Thermal Hall effect in square-lattice spin liquids: A Schwinger boson mean-field study
Rhine Samajdar, Shubhayu Chatterjee, Subir Sachdev, Mathias S. Scheurer
TThermal Hall effect in square-lattice spin liquids: A Schwinger boson mean-field study
Rhine Samajdar, Shubhayu Chatterjee,
1, 2
Subir Sachdev,
1, 3 and Mathias S. Scheurer Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, University of California, Berkeley, California 94720, USA. Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada (Dated: April 12, 2019)Motivated by recent transport measurements in high- T c cuprate superconductors in a magneticfield, we study the thermal Hall conductivity in materials with topological order, focusing on thecontribution from neutral spinons. Specifically, different Schwinger boson mean-field ans¨atze for theHeisenberg antiferromagnet on the square lattice are analyzed. We allow for both Dzyaloshinskii-Moriya interactions, and additional terms associated with scalar spin chiralities that break time-reversal and reflection symmetries, but preserve their product. It is shown that these scalar spinchiralities, which can either arise spontaneously or are induced by the orbital coupling of the mag-netic field, can lead to spinon bands with nontrivial Chern numbers and significantly enhancedthermal Hall conductivity. Associated states with zero-temperature magnetic order, which is ther-mally fluctuating at any T >
0, also show a similarly enhanced thermal Hall conductivity.
I. INTRODUCTION
The Wiedemann-Franz (WF) law is a paradigmaticproperty of a metal that relates its electrical conduc-tivity tensor ˆ σ to its thermal conductivity tensor ˆ κ attemperature T as ˆ κ/T = L ˆ σ , where L = π k B / (3 e )is the Lorenz number [1]. Recent studies of themetallic state of high- T c cuprate superconductors, suchas La . − x Nd . Sr x CuO (Nd-LSCO), obtained by sup-pressing superconductivity using magnetic fields, indi-cate a very interesting trend in the thermal Hall coeffi-cient [2] as a function of doping. On the overdoped side,with a hole doping of p > p ∗ , where p ∗ corresponds to thedoping value where the pseudogap temperature vanishes,the thermal Hall conductivity κ xy obeys the WF law forlow T . However, for hole doping p < p ∗ , correspondingto the pseudogap phase, the thermal Hall conductivitychanges sign and becomes negative, while σ xy remainspositive. Further, the magnitude of κ xy / ( T σ xy ) at lowtemperatures significantly exceeds L , thus signaling acomprehensive breakdown of the WF law.A possible explanation of this observation is the pres-ence of charge-neutral spin-carrying excitations in thepseudogap phase. By virtue of being electrically neutral,they do not couple to the external electromagnetic fieldand, by association, do not contribute to σ xy ; however,they give rise to a thermal Hall current leading to the vio-lation of the WF law in Hall conductivities. The large κ xy observed at dopings with and without N´eel order suggeststhat magnons are not responsible for this phenomenon.Further, Grissonnanche et al. [2] argue that the observedmagnitude of κ xy at low temperatures is too large to beexplained by spin-scattered phonons. This prompts therather intriguing possibility of emergent neutral excita-tions that are responsible for this unusual behavior.In this paper, we investigate the thermal Hall conduc-tivity (see Fig. 1) of phases where the electron fraction-alizes into an electrically charged gapless fermionic char-gon and a gapped charge-neutral spin-carrying spinon [3]. Such a phase of matter has topological order [4], and hasbeen previously discussed in the context of the pseudogapmetal [5–10]. Indeed, model calculations of the longitu-dinal conductivities and the electrical Hall conductivityin these fractionalized phases [5] are consistent with ex-perimental observations in the metallic phases of severalcuprates. However, Ref. 2 shows that the large negative κ xy persists even in the insulating phase as the doping p →
0. This is the extreme limit of breakdown of theWF law, as σ xy = 0. Motivated by this observation, werestrict our focus to Mott insulators with gapped char-gons and topological order, analogous to the phases dis-cussed in Refs. 6 and 9, and compute the contribution tothe thermal Hall effect from deconfined, charge-neutral,spinons.Our first set of results is related to the thermalHall conductivity in square-lattice spin-liquid states withnonzero scalar spin chiralities, χ ijk = S i · ( S j × S k ), where S i is the spin operator on site i = ( i x , i y ) ∈ Z of thesquare lattice, but without any spin-orbit coupling; theseresults are presented in Section III. Note that, by virtueof being odd under time reversal and spin rotation in-variant, χ ijk can couple to bond-current operators and,hence, these states are in general associated with nonzeroloop currents. A recent paper [9] classified four likely FIG. 1. Schematic depiction of the thermal Hall effect in aninsulator with topological order, where the heat current iscarried by fractionalized S = 1 / a r X i v : . [ c ond - m a t . s t r- e l ] A p r patterns (labeled A,B,C,D) of time-reversal and mirror-plane symmetry breaking in spin liquids with nonzero χ ijk and associated loop currents. Among these, onlypattern D has a nonzero κ xy and hence, will be the cen-ter of our attention. We will find that spin liquids ofpattern D, which breaks square-lattice and time-reversalsymmetries down to m m (cid:48) m (cid:48) , do indeed lead to valuesof κ xy /T of order k B / (cid:126) at temperatures above the spingap; below the spin gap, κ xy /T vanishes exponentiallyas T → m m (cid:48) m (cid:48) could either be spontaneous, or simply dueto the presence of an applied magnetic field. We notethat, in the latter case, no hysteresis in the thermal Hallconductance is expected. As we review in Appendix A,the orbital coupling of the field in a Hubbard-type modelinduces a coupling between the magnetic field and theuniform scalar spin chirality.We also probe the thermal Hall conductivity of theassociated magnetically ordered states which break spinrotation symmetry at T = 0. In two spatial dimensions,spin rotation invariance is restored at any nonzero tem-perature by thermal fluctuations, and this allows us totreat such states with the same formalism as that usedfor spin liquids. For such thermally disordered descen-dants of magnetically ordered states we also find valuesof κ xy /T of order k B / (cid:126) , but κ xy /T vanishes as a powerof T as T → m m (cid:48) m (cid:48) isturned off. Nevertheless, a weakly distorted N´eel state isindeed one of the possible states leading to a large κ xy /T .The second set of conclusions in this paper pertain tothe influence of the spin-orbit coupling, which inducesDzyaloshinskii-Moriya (DM) terms in the spin Hamilto-nian. We study the DM term in spin liquids connectedto the N´eel state, and find that it induces a significantlysmaller value of κ xy /T , as described in Section IV.Our starting point is a Mott insulator where the low-energy degrees of freedom are the S = 1 / H spin = 12 (cid:88) i,j (cid:0) J ij S i · S j + D m ij · S i × S j (cid:1) − (cid:88) i B Z · S i + H χ . (1)The Heisenberg couplings J ij are taken to be positive, J ij >
0, and spatially local. The orientation of the exter-nal magnetic field is assumed to be perpendicular to the lattice plane (see Fig. 1). For the Zeeman field, we have B Z = B z ˆ z , where we have absorbed the Bohr magneton µ B in the definition of B z . The associated orbital cou-pling is described by H χ which involves third-order (andhigher-order) powers in S i (see Appendix A). We also in-clude a spin-orbit-induced Dzyaloshinskii-Moriya (DM)term, which is allowed when certain spatial symmetriesare broken. The precise orientations of the DM couplingvectors D m ij will be described below.To treat H spin , we adopt a Schwinger boson mean-field approach, which is capable of describing both spin-liquid phases and ordered antiferromagnets [11, 12]. Thisapproach, as detailed later, provides us with a mean-field ansatz, and the projective action of lattice or time-reversal symmetries on the ansatz describes the particu-lar spin-liquid state under consideration [13, 14]. Amongthe different ans¨atze we consider, only one, for which allin-plane reflection symmetries are broken (pattern D inRef. 9), leads to spinon bands with nonzero Chern num-bers.In previous literature, the thermal Hall effect has beenwidely investigated on the kagom´e [15–18], pyrochlore[19], and honeycomb [20–24] lattices for insulating phaseswith and without long-range magnetic order and in thepresence of additional electric field gradients [25]. How-ever, it is strongly constrained by no-go theorems on thesquare lattice owing to the geometry thereof; the fluc-tuation of the scalar spin chirality averaged over nearbyelementary plaquettes in the square lattice vanishes for ageneric phase [26, 27]. For our model, the discrete brokensymmetries are carefully chosen such that the associatedloop current pattern corresponds to a net addition of spinchirality on neighboring triangular plaquettes [9]. Thisenables our model to overcome the symmetry barriersassociated with the square lattice. This can be achievedsince we consider a Schwinger boson mean-field ansatz(illustrated schematically in Fig. 3) that is not smoothlyconnected to that of the usual N´eel state (which has topo-logically trivial bands). Rather, our ansatz can be viewedas a perturbation to the symmetric bosonic π -flux spinliquid [14]. As we show in the paper, these perturbationscan indeed induce nonzero Chern numbers and lead to amuch larger κ xy compared to other phases with topolog-ically trivial spinon bands. At the same time, as alreadynoted above, the associated magnetically ordered phasecan still be (a small deformation of) the N´eel state.We begin in Sec. II by setting up the Schwinger-bosonmean-field formalism and its computation of the ther-mal Hall conductivity. Section III evaluates the thermalHall effect in spin liquids with nontrivial magnetic pointgroups but full SU(2) spin-rotation invariance (SRI). TheDM term is not included in these analyses, but is consid-ered separately in Sec. IV (without the additional time-reversal symmetry-breaking terms of Sec. III). Finally,Sec. V summarizes the results and four Appendices, A–E, detail our calculations. II. FORMALISM
In order to compute the thermal Hall conductivity, oneneeds to first know the nature of the low-energy excita-tions above the quantum ground state of H spin . An ap-proximate method to treat this problem is provided bySchwinger boson mean-field theory (SBMFT) in whichthe Hamiltonian is written in terms of Schwinger bosons[11, 12], whereupon an appropriate mean-field decouplingrenders it quadratic. We briefly review this formalism inthe context of the thermal Hall effect below. A. Schwinger-boson mean-field theory
The spin operator can be represented at each site i =( i x , i y ) ∈ Z of the square lattice (we set a = 1 for thelattice constant) using a pair of bosons ( b i ↑ , b i ↓ ) as S i = 12 (cid:88) σ,σ (cid:48) b † iσ σ σσ (cid:48) b iσ (cid:48) , (2)where σ = ( σ , σ , σ ) T is a vector of Pauli matrices.These operators satisfy the standard bosonic commuta-tion relations [ b iσ , b † jσ (cid:48) ] = δ ij δ σσ (cid:48) . This construction en-larges the on-site Hilbert space; to remain within thephysical space, Eq. (2) has to be supplemented with thelocal holonomic constraintˆ n i = (cid:88) σ b † iσ b iσ = 2S , (3)which enforces that S i = S (S + 1).In this fashion, the reformulated Hamiltonian H spin contains only quadratic, quartic, and sextic terms in thebosonic operators. Now, we perform a mean-field decou-pling of H spin into quadratic operators. We neglect herethe DM interactions, which will be analyzed in Sec. IVand Appendix D, and the orbital coupling H χ , which willbe discussed in Appendix A; for now, we concentrate onterms that preserve SRI. The only such operators are thespin singletsˆ A i,j = 12 (cid:88) σ,σ (cid:48) b iσ (i σ ) σσ (cid:48) b jσ (cid:48) ; ˆ A j,i = − ˆ A i,j , (4)ˆ B i,j = 12 (cid:88) σ b iσ b † jσ ; ˆ B j,i = ˆ B † i,j , (5)and their adjoints. Here and in the following, we use ito denote the imaginary unit. The expectation values, {A i,j , B i,j } , of the operators in Eqs. (4) and (5) collec-tively define the parameters of the mean-field ansatz .First, let us examine the antiferromagnetic Heisenbergexchange term [28] in a simple spin Hamiltonian: H (1) = (cid:88) i>j J ij S i · S j ; J ij > . (6) Using the identity S i · S j = : ˆ B † i,j ˆ B i,j : − ˆ A † i,j ˆ A i,j = ˆ B † i,j ˆ B i,j − ˆ A † i,j ˆ A i,j −
