Nonreciprocal directional dichroism induced by a temperature gradient as a probe for mobile spin dynamics in quantum magnets
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Nonreciprocal directional dichroism induced by a temperature gradient as a probe formobile spin dynamics in quantum magnets
Xu Yang and Ying Ran Department of Physics, Boston College, Chestnut Hill, MA 02467, USA (Dated: April 29, 2020)Novel states of matter in quantum magnets like quantum spin liquids attract considerable interestrecently. Despite the existence of a plenty of candidate materials, there is no confirmed quantumspin liquid, largely due to the lack of proper experimental probes. For instance, spectrosocopyexperiments like neutron scattering receive contributions from disorder-induced local modes, whilethermal transport experiments receive contributions from phonons. Here we propose a thermo-opticexperiment which directly probes the mobile magnetic excitations in spatial-inversion symmetricand/or time-reversal symmetric Mott insulators: the temperature-gradient-induced nonreciprocaldirectional dichroism (TNDD) spectroscopy. Unlike traditional probes, TNDD directly detectsmobile magnetic excitations and decouples from phonons and local magnetic modes.
Introduction
Quantum spin liquids(QSL), proposedby Anderson for spatial dimensions >
1, attracted con-siderable interest in the past decades (see Ref. forreviews). Although theoretically these novel states ofmatter are known to exist and have even been success-fully classified , to date there is no experimentally con-firmed QSL material. As a matter of fact, an increas-ing list of candidate QSL materials emerges recently dueto the extensive experimental efforts, including, for in-stance, Herbertsmithite , α -RuCl under a magneticfield , and quantum spin ice materials . An out-standing challenge in this field is the lack of appropriateexperimental probes. Traditional probes for magneticexcitations include thermodynamic measurements, var-ious spectroscopy measurements such as neutron scat-tering and nuclear magnetic resonance, and the thermaltransport. Ideally, one would like to directly probe themobile magnetic excitations in a QSL, such as the frac-tionalized spinons. The major limitation of traditionalprobes is from the contributions of other degrees of free-dom; e.g., the spectroscopy measurements couple to lo-cal impurity modes, while the thermal transport coupleto phonons. It is highly nontrivial to directly probe theintrinsic contribution from the mobile magnetic excita-tions. To highlight this challenge, there is no known di-rect probe to even detect the mobility gap of magneticexcitations, which is fundamentally important in the fieldof topologically ordered states.In this paper we propose a thermo-optic experimentwhich serves as a new probe for mobile magnetic excita-tions in Mott insulators respecting either the spatial in-version symmetry I or the time-reversal symmetry T ,or both: the temperature-gradient-induced nonreciprocaldirectional dichroism (TNDD). In a sense TNDD com-bines the thermal transport and optical spectroscopy to-gether, and effectively decouples from phonon and localmagnetic modes. Theory of TNDD
