Z2 spin liquid states, hence condensation of either bosonic spinons b or visons v on 55,59 establishing the correspondence between Schwinger bo- the edge of a Z2 SL. Since the bosonic spinons carry son and Abrikosov fermion representations of Z2 spin spin-1/2 each, condensing them will necessarily break liquid states. spin rotational symmetry on the edge. Therefore the only way to obtain a gapped edge without breaking the An important measurable property of topological symmetry is to condense visons, unless their symmetry phases are their edge states. Although Z2 SLs in the ab- fractionalization pattern (PSG) {ωv(g, h)} will forbid it. sence of symmetries are expected to have gapped edges55, In particular, if the symmetries (preserved on the edge) the edge may be gapless due to the protection of certain act projectively on the visons, it is impossible to con- symmetries32,56,57; or in the case of discrete symmetries, dense visons without breaking the symmetries. The rele- spontaneously break symmetry on the edge. In particu- vant symmetries here are the ones that leave the physical lar, a symmetrically gapped edge of spin-1/2 Z2 SLs with edge unchanged, e.g. at least one translation symmetry SU(2) spin rotation symmetry puts strong constraints on among T1,2 will be broken by the edge. Therefore the the vison PSG, as will be argued below. absence of symmetry protected edge states provides a strong constraint on the vison PSG. 58 The edge modes of a Z2 SL can be fermionized with Take kagome lattice for instance, on a cylinder with the same number of right (ψR,α) and left (ψL,α) movers open boundaries parallel to T1 direction (X-edge in FIG. 10
60,61 1), the remaining symmetries are translation T1, time re- state in crystal defects (such as dislocations and 3 versal T and mirror reflection Rx ≡ (Rπ/3) Ry. If there disclinations) is another signature to probe crystal sym- are no symmetry protected edge states, then the remain- metry fractionalization. Just like gapless edge modes, the ing symmetries must act trivially (i.e. not projectively) absence of defect bound states implies a trivial vison PSG on visons: of the associated crystal symmetry. In our case of kagome lattice, the crystal defect associated with Rπ/3 rotation −1 −1 ωv (T1,T ) T T T1T = e −→ = 1, 1 ωv (T,T1) is a disclination, centered on the hexagon with Frank
−1 −1 ωv (Rx,T ) angle Ω = nπ/3, n ∈ Z. Now consider a vison encir- Rx T RxT = e −→ = 1, ωv (T,Rx) cling around an elementary disclination with Ω = π/3 for 2 Rx = e −→ ωv(Rx,Rx) = 1, six times counterclockwise, and the phase factor it picks −1 ωv (T1Rx,T1Rx) up in this process equals to the vison PSG coefficient T1R T1Rx = e −→ = 1. (57) x ωv (Rx,Rx) 3 3 ωv(I,I) = ωv((Rπ/3) , (Rπ/3) ). Note that a vison only so that no backscattering term is forbidden by symmetry. acquires a trivial (+1) phase factor when it encircles any number of spinons six times. Therefore if (R )6 = −1 Another inequivalent edge is perpendicular to T1 di- π/3 rection (Y-edge in FIG.1), which preserves translation for visons, there must be a nontrivial in-gap bound state (beyond spinon or vison) localized at the Ω = π/3 discli- −1 2 Ty ≡ T1 T2 nation. Therefore the absence of in-gap bound state in the disclination indicates time reversal T and mirror reflection Ry. Similarly, ab- 6 2 sence of protected edge modes necessarily implies that (Rπ/3) = I = e −→ ωv(I,I) = 1. (61)
−1 −1 ωv (Ty ,T ) T T TyT = e −→ = 1, Notice that this argument only applies to a plaquette- y ωv (T,Ty ) centered rotation/inversion (such as Rπ/3 and I here), −1 −1 ωv (Ry ,T ) R T RyT = e −→ = 1, y ωv (T,Ry ) because the visons are located on plaquettes. For a 2 site-centered crystal rotation, similar conclusions are not R = e −→ ωv(Ry,Ry) = 1, y valid anymore. −1 ωv (Ty Ry ,Ty Ry ) TyR TyRy = e −→ = 1. (58) y ωv (Ry ,Ry ) To summarize, if a Z2 SL does not host any gapless edge states protected by symmetries or in-gap disclina- i.e. symmetry operations on visons form a linear (not tion bound states, its vison PSG must satisfy conditions projective) representation of the edge symmetry group. (57)-(61). All together this leads to the vison PSGs As summarized in TABLEI, by choosing all possible shown in TABLEI, assuming the absence of symmetry- edge configurations, one can fix all gauge-invariant vison protected gapless edge states or in-gap disclination bound PSG coefficients in TABLEI (for detailed derivations see states. AppendixA) except for the three coefficients below