14 ˆ n i , (7)with : : denoting normal ordering, Eq. (6) can be reducedto a mean-field quadratic bosonic Hamiltonian preservingSU(2) spin-rotation invariance. This is achieved by ne-glecting bond operator fluctuations and replacing (cid:104) ˆ A i,j (cid:105) and (cid:104) ˆ B i,j (cid:105) by complex bond parameters A i,j and B i,j ,respectively: H (1) mf = (cid:88) i>j,σ (cid:20) J ij (cid:16) B ∗ i,j b iσ b † jσ − A ∗ i,j σ b iσ b j − σ + H . c . (cid:17) + J ij (cid:0) |A i,j | − |B i,j | (cid:1) (cid:21) + λ (cid:88) i (cid:16) b † iσ b iσ − S (cid:17) . (8)At the mean-field level, the local constraint (3) is en-forced only on average, namely, (cid:104) ˆ n i (cid:105) = κ via the La-grange multiplier λ . One could, in principle, searchfor an optimal A i,j and B i,j by self-consistently solvingfor the stationary points of the mean-field free energy;however, for the purpose of this work, we simply treatthem as free (complex) parameters. The only constraintsthereon come from the upper bounds [29] on the moduli |A| ≤ S + 1 / |B| ≤ S, which must be obeyed for anyself-consistent ansatz in SBMFT.In the presence of a nonzero transverse magnetic field,spin-rotation invariance is broken by the additional Zee-man term in the Hamiltonian: H (2) = − B z (cid:88) i S zi = − B z (cid:88) i σ,σ (cid:48) b † iσ ( σ ) σσ (cid:48) b iσ (cid:48) = H (2) mf . (9)This term is already quadratic and thus requires no fur-ther decoupling.Since we will discuss spin liquid phases with certaindiscrete broken symmetries, to be precise, let us clar-ify when a given ansatz breaks a symmetry. The phys-ical spin operator is invariant under a local U(1) gaugetransformation b j → e i ϕ ( j ) b j . Under such a gauge trans-formation, the mean-field ansatz transforms as A i,j → e i[ ϕ ( i )+ ϕ ( j )] A i,j , B i,j → e i( ϕ ( i ) − ϕ ( j )) B i,j . (10)Therefore, a symmetry g is preserved as long as there isa gauge transformation, b j → G g ( j ) b j , G g ( j ) = e i ϕ g ( j ) ,that leaves the ansatz invariant when combined withthe action of the symmetry operation. Contrarily, if nosuch gauge transformation exists or, equivalently, there issome gauge-invariant operator that transforms nontriv-ially under g and has a finite (nonzero) expectation valuein the phase under consideration, then the symmetry g is broken. B. Diagonalization of bosonic quadraticHamiltonians
The mean-field Schwinger boson Hamiltonian can bediagonalized by the Bogoliubov-Valatin canonical trans-formation [30, 31]. For illustrative purposes, consider ageneral quadratic bosonic Hamiltonian H = 12 Ψ † M Ψ; Ψ † = (cid:16) b † , . . . , b † N , b , . . . , b N (cid:17) . (11)Generically, the index n = 1 , . . . , N on b n and b † n couldlabel momentum, spin, or some other degrees of freedom.To find the eigenmodes corresponding to M , we introducenew annihilation (creation) operators γ m ( γ † m ) such thatΨ = T Γ; Γ † ≡ (cid:16) γ † , . . . , γ † N , γ , . . . , γ N (cid:17) . (12)The standard bosonic commutation relations for both theΨ and Γ fields are conveniently encapsulated in the ma-trix equation (cid:104) Ψ i , Ψ † j (cid:105) = (cid:104) Γ i , Γ † j (cid:105) = ( ρ ) ij ; ρ ≡ (cid:18) N × N − N × N (cid:19) . (13)We choose T such that the Hamiltonian (11) becomes H = 12 Γ † T † M T Γ; T † M T = ω · · · ω . . . · · · ω N , (14)for ω i ∈ R . Meanwhile, to safeguard the bosonic statis-tics of the system, the transformation matrix must fulfillthe necessary condition T ρ T † = ρ , (15)or, in other words, T is paraunitary [32]. The elements ofthe transformation T can be obtained from the eigenvec-tors of the dynamic matrix K = ρ M , which defines theHeisenberg equation of motion for Ψ. All the eigenvaluesof the dynamic matrix (when diagonalizable) are real andappear in pairs. Then, T , conventionally referred to asthe derivative matrix, consists of all the eigenvectors of K T = [ V ( ω ) , . . . , V ( ω N ) , V ( − ω ) , . . . , V ( − ω N )] , (16)with the eigenvectors V ordered as V † ( ω i ) ρ V ( ω i ) = 1 , V † ( − ω i ) ρ V ( − ω i ) = − V ( ω i ) , V ( − ω i )). Thus, each eigenvalue of K is counted up to its multiplicity and the N dynamicmode pairs are separated and arranged sequentially ascolumns in T such that its left (right) half is filled witheigenvectors of positive (negative) unit norms [33]. Con- sequently, T − K T = diag ( ω , . . . , ω N , − ω , . . . , − ω N ) , (18) T † M T = diag ( ω , . . . , ω N , ω , . . . , ω N ) , (19)i.e. both M and K are simultaneously diagonalized. Bor-rowing fermionic terminology for Eq. (14), we refer to thebands with indices n = 1 , . . . , N ( n = N + 1 , , . . . , N ) asthe particle (hole) bands. C. Berry curvature and thermal Hall conductivity
The prescription outlined above can be straightfor-wardly applied to the Hamiltonians in the sectionshereafter, the only difference being that the matrices H ( k )—associated with the mean-field Hamiltonian H = (cid:80) k (Ψ † k H ( k ) Ψ k ) / T k therein are momentum-dependent. Suppose ε n k > n th band energyafter such a diagonalization procedure; accordingly, H = (cid:88) k N (cid:88) n =1 ε n k (cid:18) γ † n k γ n k + 12 (cid:19) . (20)Then, within SBMFT, the thermal Hall conductivity inthe clean limit is given by [34] κ xy = − k B T (cid:126) V (cid:88) k N (cid:88) n =1 (cid:26) c [ n B ( ε n k )] − π (cid:27) Ω n k , (21)where the sum on n runs only over the particle bands.Here, n B ( ε ) is the Bose distribution function, and c ( x ) ≡ (cid:90) x d t (cid:18) ln 1 + tt (cid:19) (22)= (1 + x ) (cid:18) ln 1 + xx (cid:19) − (ln x ) − ( − x ) , which is monotonically increasing with x : it has a mini-mum value of 0 at x = 0 and, in the opposite limit, tendsto π / x → ∞ . Ω n k in Eq. (21) is the Berry curvaturein momentum space [35], which, for bosonic systems, isgiven byΩ n k ≡ i (cid:15) µν (cid:34) ρ ∂ T † k ∂ k µ ρ ∂ T k ∂ k ν (cid:35) nn ; n = 1 , . . . , N. (23)The integral of the Berry curvature over the Brillouinzone (BZ) is the first Chern integer [36, 37] C n = 12 π (cid:90) bz d k Ω n k ∈ Z . (24)In addition to being integer valued, C n further obeys theconstraint N (cid:88) n =1 C n = N (cid:88) n = N +1 C n = 0 , (25)i.e., the sum of the Chern numbers over all particle andhole bands is individually zero [35]. Since the expressionin Eq. (21) for κ xy entails the summation over all par-ticle bands and the momentum sum (or integral in thethermodynamic limit) is taken over a closed surface (thefirst Brillouin zone), Eq. (25) dictates that − k B T (cid:126) V (cid:88) k N (cid:88) n =1 (cid:26) − π (cid:27) Ω n k = 0 . For this reason, we can neglect the additional − π / H ( k ) has been chosen to satisfy the particle-hole symmetry H ( k ) = ρ ( H ( − k )) T ρ ; ρ ≡ (cid:18) N × N N × N (cid:19) . (26)As it will be useful below, we point out that, as a con-sequence, the Berry curvatures of the particle and holebands are related as [38]Ω n + N, − k = − Ω n k ; 1 ≤ n ≤ N. (27)Before proceeding with the analysis of different spin-liquid states, a few general statements on the behaviorof κ xy are in order. First, if the temperature is muchlarger than the maximum energy of the m th particle bandso that n B ( ε m k ) (cid:29)
1, the contribution of this band toEq. (21) is related to its Chern number C m as (cid:2) κ xy (cid:3) m ≈ π k B T (cid:126) (cid:90) bz d k π Ω m k = π k B T (cid:126) C m . (28)Conversely, if T lies far below the minimum of the m th band, then n B ( ε m k ) ≈ n k is weighted by c [ n B ( ε n k )] in Eq. (21), thereis a nonvanishing thermal Hall conductivity at finite tem-peratures even if all bands have zero Chern numbers. Theoverall magnitude of κ xy , however, hinges on whether C n = 0 or C n (cid:54) = 0. For a trivial band, the momentum-space average of the Berry curvature is itself zero and wegenerically expect that (cid:2) κ xy (cid:3) m | C m =0 (cid:28) (cid:2) κ xy (cid:3) m | C m (cid:54) =0 .As a result, the total κ xy is expected to be much smallerfor a system with C n = 0 ∀ n than for one with nonzeroChern numbers. This is evident upon comparing Figs. 5 and 8, which correspond to conductivities arising from C (cid:54) = 0 and C = 0 bands, respectively; for a similarset of parameters, the former are a thousandfold larger.We note that, in principle, it is possible that the Berry-curvature has significant energy dependence and, hence, κ xy is large even for C n = 0; however, such a situationwas not realized for any of the ans¨atze we considered inthis work. III. SPIN LIQUID ANS ¨ATZE WITHTIME-REVERSAL SYMMETRY BREAKING
Having established the necessity of Chern numbers fora sizable thermal Hall conductivity, we study spin liquidmodels that can yield such topologically nontrivial bandstructures within SBMFT. Inspired by the recent workof Ref. 9 in the context of possible broken symmetriesin cuprates, we examine states with nontrivial magneticpoint groups. By breaking time-reversal symmetry whilepreserving SRI, the ans¨atze we discuss are naturally as-sociated with nonzero scalar spin chiralities.The simplest class of symmetry-breaking spin liquids ofRef. 9 are described by ans¨atze that, while preserving alltranslational symmetries of the square lattice, have mag-netic point group m (cid:48) mm ; this means that two-fold ro-tation perpendicular to the plane, C , and time-reversalsymmetry, Θ, are broken, but the product Θ C is pre-served. Depending on whether the reflection symmetryalong a Cu-O bond or along a diagonal Cu-Cu bond ispresent, these states are referred to as patterns A and Bin Ref. 9; they also appeared in studies of Z spin liquidsusing bosonic [6, 8] and fermionic [39] spinons. How-ever, as will be shown below, both these ans¨atze lead tospinon bands which are topologically trivial, promptingthe consideration of other patterns to procure nonzeroChern numbers.To this end, we analyze a translationally invariant spinliquid phase, referred to as pattern D in Ref. 9, thathas magnetic point group m m (cid:48) m (cid:48) ; this means that time-reversal symmetry and the point group C v have beenbroken down to the symmetry group generated by four-fold rotation perpendicular to the plane, C , and Θ R x (the product of time-reversal Θ and reflection symmetry R x at the xz plane). Unlike the earlier cases, all mir-ror symmetries are broken by this ansatz and the sumof all scalar spin chiralities within the unit cell does notadd up to zero. As evidenced in this section, we find thatnonzero Chern integers can indeed be realized. Note thatthe magnetic symmetries of the state we consider are thesame as those of an orbital magnetic field. Consequently,if the ansatz emerges spontaneously, we find an anoma-lous contribution to κ xy , i.e., a thermal Hall response inthe absence of an external magnetic field. This, how-ever, also means that the symmetry-breaking terms ofthe ansatz can be induced by the orbital coupling H χ . Inthe latter case, there is no anomalous contribution. A. One-orbital model with trivial bands
Throughout this section, we direct our attention tothe one-orbital model of the cuprate superconductors,which only involves the Cu- d orbitals forming a squarelattice as shown in Fig. 3. The general form of the mean-field Hamiltonian, only involving spin-rotation invariantterms, reads as H mf = J (cid:88) i,j, σ (cid:16) B i,j b † iσ b jσ − A ∗ i,j σ b iσ b j − σ + H.c. (cid:17) + λ (cid:88) iσ (cid:16) b † iσ b iσ − S (cid:17) . (29)One can write down a suitable ansatz consistent with allthe m (cid:48) mm symmetries to describe pattern A as A i,i +ˆ x = A i,i +ˆ y = A , B i,i +ˆ x = B i,i +ˆ y = i B , A i,i +ˆ x +ˆ y = A i,i − ˆ x +ˆ y = A , (30a)and all others terms set to zero, where ˆ x = (1 ,
