Enhanced thermal Hall effect in the square-lattice Néel state

Common wisdom about conventional antiferromagnets is that their low-energy physics is governed by spin–wave excitations. However, recent experiments on several cuprate compounds have challenged this concept. An enhanced thermal Hall response in the pseudogap phase was identified, which persists even in the insulating parent compounds without doping. Here, to explain these surprising observations, we study the quantum phase transition of a square-lattice antiferromagnet from a confining Néel state to a state with coexisting Néel and semion topological order. The transition is driven by an applied magnetic field and involves no change in the symmetry of the state. The critical point is described by a strongly coupled conformal field theory with an emergent global SO(3) symmetry. The field theory has four different formulations in terms of SU(2) or U(1) gauge theories, which are all related by dualities; we relate all four theories to the lattice degrees of freedom. We show how proximity of the confining Néel state to the critical point can explain the enhanced thermal Hall effect seen in experiments. The remarkably large thermal Hall response recently observed in the copper oxides challenges our understanding of the excitations in an insulating antiferromagnet. Here, a possible explanation of the underlying physics is provided.

T he thermal Hall effect has attracted much attention in recent years as a powerful tool to gain information about the nature of excitations in exotic materials, as, for instance, in the spin-liquid candidate system α-RuCl 3 (ref. 1 ). Grissonnanche et al. 2 measured the thermal Hall effect in the normal state of four different copper-based superconductors. A strong signal is found from optimal doping, where the pseudogap phase ends, all the way to the insulating parent compounds. These observations are surprising, as the insulator is expected to be a conventional Néel state, and spinwave theory shows that this state has a much smaller thermal Hall response in an applied magnetic field than that observed 3 . There is no sign of a quantized thermal Hall response, so the insulator is not in a state with topological order and protected edge excitations.
Here, we study the possibility that the orbital coupling of the applied magnetic field can drive the conventional, confining Néel insulator to a state that has semion topological order 4 coexisting with Néel order N (see Fig. 1). We assume that the current experiments are at a field where the ground state is a conventional Néel state in which the only low-energy excitations are spin waves, and we describe how the proximity to the lower quantum phase boundary in Fig. 1 can enhance the thermal Hall response of such a conventional state. The applied field and N break spin-rotation, time-reversal and mirror-plane symmetries, and the states on both sides of the transition have an identical pattern of symmetries. So, the quantum phase transition involves only the onset of topological order. We obtain the universal critical field theory describing the vicinity of the lower phase boundary in Fig. 1