ωv(T2,T1) IV. UNIFICATION OF PARTON MEAN-FIELD , ωv(T,T ), ωv(I,I). ωv(T1,T2) THEORIES ON THE KAGOME LATTICE In a Mott insulator (with an odd number of spin 1/2 moments per unit cell) like the spin-1/2 kagome lattice, With all results established in previous sections, now any unfractionalized featureless phase with a spin gap we are ready to establish the vison PSG {ωv(g, h)} and must double the unit cell. Therefore the vision PSG for fermionic spinon PSG {ωf (g, h)} in any SBMF Z2 spin translations T must satisfy: liquid state. This allows us to unify Z2 spin liquid states 1,2 in Schwinger boson (bSR) and Abrikosov fermion (f SR)
ωv(T2,T1) representations, as summarized in TABLEII. T −1T −1T T = e −→ = −1 (59) 52 2 1 2 1 ω (T ,T ) In bSR or Schwinger boson construction , a spin-1/2 v 1 2 on lattice site r is decomposed into two species of bosonic This may be thought of as visons acquiring an extra −1 spinons {br,α|α =↑ / ↓}: phase factor on going around a unit cell containing an odd −1 −1 1 number of spinons, under the operations of T T T2T1. X † 2 1 S~r = b ~σα,βbr,β (62) Meanwhile, since visons are spinless particles, they 2 r,α α,β=↑/↓ transform trivially as a Kramers singlet under time re- versal T and hence where ~σ are Pauli matrices. Meanwhile in f SR or 40 2 Abrikosov-fermion approach spin-1/2 is represented by T = e −→ ωv(T,T ) = 1. (60) 43 two flavors of fermionic spinons {fr,α|α =↑ / ↓} Finally we present one argument to fix ω (I,I) associ- v 1 X ated with algebraic identity I2 = (R )6 = e. In addi- S~ = f † ~σ f (63) π/3 r 2 r,α α,β r,β tion to edge states, symmetry-protected in-gap bound α,β=↑/↓ 11
Abrikosov fermion representation48 (f SR) Schwinger boson representation39(bSR)
# η12 ησ ησT ησC6 ηC6T ηC6 Label Perturbatively (p1, p2, p3) Label gapped?
1 +1 +1 +1 +1 +1 +1 Z2[0, 0]A Yes (1,1,0)
2 -1 +1 +1 +1 +1 -1 Z2[0, π]β Yes (0,1,0) Q1 = Q2 state
5 +1 +1 +1 -1 -1 -1 Z2[0, 0]B Yes (1,0,1)
6 -1 +1 +1 -1 -1 +1 Z2[0, π]α No (0,0,1) Q1 = −Q2 state
13 +1 -1 -1 -1 -1 -1 Z2[0, 0]D Yes (1,1,1)
14 -1 -1 -1 -1 -1 +1 Z2[0, π]γ No (0,1,1)
15 +1 -1 -1 +1 +1 +1 Z2[0, 0]C Yes (1,0,0)
16 -1 -1 -1 +1 +1 -1 Z2[0, π]δ No (0,0,0)
TABLE II: Correspondence between Schwinger boson mean-field (SBMF) states39 and Abrikosov fermion states48 of spin-1/2 symmetric Z2 spin liquids on the kagome lattice. All 8 different SBMF (bSR) states have their counterparts among 20 distinct 38 48 Abrikosov fermion (f SR) states. Q1 = Q2 state in Schwinger boson representation is equivalent to Z2[0, π]β state in Abrikosov fermion representation, which is the only gapped Z2 spin liquid in the neighbor of energetically favorable U(1) Dirac 45 38 state . On the other hand Q1 = −Q2 state or (0, 0, 1) state in Schwinger boson representation corresponds to Z2[0, π]α state in Abrikosov fermion representation. “Perturbatively gapped” means that fermion spinons can have a fully-gapped mean-field spectrum by perturbing the nearest neighbor (NN) hopping ansatz.
as discussed in (9), a Schwinger-boson ansatz corresponds to the same phase as an Abrikosov-fermion ansatz if and To faithfully reproduce the 2-dimensional Hilbert space only if they share the same vison PSG {ωv(g, h)} and for spin-1/2, there is a single-occupancy constraint: the same fermionic spinon PSG ω (g, h). As mentioned P b† b = P f † f = 1 on every lattice site f α r,α r,α α r,α r,α earlier, vison PSGs of a Z SL can be probed by checking ∀ r. The variational wavefunctions are obtained by 2 62 whether symmetry-protected edge states or in-gap discli- implementing Gutzwiller projections on spinon mean- nation bound states exist or not. One important obser- field ground state |MF i, in order to enforce the single- vation is that none of the SBMF states of Z SLs con- occupancy constraint. Here |MF i is the ground state of 2 38,39 structed in Schwinger-boson approach supports any gap- (quadratic) mean-field ansatz for bosonic spinons : less edge state or in-gap disclination bound state. This can be verified by computing the edge spectrum or de- ˆ b X X † αβ HMF = Ax,ybx,αby,α + Bx,ybx,α by,β + h.c. fect spectrum in a Schwinger boson mean-field ansatz. x,y α,β Any gapped Schwinger-boson Z2 SL ansatz can be tuned continuously to a limit that on-site chemical potential and similarly dominates over pairing/hopping terms, where it is clear T no in-gap modes exist in edge/defect spectra. Therefore † ! ! ! X f t ∆ fy,↑ the vison PSG {ω (g, h)} in any Schwinger-boson ansatz Hˆ f = x,↑ x,y x,y + h.c. v MF ∗ ∗ f † is fully fixed as in TABLEI. Amazingly this result (last x,y fx,↓ ∆x,y −tx,y y,↓ column in