0) andˆ y = (0 ,
1) have been introduced. Similarly, for patternB, A i,i +ˆ x = A i,i +ˆ y = A , B i,i +ˆ x = B i,i +ˆ y = i B , A i,i +ˆ x +ˆ y = A . (30b) (a) (b) FIG. 2. Schwinger boson band structure (in units of J A )for the ansatz of (a) Eq. (30a) (pattern A), and (b) Eq. (30b)(pattern B), with A = 0 . B = 0 . B z = 0, and λ = 3.For clarity, the eigenvalues of the dynamic matrix are shown;the energies of the actual bosonic bands are just the abso-lute values of the same and are strictly positive. The differ-ent lines for each of the two colors refer to distinct valuesof k y = − π, − π + π/ , . . . , π . The dispersion minima are at ± ( π/ , π/
2) for A = 0, but shift to ± ( K , K ), with K incom-mensurate, when A (cid:54) = 0. The states can thus be smoothlyconnected to the antiferromagnet by tuning A . By tuning |B | and |A | to sufficiently small values,the ans¨atze in Eq. (30) can be brought arbitrarily closeto that of the conventional two-sublattice N´eel state andits quantum-disordered partner [for which only A isnonzero in Eq. (30)]. Accordingly, the concomitant mag-netically ordered state is a smooth deformation of theN´eel state and happens to be a conical spiral [6, 40].Since the spectrum for |B | , |A | (cid:54) = 0, illustrated in FIG. 3. Schwinger-boson mean-field ansatz for the one-orbitalmodel defined by Eqs. (31) and (33). The Cu atoms in theCuO plane are depicted here as dark blue circles. The ar-rows indicate the sign conventions: along the (next-)nearest-neighbor bond from site i to site j , the bond operators havethe expectation values (cid:104) ˆ A i,j (cid:105) = A , (cid:104) ˆ B i,j (cid:105) = i B ; dueto ˆ A j,i = − ˆ A i,j and ˆ B j,i = ˆ B † i,j , the bonds are directed andassociated with blue (red) arrows in the figure. Fig. 2, retains its gap upon continuously tuning B and A to zero, the Chern numbers must be C n = 0 (exactlylike those of the N´eel state), wherefore these ans¨atze arenot expected to be a good starting point for obtaining asizable thermal Hall response. B. Chern numbers and thermal Hall conductivity
The considerations above seem to suggest looking in-stead at ans¨atze that are not adiabatically connected tothat of the conventional antiferromagnet with only A nonzero. Motivated by the recent study [9] of spin-liquidstates with orbital loop currents, we next consider anansatz with magnetic point group m m (cid:48) m (cid:48) . A minimalchoice, yielding this point group while preserving trans-lations, T x , T y , is A i,i +ˆ x = A , A i,i +ˆ y = ( − i x + i y A , (31a) B i,i + η µ = i s µ ( − i x + i y B , (31b)with second-nearest-neighbor vectors η µ = ˆ x + ( − µ ˆ y .The relative signs of s µ ∈ { +1 , − } can be read off Fig. 3,and are chosen so as to attain the correct magnetic pointgroup. Obviously, the ansatz is not explicitly invariantunder the symmetry generators T x , T y , C , and Θ R x .However, since the symmetries act projectively, it is in-variant under the respective symmetry operations whenthey are applied in conjunction with the following gaugetransformations: G T µ ( j ) = ( − j y , ; µ = x, y, (32a) G Θ R µ ( j ) = i( − j x + j y , (32b) G C ( j ) = ( − j x , (32c) G C ( j ) = cos (cid:16) π j x + j y ) (cid:17) ; j ∈ α, sin (cid:16) π j x + j y ) (cid:17) ; j ∈ β. (32d)At the same time, one can indeed construct explicitgauge-invariant fluxes which are odd under Θ or R µ [9],and our ansatz does break these symmetries.It turns out that the ansatz of Eq. (31a–b) alone provesto be insufficient to yield bands with nonzero Chern num-bers, so we add on top the additional operator expecta-tion values: B i,i +ˆ x = i B , B i,i +ˆ y = i( − i x + i y B , (33a) A i,i + η µ = s µ ( − i x + i y A . (33b)It is straightforward to check that Eqs. (31) and (33),in totality, preserve both translation and m m (cid:48) m (cid:48) by ap-plying the gauge transformations in Eq. (32). From thispoint onward, the term “one-orbital model” always im-plicitly refers to this combined ansatz for pattern D. Forcompleteness, the three-orbital model of the cuprates,also taking into account the oxygen p orbitals, is dis-cussed in Appendix E; the conclusions are similar inspirit.The generalization in (33) results in topologically non-trivial bosonic bands and, hence, a considerable thermalHall response as we show below. As long as the inter-band gaps remain open, the Chern integers are invariantunder smooth variations of the mean-field parameters {A µ , B µ } in the Hamiltonian. Consequently, this stateis not smoothly connected to the SBMFT of the con-ventional square-lattice antiferromagnet, for which theChern numbers of all the bands are identically zero.A useful characterization of spin-liquid phases can beobtained by gauge invariant fluxes. Of particular impor-tance for our study is the flux φ = A , A ∗ , A , A ∗ , ,where 1 , , , and 4 label the four sites of any elementarysquare plaquette in counterclockwise order. The limit-ing case A = B = B = 0 of the ansatz in Fig. 3corresponds to the π -flux states of Yang and Wang [14],which have full square-lattice and time-reversal symme-tries; turning on nonzero values of A , B , and B reducesthe symmetry to m m (cid:48) m (cid:48) , and leads to spinon bands withnonzero Chern numbers. On the other hand, the CP model [41], a low-energy effective field theory of quan-tum antiferromagnets on a square lattice, describes themore familiar zero-flux Schwinger boson state [14]. It wasshown in Ref. 9 that there is no quadratic perturbationto the CP theory which breaks the symmetry down to m m (cid:48) m (cid:48) , and we discuss the needed perturbations furtherin Appendix B. Our results here are consistent with theseearlier results: we need to perturb a π -flux state to havenonzero Chern numbers of spinon bands in SBMFT; suchnontrivial bands cannot be obtained as a perturbation ofthe zero-flux state. Further, the CP theory can natu-rally describe low-energy excitations close to Q = (0 , π, π ); in contrast the spin-liquid phase we considerhas low energy excitations at (0 , π ) and ( π,
0) as well.Yang and Wang [14] also analyzed the magnetic or-dered states that appeared upon condensing bosonicspinons from the π -flux state. They found a variety ofpossibilities with ordered moments at wavevectors (0 , π ), ( π, π, π ): this included cases where the dominantmoment was at the ( π, π ) wave vector of the N´eel state.Nonzero values of A , B , and B distort these states toalso allow for a (possibly small) ferromagnetic momentat (0 , (cid:104) S ( j ) (cid:105) = n (0 , + ( − j x n ( π, + ( − j y n (0 ,π ) (34)+ ( − j x + j y n ( π,π ) . Note that this ferromagnetic moment arises without aZeeman term in the Hamiltonian, and is a consequence ofeither spontaneous breaking of the symmetry to m m (cid:48) m (cid:48) ,or one induced by the orbital coupling to the externalfield (see Appendix A).One might wonder whether adding the orbital couplingof the magnetic field, H χ , described in leading order in t/U by terms involving the triple products S i · ( S j × S k )[42], can be used to describe the symmetry reductionto the magnetic point group m m (cid:48) m (cid:48) within SBMFT.We consider the decoupling of this triple-product termin Appendix A. Although we do not include this self-consistently in our analysis, we verify that spin-liquidstates with symmetry broken to m m (cid:48) m (cid:48) do indeed leadto a nonzero expectation value for the triple products inthe Hamiltonian, in the quadratic approximation.
1. Spectrum and symmetries
In spite of the final thermal Hall conductivity itselfbeing a gauge-invariant quantity, any intermediate cal-culations require the explicit choice of a gauge. Owingto the alternating factor of ( − i x + i y , the ansatz (31) istranslationally invariant only modulo a gauge transfor-mation or, in other words, it is invariant under two -sitelattice translations when working in a fixed gauge. Wetherefore choose a two-sublattice unit cell with sublat-tice indices defined by the parity of i x + i y . In eachunit cell, we denote the Schwinger boson operators by α (even parity) and β (odd parity). The basis vectorsfor this new bipartite lattice are η µ , and the reciprocallattice vectors are G µ = π η µ , so the BZ can be chosento be the conventional antiferromagnetic Brillouin zone, { ( k x , k y ) | k x , k y ∈ [ − π, π ); | k x | + | k y | ≤ π } .As sketched in Appendix C, the mean-field Hamilto-nian can be represented in terms of the eight-componentspinor Ψ † k = ( α † k ↑ β † k ↑ α † k ↓ β † k ↓ α − k ↑ β − k ↑ α − k ↓ β − k ↓ ) with H mf = (cid:80) k (Ψ † k H ( k ) Ψ k ) /
2. The associated band struc-tures upon diagonalization are plotted in Fig. 4. At eachmomentum k , the dynamic matrix K has eight eigenval-ues, four positive and four negative; we label the former(latter) by n = 1 , . . . , n = 5 , . . . ,
8) in ascending (de-scending) order. The energies of the actual bosonic bandsare simply the absolute values of these and are necessarilypositive.Additionally, the Hamiltonian H ( k ) harbors anothersymmetry that is somewhat less apparent. Although the (a) (b) (c) (d) (e) - - - - - (f) FIG. 4. (a) Dispersion of the Schwinger boson particle bands ε n k , n = 1 , . . . ,
4, shown in blue, orange, green, and red,respectively, along the line k x = 0, for the one-orbital model with A = 0, B = 0, B = 0 . λ = 2, and B z = 0 .
5, measuredin units of J A . The bands touch along lines in the BZ, as underscored by the density plot of ε k − ε k in (b), and thuslack well-defined Chern numbers. (c) The intersection of the bands persists even with A = 0 .
75 on top of the parametersin (a,b). (d) The addition of a nonzero B (taken to be 0.5 here) is required to prevent the touching of two particle bands,necessitating the addition of Eq. (33) to the minimal ansatz. With B (cid:54) = 0, the bands acquire a nontrivial Chern number. (e)The dispersion of the lowest-energy band in (d) exhibits minima at k = ( ± π/ , k = 0at the global minima of the dispersion. The first Chern integers are C n = − n = 1 , n = 3 ,
4) bands. Thecurvatures are ill-defined at B z = 0, for which all the particle bands are pairwise degenerate. particle bands are generically distinct, they become pair-wise degenerate when there is no Zeeman field, B z = 0.We emphasize that this degeneracy is not the same as thetrivial redundancy described in Eq. (19), which arises dueto the pairwise occurrence of the eigenvalues of the dy-namic matrix. Despite the seeming lack of an a priori reason, the degeneracy of these eigenvalues stems froman effective antiunitary symmetry, which we scrutinizemore carefully later in Appendix C 1.From the paraunitary matrix T k , one can calculate theBerry curvatures of the bands. However, the Berry con-nection, defined as A j,µ ( k ) ≡ i Tr (cid:104) Γ j ρ T † k ρ (cid:0) ∂ k µ T k (cid:1)(cid:105) , (35)where Γ j is a diagonal matrix with (Γ j ) ab = δ ja δ jb , can-not be smoothly specified over the entire BZ and thephases of the eigenvectors that constitute T k must bechosen accordingly. The resolution lies in decomposing the BZ into two overlapping regions H and H with H ∪ H = BZ, and H ∩ H = ∂H = − ∂H [35]. Theseregions are chosen such that [ T k ] m ν ,j is never zero withinthe region H ν , where ν = 1 ,
2, and m ν = 1 , . . . ,
8. Thephase of the j th eigenvector can then be uniquely de-fined by choosing a gauge in region H ( H ) such that[ T k ] m ,j ([ T k ] m ,j ) is always real and positive. The twogauge choices, which are related by a U(1) transforma-tion, are patched together to cover the entire BZ. Thisconstruction enables us to unambiguously calculate theChern number [43, 44] as C j = 12 π (cid:73) ∂H d k · (cid:16) A (1) j − A (2) j (cid:17) , (36)where ( A ( ν ) j ) µ is the gauge field [Eq. (35)] of band j in thepatch ν . Inspecting the eigenstructure of T k , we find asuitable partition to be H = { k : k y ≤ , | k x | + | k y | ≤ π } and H = BZ \ H . The resultant Berry curvatures forthe particle bands are illustrated in Fig. 4(f). The finalthermal Hall conductivity, which involves contributionsfrom all four bands, is plotted in Fig. 5.