at low temperatures (T).
We find that the critical theory is one that has been carefully studied 5 in the context of the recent advances in dualities of non-Abelian conformal gauge theories in 2 + 1 spacetime dimensions [6][7][8] . The theory of interest has four different dual formulations in terms of relativistic field theories, and we relate all of them to theories of the lattice antiferromagnet; the assumption of universality at the quantum phase transition then provides a new route to obtaining the dualities.
We are interested in spin S = 1/2 antiferromagnets with spin operators S i on the sites i of the square lattice and Hamiltonian H = H 1 + H B . The first term has the form which describes near-neighbour exchange interactions and possible ring-exchange terms, all of which preserve the global SU(2) spinrotation, time-reversal and square-lattice symmetries; here, i and j run over all lattice sites. The second term, induced by the applied magnetic field, is where J χ is the coupling to the scalar spin chirality and the first summation is over all elementary triangular plaquettes, denoted by Δ. This term is induced by the orbital coupling of the applied magnetic field to the underlying electrons 9 . It preserves lattice translations and rotations, but explicitly breaks time-reversal and mirror-plane symmetries while preserving their product. The value of J χ itself is proportional to the small magnetic flux penetrating the square lattice. The B Z term in equation (2) is the Zeeman term, and the electron magnetic moment has been absorbed in the definition of B Z . We do not include spin-orbit interactions; we note that with B Z ≠ 0, spin-orbit interactions can enhance the stability of chiral topological phases similar to those discussed here 10 , and we do not expect such interactions to modify the universal critical theories presented below.
Numerical studies of H at B Z = 0 and J χ ≠ 0 on the kagome 11-13 and triangular [14][15][16][17] lattices have found convincing evidence above very small values of J χ (values as small as J χ /J 1 = 0.0014 in Fig. 19 of ref. 16 ) for a 'chiral spin liquid': a gapped state with semion topological order, but no antiferromagnetic order. More recently, a study of the Hubbard model on the triangular lattice 18 found evidence for the same chiral spin liquid even at J χ = 0. On the square lattice, Nielsen et al. 19 studied the antiferromagnet with first-(J 1 ) and second-(J 2 ) neighbour exchange and a non-zero J χ , and found evidence for the chiral spin liquid at quite small values of J χ , but in relatively small system sizes. These strong effects of a small J χ can be understood by the proximity to a critical spin liquid at which an infinitesimal J χ is a relevant perturbation. The phase diagram we propose for the square-lattice J 1 −J 2 −J χ antiferromagnet is summarized in Fig. 1, and the critical spin liquid is realized by the deconfined critical point at J χ = 0 between the Néel and valence bond solid (VBS) states. Recent analyses 20 have shown that a relevant J χ at this critical point does indeed lead to semion topological order. At such a critical point, there is a discontinuous jump in the thermal Hall conductivity κ xy at low T from κ xy /T = 0 at J χ = 0 to jκ xy =Tj ¼ ðπ=6Þðk 2 B =ℏÞ I (ref. 21 ; where k B is the the Boltzmann constant) at infinitesimal J χ , and we use proximity to this discontinuity to obtain the enhanced thermal Hall response in the Néel state. We show that turning on J χ at values of J 2 /J 1 smaller than at the deconfined critical point (for example, along the red arrow in Fig. 1) leads to a state with coexisting Néel and semion topological order across a new quantum critical phase boundary whose universal theory is obtained below. See the Methods for further discussion of Fig. 1.
Our analysis starts from a model 20 of the square-lattice Néel state as the confining phase of an SU(2) gauge theory of fluctuations about a 'π-flux' mean-field state 22 . In this formulation, the spins are represented by fermionic spinons f iα , via S i ¼ ð1=2Þf y iα σ αβ f iβ I , where σ are the Pauli matrices, and summation over α and β (α,β = ↑,↓) is implied. This spinon representation induces an SU(2) gauge symmetry 23 , and a full treatment requires careful consideration of the associated SU(2) gauge field. However, much can be learnt from a mean-field theory in which we ignore the SU(2) gauge fluctuations; we analyse such a mean-field theory now, and turn to the gauge fluctuations later.