TABLEI) agrees with the vison PSG com- 54 for fermionic spinons3,27. Proper on-site chemical poten- puted microscopically from Schwinger boson ansatz . tials guarantee single-occupancy in |MF i on average. The physical properties of a gapped Z2 SL described by a projected wavefunction can be understood in terms As discussed previously, symmetry fractionalization of its mean-field ansatz. Specifically in bSR and f SR of patterns {ωa(g, h)} (or PSGs) of all three anyons {a = Z SLs, different PSGs for bosonic and fermionic spinons 2 b, f, v} in a Z2 spin liquid are restricted by fusion rules lead to distinct hopping/pairing patterns in mean-field (8). They must satisfy relation (10) due to fusion rules, b f ansatz HMF anf HMf . As pointed out in Ref. 39 there i.e. the product of fermionic spinon PSG and vison PSG are 8 different Schwinger-boson (bSR) mean-field ansatz equals the bosonic spinon PSG, up to nontrivial twist fac- of Z2 SLs on kagome lattice, while 20 distinct mean-field tors discussed in sectionIIC. These nontrivial twist fac- ansatz of Z2 spin liquid exists in f SR as shown in Ref. 48. tors take place in row 6-8 and 11-12 in TABLEI, for alge- 2 2 −1 −1 A natural question is: what is the relation between the braic relations I = e, Rx,y = e and Rx,yT Rx,yT = e. Z2 SLs in bSR and those in f SR? Can they describe the Therefore from TABLEI, we can determine the cor- same Z2 SL phase or not? respondence between a Schwinger-boson ansatz and an To answer this question, we use their vison symme- Abrikosov-fermion ansatz, if they describes the same try fractionalization pattern (or PSG) to determine the SET phase with the same PSGs for all three anyons. (in)equivalence of the two representations. To be precise, More precisely, fusion rule constraint (10) with associ- 12
perconductor of fermionic spinons near the energetically ated twist factors leads to the following conditions: favorable U(1) Dirac SL45. The s-wave pairing opens up
p1+1 a gap at the Dirac point of the underlying U(1) Dirac η12 = (−1) , SL and this implies that the Z2[0, π]β state could be p2+p3+1 ησ = (−1) , energetically competing with the U(1) Dirac SL48. Al- p3 ησC6 = (−1) , though variational wavefunctions that include pairing are 46 p2+p3+1 often found to have higher energy than the underlying ησT = (−1) , U(1) Dirac SL, we note that this is a restricted class η = (−1)p1+p3+1, C6 of states accessible via the parton construction, and a p3 ηC6T = (−1) . (64) more complete search may land in the superconducting (i.e. Z spin liquid) phase. For our purposes we will It turns out all 8 distinct gapped SBMF Z SL states 2 2 be content that it is proximate to the energetically fa- described by Schwinger-boson approach (bSR) can also vorable U(1) spin liquid state. One can also describe represented by Abrikosov-fermion approach (f SR), as a continuous transition from this ztsl to the coplanar summarized in TABLEII. Moreover, the most promis- q = 0 magnetically ordered state, although the argument ing Schwinger-boson state for kagome Heisenberg model, here is more involved than in the case of the Schwinger- i.e. Q = Q state38, describes the same gapped symmet- 1 2 boson representation. We make this argument47 in two ric Z spin liquid phase as the most promising Z [0, π]β 2 2 parts - first by ignoring the effects of the gauge field and state48 in Abrikosov-fermion approach. We notice that 4 recalling65 a seemingly unrelated transition between an of these 8 correspondences (the top 4 rows in TABLEII) s-wave superconductor and a quantum spin Hall phase of have been obtained in Ref. 51. the fermionic partons. The latter spontaneously breaks the SU(2) spin rotation symmetry down to U(1) that de- fines the direction of the conserved spin component. On V. CANDIDATE Z2 SPIN LIQUIDS FOR KAGOME HEISENBERG ANTIFERROMAGNET including gauge fluctuations one can argue that the quan- tum spin Hall phase is to be identified with the q = 0 magnetic order47. We have noted that there are 20 different Abrikosov- fermion mean-field (f SR) states and 8 different SBMF Starting with the U(1) Dirac spin liquid, the s-wave (bSR) states of symmetric spin-1/2 Z2 spin liquids on the superconductor representing the Z2 spin liquid is ob- kagome lattice. Here we will argue that one state is the tained by including a superconducting ‘mass’ term that most promising candidate for kagome Heisenberg model gaps the Dirac dispersion. Similarly, the quantum spin from the following two criteria (i) energetics, as elabo- Hall phase is obtained by introducing a