2. Parameter dependence of κ xy In this subsection, we discuss the parameter depen-dence of the thermal Hall conductivity in Fig. 5 in detailand compare with asymptotic analytical considerations.First, note that while κ xy is always positive in the plotsof Fig. 5, its sign is actually determined by that of theparameters A µ and B µ of the ansatz; under the simulta-neous reversal of A µ → −A µ and B µ → −B µ , the Hallconductivity also changes sign as κ xy → − κ xy . This is re-quired by symmetry as the global sign reversal of A µ and B µ is equivalent [modulo gauge transformation G ( j ) = i]to performing a time-reversal transformation.Next, we turn to the temperature and field depen-dence. κ xy /T tends to zero at high temperatures, whereall bands are equally occupied, as well as very low tem-peratures, below the spinon gap, when all bands arenearly empty: intuitively, c ( n B ) is the same constantfor any band for both high and low T ; factoring it out,we are left with the sum of the Chern numbers of all theparticle bands and these add up to zero. To determinehow κ xy /T decays for low and high T , we use the asymp-totic expansions for the c function defined in Eq. (22): c ( x ) → π − x + 12 x + O (cid:18) x (cid:19) ; for x → ∞ , (cid:0) − ln( x ) + ln ( x ) (cid:1) x + O ( x ln( x )); for x → . (37)For simplicity, consider the contribution to κ xy for a sin-gle pair of particle bands that have equal Berry curva-tures (ergo, Chern numbers)—the existence of such a pairis guaranteed by the effective antiunitary symmetry inthe one-orbital model discussed above. Without loss ofgenerality, let these be labeled by n = 1 ,
2; the discus-sion here can be easily extended to include the n = 3 , ε k = ε k ≡ E k , and havethe same curvatures Ω k = Ω k . A finite uniform Zee-man field splits their energies to E k ± B z /
2. The Zeemanterm is proportional to the identity in the dynamical ma-trix K of Eq. (C5). Therefore, it leaves the spinon wavefunctions, which are determined by the dynamic ma-trix K rather than the Hamiltonian, unchanged. Hence,the Berry curvature remains unaffected, whereby we stillhave Ω k = Ω k .At temperatures much larger than the band maxi-mum, it is reasonable to approximate the Bose distribu-tion function by n B ( E ) ∼ k B T /E for k B T (cid:29) E . Using Eq. (21), the thermal Hall conductivity then follows as κ xy T = − k B (cid:126) V (cid:88) k (cid:88) n =1 , (cid:26) c [ n B ( ε n k )] − π (cid:27) Ω n k , ≈ k B (cid:126) V (cid:88) k (cid:18) Ω k n B ( ε k ) + Ω k n B ( ε k ) (cid:19) + O (cid:18) n B ( ε n k ) (cid:19) = k B (cid:126) V (cid:88) k Ω k (cid:18) E k − B z / k B T + E k + B z / k B T (cid:19) = (cid:18) T (cid:19) k B (cid:126) V (cid:88) k Ω k E k ≈ k B ζ C π (cid:126) T , (38)where C is the Chern number of the n = 1 band, and ζ is a measure of the average band energy without themagnetic field. We stress that Eq. (38) is a consequenceof the effective antiunitary symmetry explicated in Ap-pendix C 1, and, in particular, of Eq. (C10), which en-sures the equality of the Berry curvatures for the twobands. Therefore, to first order, κ xy is independent of B z at high temperatures, in consistence with Fig. 5(b). Inparticular, there is an anomalous thermal Hall response,i.e., κ xy (cid:54) = 0 for B z = 0. This is expected based on thesymmetries of the ansatz that are identical to those ofthe orbital magnetic field.Going beyond leading order in the 1 /T expansion in-corporates a subleading term κ xy T = k B ζ C + π (cid:126) T (cid:18) − B z + 4 ζ k B T (cid:19) + O (cid:18) T (cid:19) . (39)This term is of the opposite sign but it is parametri-cally small, and being of O ( B z /T ), negligible at high T .Hence, the decrease of κ xy with B z is hardly observable inFig. 5(b). Note, however, that in reality, the parametersof the ansatz itself might be magnetic field dependent;this is not accounted for in the present calculation, andmight yield a rather different dependence of κ xy on themagnetic field.Equation (39) also specifies that κ xy /T goes to zero as1 /T at large temperatures (with 1 /T corrections), whichis indeed confirmed by Fig. 5(c) for T (cid:38) . T much smaller than the spinongap ∆, the bosonic band occupancies are almost zero,and we can approximate n B ( E ) ≈ e − E/k B T for all bands.For the leading contribution, we need only consider thedominant term in the small- x expansion of c ( x ) fromEq. (37), which goes as x ln ( x ). The net result in the T (cid:28) ∆ limit is κ xy T = − k B (cid:126) V (cid:88) k (cid:88) n =1 , c [ n B ( ε n k )] Ω n k ≈ − k B (cid:126) V (cid:88) k (cid:16) ε k e − ε k /k B T + ε k e ε k /k B T (cid:17) Ω k ( k B T ) Printed by Wolfram Mathematica Student Edition T = 0 . T = 0 . T = 0 . T = 0 . (a) Printed by Wolfram Mathematica Student Edition T = 3 . T = 2 . T = 2 . T = 1 . T = 1 . (b) (c) (d) FIG. 5. Thermal Hall conductivity in the one-orbital model with the parameters A = 0 . B = 0 . B = 0 .
25, and λ = 2,as a function of Zeeman field at (a) low, and (b) high temperatures. In the second case, there is almost no dependence on B z . We emphasize that we only show the dependence of κ xy /T on the Zeeman field at constant orbital coupling. The latterenters indirectly through the parameters, A , , B , , of the ansatz. (c) The variation of κ xy /T with temperature at a constant B z = 0 .
25 for which the spinon gap (inset) is ∆ = 0 . A can be used to tune the strength of the response. κ xy /T decays as (∆ /T ) exp( − ∆ /T ) and 1 /T (with 1 /T corrections) at low and high temperatures, respectively. (d) The sameas in (c) but with the gap now varied as ∆( T ) = T exp( − m/T ); m = 0 . π , so that it is exponentially small with temperature.As in the figures above, all energies are measured in units of J A . ≈ C π (cid:126) T e − ∆ /k B T (cid:16) ∆ + e − B z /k B T (∆ + B z ) (cid:17) . (40)In moderate magnetic fields B z > T , κ xy /T decays expo-nentially as (∆ /T ) exp( − ∆ /k B T ) at low temperatures,in agreement with the regime of T (cid:46) . κ xy on the external magnetic field. Recognizing that thespinon gap ∆ at a finite field B z is related to the zero-fieldgap ∆ as ∆ = ∆ − B z /
2, we find that κ xy T ≈ C π (cid:126) T e − ∆ /k B T (cid:32) (cid:88) n = ± e − nB z / k B T (∆ + nB z / (cid:33) ≈ C π (cid:126) T cosh (cid:18) B z k B T (cid:19) e − ∆ /k B T (41)for small B z (cid:28) ∆ , thereby justifying the nonlinear be- havior observed in Fig. 5(a).Another interesting limit is the intermediate temper-ature range when ∆ < max ε , k (cid:46) T (cid:46) min ε , k . Fromthe aforementioned calculations, we notice that the ther-mal Hall conductivity is the largest in this two-band pic-ture when the magnetic field splits the particle and holebands—both of which have nonzero Chern numbers—such that the temperature T is greater than the lower-band maximum, but smaller than the upper-band mini-mum.With our formalism, we can also study phases withmagnetic order at T = 0, but with restored SRI dueto thermal fluctuations at nonzero temperature. To thisend, we vary the gap such that it is exponentially smallwith temperature; in practice, this is achieved by tun-ing the Lagrange multiplier λ . Instead of performing aself-consistent calculation, we assume a functional form∆( T ) = T exp( − m/T ), m = 2 πρ s (with spin stiffness1 ρ s ), in analogy with the two-dimensional (2D) antifer-romagnetic Heisenberg model [45–47]. The variation of κ xy /T with this choice of ∆( T ) is conveyed by Fig. 5(d).Despite always being in the regime ∆ < T , κ xy /T doesnot diverge as T →
0, but instead tends to zero. Tounderstand this, we focus on the contribution from thelowest band and momenta close to the dispersion min-ima ± k . Near ± k , the momentum dependence of theenergy is quadratic, while that of the Berry curvature isempirically observed to be quartic. Accordingly, assum-ing ∆ = 0, κ xy T ≈ − k B (cid:126) V (cid:88) k Ω k n B ( ε k ) (42) ≈ − k B (cid:126) (cid:90) | k − k | < Λ d k (2 π ) Ω ( k − k ) c (cid:18) ( k − k ) / m ∗ T − (cid:19) − k B (cid:126) (cid:90) | k + k | < Λ d k (2 π ) Ω ( k + k ) c (cid:18) ( k + k ) / m ∗ T − (cid:19) . As T →
0, we may rescale k ± k = y √ m ∗ T and extendthe upper limit of y integration to infinity, to obtain κ xy T = − k B (2 m ∗ T ) Ω (cid:126) (cid:90) ∞ y dy π c (cid:18) e y − (cid:19) = − k B (2 m ∗ T ) Ω (cid:126) (5 . . . . ) (43)So we find that κ xy /T ∼ T as T → (cid:28) T . IV. ANTIFERROMAGNET WITHDZYALOSHINSKII-MORIYA INTERACTIONS
So far, our discussion has been confined exclusively tospin-rotation-invariant spin liquids. In this section, wewill extend the analysis to include spin-orbit coupling,i.e. spin-rotations are not independent symmetry oper-ations any more, but are coupled with real-space sym-metry transformations. In terms of the underlying spinmodel, this corresponds to including DM interactions[48–50] as described by the term proportional to D m ij inEq. (1), H (3) = (cid:88) (cid:104) i,j (cid:105) D m ij · ( S i × S j ) . (44)The thermal transport properties of a spin Hamilto-nian with DM coupling were studied on the kagom´e lat-tice in Ref. 51 for the magnetically ordered phase usingboth Holstein-Primakoff bosons and Schwinger bosons;in particular, the latter approach featured a large ther-mal Hall coefficient at B z , T ∼ J . On the square lat-tice, however, it is strongly constrained by no-go theo-rems [26, 27]. In a recent spin-wave analysis, Ref. 52demonstrated that a thermal Hall effect can be realized FIG. 6. Illustration of the DM coupling vectors [53, 54]for (a) orthorhombic La CuO and (b) YBCO, where theblack dots represent the Cu atoms of the CuO planes and D = ( d , d , T , D = ( − d , − d , T , D = ( d , , T , D = (0 , d , T with real constants d j (not determined bysymmetry). Given that D m ij = − D m ji , the DM coupling vec-tor D m ij corresponds to a directed bond, which is indicatedby the arrows in the figure. The different DM textures aredue to the different symmetries: in YBCO, the Cu atomsare not centers of inversion, which allows a spatially constantDM coupling vector; in La CuO , it must alternate in signsince the Cu atoms are inversion centers, which is permittedbecause of the broken translational symmetry. in an inversion-symmetry-broken square-lattice antifer-romagnet with DM couplings. Here, we move away fromthe magnon description, which necessarily requires long-range magnetic order, and probe the influence of the DMinteractions relevant to the cuprate superconductors in aspin-liquid phase using Schwinger bosons. We will showthat some of these DM vectors can lead to a nonzeroBerry curvature, Ω n k (cid:54) = 0, and, in turn, a nonzero ther-mal Hall coefficient, albeit with much smaller magnitudethan in the ansatz of Sec. III B. This is related to thefact that the Chern number vanishes for each band inthe models with DM interactions that we study here.We will focus here on the Zeeman coupling of themagnetic field and neglect orbital effects. In this case,only a certain class of DM coupling vectors can lead to κ xy (cid:54) = 0 due to symmetry constraints. For instance, con-sider global spin rotations by angle | ϕ | along axis ϕ / | ϕ | .Under these transformations, it holds that J ij → J ij , B Z → R ϕ B Z , and D m ij → R ϕ D m ij , where R ϕ is thevector representation of the spin rotation. As for anyspin-rotation-invariant observable, the thermal Hall con-ductivity κ xy satisfies κ xy [ J ij , D m ij , B Z ] = κ xy [ J ij , R ϕ D m ij , R ϕ B Z ] . (45)Being odd under time-reversal, it further obeys κ xy [ J ij , D m ij , B Z ] = − κ xy [ J ij , D m ij , − B Z ] . (46)Consequently, if the DM coupling vectors are collinear,i.e. D m ij ∝ ˆ d , and ˆ d · B Z = 0, the combination of Eqs. (45)and (46), with ϕ = π ˆ d , implies κ xy = 0. To wit, thisis the case for D m ij = D ˆ x , or for the potentially morerelevant (spatially alternating) DM coupling vector of the2tetragonal phase of La CuO [54].It is also easily seen that κ xy vanishes for the DMcoupling vector in the orthorhombic phase of La CuO [Fig. 6(a)]: the spatial reflection symmetry R y with ac-tion ( x, y ) → ( − x, y ), not combined with any rotation inspin space, remains a symmetry of the system also in thepresence of Zeeman field along ˆ z . Being odd under R y , κ xy has to vanish.This is different for the DM coupling vector expected toarise in the tetragonal phase of YBa Cu O x (YBCO)[54], shown in Fig. 6(b), which analytically correspondsto D m ij = D (cid:107) ˆ d ij , ˆ d ij = d ij (cos θ ij ˆ x + sin θ ij ˆ y ) , (47)where ˆ d ij is a unit vector, d ij = − d ji = ± j = i ± ˆ e µ ( µ = x, y ), and θ ij = 0 ( π/
2) on all x ( y ) bonds.Note that this form of D m ij respects the translationaland fourfold-rotational ( C ) symmetries of the underly-ing square lattice (when accompanied by an appropriaterotation in spin space). It is not collinear and does breaktime-reversal symmetry [the argument in Eqs. (45)–(46)does not apply]; furthermore, it also breaks all in-planereflection symmetries in combination with a Zeeman fieldand will indeed give rise to a nonzero thermal Hall re-sponse as we will show next.To proceed with the Schwinger boson description ofthe DM interactions, we define the additional operatorsˆ C † i,j = 12 (cid:88) µ ν b † iµ (cid:16) i ˆ d ij · σ (cid:17) µν b jν = i2 d ij e − i σ θ ij b † iσ b j − σ , ˆ D i,j = 12 (cid:88) µ ν b iµ (cid:16) σ ˆ d ij · σ (cid:17) µν b jν = − i2 σ d ij e i σ θ ij b iσ b jσ , whereupon the DM term can be decomposed as [55]ˆ d ij · ( S i × S j ) = 12 (cid:18) : ˆ B † i,j ˆ C i,j + ˆ C † i,j ˆ B i,j :+ ˆ A † i,j ˆ D i,j + ˆ D † i,j ˆ A i,j (cid:19) . (48)Assuming only SU(2) spin-rotation-invariant operatorsacquire nontrivial expectation values in the mean-fielddecoupling (e.g., ˆ B † i,j ˆ C i,j → (cid:104) ˆ B † i,j (cid:105) ˆ C i,j + const.), theSBMFT analysis is carried out in Appendix D to obtainthe dispersion of the bosonic bands for a zero-flux ansatzappropriate to a conventional N´eel state [14], A i,i + µ = A , B i,i + µ = B ∀ i , µ = ˆ x, ˆ y , taking the DM coupling vectordefined in Eq. (47) and Fig. 6(b). Note that consideringonly SU(2)-invariant operators does not mean that theresulting mean-field Hamiltonian preserves SRI since theDM term in Eq. (48) couples the operators, ˆ A i,j and ˆ B i,j ,that are spin rotation invariant, to ˆ C i,j and ˆ D i,j , whichare not.Unlike previously, there is no effective antiunitary sym-metry and therefore, the bands are nondegenerate evenin zero fields. Nonetheless, in the absence of a magnetic - - - (a) - - - (b)(c) (d)(e) (f) FIG. 7. (a–b): Dispersion of the Schwinger boson bandsfor the mean-field approximation to H spin (1), with J x A = J y A = 1, B = 0 . D (cid:107) = 0 .