Nonreciprocal directional dichro-ism (NDD) is a phenomenon referring to the difference inthe optical absorption coefficient between counterpropa-gating lights . From the Fermi’s golden rule, NDD for linearly polarized lights is due to the interference betweenthe electric dipole and magnetic dipole processes : δ ˆ n α ( ω ) ≡ α ˆ n ( ω ) − α − ˆ n ( ω ) = 2 µ r ǫ c π ~ · ~ ωV X i,f ( ρ i − ρ f ) · · Re[ h i | ~P · ˆ E| f ih f | ~M · ˆ B| i i ] · δ ( E f − E i − ~ ω ) (1)where α ± ˆ n ( ω ) is the optical absorption coefficient ofcounterpropagating lights (along ± ˆ n ) at frequency ω ,which are I (or T ) images of each other. ~P ( ~M ) is theelectric polarization (magnetic moment) operator. ˆ E ( ˆ B )is the direction of the electric field (magnetic field) andˆ n ∼ ˆ E × ˆ B . | i i , | f i label the initial and final states inthe optical transition ( ρ i and ρ f are their density ma-trix elements), ǫ and c are the vacuum permittivity andthe speed of light, V is the volume of the material, and µ r is the material’s relative permeability. Clearly both I and T need to be broken to have a nonzero NDD be-cause Re[ h i | P | f i · h f | M | i i ] is odd under either symmetryoperation. NDD has been actively applied in the fieldof multiferroics to probe the dynamical coupling be-tween electricity and magnetism.The TNDD spectroscopy essentially detects the jointdensity of states of mobile magnetic excitations, and canbe intuitively understood as follows (see Fig.1(a)). Con-sider a Mott insulator respecting I and/or T so thatNDD vanishes in thermal equilibrium. In the presence ofa temperature gradient, the system reaches a nonequilib-rium steady state with a nonzero heat current carried bymobile excitations. For simplicity one may assume thatexcitations of the system are well-described by quasipar-ticles, e.g., spinons or magnons, phonons, etc. The lead-ing order nonequilibrium change of ρ i and ρ f in Eq.(1)satisfies δρ i , δρ f ∝ ∇ T · τ from a simple Boltzmann equa-tion analysis, where τ is the relaxation time. The crucial observation is that this nonequilibriumstate breaks both the inversion symmetry (by ∇ T )and the time-reversal symmetry (by τ ). Consequentlyone expects a NDD signal proportional to ∇ T · τ .Precisely speaking TNDD is a second-order thermo-electromagnetic nonlinear response: it is a change ofoptical absorption (a linear response) due to a temper-ature gradient. The factor ∇ T · τ in TNDD indicatesthat it is a generalization of Drude-phenomenon to non-linear responses. Notice that the Drude-phenomenon isindependent of whether the system has a quasiparticledescription or not. Even in the absence of quasiparticledescriptions, strongly interacting liquids may have nearlyconserved momentum. The relaxation time τ in Drudephysics should be interpreted as the momentum relax-ation time . This indicates that TNDD discussed herecan be generalized to systems without quasiparticle de-scriptions such as the U(1)-Dirac spin liquid and thespinon Fermi surface state . Advantages of TNDD spectroscopy
Now we com-ment on the major advantages of TNDD as a probeof spin dynamics. First, TNDD is a dynamical spec-troscopy with the frequency resolution in contrast tothe DC thermal transport, and essentially probes thejoint density of states of magnetic excitations. Second,the fact that TNDD only receives contributions fromRe[ h i | P | f i · h f | M | i i ] dictates that the phonons ’ contri-bution can be safely ignored: The natural unit for themagnetic moment of phonon, the nuclear magneton, ismore than three orders of magnitudes smaller than thatof the electron, the Bohr magneton.In addition, at the intuitive level, a local magneticmode (e.g. from a magnetic impurity atom) can onlycouple to a local temperature instead of a temperaturegradient. A local temperature respects both I (after