distinct ‘mass’ rated below; and (ii) the requirement that the phase to term that also gaps out the Dirac nodes. There are be connected to a q = 0 magnetically ordered state via a three such mass terms indexed by the direction of the continuous transition. This candidate is Q1 = Q2 SBMF conserved spin in the quantum spin Hall state. All state in Schwinger boson representation, or equivalently of them anti-commute with the superconducting mass Z2[0, π]β state in Abrikosov fermion representation. term, which implies that a continuous transition is Numerical studies on the kagome lattice Heisenberg possible between these phases65. On integrating out antiferromagnet supplemented by a 2nd neighbor anti- the fermions, the coefficients of the mass terms form 63 ferromagnetic coupling (J2) reveal that on increasing an O(5) vector of order parameters (real and imaginary J2, the quantum spin liquid phase is initially further sta- parts of the pairing and the three quantum spin Hall bilized, before undergoing a transition63 into a q = 0 mass terms), described by O(5) non-linear sigma model 47,65–68 magnetic order around J2 ' 0.2J1. Since the correla- with a Wess-Zumino-Witten (WZW) term . The tion length increases as the transition is approached, it presence of the WZW term implies a continuous phase is likely to be second order. The q = 0 magnetic order transition from the superconductor to the quantum is a coplanar magnetic ordered state in which the three spin hall state with a spontaneously chosen orientation. sublattices have spins aligned along three directions at This is most readily seen by noting that skyrmions of 120 degrees to one another. The Schwinger-boson state the quantum spin Hall director carry charge 2, which that naturally accounts for this is the Q1 = Q2 state in when condensed leads to a superconductor with SU(2) Ref. 38, where under condensation of bosonic spinons the spin rotation symmetry65. Now, on including gauge q = 0 magnetic order develops via a continuous transi- couplings the superconductor is converted into the 64 tion in the O(4) universality class . Furthermore, varia- gapped Z2[0, π]β SL state. The quantum spin Hall 44 tional permanent wavefunctions based on the Q1 = Q2 state is a gapped insulator coupled to a compact U(1) state were shown to give very competitive energies in gauge field, which is expected to confine and lead to particular for small and positive J2, establishing it as a a conventional ordered state. This is seen to be the contending state. non-collinear magnetically ordered phase with the vector 48 The Z2[0, π]β mean-field state in Abrikosov fermion chirality at q = 0 (for details see AppendixD). The representation (f SR), also satisfies the desirable proper- photon is identified as the additional Goldstone mode ties above. It can be thought as the s-wave paired su- that appears since the q = 0 state completely breaks 13
ˆ P † where F = r,σ fr,σfr,σ stands for the total fermion number. As shown in AppendixC, only when spin rotation symmetry. This further confirms the identification between Q1 = Q2 Schwinger-boson state ησ = −1, ησT = +1. (67) and Z [0, π]β Abrikosov-fermion state, since they are 2 will there be a nontrivial integer index for topolog- both in proximity to the same magnetic order via a Z ical superconductors. As summarized in TABLEIII, continuous phase transition. there are 6 fermionic-spinon PSGs (#7 ∼ #12) that may support such a topological superconductor of fermonic spinons.
VI. FERMIONIC Z2 SPIN LIQUIDS WITH SYMMETRY PROTECTED EDGE STATES B. Reflection protected Y-edge states In the Schwinger-boson representation, due to the ab- On a cylinder with open Y-edge the symmetry group sence of protected gapless edge states (or in-gap discli- is generated by {T,T ≡ T −1T 2,R } and SU(2) spin nation bound states), the vison PSG is completely fixed. y 1 2 y rotations. Again one can easily show that Therefore the PSG of bosonic spinons (b) fully deter- mines the SET phase in any SBMF state. In other words, ωf (Ty,T ) ωf (TyRy,TyRy) two SBMF ansatz correspond to the same Z SL phase = = 1 (68) 2 ωf (T,Ty) ωf (Ry,Ry) if and only if they share the same bosonic-spinon PSG {ωb(g, h)}. in all Abrikosov-fermion states (see TABLEII), and we However this is not true in the Abrikosov-fermion rep- can disentangle translation Ty from other symmetries. resentation: i.e. two distinct Z2 spin liquid phases can Reflection Ry and time reversal T on Y-edge act on share the same fermionic-spinon PSG in their Abrikosov- Abrikosov fermions with