10, in (a) zero and (b)large ( B z = 2) magnetic fields. Shown are the eigenval-ues of the dynamic matrix—the bosonic bands have ener-gies given by the absolute values of the same, which arealways positive. In a finite magnetic field, the individualparticle and hole bands become progressively well separated.Exactly as in Fig. 2, the lines refer to different values of k y = − π, − π + π/ , . . . , π . (c–d): Same as above but nowplotted in the k x – k y plane for the (c) n = 1 (blue; particle)and (d) n = 3 (yellow; hole) bands, at B z = 0—the two bandsare nonidentical. At each point in k -space, min ( ε k , ε k ) cor-responds to the lowest energy eigenmode and the band min-ima are at { ( π/ , π/ , ( − π/ , − π/ } . Condensation of theseSchwinger bosons generally leads to long-range antiferromag-netic order. (e–f) Berry curvatures of the particle bands withthe same parameters as before, and a magnetic field B z = 0 . field, the two particle (and hole) bands intersect at afinite number of points as can be seen in Fig. 7(a), sothe Berry curvatures are well-defined only for B z (cid:54) = 0.These are plotted for the Schwinger-boson particle bandsin Figs. 7(e) and 7(f); the curvatures of the hole bands arerelated by Eq. (27). Despite a nonvanishing Berry cur-vature, each bands is actually topologically trivial with3 Printed by Wolfram Mathematica Student Edition
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FIG. 8. The thermal Hall conductivity in an antiferromag-netic Heisenberg spin model with Dzyaloshinskii-Moriya in-teractions, as a function of magnetic field for different con-stant temperatures (top) and as a function of temperatureat a constant magnetic field B z = 0 . κ xy has to vanishexactly at zero field (no anomalous contributions) as dictatedby symmetry. The couplings considered are J x A = J y A = 1(solid lines in both plots) and J x A = 1 . , J y A = 0 .
95 (reddots), with all other parameters the same as in Fig. 7. When J x (cid:54) = J y , C rotational symmetry is broken. The Schwinger-boson bands do not acquire nontrivial Chern numbers in themodel considered, and κ xy is thus much smaller than for thespin-liquid ans¨atze in Sec. III. zero Chern number.The ensuing thermal Hall conductivities, which can becalculated directly using the formalism of Sec. II C, arefound to be more than two orders of magnitude smallerthan for the earlier spin-liquid ans¨atze that result innonzero Chern numbers. Although the Hall coefficientsare nonzero, as displayed in Fig. 8, this is a purely ther-mal effect in the sense that the main contribution to κ xy comes from asymmetric weighting of the Berry cur-vature by the thermal distribution function n B ( ε n k ) inEq. (21) because the integral of Ω n k over the Brillouinzone alone is identically zero. We also remark that thereis no anomalous contribution as time-reversal symme-try is preserved at zero Zeeman field, guaranteeing that κ xy = 0.Since the CuO square plaquettes in YBCO are slightly distorted and form a rectangular lattice [56, 57], we havealso studied the impact of anisotropic Heisenberg ex-changes J x and J y along the ˆ x and ˆ y directions, respec-tively; this breaks the C rotation symmetry down to C . As demonstrated by Fig. 8, even a moderately largeanisotropy has no significant impact on κ xy . V. CONCLUSION
Our primary collection of results concerns the ther-mal Hall effect of spin liquids on the square lattice us-ing SBMFT in the absence of spin-orbit coupling. Wehave discussed different spin-rotation and translation-invariant ans¨atze that break time-reversal and certainpoint group symmetries; these phases exhibit nonzeroscalar spin chiralities. Among the ans¨atze considered,only one, with magnetic point group m m (cid:48) m (cid:48) and definedin Fig. 3, yields spinon bands with nonzero Chern num-bers. As seen in Fig. 5, where the Zeeman field, B z ,and temperature, T , dependence of the resulting ther-mal Hall conductivity κ xy are shown, the nonzero Chernnumbers lead to a sizable κ xy , of order one in units of k B / (cid:126) . We derived asymptotic expressions for the depen-dence of κ xy on T and B z , and established that κ xy /T vanishes as ∼ exp( − ∆ /T ) at low T for a spin liquid witha nonzero energy gap ∆ .Our formalism also enables us to consider states inwhich spin-rotation symmetry is broken and there is mag-netic order as T →
0. Any broken spin rotation sym-metry is restored at infinitesimal temperatures in twospatial dimensions, and within SBMFT, this can be cap-tured by a spin liquid with a gap, ∆, which vanishes as∆ ∼ exp( − m/T ). In this case, we found that κ xy /T ac-quired similarly large values (Fig. 5), and vanished onlyas a power of T as T → m m (cid:48) m (cid:48) symmetry descendfrom the time-reversal-preserving π -flux SBMFT statesof Yang and Wang [14]. As such, they do not have a spe-cial connection to the N´eel state in the limit of a vanish-ing spin gap. However, our spin liquids do include casesin which they condense to small distortions of the N´eelstate, although there is no natural selection mechanismfor such states, at least in mean-field theory. With such aselection mechanism, our results yield an attractive pro-posal to explain recent observations in the cuprates [2].The breaking of square-lattice and time-reversal sym-metries to m m (cid:48) m (cid:48) in our states could either be spon-taneous, or simply induced by the orbital coupling ofthe applied magnetic field (see Appendix A). Only forthe case when the symmetries are spontaneously broken,there is an anomalous contribution to the thermal Halleffect, i.e. κ xy (cid:54) = 0 even when B z = 0.Finally, we also discussed whether the DM interactionsrelevant to the cuprates can give rise to a thermal Halleffect within a SBMFT treatment of the spin model inEq. (1). We identify one DM coupling vector, defined4in Eq. (47) and in Fig. 6(b), which not only is expectedto be realized in YBCO [54] but also produces a nonzero κ xy . However, as evinced by Fig. 8, the thermal Hall con-ductivity is much weaker than that of the ansatz in Fig. 3with m m (cid:48) m (cid:48) symmetry, due to the absence of bands withnontrivial Chern numbers. Notes added: ( i ) In a recent paper with others [58], wehave discussed the thermal Hall response of antiferromag-nets using fermionic spinons. ( ii ) Han et al. [59] havedescribed the thermal Hall response of the cuprates us-ing a quantum spin Hall state, that could be favored byspin-orbit interactions. ACKNOWLEDGMENTS
We acknowledge many insightful discussions withG. Grissonnanche and L. Taillefer, and thank them forsharing their results before publication. We also thankA. Rosch and A. Vishwanath for useful discussions. Thisresearch was supported by the National Science Foun-dation under Grant No. DMR- 1664842. Research atPerimeter Institute is supported by the Government ofCanada through Industry Canada and by the Provinceof Ontario through the Ministry of Research and Innova-tion. SS also acknowledges support from Cenovus Energyat Perimeter Institute. MS acknowledges support fromthe German National Academy of Sciences Leopoldinathrough grant LPDS 2016-12. SC acknowledges supportfrom the ERC synergy grant UQUAM.
Appendix A: Coupling to an orbital magnetic field
Aside from the Zeeman coupling (9), which we focusedon in the main text, there is also an orbital coupling ofthe magnetic field. Being odd under time reversal andspin-rotation invariant, its leading contribution in a t/U expansion of the underlying Hubbard model involves thetriple product of neighboring spins and is of order t /U .Explicitly, it reads as [42] H χ = − Υ (cid:88) (cid:52) sin(Φ) S i · ( S j × S k ) , Υ = 24 t t (cid:48) U , (A1)where the sum involves the triangular plaquettes (cid:52) formed by nearest-neighbor (with hopping t ) and next-nearest-neighbor bonds (hopping t (cid:48) ), and Φ is the flux ofan applied magnetic field through a single triangular pla-quette. We see from Eq. (A1) that this orbital couplinginduces uniform scalar spin chiralities and, as mentionedearlier, breaks the symmetry of the system to m m (cid:48) m (cid:48) .In this appendix, we prove that the different terms inEq. (A1) cancel out exactly on the square lattice afterperforming a Schwinger-boson mean-field decoupling, aslong as there exists a gauge where the ansatz is explicitlytranslation invariant. This is, for instance, certainly the FIG. 9. Convention for the spin chirality term S i · ( S j × S k )in the Hamiltonian. For each triangular plaquette, the sites i , j , and k are the vertices of the corresponding dashed triangle,taken succesively in a clockwise fashion. The net interaction H χ involves the sum over all C rotated copies of such trian-gles. case for the conventional ansatz of the antiferromagneticstate (with only A i,i +ˆ x = A i,i +ˆ y = A ), but not for theone with m m (cid:48) m (cid:48) symmetry defined in Sec. III B. As weoutline below, H χ in Eq. (A1) will lead to a nonzerocontribution at the mean-field level when decoupled withthe parameters in Eqs. (31) and (33).As a means of decoupling H χ within SBMFT, we usethe identity [29],4 : ˆ B i,j ˆ B j,k ˆ B k,i : = 12 (ˆ n i S j · S k + ˆ n j S k · S i + ˆ n k S i · S j )+ ˆ n i ˆ n j ˆ n k − i S i · ( S j × S k ) , (A2)from which, it follows that S i · ( S j × S k ) = 2i (cid:16) ˆ B i,j ˆ B j,k ˆ B k,i − ˆ B † k,i ˆ B † j,k ˆ B † i,j (cid:17) . (A3)In a mean-field approximation,ˆ B i,j ˆ B j,k ˆ B k,i (cid:39) (cid:104) ˆ B i,j (cid:105) (cid:104) ˆ B j,k (cid:105) ˆ B k,i + (cid:104) ˆ B i,j (cid:105) ˆ B j,k (cid:104) ˆ B ki, (cid:105) (A4)+ ˆ B i,j (cid:104) ˆ B j,k (cid:105) (cid:104) ˆ B k,i (cid:105) − (cid:104) ˆ B i,j (cid:105) (cid:104) ˆ B j,k (cid:105) (cid:104) ˆ B k,i (cid:105) . Based off this simplification, we can now evaluate thequadratic terms for each individual bond. As an example,consider a bond linking sites i and i + ˆ x ; following thelabeling scheme of Fig. 9, let this be numbered 1–3. Theonly spin chirality terms in the Hamiltonian that involvethis bond are S · [( S × S ) + ( S × S ) + ( S × S ) + ( S × S )](A5)5 ≈ (cid:104) ˆ B , (cid:16) B ∗ + |B| (cid:17) + ˆ B † , (cid:0) B + |B| (cid:1)(cid:105) − H . c . + . . . , where we have isolated the terms proportional to ˆ B , orˆ B , , and those from all other bonds are grouped togetherin the ellipsis. However, the term enclosed in the bracketsis already Hermitian so the total contribution from the1–3 (and more generally, any horizontal or vertical) edgeis always zero. An analogous statement holds for anybond in the diagonal direction as well. In this regard,let us survey the 1–4 link, which connects sites i and i + ˆ x + ˆ y . The relevant spin interactions in which thisbond participates are S · ( S × S ), and S · ( S × S ),and collecting the quadratic terms for Eq. (A4), we finallyhave ˆ B , B ∗ + ˆ B † , B − H . c . = 0 . (A6)Since this cancellation occurs on any bond on the squarelattice, H χ in Eq. (A1) does not contribute to the Hamil-tonian to quadratic order and the orbital coupling tothe magnetic flux necessarily vanishes in the mean-fieldframework.If, instead, we use the parameters of the ansatz withsymmetry m m (cid:48) m (cid:48) in Eqs. (31) and (33), there is no can-cellation using SBMFT. In fact, as expected from a sym-metry point of view, the resultant mean-field contribu-tion of H χ can be absorbed by rescaling of the ansatzper se as B −→ B −
4Υ sin Φ B B , (A7) B −→ B −