Mean-field theory
After inserting the spinon representation of S i in H, and a meanfield factorization respecting lattice and gauge symmetries, we obtain the quadratic spinon Hamiltonian 20,24-26 The pattern of the hopping matrix elements t ij is shown in Fig. 2a.
The first-neighbour hopping, t 1 , arises from factoring the exchange couplings in H 1 . The second-neighbour hopping, ±it 2 , arises from J χ , and has the same symmetry as the orbital coupling of the underlying electrons to the magnetic field orthogonal to the plane of the square lattice. We have assumed a non-zero Néel order, and this leads to the N term after factoring H 1 ; η = ±1 has opposite signs on the two checkerboard sublattices of the square lattice. The Zeeman term minimizes the energy of the square-lattice antiferromagnet when the Néel order is orthogonal to the magnetic field, so we take B Z ⋅ N = 0; the B Z term is not essential to the topological and field-theoretic considerations below, but could be important in understanding the experimental role of the applied field. Many key results follow from a consideration of the topology of the spinon band structure implied by H f . Our choice of t ij in Fig. 2a and η i leads to a unit cell with two sites. Combined with the spin label α, we obtain a total of four spinon bands, which are half-filled. The key discriminant is the net Chern number of the occupied bands. When this is zero, there will be no Chern-Simons term in the theory for gauge fluctuations, leading to confinement and a conventional Néel state. However, when the net Chern number is 2, we obtain a Chern-Simons term and a state with semion topological order (as argued in ref. 20 ) coexisting with the Néel order here, because N ≠ 0; this state has gapped excitations with semionic statistics, along with the conventional spin-wave modes of the Néel state. In this manner, we obtain the mean-field phase diagram shown in Fig. 3a.
The evolution of the band structure across the phase boundary in Fig. 3a is shown in Fig. 2b-e. Note the appearance of two massless Dirac fermions at the critical point. Away from the critical point, these fermions acquire a common Dirac mass, which has opposite signs in the two phases.
Next, we compute κ xy across the phase boundary in Fig. 3a. The results are shown in Fig. 3b. Denoting the Berry curvature by Ω nk for each band n = (1, …, 4), κ xy is given by 27 is ħ/e 2 times the Hall conductivity, and f(ε) is the Fermi distribution function at energy ε. The corresponding Chern number is As T approaches 0, Consequently, κ xy =T ! ðπ=3Þðk 2 B =ℏÞ I as T approaches 0 in the phase with topological order; quantum gauge fluctuations, to be discussed, will change the prefactor (π/3) to the exact quantized value (π/6) in this phase. In the other phase, κ xy /T varies non-monotonically as T is lowered, and eventually vanishes as T approaches  Fig. 3a for a phase diagram with non-zero B Z .) By varying the first, J 1 , and second, J 2 , nearest-neighbour exchange interactions and the orbital coupling J χ in equation (2), the antiferromagnet on the square lattice shows phases with combinations of Néel, VBS and chiral spin liquid (CSL) topological order. The phase boundaries are presumed to meet at an SO(5)-symmetric (near) critical point at which J χ is a relevant perturbation, and the phase boundaries all scale as J χ  jJ 2 � J 2c j λχ =λ2 I ; we expect λ χ /λ 2 > 1. Here, we imagine starting from the Néel state at zero magnetic field (J χ = 0), close to the boundary of the VBS order such that a small value of field-induced J χ can already drive the system close to the phase boundary with Néel + CSL (indicated by the red arrow). The existence of an SO(5) critical point is not a precondition for a continuous Néel to Néel + CSL transition.
We plot the field dependence of κ xy within the Néel phase in Fig. 3c-e along the dashed lines in Fig. 3a. Note that κ xy is a nearly linear function of the field, with a slope that is enhanced as we approach the phase boundary of the state with semion topological order.

Gauge theories and dualities
We now discuss universal properties of the quantum phase transition in Fig. 3a and the lower phase boundary in Fig. 1. This critical theory has four different dual formulations, summarized in Fig. 4.   (3) are shown as a function of it 2 (see Fig. 2a) and |B Z |. Here, we take N ¼ 0:5ẑ I and measure all energies in units of t 1 . As discussed in the main text, it 2 is induced by the orbital coupling of the magnetic field. Both t 2 and |B Z | are linear functions of the applied magnetic field, and the dashed purple lines show three possible trajectories for which we plot the field dependence of κ xy in c-e for different T. b, Temperature dependence of the mean-field κ xy as t 2 is tuned across the phase boundary; the corresponding discrete values of |B Z | and t 2 are indicated by green dots in a. The quantized value of the ordinate in the topological phase is π/3, and the bifurcation point as T approaches 0 is at π/6. Both values are corrected by gauge fluctuations (the exact quantized value in the topological phase is π/6). c-e, Field dependence of κ xy for different T for the trajectories noted in a when |B Z | ≡ 50t 2 (c), 7t 2 (d) and t 2 (e).
-π/2 π/2 0 π -π/2 π/2 -π 0 π -π -π/2 π/2 0 π -π/2 π/2 -π ε nk ε nk ε nk ε nk The first of these formulations, labelled SU(2) −1/2 in Fig. 4, is obtained by reinstating gauge fluctuations to the free-fermion mean-field theory described above. The resulting field theory turns out to have an emergent global SO(3) symmetry, which must then also be a property of the other dual formulations. We discuss these field theories and their connections to the lattice antiferromagnet, in turn. We refer the reader to recent reviews 29-32 for subtle aspects of gauge and gravitational anomalies, which we do not discuss here. Fig. 3a, we can focus on the effective theory of the nearly massless Dirac fermions. These form a single doublet ψ under the SU(2) gauge symmetry; a low-energy theory will therefore have an SU(2) gauge field A μ (μ = 0, x, y) coupled minimally to ψ. However, we cannot entirely neglect the single filled fermionic band far from the Fermi level (see Fig. 2b-e). This band has a non-zero Chern number, and integrating out these fermions yields a Chern-Simons term for A μ at level −1/2. In this manner, we obtain the low-energy 2 + 1 dimensional Lagrangian