fermion mean-field ansatz. This is due to the band 2 Fˆ −1 −1 ∗ Fˆ (UR ) = (ησησC ) ,U U U UT = (ησT ηC T ) (69). topology69–71 in an Abrikosov-fermion mean-field ansatz, y 6 Ry T Ry 6 which can lead to symmetry protected edge states in Similarly a nontrivial integer index Z for protected edge a Z2 spin liquid, not captured by the fermionic-spinon states can only happen when PSG {ωf (g, h)}. In particular, certain fermionic-spinon PSGs allow for a nontrivial band topology, manifested ησησC6 = −1, ησT ηC6T = +1. (70) by gapless edge states protected by mirror reflection It turns out that only 6 fermionic-spinon PSGs symmetry in a topological superconductor of Abrikosov- (#3, #4, #9, #10, #19, #20) among all 20 cases in TA- 72,73 fermions . BLEIII may support topological superconductors with protected gapless modes on Y-edge, while the other 14 are not allowed. A. Reflection protected X-edge states
There are two types of open edges in a cylinder ge- C. An example from minimal Dirac model ometry, i.e. X-edge and Y-edge in FIG.1. Other translational-symmetric open edge directions can be ob- As one example, we present a continuum model tained by Rπ/3 rotations on these two prototype edges. based on the minimal Dirac Hamiltonian for the mirror- As mentioned earlier, a cylinder with open X-edge pre- protected topological superconductor in symmetry class serves a symmetry group generated by {T,T1,Rx ≡ CI74–76. In the root state (ν = 1) which generates the (R )3R } and SU(2) spin rotations. It’s straightfor- π/3 y integer (ν ∈ Z) topological index, the low-energy edge ex- ward to see that citations are described by 2 pairs of counter-propagating ω (T ,T ) ω (T R ,T R ) fermion modes (see AppendixC for derivations) f 1 = f 1 x 1 x = 1 (65) ωf (T,T1) ωf (Rx,Rx) 0 X † † Ledge = iψR,a(∂t − ∂x)ψR,a − iψL,a(∂t + ∂x)ψL,a for all Abrikosov-fermion states in TABLEII. Since a=↑,↓ translation T1 also commutes with spin rotations, it can (71) be disentangled from other symmetries. As discussed in where the velocity is normalized as unity. The fermion AppendixC, one can futher show that translational sym- modes transform under symmetries (time reversal T , re- metry T1 won’t give rise to any nontrivial topological in- flection R and spin rotations) in the following way: dex. Focusing on time reversal T , mirror reflection Rx T P (and SU(2) spin rotations which commute with both T ψα,a −→ β,b[τx]α,β[iσy]a,bψβ,b (72) and Rx), when acting on Abrikosov fermions {fr,σ} they R P † satisfy ψα,a −→ β,b[τx]α,β[iσy]a,bψβ,b (73)
exp( i θnˆ·S~) θ 2 Fˆ −1 −1 ∗ Fˆ P i nˆ·~σ (U ) = η ,U U U U = η . (66) ψα,a −→ e 2 ψα,b (74) Rx σ Rx T Rx T σT b a,b 14
# η12 ησ ησT ησC6 ηC6T ηC6 Label Perturbatively X-edge Y-edge (p1, p2, p3) Label gapped?
1 +1 +1 +1 +1 +1 +1 Z2[0, 0]A Yes 0 0 (1,1,0)
2 -1 +1 +1 +1 +1 -1 Z2[0, π]β Yes 0 0 (0,1,0) Q1 = Q2 state
3 +1 +1 +1 -1 +1 -1 Z2[π, π]A No 0 allowed
4 -1 +1 +1 -1 +1 +1 Z2[π, 0]A No 0 allowed
5 +1 +1 +1 -1 -1 -1 Z2[0, 0]B Yes 0 0 (1,0,1)
6 -1 +1 +1 -1 -1 +1 Z2[0, π]α No 0 0 (0,0,1) Q1 = −Q2 state 7 +1 -1 +1 -1 +1 -1 - - allowed 0 8 -1 -1 +1 -1 +1 +1 - - allowed 0 9 +1 -1 +1 +1 +1 +1 - - allowed allowed 10 -1 -1 +1 +1 +1 -1 - - allowed allowed 11 +1 -1 +1 +1 -1 -1 - - allowed 0 12 -1 -1 +1 +1 -1 +1 - - allowed 0
13 +1 -1 -1 -1 -1 -1 Z2[0, 0]D Yes 0 0 (1,1,1)
14 -1 -1 -1 -1 -1 +1 Z2[0, π]γ No 0 0 (0,1,1)
15 +1 -1 -1 +1 +1 +1 Z2[0, 0]C Yes 0 0 (1,0,0)
16 -1 -1 -1 +1 +1 -1 Z2[0, π]δ No 0 0 (0,0,0)
17 +1 -1 -1 +1 +1 -1 Z2[π, π]B No 0 0
18 -1 -1 -1 +1 +1 +1 Z2[π, 0]B No 0 0
19 +1 -1 -1 +1 -1 -1 Z2[π, π]C No 0 allowed
20 -1 -1 -1 +1 -1 +1 Z2[π, 0]C No 0 allowed
TABLE III: 20 different Abrikosov-fermion Z2 SLs on a kagome lattice in the notation of Ref. 48. Among them state #2 38 39 or Z2[0, π]β state corresponds to the same phase as Q1 = Q2 state in Schwinger-boson representation with (p1, p2, p3) = (0, 1, 0). Meanwhile #6 or Z2[0, π]α state belongs to the same phase as the so-called Q1 = −Q2 state in Schwinger-boson representation with (p1, p2, p3) = (0, 0, 1). “Perturbatively gapped” means that fermion spinons can reach a fully-gapped superconducting ground state by perturbing the nearest neighbor (NN) hopping ansatz. “0” is a trivial topological index, indicating the absence of symmetry protected gapless modes on the edge. Among the 20 Abrikosov-fermion states, 6 may host protected edge states on X-edge, while 6 may support protected edge modes on Y-edge. None of the 8 Abrikosov-fermion states that have counterparts in Schwinger-boson representation can support gapless edge states.