2Υ sin Φ B . (A8)This conveys that the parameters B and B can alsobe induced or enlarged by the orbital coupling to theexternal magnetic field. Appendix B: Perturbations in the CP theory Quantum fluctuations about the conventional square-lattice N´eel state are conveniently described in theSchwinger boson theory using a continuum formulationbased on the CP model [60]. Here, we discuss, follow-ing Ref. 9, the additional perturbations that are intro-duced into this theory from the three-spin interaction inEq. (A1), which is induced by the orbital effect of theapplied magnetic field, and which breaks the symmetrydown to m m (cid:48) m (cid:48) .The CP model is expressed in terms of a bosonicspinor z σ which is coupled to a U(1) gauge field a µ ( µ = τ, x, y ) with Lagrangian L CP = 1 g | ( ∂ µ − ia µ ) z σ | (B1)Perturbations with symmetry of H χ are most conve-niently expressed in terms of the gauge field a µ . In arelativistic formulation, the leading perturbation is theterm [9, 61] (cid:15) µνλ f µν ∂ ρ f ρλ . But, more generally, withoutrelativistic invariance, there are two independent termswhich are expressed in terms of the internal electric andmagnetic fields derived from a µ (these are unrelated tothe applied external electromagnetic field): e i = ∂ τ a i − ∂ i a τ , b = ∂ x a y − ∂ y a x . (B2)Analysis of symmetries leads to the perturbation L χ = iλ ( e x ∂ τ e y − e y ∂ τ e x ) + iλ b ∂ i e i (B3)with couplings λ , which are expected to be proportionalto Υ sin(Φ) in Eq. (A1).In terms of the underlying spin-wave fluctuations, thegauge field a µ involves terms with one gradient, and so L χ has five spatiotemporal gradients [9]. As such, itseffects can be expected to be quite weak. Appendix C: Mean field Hamiltonian for the one-orbital model
The mean-field Hamiltonian for the one-orbital model presented in Sec. III is described by Eq. (29). We first expandout the different terms therein with the ansatz of Eqs. (31) and (33). Labeling the two kinds of sites for a fixed gaugechoice by α and β , this can be written as H mf = J (cid:88) ( u,v ) ∈ α, σ (cid:18) i B α † ( u,v ) σ β ( u,v )+ˆ x σ + i B α † ( u,v ) σ β ( u,v )+ˆ y σ + i B α † ( u,v ) σ α ( u,v )+ η σ − i B α † ( u,v ) σ α ( u,v )+ η σ − A σ α ( u,v ) σ β ( u,v )+ˆ x − σ − A σ α ( u,v ) σ β ( u,v )+ˆ y − σ − A σ α ( u,v ) σ α ( u,v )+ η − σ + A σ α ( u,v ) σ α ( u,v )+ η − σ (cid:19) + H . c . + J (cid:88) ( u,v ) ∈ β, σ (cid:18) i B β † ( u,v ) σ α ( u,v )+ˆ x σ − i B β † ( u,v ) σ α ( u,v )+ˆ y σ − i B β † ( u,v ) σ β ( u,v )+ η σ + i B β † ( u,v ) σ β ( u,v )+ η σ − A σ β ( u,v ) σ α ( u,v )+ˆ x − σ + A σ β ( u,v ) σ α ( u,v )+ˆ y − σ + A σ β ( u,v ) σ β ( u,v )+ η − σ − A σ β ( u,v ) σ β ( u,v )+ η − σ (cid:19) + H . c . + λ (cid:88) ( u,v ) , σ (cid:16) α † ( u,v ) σ α ( u,v ) σ + β † ( u,v ) σ β ( u,v ) σ − S (cid:17) , (C1)with ( u, v ) running exclusively over all α ( β ) sites in the first (second) summation above. Fourier transforming tomomentum space, with the convention b iσ = (cid:80) k b k σ exp(i k · r i ) / √ N , we find H mf = (cid:20) J (cid:88) k σ (cid:18) i B E + α † k σ β k σ + 2 B e i k x S y α † k σ α k σ − A σ ( E + ) ∗ α k σ β − k − σ − A σ e − i k x S y α k σ α − k − σ (cid:19) + J (cid:88) k σ (cid:18) i B E − β † k σ α k σ − B e i k x S y β † k σ α k σ − A σ ( E − ) ∗ β k σ α − k − σ + 2i A σ e − i k x S y β k σ β − k − σ (cid:19)(cid:21) + H . c . + λ (cid:88) k σ (cid:16) α † k σ α k σ + β † k σ β k σ − S (cid:17) , (C2)where we have adopted the shorthand C µ ≡ cos( k µ ), S µ ≡ sin( k µ ), and E ± ≡ exp(i k x ) ± exp(i k y ). In real space, thepositions of the α and β states within the same unit cell are spatially separated, so the second-quantized Hamiltonianis invariant under k → k + G µ only up to a gauge transformation [62]. The presence of an external magnetic field B z now appends the Zeeman term (9) to H mf . Equation (C2) is easily converted into the form H mf = (cid:80) k (Ψ † k H ( k ) Ψ k ) / † k = ( α † k ↑ β † k ↑ α † k ↓ β † k ↓ α − k ↑ β − k ↑ α − k ↓ β − k ↓ ). This can be diago-nalized in accordance with the process sketched in Sec. II B to calculate the Berry curvatures and conductivities.More compactly though, H mf can equivalently be expressed using the reduced four -component spinor ψ † =( α † k ↑ β † k ↑ α − k ↓ β − k ↓ ). Up to a constant, the bosonic mean-field Hamiltonian reads as H ( k ) = 12 − B + 4 B J C x S y + 2 λ B J ( C y + i S x ) 4i A J C x S y − A J ( C y + i S x ) − B J ( C y − i S x ) − B − B J C x S y + 2 λ A J ( C y − i S x ) − A J C x S y − A J C x S y A J ( C y + i S x ) B − B J C x S y + 2 λ B J ( − i C y + S x ) − A J ( C y − i S x ) 4i A J C x S y B J (i C y + S x ) B + 4 B J C x S y + 2 λ . (C3)Denoting the Pauli matrices acting in spin and sublattice space by σ and τ , respectively, H = λσ τ + Jσ ( A S x τ + A C y τ − A C x S y τ ) − B σ τ − Jσ ( B S x τ + B C y τ − B C x S y τ ) . (C4)This form of the kernel H contains the same information as the 8 × × H ( k ). In this language,the dynamic matrix K = ρ H = σ τ H is K = − B σ τ − J B S x σ τ − J B C y σ τ + 2 J B C x S y σ τ − i J A S x σ τ − i J A C y σ τ + 2i J A C x S y σ τ + λσ τ . (C5)Diagonalizing this dynamic matrix results in two particle bands, which we list as m = 1 ,
2, and two hole bands( m = 3 , ψ and the correspondence betweenthese bands and our previous indexing scheme is m = { , , , } ↔ n = { , , , } . For the remaining n bands,associated with n = { , , , } ≡ n (cid:48) , the energies and curvatures are simply related as ε n (cid:48) , k = ε ( n (cid:48) +4) mod 8 , k andΩ n (cid:48) , k = − Ω ( n (cid:48) +4) mod 8 , − k , but ( n (cid:48) + 4) mod 8 ∈ { , , , } , closing the loop between the four- and eight-componentformulations.7
1. Effective antiunitary symmetry
As mentioned in Sec. III B 1, the pairwise degeneracy of the particle bands in the one-orbital model (at zero Zeemanfields) is due to an effective symmetry of the Hamiltonian, which we single out here. To begin with, we identify ananti-unitary operator O = U C , where U is unitary and C is complex conjugation such that O K ( k ) O † = − K ( k ) = ⇒ U K ∗ ( k ) U † = − K ( k ) . (C6)This implies that if Φ m is an eigenvector of K with eigenvalue ω m , then so is U Φ ∗ m but with eigenvalue − ω m , whichis precisely the particle-hole symmetry that must be broken to lift the degeneracy of the bosonic bands. The onlysuch operator (unique up to an additional phase factor) is O = σ τ C , i.e. U = σ τ . Equation (C6) then states that σ U σ H ∗ ( k ) U † = −H ( k ) . (C7)As σ and U = σ τ anticommute, this yields an effective “time-reversal symmetry”, i.e. H ( k ) and the anti-unitaryoperator O commute, O H ( k ) O † = H ( k ) . (C8)Since O = +1, this does not translate to a Kramers degeneracy (in general, all eigenvalues of H are indeed nonde-generate) whereas Eq. (C6) does force the spectrum of K to be symmetric with respect to zero energy. It then followsthat the resulting degenerate bands have opposite Chern numbers. The wave functions are the eigenvectors of K and,by virtue of Eq. (C6), may be grouped according to the eigenvalues as T k = [ v ( k ) v ( k ) ( U v ∗ ( k )) ( U v ∗ ( k ))]. Moreconcisely, T k = U T ∗ k σ ; U = σ τ . (C9)The implication for the Berry curvature is thatΩ m k = i (cid:15) µν (cid:34) σ ∂ T † k ∂ k µ σ ∂ T k ∂ k ν (cid:35) mm = i (cid:15) µν (cid:20) σ σ ∂ T T k ∂ k µ U † σ U ∂ T ∗ k ∂ k ν σ (cid:21) mm = i (cid:15) µν (cid:20) σ ∂ T T k ∂ k µ σ ∂ T ∗ k ∂ k ν (cid:21) m m = − Ω ∗ m k = − Ω m k , (C10)where m = 3 ( m = 4) for m = 1 ( m = 2), and we have used the fact that Ω m k is real in the last step. Translatingback to the band index n , this proves that the pairs n = (1 ,
2) and (3 ,
4) are indeed degenerate and also have thesame curvatures modulo k → − k . The degeneracy is split at any temperature by a uniform Zeeman field B Z , whichcreates a constant gap between the two bands at each momentum.
2. Magnetic order
Within the Schwinger boson framework, magnetic order is obtained via the condensation of bosons, which occurswhen the bosonic modes have at least one zero eigenvalue [63, 64]. The minima of the spinon bands are found fromdiagonalizing K = σ τ H ( k ), with H ( k ) as in Eq. (C3), and lie at ± k , where k = ( π/ , B z = 0), the eigenvalues, each doubly degenerate at these momenta, are ε ± = (cid:12)(cid:12)(cid:12)(cid:12)(cid:113) λ − A J ± √ B J (cid:12)(cid:12)(cid:12)(cid:12) . (C11)For B >
0, the spinon gap is set by ε − and closes when (cid:112) λ − A J = √ B J ; B appears neither in this equationnor in the eigenstates below. Eliminating B in favor of A , λ , and setting ξ ≡ λ/ ( √ A J ) for notational convenience,we find that the two zero energy eigenvectors at k = k = ( π/ ,
0) areΨ = (cid:16) e i π/ ξ, i (cid:112) ξ − , , (cid:17) T , Ψ = (cid:16) i (cid:112) ξ − , − e − i π/ ξ, , (cid:17) T , (C12)8where the superscript T denotes transpose. Likewise, at k = − k = ( − π/ , = (cid:16) e − i π/ ξ, i (cid:112) ξ − , , (cid:17) T , Ψ = (cid:16) i (cid:112) ξ − , − e i π/ ξ, , (cid:17) T . (C13)The condensate in real space is a linear combination of those at ± k . Introducing arbitrary complex numbers z i torepresent the strength thereof, we have (cid:104) α r ↑ (cid:105)(cid:104) β r ↑ (cid:105)(cid:104) α † r ↓ (cid:105)(cid:104) β † r ↓ (cid:105) = ( z Ψ + z Ψ )e i k · r + ( z Ψ + z Ψ )e − i k · r , (C14)whereafter the condensate on each sublattice can be written as X α = (cid:18) (cid:104) α r ↑ (cid:105)(cid:104) α r ↓ (cid:105) (cid:19) = (cid:18) e i π/ z ξ + i z (cid:112) ξ − − i π/ z ξ + i z (cid:112) ξ − z ∗ z ∗ (cid:19) (cid:18) e i k · r e − i k · r (cid:19) , (C15) X β = (cid:18) (cid:104) β r ↑ (cid:105)(cid:104) β r ↓ (cid:105) (cid:19) = (cid:18) i z (cid:112) ξ − − e − i π/ z ξ i z (cid:112) ξ − − e i π/ z ξz ∗ z ∗ (cid:19) (cid:18) e i k · r e − i k · r (cid:19) . The spinor (e i k · r , e − i k · r ) T is proportional to (1 , T for even x , whereas for odd x coordinate, it is ∝ (1 , − T [theoverall U(1) phase is redundant for calculating physical spin expectation values]. This calls for further classificationof the sites on the α and β sublattices—defined by ( − j x + j y = 1 and −
1, respectively, according as whether x iseven ( e ) or odd ( o ), creating a four-sublattice structure for the magnetic order. The expectation value of the spin ateach site can then be evaluated as (cid:104) S µa ( r ) (cid:105) = X † µa σ X µa for µ = { α, β } and a = { e, o } .At this point, we note that the spin-liquid state described by the ansatz (31) has a gauge-invariant flux φ of π (modulo 2 π ) through each elementary square plaquette [65] (see main text for definition). Similar π -flux stateson the square lattice were studied by Yang and Wang [14]; the latter states are all identical in the limit of only A (cid:54) = 0. The corresponding magnetically ordered state was found to be a subset of the classical ground state for the J /J = 1 / π -flux ansatz is to construct a local gauge transformation mapping the one-orbital model onto it.Recall that under such a transformation, one generically has b jσ → e i ϕ ( j ) b jσ , A i,j → e i( ϕ ( i )+ ϕ ( j )) A i,j B i,j → e i( ϕ ( i ) − ϕ ( j )) B i,j . (C16)The ansatz (31) is characterized by A i,i +ˆ x = A and A i,i +ˆ y = ( − i x + i y A , whereas that of Ref. 14 has A i,i +ˆ x =( − i y A and A i,i +ˆ y = −A . If the two are to be related by a gauge transformation, then the phase ϕ ( j ) must satisfy ϕ ( j x , j y ) + ϕ ( j x + 1 , j y ) = πj y , ϕ ( j x , j y ) + ϕ ( j x , j y + 1) = π ( j x + j y + 1) . (C17)Both these equations hold modulo 2 π and their solution is ϕ ( j x , j y ) = π (cid:0) − j x + 2 j y + 1 (cid:1) /
4. Applying this transfor-mation shifts the one-orbital dispersion minima, which are inherently gauge dependent: with the earlier gauge choice,the minima were positioned at ( ± π/ ,
0) but in the new gauge, they are at ± ( π/ , π/ A are zero. Proceeding beyond this special case, we can similarly transform theremaining ( A , B , B ) terms in Eq. (33) according to Eq. (C16), and the minimal ansatz which gives quantizedChern bands in this gauge reads: A i,i +ˆ x = ( − i y A , A i,i +ˆ y = −A , B i,i +ˆ x = − ( − i x B , B i,i +ˆ y = B ( − i x + i y , B i,i +ˆ x +ˆ y = B i,i +ˆ x − ˆ y = − i B ( − i y . (C18)As Fig. 10(a) corroborates, the minima for the lowest-energy spinon band remain at ± ( π/ , π/
2) even on turning on B and B [cf. Fig. 4(e)].We return to our original gauge choice where the computation of magnetic order is more tractable. Noting thatthe spin-liquid state for the one-orbital model reduces to that in Ref. 14 in the limit of only A (cid:54) = 0, we first set9 (a) (b) (c)(d) (e) FIG. 10. (a) Dispersion of the n = 1 spinon band corresponding to the Schwinger boson ansatz for the one-orbital model inEq. (C18) with J = 1, A = 1, B = 0 . B = 0 . λ = 2 .