SU(2) −1/2 with a fermion doublet. Near the phase boundary in
Here, γ μ are the Dirac matrices, m is the mass term, which changes sign across the phase transition, and CS represents the SU(2) Chern-Simons term. When m is non-zero, we can safely integrate out the fermions. For one sign of m, the net Chern-Simons term vanishes, and we obtain a 'trivial' confining phase with κ xy = 0. For the other sign of m, we obtain a Chern-Simons term at level 1, and the SU(2) 1 theory describes a chiral spin liquid with κ xy =T ¼ ðπ=6Þðk 2 B =ℏÞ I U(1) 2 with a charged scalar. The second dual theory has a complex scalar ϕ coupled to a U(1) 2 gauge field, a μ : However, the SO(3) global symmetry is not manifest in this formulation, and its description requires consideration of monopole operators [5][6][7][8] . The coupling s tunes across the phase transition at the critical point s = s c , while the quartic nonlinearity u is assumed to flow to a fixed-point value, analogous to that in the Wilson-Fisher theory without the Chern-Simons term. For s < s c , the ϕ field forms a Higgs condensate, which quenches a μ and all topological effects; we thus obtain the conventional Néel state. This maps to the positive-mass phase of the SU(2) −1/2 fermion theory discussed above. For s > s c , we obtain the state with semion topological order. The gapped ϕ quasiparticles have mutual semion statistics induced by the Chern-Simons term. Below the quasiparticle gap, this phase is described by U(1) 2 and maps to the negative-mass phase of the SU(2) −1/2 fermion theory. We can connect the field theory L 2 I to the lattice antiferromagnet by viewing the latter as a theory of hard-core bosons S + = S x + iS y ; then, assuming the bosons form a ν = 1/2 fractional quantum Hall state, as in the chiral spin liquid 4 , we identify ϕ as the quasiparticle (vortex) operator in the Chern-Simons-Landau-Ginzburg theory 33,34 .

U(1) −3/2 with a charged fermion.
The third dual theory of ref. 5 is a theory that had been discussed in ref. 35 : it has a single Dirac fermion coupled to a U(1) −3/2 gauge field. The SO(3) symmetry is not manifest. Such a field theory can be related to a fractionalization of the hard-core boson S + into two fermions ∼f 1 f 2 (ref. 35 ). In the state with topological order, which is a ν = 1/2 fractional quantum Hall state of the bosons (as above), both fermions fill bands with unit Chern numbers, as in composite fermion theory 36; the phase transition maps to a change of the Chern number of one band to zero 34,35,37 . SU(2) 1 with a scalar doublet. The fourth dual theory 5-8 has a complex scalar doublet transforming as the fundamental of an SU(2) 1 gauge field. This can be connected to the SU(2) gauge theory obtained by transforming to a rotating reference frame in spin space 26,38,39 , in which we write the electrons as c α = R αβ ψ β . Here ψ β is a fermion (the 'chargon') and R αβ is an SU(2) matrix ðR y R ¼ 1Þ I , which can be expressed in terms of the aforementioned complex scalar doublet; in the renormalized continuum theory, the unit-length constraint on the scalar doublet can be replaced by a quartic self-interaction. The SU(2) gauge symmetry corresponds to right multiplication of R (and left multiplication of ψ), while the SO(3) global symmetry corresponds to left multiplication of R. We assume that the band structure of the ψ β fermions is such that both species are in a filled band with unit Chern number; integrating out these gapped fermions then yields the Chern-Simons terms for the SU(2) 1 gauge field. The needed transition is now obtained by the Higgs transition of the scalar: the topological phase has R gapped, while the trivial phase has R condensed. We also need spectator SU(2) gauge-neutral electrons in filled Chern bands to match the thermal and electrical Hall conductivities of the two phases.