Abrikosov-fermion representation (f SR). In the presence of physical symmetries, symmetry fractionalization pat- where we use index α = R/L and ~τ matrices for chirality terns (manifested as projective symmetry groups, or (right/left movers), index a =↑ / ↓ and ~σ matrices for PSGs in slave-particle/parton constructions) of anyons spin. It’s straightforward to see that TR = RT and 2 characterizes a symmetric Z2 spin liquid, or more gen- R = −1. There are two kinds of backscattering terms erally a SET phase. We show that the vison PSGs between right and left movers, which preserve SU(2) spin can be determined by the absence of symmetry pro- rotational symmetry: tected edge modes or in-gap bound states localized at crystal defect in Z2 SLs. Observing that there are no P † ∗ † Hhop = a t ψR,aψL,a + t ψL,aψR,a , symmetry-protected edge modes or defect bound states in any Schwinger-boson mean-field state, we show that P ∗ † † Hpair = a,b ∆ψR,a[iσy]a,bψL,b + ∆ ψL,b[iσy]a,bψR,a . vison PSG in any SBMF state is fully fixed. Utilizing the fusion rule constraint (10) relation between PSGs of three Among them, imaginary pairing is forbidden by time re- anyon types in a Z2 spin liquid, we obtain the fermion versal T , while hopping and real pairing are both forbid- PSG in any SBMF state, and hence a correspondence be- den by mirror reflection R. tween Schwinger-boson and Abrikosov-fermion states is achieved. Applying this general framework to kagome lattice VII. CONCLUSION AND OUTLOOK Z2 SLs, we showed that all 8 distinct Schwinger- boson (bSR) mean-field states have their counterparts In this work we systematically establish a con- in Abrikosov-fermion (f SR) representation. In partic- nection between two different representations of Z2 ular we found that two energetically favorable states, 38 SLs, i.e. Schwinger-boson representation (bSR) and Schwinger-boson Q1 = Q2 state and Abrikosov-fermion 15
Appendix A: Deriving vison PSGs in TABLEI from 48 Z2[0, π]β state , in fact belong to the same gapped sym- edge states metric Z2 spin liquid phase in proximity to q = 0 non- collinear magnetic order. We argue that this phase is the In this section we explicitly show how to determine the most promising candidate for the observed Z2 SL ground vison PSGs in the last column of TABLEI. First of all state in kagome Heisenberg model. we can always choose a proper gauge by multiplying a With this connection in hand, we can potentially have proper ±1 sign to symmetry actions T1,2, so that a full understanding of the possible proximate phases −1 T2R = R T1,T1R = R T T1. (A1) and quantum phase transitions out of a Z2 spin liquid. π/3 π/3 π/3 π/3 2 It is well-known that the Schwinger-boson approach al- In other words both the 2nd and 3rd rows of TABLEI lows one to identify quantum phase transitions (QPT) are +1. Meanwhile as discussed in the end of sectionIII, into neighboring magnetic-ordered phases from a Z2 SL, we have through condensation of bosonic spinons38. Meanwhile, knowing the vison PSGs one can study possible QPTs T1T2 = −T2T1. (A2) between paramagnetic valence-bond-solid (VBS) phases for visons in a spin-1/2 Z SL on kagome lattice. and Z2 SLs. On the other hand, in f SR it’s straightfor- 2 ward to track down gapless spin liquid phases connected The absence of symmetry protected edge states along X-edge leads to conditions (57). In particular we have to a gapped Z2 SL through a phase transition, as well as proximate superconducting ground states upon doping T −1T −1T T = 1, a quantum SL77. Therefore the identification between 1 1 2 2 f SR and bSR can point to a full phase diagram near a Rx = (Rπ/3Ry) = 1. gapped Z2 SL. and The correspondence obtained here also serves as impor- R−1T −1R T = (R−1 T −1R T ) · (R−1T −1R T ) = 1. tant guidance towards a complete specification of sym- x x π/3 π/3 y y metric Z spin liquids on the kagome lattice. If one of the 2 and two promising states we identified is indeed the ground state of kagome Heisenberg model, then it provides a −1 −1 −1 −1 −1 T1Rx T1Rx = (T1 T2Ry T2Ry) · (T2 T1 T2T1) clear target for future studies to look for “smoking gun” −1 −1 signatures of these two states. Finally, we point out that = −T1 T2Ry T2Ry = 1. similar studies can be applied to Z2 SLs on the square At the same time, conditions (58) come from the ab- 17,18 lattice , which is a direction for future works. sence of protected edge states along Y-edge. Therefore we have