0, and B z = 0. In this gauge, the minima are always at ± ( π/ , π/ ξ = 1 (implying B = 0) and only one of the z i (taken to be z here) nonzero. (c) This state can be perturbed by increasing ξ which equals 1 .
05 here, thus setting B ≈ . A . (d)The magnetically ordered state with the complex coefficients chosen to be { z , z , z , z } = { z, i z, , } , and (e) { z, − i z, , } .The magnetic moment is uniform at all sites for the states exhibited, and the vector plotted in each figure is, for clarity,( S x / , S y / , S z ). ξ = 1 (as dictated by the gap-closing condition with B = 0). Upon calculating the spin expectation values using theboson-condensation procedure, we find that the ordered moments on the four sites of a plaquette add to zero, i.e. (cid:88) µ = α,β (cid:88) a = e,o (cid:104) S µa (cid:105) = 0 , (C19)which is precisely the four-sublattice ordered state in Ref. 14. A particular instance thereof is the N´eel state which isobtained when the coefficients are chosen such that only one of the four z i is nonzero. For general ξ >
1, the spinors X µa are X αe = (cid:18) ξ (e i π/ z + e − i π/ z ) + i (cid:112) ξ − z + z ) z ∗ + z ∗ (cid:19) , X αo = (cid:18) ξ (e i π/ z − e − i π/ z ) + i (cid:112) ξ − z − z ) z ∗ − z ∗ (cid:19) , (C20) X βe = (cid:18) − ξ (e − i π/ z + e i π/ z ) + i (cid:112) ξ − z + z ) z ∗ + z ∗ (cid:19) , X βo = (cid:18) ξ ( − e − i π/ z + e i π/ z ) + i (cid:112) ξ − z − z ) z ∗ − z ∗ (cid:19) . Akin to the analysis above, we again compute the values of the ordered moment at each site but the analyticalexpressions in this case prove to be unwieldy. Specifically, S zµa takes the form S zαe = ξ (cid:0) | z − i z | + | z + z | (cid:1) − | z + z | + 2 ξ (cid:112) ξ − (cid:104) ( z ∗ + z ∗ )( z e i π/ + z e − i π/ ) (cid:105) , (C21)0 S zαo = ξ (cid:0) | z + i z | + | z − z | (cid:1) − | z − z | + 2 ξ (cid:112) ξ − (cid:104) ( z ∗ − z ∗ )( z e i π/ − z e − i π/ ) (cid:105) ,S zβe = ξ (cid:0) | z + i z | + | z + z | (cid:1) − | z + z | + 2 ξ (cid:112) ξ − (cid:104) ( z + z )( z ∗ e i π/ + z ∗ e − i π/ ) (cid:105) ,S zβo = ξ (cid:0) | z − i z | + | z − z | (cid:1) − | z − z | + 2 ξ (cid:112) ξ − (cid:104) ( z − z )( z ∗ e i π/ − z ∗ e − i π/ ) (cid:105) . As can be seen, for general complex values z i , there is no simple relation between the z components. Further, (cid:88) µ = α,β (cid:88) a = e,o (cid:104) S zµ,a (cid:105) = 4( ξ − (cid:88) i =1 | z i | + 4 ξ (cid:112) ξ − (cid:104) z z ∗ e i π/ + z z ∗ e − i π/ (cid:105) (C22)vanishes only for ξ = 1. Therefore, the sum of ordered moments on the four sites of a plaquette is nonzero, and thespin order parameter can be parametrized as (cid:104) S ( j ) (cid:105) = n (0 , + ( − j x n ( π, + ( − j y n (0 ,π ) + ( − j x + j y n ( π,π ) , (C23)where we have defined n (0 , = 14 (cid:18) (cid:104) S αe (cid:105) + (cid:104) S αo (cid:105) + (cid:104) S βe (cid:105) + (cid:104) S βo (cid:105) (cid:19) , n ( π, = 14 (cid:18) (cid:104) S αe (cid:105) − (cid:104) S αo (cid:105) + (cid:104) S βe (cid:105) − (cid:104) S βo (cid:105) (cid:19) , n (0 ,π ) = 14 (cid:18) (cid:104) S αe (cid:105) − (cid:104) S αo (cid:105) − (cid:104) S βe (cid:105) + (cid:104) S βo (cid:105) (cid:19) , n ( π,π ) = 14 (cid:18) (cid:104) S αe (cid:105) + (cid:104) S αo (cid:105) − (cid:104) S βe (cid:105) − (cid:104) S βo (cid:105) (cid:19) . (C24)It is noteworthy that n (0 , = 0 exactly corresponds to the solution of Ref. 14 with zero average moment on aplaquette. The most general ordered state breaks C and lattice translation ( T x and T y ) symmetries but preservesthe reflections R x and R y ; of course, it also breaks time reversal and SRI. While the moments on the four sites ofeach plaquette are generically distinct, previously studied states on the square lattice, like the N´eel, the canted N´eel,or the tetrahedral umbrella state [66] are not necessarily ruled out. If the structure of the condensate is such that n ( π,π ) is large in magnitude compared to n (0 , , n ( π, , and n (0 ,π ) , the magnetically ordered state can be thought ofas a perturbation to the N´eel state, an example of which is sketched in Fig. 10(c) for z i = 0 ∀ i (cid:54) = 1. The magnitudeof the ordered moments is uniform at all lattice sites, i.e. X † µa X µa = constant, if we choose such a solution for the z i . One can also impose this requirement of uniformity when more than one coefficient is nonzero. Endowed withthis constraint, there are four solutions, which are { z , z , z , z } = { z, ± i z, , } or { , , z, ± i z } . The two associatedsymmetry-inequivalent ordered states are shown in Figs. 10(d) and 10(e). Appendix D: SBMFT with Dzyaloshinskii-Moriyainteractions
In this appendix, we continue along the lines ofSec. IV to develop the mean-field Hamiltonian for thenearest-neighbor Heisenberg antiferromagnet with ad-ditional Dzyaloshinskii-Moriya couplings. Pursuant toEq. (48), the mean-field approximation for the in-planeDM term is H (3) mf = D (cid:107) (cid:88) (cid:104) i,j (cid:105) (cid:16) B ∗ i,j ˆ C i,j + ˆ C † i,j B i,j + A ∗ i,j ˆ D i,j + ˆ D † i,j A i,j (cid:17) = D (cid:107) (cid:88) (cid:104) i,j (cid:105) − i2 d ij e i σθ ij (cid:16) B ∗ i,j b † j − σ b iσ + σ A ∗ i,j b iσ b jσ (cid:17) + H . c .. (D1)The total mean-field Hamiltonian H mf is now a sum ofEqs. (8), (9), and (D1). All things considered, H spin bears the mean-field momentum-space representation: H (1) mf = (cid:88) k σ µ (cid:20) J µ e − i k µ (cid:18) B ∗ b † k σ b k σ − σ A ∗ b k σ b − k − σ (cid:19) + H . c . + λ b † k σ b k σ (cid:21) − N s λS + 2 N s J (cid:0) |A| − |B| (cid:1) ,H (2) mf = − B (cid:88) k σ σ b † k σ b k σ , (D2) H (3) mf = D m (cid:88) k σ (cid:104) − i ¯ E σ (cid:16) B ∗ b † k − σ b k σ + σ A ∗ b k σ b − k σ (cid:17) + H . c . (cid:105) . For the sake of notational brevity, we work with theshorthand E σ ≡ (e i k x + i σ e i k y ) and overhead bars con-note the same expressions but with the replacement k →− k ; thus, ( E + ) ∗ = ¯ E − . Upon expanding and explicitlysumming over σ = ↑ , ↓ in Eq. (D2), the full Hamiltonian isexpressible, as before, as H mf = (cid:80) k (Ψ † k H ( k ) Ψ k ) / † k ≡ (cid:16) b † k ↑ b † k ↓ b − k ↑ b − k ↓ (cid:17) , and the kernel H ( k ) = (cid:0) B J µ e i k µ (cid:1) r + (cid:0) λ − B (cid:1) i4 D (cid:107) (cid:0) B E − − B ∗ ¯ E − (cid:1) i2 D (cid:107) A E − − J A E +i4 D (cid:107) (cid:0) B E + − B ∗ ¯ E + (cid:1) (cid:0) B J µ e i k µ (cid:1) r + (cid:0) λ + B (cid:1) J A E + − i2 D (cid:107) A E + − i2 D (cid:107) A ∗ ¯ E + J A ∗ E + (cid:0) B J µ e − i k µ (cid:1) r + (cid:0) λ − B (cid:1) i4 D (cid:107) (cid:0) B ¯ E + − B ∗ E + (cid:1) − J A ∗ E + i2 D (cid:107) A ∗ ¯ E − i4 D (cid:107) (cid:0) B ¯ E − − B ∗ E − (cid:1) (cid:0) B J µ e − i k µ (cid:1) r + (cid:0) λ + B (cid:1) , (D3)where the superscript r stands for the real part; H mf fur-ther includes another constant piece, which we ignore.Diagonalizing with the paraunitary matrix T k gives the full information of the dispersions for the volume-modebands and some representative energy dispersions areshown in Fig. 7. Appendix E: Three-orbital model
The three-orbital CuO model—with the broken time-reversal and reflection symmetries of pattern D—allows fornonzero loop currents unlike its one-orbital counterpart [9] studied in Sec. III, and offers the added advantage ofan explicitly translation-invariant ansatz. In this appendix, we illustrate that the three-orbital model also shows alarge thermal Hall conductivity in the presence of a magnetic field, analogous to the one-orbital model, with identicalbroken symmetries as in Sec. III B. A1, i B1A2, i B2 (a) ˆ e y /
00 2i J A S x J A S y B z + 2 λ J B S x J B S y J A S x J A C x C y J B S x B z + 2 λ − J B C x C y J A S y − J A C x C y J B S y J B C x C y B z + 2 λ . (E5) (a) (b) (c) (d) FIG. 12. Schwinger boson band structure for three of the six different particle bands with J = 1, A = 1, B = 0 . λ = 2 . B z = 0; the other bands are degenerate at zero field and are not shown. The remaining parameters are chosen as follows:(a) A = 0, B = 0; (b) A = 0 . B = 0; (c) A = 0 . B = 0 .
5. Only with B (cid:54) = 0 are the upper bands prevented fromtouching; all the bands then acquire well-defined Chern numbers. The bands that are the degenerate counterparts of the onesshown have the same Chern numbers. (d) The dispersion for the lowest-energy band exhibits minima at k = (0 , (a) (b) Printed by Wolfram Mathematica Student Edition (c)
Printed by Wolfram Mathematica Student Edition (d)
FIG. 13. (a) Berry curvatures of the particle bands with nonzero Chern numbers in the three-orbital model, with the parameters J = 1, A = 1, A = 0 . B = B = 0 . B z = 0, and λ = 2 .