tak-ing disorder-average) and T . Consequently, such localmodes are not expected to contribute to TNDD either.From a more careful estimate (see App.A for detailed dis-cussions), we find that the contribution to TNDD fromlocalized modes with a localization length ξ , comparingto the contribution from the intrinsic mobile magneticmodes, is at least down by a factor of ξ/l m , where l m isthe mean-free path of the mobile magnetic excitations.We have assumed that ξ ≪ l m : for local magnetic modescarried by magnetic impurity atoms or crystalline defects,typically ξ is comparable with the lattice spacing a , whileusually l m ≫ a in a reasonably clean Mott insulator atlow temperatures. Estimate of the TNDD response
One may esti-mate the size of TNDD signal in a spin-orbital coupledMott insulator. The relevant dimensionless ratio limitingthe experimental resolution is:
T N DD ( ω ) ≡ δ ˆ n α ( ω ) α ˆ n ( ω ) + α − ˆ n ( ω ) . (2)In a Mott insulator, the polarization carried by a mag-netic excitation can be estimated as ζ · e · a , where a isthe lattice spacing and ζ is dimensionless. Assuming theaverage temperature of the system k B T to be compara-ble to the magnetic excitation energy , we find that (see App.B for details): T N DD ( ω ) ∼ (cid:18) DJ (cid:19) ζ α · |∇ T | · l m T ∼ (cid:18) DJ (cid:19) · |∇ T | · l m T , (3)in the limit of a weak spin-orbit coupling. Here α ≈ /
137 is the fine-structure constant and we used ζ ∼ − ∼ α in typical transition metal Mott insulators .Notice that in the absence of spin-orbit coupling, TNDDvanishes since the spin magnetic moment M is a spin-triplet . D and J are the Dzyaloshinskii-Moriya(DM)interaction and the exchange interaction respectively. Ina system with a strong spin-orbit coupling one may set D/J ∼
1, and
T N DD ( ω ) is proportional to the ra-tio of the temperature change across l m and the tem-perature. To optimize signal, one may choose a largetemperature gradient such that ∇ T · w ∼ T where w is the linear system size along the ∇ T direction, and T N DD ( ω ) ∼ l m / w . For instance, l m of magnetic excita-tions in a quantum spin ice material was reported to be ofthe order of a micron . For a typical millimeter samplesize, T N DD ( ω ) can be as large as 10 − , well detectablewithin the currently available experimental technology. Crystal symmetry analysis
TNDD can be phe-nomenologically described by a tensor η : δ ˆ n α ( ω ) = X a,b,c η abc ( ω ) ˆ E a ˆ B b ∇ c T (4)The symmetry condition for η abc ( ω ) is determined by thefusion rule of two vectors ( ˆ E , ∇ T ) and one pseudovector( ˆ B ) into a trivial representation under the point group.For any point group, symmetry always allows nonzero η abc : one may always consider the case ˆ n ∼ ˆ E × ˆ B to beparallel to ∇ T .As an example, we find that there are four independentresponse coefficients for the D d point group: δ ˆ n α = η ∇ z T ( ˆ E × ˆ B ) z + η ˆ B z ( ˆ E × ∇ T ) z + η ˆ E z ( ˆ B × ∇ T ) z + η (cid:2) ( ˆ E x ˆ B y + ˆ E y ˆ B x ) ∇ y T − ( ˆ E x ˆ B x − ˆ E y ˆ B y ) ∇ x T (cid:3) (5)Here the x-axis is a C -axis and the yz-plane is a σ d mirror-plane in the D d group. The D d point groupis realized in the QSL candidate Herbertsmithite, in theHeisenberg model on the Kagome lattice with DM in-teractions (see below and Fig.1(b)), as well as in thegeneralized Kitaev-Heisenberg model on the honeycomblattice , relevant for Na IrO and RuCl . Microscopic model
We present a concrete micro-scopic calculation for the TNDD spectrum. The nearestneighbor spin-1/2 Hamiltonian under consideration is onthe kagome lattice: H = J X