Discussion
For thermal Hall measurements in the cuprates, our main mean-field results are in Fig. 3. Note the large rise in the thermal Hall response in the conventional Néel state proximal to the phase boundary, before it eventually vanishes at low enough T: this rise is our proposed explanation for the observations of Grissonnanche et al. 2 . The field and temperature dependences of κ xy in Fig. 3 match well with observations. It is possible that stronger fields will drive the cuprates across the quantum phase transition into a state with semion topological order, but the stronger field also enhances the Zeeman term, and Fig. 3a shows that this term is detrimental to such a transition. We also discussed gauge-field fluctuation corrections to the results in Fig. 3. In the topological phase, such corrections renormalize the thermal Hall conductivity from ðπ=3Þk 2 B T=ℏ I to ðπ=6Þk 2 B T=ℏ I as T approaches 0. Computing the analogous corrections at higher T and across the phase boundary in Fig. 3 is more challenging. The critical theory of the phase boundary was shown to be a central actor in recent studies of dualities of strongly interacting conformal field theories in 2 + 1 dimensions [5][6][7][8] . The theory of interest has four different formulations, which we summarize in Fig. 4; we also provided lattice interpretations of all four field theories in terms of the degrees of freedom of the square-lattice antiferromagnet. An expansion in the inverse number of matter flavours (analogous to ref. 28 ) is a promising route to computing the universal non-zero temperature thermal Hall effect in these gauge theories near the quantum critical point in Fig. 3a.
Finally, let us comment on the role of fluctuations of the Néel order parameter. Spin waves make only a small contribution to the thermal Hall effect 3 . In two spatial dimensions, thermal fluctuations of the Néel order restore spin rotation symmetry at all non-zero T (ref. 40 ), but these classical fluctuations are not expected to substantially modify the quantum criticality of the topological quantum phase transition in Fig. 3a, which involves no change in symmetry.

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Methods
Deconfined criticality and phase diagram. The square-lattice antiferromagnet with irst (J 1 ) and second (J 2) neighbour exchange interactions has been the focus of many numerical studies in the past decades. There appears to be general agreement that increasing J 2 /J 1 destroys the Néel state and leads to a state with VBS order 41,42 . There is also significant evidence that the transition region between these states is described by a deconfined critical field theory 43 over a large intermediate length scale 41 . Furthermore, there is strong support for a global SO(5) symmetry between the Néel and VBS orders 44 over this scaling region, as is expected for the deconfined critical theory 20,45,46 . The ultimate fate of the phase transition at the longest distances remains unsettled, but it is plausible that it is described by a complex fixed point, very close to the real physical axis 20,47,48 .
We now consider the phase diagram of the J 1 −J 2 −J χ antiferromagnet on the square lattice by starting from a theory in which the SO(5) symmetry is initially explicit. This is the fermionic spinon representation used in equation (3). In the absence of Néel order (N = 0) and an applied field (B Z = 0), the continuum limit of equation (3) yields two flavours of two-component Dirac fermions, which are then coupled to an SU(2) gauge field. Instead of equation (7), we now have 20 where a = 1, 2 is the flavour index. The SO(5) symmetry is apparent after we express L SOð5Þ I in terms of Majorana fermions. The fermion mass m χ ∝ J χ is also SO(5) invariant and is a perturbation on the putative SO(5)-invariant Néel-VBS critical point at m χ = 0. It is plausible that m χ is a relevant perturbation on such a critical point (with scaling dimension λ χ > 0); then an infinitesimal J χ will be sufficient to drive the critical antiferromagnet into the chiral spin liquid phase. If the Néel-VBS transition is weakly first-order, then a very small value of J χ will be sufficient. Tuning away from the critical point by changing the value of J 2 /J 1 yields a second relevant perturbation to the critical point (with scaling dimension λ 2 > 0), which explicitly breaks SO(5) symmetry, but is allowed by the symmetries of the underlying antiferromagnet. We obtain the phase diagram proposed in Fig. 1 by considering the interplay of these perturbations; all phase boundaries scale as J χ  jJ 2 � J 2c j λχ =λ2 I for an SO (5) critical point at J χ = 0, J 2 = J 2c . For λ χ > λ 2 , we obtain the onset of semion topological order at small values of J χ , even away from the SO(5) point. In the limit of a large number of fermion flavours, λ χ = 1 and λ 2 = −1, λ χ > λ 2 is thus plausible. This phase diagram is compatible with the small-system size studies of ref. 19 . We emphasize that the existence of an SO(5) critical point is not a requirement for the existence of a continuous Néel to Néel + CSL transition with SO(3) symmetry described in the main part of the paper.

Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.