5. (b) The thermal Hall conductivity as a function of temperatureat fixed B z = 0 . A , indicating that the strength of the thermal Hall signal does vary with A even though the Chern numbers do not. (c–d) The magnetic field dependence of the conductivity for different temperatureswith A = 0 . In a like manner, from the paraunitary matrix T k , one can once again calculate the Berry curvature for these bands[Fig. 13(a)] using the partition H = { k : k y < } and H = { k : k y ≥ } . The caveat is that the expression for thethermal Hall conductivity in Eq. (21) is formulated exclusively in terms of particle bands whereas our choice of thesix-component spinor in Eq. (E4) eliminates the trivial particle-hole duplication, leaving us with three particle andthree hole bands. Exploiting the relation (27) between the curvatures of the particle and hole bands, Eq. (21) can bebrought to the more implementable form κ xy = − k B T (cid:126) V (cid:88) k (cid:88) n ∈ particle (cid:26) c [ n B ( ε n k )] − π (cid:27) Ω n k − (cid:88) n ∈ hole (cid:26) c [ n B ( ε n − k )] − π (cid:27) Ω n − k . (E6)Summing over all six bands, the net conductivity in Fig. 13 is observed to be three orders of magnitude greater than inthe model with Dzyaloshinskii-Moriya interactions alone. The behaviors at both high and low temperatures resemblethat for the one-orbital model in Fig. 5 and is owed to origins similar to the discussion in Sec. III B 2. Furthermore,we again find an anomalous contribution. [1] J. Ziman, Electrons and Phonons: The Theory of Trans-port Phenomena in Solids , Oxford Classic Texts in the Physical Sciences (Oxford University Press, New York, ,014001 (2018).[4] M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero,A. Georges, and S. Sachdev, “Topological order in thepseudogap metal,” Proc. Natl. Acad. Sci. U.S.A. ,E3665 (2018).[5] S. Chatterjee, S. Sachdev, and A. Eberlein, “Thermaland electrical transport in metals and superconductorsacross antiferromagnetic and topological quantum tran-sitions,” Phys. Rev. B , 075103 (2017).[6] S. Chatterjee, S. Sachdev, and M. Scheurer, “Intertwin-ing topological order and broken symmetry in a theoryof fluctuating spin density waves,” Phys. Rev. Lett. ,227002 (2017).[7] S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, “Fluc-tuating spin density waves in metals,” Phys. Rev. B ,155129 (2009).[8] S. Chatterjee and S. Sachdev, “Insulators and metalswith topological order and discrete symmetry breaking,”Phys. Rev. B , 205133 (2017).[9] M. S. Scheurer and S. Sachdev, “Orbital currents in in-sulating and doped antiferromagnets,” Phys. Rev. B ,235126 (2018).[10] W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev,A. Georges, and M. Ferrero, “Pseudogap and fermi-surface topology in the two-dimensional hubbard model,”Phys. Rev. X , 021048 (2018).[11] A. Auerbach, Interacting Electrons and Quantum Mag-netism , Graduate Texts in Contemporary Physics(Springer-Verlag, New York, 1994).[12] A. Auerbach and D. P. Arovas, “Schwinger bosons ap-proaches to quantum antiferromagnetism,” in
Introduc-tion to Frustrated Magnetism: Materials, Experiments,Theory , edited by C. Lacroix, P. Mendels, and F. Mila(Springer Berlin Heidelberg, Berlin, Heidelberg, 2011)pp. 365–377.[13] X. G. Wen, “Mean-field theory of spin-liquid states withfinite energy gap and topological orders,” Phys. Rev. B , 2664 (1991).[14] X. Yang and F. Wang, “Schwinger boson spin-liquidstates on square lattice,” Phys. Rev. B , 035160 (2016).[15] S. A. Owerre, “Topological thermal Hall effect in frus-trated kagome antiferromagnets,” Phys. Rev. B ,014422 (2017).[16] S. A. Owerre, “Topological magnetic excitations on thedistorted kagom´e antiferromagnets: Applications to vol-borthite, vesignieite, and edwardsite,” EPL , 37006(2017).[17] A. Mook, J. Henk, and I. Mertig, “Spin dynamics simula-tions of topological magnon insulators: From transversecurrent correlation functions to the family of magnonHall effects,” Phys. Rev. B , 174444 (2016).[18] R. Seshadri and D. Sen, “Topological magnonsin a kagome-lattice spin system with XXZ andDzyaloshinskii-Moriya interactions,” Phys. Rev. B , 134411 (2018).[19] F.-Y. Li, Y.-D. Li, Y. B. Kim, L. Balents, Y. Yu, andG. Chen, “Weyl magnons in breathing pyrochlore anti-ferromagnets,” Nat. Commun. , 12691 (2016).[20] S. A. Owerre, “Magnon Hall effect in AB-stacked bilayerhoneycomb quantum magnets,” Phys. Rev. B , 094405(2016).[21] S. A. Owerre, “Topological honeycomb magnon Halleffect: A calculation of thermal Hall conductivity ofmagnetic spin excitations,” J. Appl. Phys. , 043903(2016).[22] S. A. Owerre, “Topological magnon bands and unconven-tional thermal hall effect on the frustrated honeycomband bilayer triangular lattice,” J. Phys.: Condens. Mat-ter , 385801 (2017).[23] Y. Lu, X. Guo, V. Koval, and C. Jia, “Topologicalthermal hall effect driven by fluctuation of spin chiralityin frustrated antiferromagnets,” arXiv preprint (2018),arXiv:1811.07319 [cond-mat.str-el].[24] Y. Zhang, S. Okamoto, and D. Xiao, “Spin-Nernst ef-fect in the paramagnetic regime of an antiferromagneticinsulator,” Phys. Rev. B , 035424 (2018).[25] K. Nakata, J. Klinovaja, and D. Loss, “Magnonic quan-tum Hall effect and Wiedemann-Franz law,” Phys. Rev.B , 125429 (2017).[26] H. Katsura, N. Nagaosa, and P. A. Lee, “Theory of theThermal Hall Effect in Quantum Magnets,” Phys. Rev.Lett. , 066403 (2010).[27] T. Ideue, Y. Onose, H. Katsura, Y. Shiomi, S. Ishiwata,N. Nagaosa, and Y. Tokura, “Effect of lattice geometryon magnon Hall effect in ferromagnetic insulators,” Phys.Rev. B , 134411 (2012).[28] D.-V. Bauer and J. O. Fjærestad, “Schwinger-bosonmean-field study of the J − J Heisenberg quantum anti-ferromagnet on the triangular lattice,” Phys. Rev. B ,165141 (2017).[29] L. Messio, C. Lhuillier, and G. Misguich, “Time reversalsymmetry breaking chiral spin liquids: Projective sym-metry group approach of bosonic mean-field theories,”Phys. Rev. B , 125127 (2013).[30] N. N. Bogoliubov, “On the theory of superfluidity,” J.Phys , 23 (1947).[31] J. G. Valatin, “Comments on the theory of superconduc-tivity,” Nuovo Cim. , 843 (1958).[32] J. H. P. Colpa, “Diagonalization of the quadratic bosonhamiltonian,” Physica A , 327 (1978).[33] M.-w. Xiao, “Theory of transformation for the diago-nalization of quadratic Hamiltonians,” arXiv preprint(2009), arXiv:0908.0787 [math-ph].[34] R. Matsumoto, R. Shindou, and S. Murakami, “ThermalHall effect of magnons in magnets with dipolar interac-tion,” Phys. Rev. B , 054420 (2014).[35] R. Shindou, R. Matsumoto, S. Murakami, and J.-i. Ohe,“Topological chiral magnonic edge mode in a magnoniccrystal,” Phys. Rev. B , 174427 (2013).[36] D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phys. Rev. Lett. , 405(1982).[37] M. Kohmoto, “Topological invariant and the quantiza-tion of the Hall conductance,” Ann. Phys. , 343(1985).[38] S. Murakami and A. Okamoto, “Thermal hall effect ofmagnons,” J. Phys. Soc. Jpn. , 011010 (2016). [39] A. Thomson and S. Sachdev, “Fermionic spinon theoryof square lattice spin liquids near the N´eel state,” Phys.Rev. X , 011012 (2018).[40] Y. Yoshida, S. Schr¨oder, P. Ferriani, D. Serrate, A. Ku-betzka, K. von Bergmann, S. Heinze, and R. Wiesen-danger, “Conical Spin-Spiral State in an Ultrathin FilmDriven by Higher-Order Spin Interactions,” Phys. Rev.Lett. , 087205 (2012).[41] S. Sachdev and R. Jalabert, “Effective lattice modelsfor two-dimensional quantum antiferromagnets,” Mod.Phys. Lett. B , 1043 (1990).[42] D. Sen and R. Chitra, “Large- U limit of a Hubbard modelin a magnetic field: Chiral spin interactions and param-agnetism,” Phys. Rev. B , 1922 (1995).[43] T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern numbersin discretized Brillouin zone: efficient method of com-puting (spin) Hall conductances,” J. Phys. Soc. Jpn. ,1674 (2005).[44] T. Fukui and Y. Hatsugai, “Quantum spin Hall effect inthree dimensional materials: Lattice computation of Z topological invariants and its application to Bi and Sb,”J. Phys. Soc. Jpn. , 053702 (2007).[45] S. Chakravarty, B. I. Halperin, and D. R. Nelson, “Low-temperature behavior of two-dimensional quantum anti-ferromagnets,” Phys. Rev. Lett. , 1057 (1988).[46] S. Chakravarty, B. I. Halperin, and D. R. Nelson, “Two-dimensional quantum Heisenberg antiferromagnet at lowtemperatures,” Phys. Rev. B , 2344 (1989).[47] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, New York, 2011).[48] I. Dzyaloshinsky, “A thermodynamic theory of “weak”ferromagnetism of antiferromagnetics,” J. Phys. Chem.Solids , 241 (1958).[49] T. Moriya, “Anisotropic superexchange interaction andweak ferromagnetism,” Phys. Rev. , 91 (1960).[50] T. Moriya, “New mechanism of anisotropic superex-change interaction,” Phys. Rev. Lett. , 228 (1960).[51] H. Lee, J. H. Han, and P. A. Lee, “Thermal Hall effect ofspins in a paramagnet,” Phys. Rev. B , 125413 (2015).[52] M. Kawano and C. Hotta, “Thermal hall effect and topo-logical edge states in a square lattice antiferromagnet,”arXiv preprint (2018), arXiv:1805.05872 [cond-mat.mtrl-sci].[53] D. Coffey, K. S. Bedell, and S. A. Trugman, “Effectivespin Hamiltonian for the CuO planes in La CuO andmetamagnetism,” Phys. Rev. B , 6509 (1990).[54] D. Coffey, T. M. Rice, and F. C. Zhang, “Dzyaloshinskii-Moriya interaction in the cuprates,” Phys. Rev. B ,10112 (1991).[55] L. Manuel, C. J. Gazza, A. E. Trumper, and H. A. Cec-catto, “Heisenberg model with Dzyaloshinskii-Moriya in-teraction: A mean-field Schwinger-boson study,” Phys.Rev. B , 12946 (1996).[56] H. A. Mook, P. Dai, F. Dogan, and R. D. Hunt,“One-dimensional nature of the magnetic fluctuations inYBa Cu O . ,” Nature , 729 (2000).[57] V. Hinkov, S. Pailhes, P. Bourges, Y. Sidis, A. Ivanov,A. Kulakov, C. T. Lin, D. P. Chen, C. Bernhard, andB. Keimer, “Two-dimensional geometry of spin excita-tions in the high-transition-temperature superconductorYBa Cu O x ,” Nature , 650 (2004).[58] S. Chatterjee, H. Guo, S. Sachdev, R. Samajdar, M. S.Scheurer, N. Seiberg, and C. Xu, “Field-induced transi-tion to semion topological order from the square-lattice Nel state,” (2019), arXiv:1903.01992 [cond-mat.str-el].[59] J. H. Han, J.-H. Park, and P. A. Lee, “Considerationof Thermal Hall Effect in Undoped Cuprates,” arXiv e-prints (2019), arXiv:1903.01125 [cond-mat.str-el].[60] N. Read and S. Sachdev, “Valence-bond and spin-peierlsground states of low-dimensional quantum antiferromag-nets,” Phys. Rev. Lett. , 1694 (1989).[61] C. Wang, A. Nahum, M. A. Metlitski, C. Xu, andT. Senthil, “Deconfined quantum critical points: Sym-metries and dualities,” Phys. Rev. X , 031051 (2017).[62] D. Singh, M. S. Scheurer, A. Hillier, and R. P.Singh, “Time-reversal-symmetry breaking and unconven-tional pairing in the noncentrosymmetric superconduc-tor La Rh probed by µ SR,” arXiv preprint (2018),arXiv:1802.01533 [cond-mat.supr-con].[63] S. Sarker, C. Jayaprakash, H. R. Krishnamurthy, andM. Ma, “Bosonic mean-field theory of quantum Heisen-berg spin systems: Bose condensation and magnetic or-der,” Phys. Rev. B , 5028 (1989).[64] S. Sachdev, “Kagom´e-and triangular-lattice Heisenbergantiferromagnets: Ordering from quantum fluctuationsand quantum-disordered ground states with unconfinedbosonic spinons,” Phys. Rev. B , 12377 (1992).[65] O. Tchernyshyov, R. Moessner, and S. L. Sondhi, “Fluxexpulsion and greedy bosons: Frustrated magnets atlarge N ,” EPL , 278 (2005).[66] C. Hickey, L. Cincio, Z. Papi´c, and A. Paramekanti,“Emergence of chiral spin liquids via quantum melting ofnoncoplanar magnetic orders,” Phys. Rev. B96