2 −1 −1 Ry = Ry T RyT = 1. (A3) and Acknowledgments −1 2 −1 −1 2 T1 T2 Ry T1 T2 Ry = −1 −1 −1 −1 −1 −1 We thank M. Hermele, Y. Ran, C. Xu, Y.B Kim, (T1 Ry T1Ry) · (T2 T1 T2T1) = −T1 Ry T1Ry = 1. T. Grover, M. Lawler, P. Hosur, F. Wang, S. White, These conditions fix all the vison PSGs except for T. Senthil, P.A. Lee, D.N Sheng and S. Sachdev for 6 −1 −1 helpful discussions. We thank Mike Zaletel for pene- (Rπ/3) and T2 T T2T . The latter one is easily de- trating comments and for collaborations on a related termined as paper49. GYC thanks Joel E. Moore for support and −1 −1 T T T2T = 1. (A4) encouragement during this work. YML is indebted to 2 Shenghan Jiang for pointing out a typo in TABLEI by the absence of protected edge states in a cylinder and for bringing Ref. 51 to our attention. The au- whose edges are parallel to the direction of translation 6 thors acknowledge support from Office of BES, Materials T2. As discussed in sectionIII,( Rπ/3) = 1 is deter- Sciences Division of the U.S. DOE under contract No. mined by the absence of protected mid-gap states in a DE-AC02-05CH11231 (YML,AV), NSF DMR-1206515, disclination. DMR-1064319 and ICMT postdoctoral fellowship at UIUC (GYC) and in part from the National Science Foundation under Grant No. PHYS-1066293(YML). Appendix B: Vison PSGs obtained by explicit 54 YML thanks Aspen Center for Physics for hospitality calculations where part of the work is finished. GYC is especially thankful to M. Punk, Y. Huh and S. Sachdev for bring- In this section we deduce the vison PSGs from the dual ing Ref. 54 to our attention and for the helpful discussions frustrated Ising model obtained in Ref. 54, which de- and suggestions. scribes vison fluctuations of Schwinger-boson Z2 SLs on 16
perconductor with mirror reflection symmetry in two di- kagome lattice. The 4-component vison modes {vn|1 ≤ mensions. In a cylinder geometry, the symmetry group n ≤ 4} in section III A of Ref. 54 transform under sym- is generated by translation Te along the open edge (or metry g as cylinder circumference), mirror reflection R, time rever- 4 sal T and SU(2) spin rotations. Without loss of general- g X ity, let’s assume that SU(2) spin rotations commute with vm −→ vn · Oφ(g) n,m (B1) n=1 all other symmetries {T,Te,R}. We further assume that translation Te acts as where the matrices {Oφ(g)} are given by T −1T −1T T = T R−1T R = +1. (C1) 0 0 0 1 e e e 3 for fermions. 0 0 −1 0 Oφ(T1) = − , (B2) 0 1 0 0 −1 0 0 0 1. Classification 0 0 −1 0 0 0 0 −1 Since translation Te has trivial commutation relation Oφ(T2) = , (B3) 1 0 0 0 with other symmetry operations, it can be disentangled from the full symmetry group. Are there any gapless 0 1 0 0 edge states protected by translation symmetry? This 1 0 0 0 corresponds to “weak index”78 of 2d topological super- 0 0 1 0 conductors in class CI (with time reversal and SU(2) spin Oφ(Ry) = , (B4) 0 1 0 0 rotations), which is nothing but 1d topological index of the same symmetry class. Class CI has trivial classi- 0 0 0 −1 fication (0) in 1d, therefore we don’t have translation- 0 0 1 0 protected edge states. Due to absence of 2d topolog- 1 0 0 0 ical index in class CI, any protected edge states must Oφ(Rπ/3) = . (B5) come from mirror reflection symmetry R. The classi- 0 1 0 0 fication of mirror reflection protected topological insu- 0 0 0 1 lators/superconductors is resolved in Ref. 72,73 in the 76 Note that Ref. 54 considers mirror reflection Ix = framework of K-theory . The classification of non- Rπ/3Ry with interacting topological phases of fermions in class CI with mirror reflection R depends on the commutation relation 0 1 0 0 2 −1 −1 R = s1,R T RT = s2; si = ±1. (C2) 1 0 0 0 Oφ(Ix) = Oφ(Rπ/3)Oφ(Ry) = 0 0 1 0 with time reversal T . When s1 = s2 = +1, the K-theory 0 0 0 −1 classification is given by π0(R6) = 0, i.e. no topologi- cal superconductors with protected edge states. When It’s straightforward to check the vison PSGs s1 = s2 = −1, the classification is π0(C5) = 0 i.e. no −1 −1 Oφ(T1)Oφ(T2)Oφ(T1) Oφ(T2) = −1; topological superconductors. When s1 = +1, s2 = −1 2 2 −1 the classification is π0(R5) = 0 = 0 i.e. no topologi- Oφ(R ) Oφ(T1)Oφ(R )Oφ(T2) = 1; π/3 π/3 cal superconductors. Only when −1 −1 Oφ(Rπ/3) Oφ(T2)Oφ(Rπ/3)Oφ(T2) Oφ(T1) = 1; −1 −1 −1 s1 = −1, s2 = +1. (C3) Oφ(T1)Oφ(T2) Oφ(Ry)Oφ(T2) Oφ(Ry) = −1; the classification is given by π (R ) = and there are O (T )O (R )O (T )−1O (R )−1 = −1; 0 4 Z φ 1 φ y φ 1 φ y topological superconductors with an integer index ( ). 6 Z Oφ(Rπ/3) = 1 So far we’ve only considered global symmetries to- 2 gether with spatial mirror reflections to arrive at this Oφ(Ry) = 1 −1 integer Z classification for non-interacting electrons. In Oφ(Rπ/3)Oφ(Ry)Oφ(Rπ/3)Oφ (Ry) = 1 (B6) a symmetric spin liquid of spin-1/2 particles on a kagome which agree with the last column of TABLEI. lattice, all space group symmetries need to to taken into account, and the interactions between fermionic spinons are also important. Other space group symmetries and Appendix C: Two-dimensional TRI singlet the single-occupancy constraints for Abrikosov fermions superconductors (Class CI) with mirror reflection will impose extra conditions and reduce this integer clas- symmetry sification. Interaction effects may further reduce the clas- sification. Therefore in TABLEIII we’ve denoted the 6 In this section we discuss possible symmetry-protected states (for X- and Y-edge each) as “allowed” to support edge states of a time-reversal-invariant (TRI) singlet su- protected gapless edge states. 17