1, ˆ r ij is the unit vector along the directionfrom the site- j to the site- i . As shown in Fig.1(b), in eachbow-tie: d = d = d = 1 , d = d = d = − . Dipole-coupling with an external electric field δH = − ~E · ~P , the electric polarization ~P has the following formfor the nearest neighbor terms : P y = ζea √ ~S · ( ~S + ~S − ~S − ~S ) − ~S · ~S + 2 ~S · ~S ] ,P x = ζea · [ ~S · ( ~S − ~S + ~S − ~S )] , (8)where e < a is the nearest neigh-bor distance, and ζ is a dimensionless coupling constant(in this paper ~S = ~σ/ ζ can be generated via a t/U expansion in a Hubbard model . In the leading order J = t U and ζ = t U . Q = Q Z spin liquid: Schwinger boson mean-field treatment There are extensive numerical evi-dences that the Heisenberg model on the kagome latticemay realize a QSL ground state, although the natureof the QSL is under debate . The present workdoes not attempt to resolve this long-standing puzzle.Instead, we will focus on one candidate spin liquid state,which may be realized in the model Eq.(6): Sachdev’s Q = Q Z QSL . The Q = Q QSL is a gappedstate and can be described using the Schwinger bosonmean-field theory , in which spin is represented bybosonic spinons: ~S i = b † iα ~σ αβ b iβ , while boson numberper site is subject to the constraint b † iα b iα = 2 S. Wethen do the usual mean-field decoupling and diagonalize - - FIG. 2. The Schwinger boson band dispersion (blue solidlines) for the mean-field Hamiltonian Eq.(9) of Sachdev’s Q = Q Z QSL with parameters A = 1, D z = D p = 0 . J ,and µ = − . J . The low energy band-1 near the Γ pointis well described by the relativistic dispersion Eq.(10) withgap ∆ = 0 . J (red line). The two-spinon (red dots at ± ~k )contribution to the TNDD response computed in Eq.(11) andApp.C is illustrated. the quadratic mean-field spinon Hamiltonian to obtainthree spinon bands. We treat DM interaction as a per-turbation and keep contributions up to the linear orderof D/J . Under this approximation we arrive at the fol-lowing mean-field Hamiltonian. H MF = − µ X i ( b † iα b iα − S ) − J X h ij i ( A ∗ ij ˆ A ij + h.c. )+ X h ij i ( ~D ij · A ij ˆ ~C † ij + h.c. ) , (9)where operators ˆ A ij ≡ b iα ǫ αβ b jβ and ˆ C ij ≡− ib iα ( ǫ~σ ) αβ b jβ . H MF may be viewed as an ansatz toconstruct variational spin-liquid wavefunctions with pa-rameters A ij , µ . In Sachdev’s Q = Q state, A ij havethe following spatial pattern: A ij = d ij A , and A can bechosen to be real. See Appendix. C for more details.After Bogoliubov diagonalization, three bands arefound: H MF = P α = ↑ , ↓ ~k,u =1 , , E u,~k γ α † u,~k γ αu,~k as shown inFig.2, where ↑ , ↓ label the Kramers degeneracy. Tun-ing chemical potential µ so that the band structure isnear the boson condensation at Γ, the lowest energy band u = 1 is well described by a relativistic boson disperson: E ,~k ≈ p ∆ + ~ k v , (10)where ∆ is the bosonic spinon gap. TNDD contributed from the bosonic spinons
Inthe low temperature limit, the two-spinon contributiondominates TNDD with | f i ∼ γ α, † u,~k γ β † v, − ~k | i i in Eq.(1). . ρ i , ρ f in Eq.(1) is related to the nonequilibrium bosonicspinon occupation g u,~k . From a simple Boltzmann equa-tion analysis with a single relaxation time τ , g u,~k deviatesfrom the equilibrium occupation g u,~k = e β ( ~r ) Eu,~k − by Δ Δ FIG. 3. The bosonic two-spinon contribution to TNDD spec-tra of Sachdev’s Q = Q Z QSL Eq.(9) at the tempera-ture k B T = 0 .
7∆ (solid black line) and k B T = 0 .