in Dirac spectrum and a strongly fluctuating U(1) gauge field aµ:
2. Minimal Dirac model and protected edge states X † 0 0 H = ψkσ µ (τxkx + τyky)ψk. (D2) k Here we construct a Dirac model for the root state (ν = 1) of topological superconductors with integer index where τ ν , σν and µν are the Pauli matrices acting on ν ∈ Z and mirror reflection R satisfying Dirac, spin and valley indices of ψk (there are two nodes 2 or two “valleys” : hence ψk is a 8-component spinor). R = −1,RT = TR. (C4) Here we temporarily ignore the compact U(1) gauge the- ory for clarity of discussion. Writing spin-1/2 electrons in the Nambu basis ψk ≡ (c , c† )T , we use Pauli matrices ~τ for Nambu index To obtain a Z2 spin liquid from this U(1) Dirac spin k,↑ −k,↓ liquid, a BCS-type pairing term is introduced for the and ~µ,~ρ for orbital index. The 8-band massless Dirac fermion ψ. We require the pairing term to be invariant Hamiltonian is given by under spin rotational symmetry, lattice symmetry opera- X tions (see Fig.1 for the symmetries of kagome lattice) and H = (k µ + k µ )ρ τ (C5) Dirac x x y z y x time-reversal symmetry, because the spin liquid found in k DMRG study does not break any of the symmetries. Fur- The Dirac fermion transforms as thermore, the pairing should gap out the Dirac spectrum as the Z2 spin liquid found numerically is fully gapped. T ∗ ψk −→ iτyψk, R δH = ∆ψ†(σyµyτ y)ψ∗ + h.c., (D3) ψ(kx,ky ) −→ iµxρyψ(kx,−ky ) (C6) The only symmetry-allowed mass term is which is a singlet of spin, valley and Dirac spinor indices. Being a singlet under spin and valley indices gauran- M = τz (C7) tees that the pairing is invariant under spin rotational symmetry and lattice symmetry operations. The low- Now let’s create a mass domain wall across an open energy physics of Z2[0, π]β state is described by H + δH edge at x = 0 along y-axis. The gapless edge states is in Eq.(D2) and Eq.(D3), i.e. a gapped singlet supercon- captured by zero-energy solution of differential equation ductor of fermionic spinons coupled to a dynamical Z2 gauge field. −i∂xµxρyτx + m(x)τz = 0, m(x) = |m(x)| · Sgn(x) It is known that Dirac fermions are unstable (with suf- − R x m(λ)dλ which is |edgei ∼ e 0 |µxρyτy = +1i. Therefore ficiently large interactions) to open up gap in various ρy = µxτy for the protected gapless edge modes, localized channels. Each channel is called a “mass” and is repre- on the edge between the root topological superconductor sented by a constant matrix in the Dirac spinor represen- and the vacuum. The Hamiltonian for the protected edge tation. For example in Dirac Hamiltonian (D2), τ zµασβ states is given by are all mass terms, with α, β = 0, x, y, z. For a 2+1-D Dirac fermion, when we can find five such mass matri- 66,68,81 Hk = kyµzρyτx → −kyµyτz (C8) ces anti-commuting with each other , we obtain a non-linear sigma model supplemented with a topological Apparently there are 4 gapless modes: 2 right movers WZW term82,83 after integrating out massive fermions. and 2 left movers. It’s straightforward to simplify such This non-linear sigma model describes fluctuating order edge states to the form in sectionVIC. parameters of the Dirac fermions. Most importantly, the theory can describe a Landau-forbidden second-order transition between the two phases84, where the transition Appendix D: Z2[0, π]β mean-field state and its is driven by condensing the topological defects47,65,67,68. proximate phases The mass terms associated to the Z2 spin liquid are real and imaginary parings in (D3). We begin by revisiting U(1) Dirac spin liquid45,79,80 48 Because we seek for the nearby phases of the Z2 spin and Z2[0, π]β state and on a kagome lattice. The liquid, the relevant mass terms should anti-commute with spin liquid states are proposed to be the ground state of the pairing term (D3) and the kinetic term (D2). We nearest-neighbor spin-1/2 Heisenberg model on kagome immediately find two O(3) vector mass terms among lattice: 26 mass terms of the U(1) Dirac spin liquid80, anti- X commuting with the pairing term (D3). H = J Sˆi · Sˆj (D1) Among the two O(3) vector mass terms, we con-
Goldstone modes in the magnetically ordered phase, one spin-triplet and time-reversal symmetric. This order pa- photon mode from the non-compact U(1) guage field aµ rameter represents the sum of the vector chirality around and two Goldstone modes from the ordering of the O(3) honeycomb plaquette H on Kagome lattice. vector Vˆ . Meanwhile accompanying the proliferation of a photons, fermionic spinons will be confined86 due to ˆ a X ~ ~ a µ V ∼ (Si × Sj) , (D4) instanton effect of 2+1-D U(1) gauge theory. Therefore
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work.