4∆ (solidred line), together with the two-spinon joint density of states(dashed blue line). δg u,~k = ∂g u,~k ( ~r ) ∂E E u,~k τ~v u,~k ·∇ TT ( ~r ) , where ~v u,~k = ∂E u,~k ~ ∂~k . This δg u,~k is responsible for TNDD.Since TNDD is a bulk response we consider a 3D sys-tem consisting of stacked 2D layers each described by themodel Eq.(6) with an interlayer distance d . Using theelectric polarization Eq.(8) and spin magnetic moment ~M = g s µ B ~S , in App.(C we compute the low temper-ature/energy TNDD response tensor defined in Eq.(4)within our mean-field treatment (corresponding to η in Eq.(5)). As plotted in Fig.3, we find that ( x, y, z -directions are illustrated in Fig.1) η xzy ( ω ) = C · (cid:2) g ( ~ ω/ (cid:3) · ( k B T ) · (cid:2) G ( z ) − z · G ( z ) + (ln z ) G ( z ) (cid:3) · e − √ ( ~ ω/ − ∆ / ∆ · ~ ω · JDOS ( ~ ω ) · τ · vT . (11)Here the constant C ≡ π u α ζaa · µ r g s a ~ v , where a is the Bohr radius. u ∝ ( D/J ) is a dimension-less constant related to the mean-field band structureand can be determined numerically. For the parame-ters D z = D p = 0 . J and µ = − . J we find that u = 0 . JDOS ( ~ ω ) ≡ D · ~ ω · Θ( ~ ω − D ≡ π ~ v d . g ( ~ ω/
2) = e ~ ω/ kBT − , z ≡ e − ∆ kBT , and G ν ( z ) ≡ ν ) R ∞ x ν − dxz − e x − is the Bose-Einstein integral. Eq.(11)holds when the temperature and the photon energy arewithin the regime of the relativistic dispersion Eq.(10).In the limit k B T ≪ ∆, Eq.(11) can be simplified andwe have η xzy ( ω ) ∝ e − ∆ /k B T , where the thermal acti-vation factor can be traced back to δg ~k . Importantly,beyond the mean-field treatment, TNDD is generally ∝ δρ i , δρ f ∝ ∇ T · τ in Eq.(1), and a thermal activationfactor e − ∆ /k B T in TNDD is always due to the energydiffusion near the mobility gap ∆. Therefore TNDD canserve as a sharp measurement of the mobility gap ∆ ofthe magnetic excitations. Discussion
Bosonic vs. fermionic spinons
We com-puted the TNDD response contributed from bosonic spinons in the Sachdev’s Q = Q Z QSL. Fermionicspinons also exist in this Z QSL and their contribu-tion to TNDD can be similarly computed in a dualAbrikosov fermion approach . Without pursuing thiscalculation in details, one expects that the bosonic factor[1 + 2 g ( ~ ω/ − f ( ~ ω/ f ( ~ ω/
2) =1 / ( e ~ ω/ k B T + 1). The contributions from the bosonicspinons and fermionic spinons have different temperaturedependence, which, in principle, may be used to detectthe statistics of quasiparticles in certain situations. Magnetically ordered states
It is also interesting to con-sider the TNDD response in a conventional magneticallyordered state respecting either I , or T combined with alattice-translation symmetry (as in the case of an anti-ferromagnet), or both. One may similarly consider thetwo-magnon contribution to the TNDD response, whichprobes the joint density of states of magnons. Our esti-mate Eq.(3) will be modified as follows (see Appendix Bfor details). If the magnetic order is non-collinear, whichbreaks spin-rotational symmetry completely, the ( D/J ) factor in Eq.(3) is replaced by ∼
1. If the magnetic orderis collinear, which only breaks the spin-rotation symme-try down to U (1), the ( D/J ) factor is replaced by D/J . Conclusion
In this paper we propose thetemperature-gradient-induced nonreciprocal direc-tional dichroism (TNDD) spectroscopy experiment inMott insulators. Comparing with traditional probesfor magnetic excitations, TNND spectroscopy hasunique advantages: it directly probes mobile magneticexcitations and decouples from local impurity modesand phonon modes. For instance, an activation behavior ∝ e − ∆ /k B T in the temperature dependence of TNDDsharply measures the mobility gap ∆ of the magneticexcitations, a quantity challenging to measure usingtraditional probes but of fundamental importance in thefield of topologically ordered QSL.The present work can be viewed as one example ina large category of nonlinear thermo-electromagnetic re-sponses. There are other interesting effects. For instance,a temperature gradient also induces a circular dichroismin a system respecting both T and I . We leave theseother responses as topics of future studies.We thank Kenneth Burch and Di Xiao for helpful dis-cussions. XY and YR acknowledge support from theNational Science Foundation under Grant No. DMR-1712128. Appendix A: Localized modes
Let us consider the situation of a Mott insulator in thepresence of impurities/disorders, which could introducelocalized magnetic modes. Below we consider the contri-bution to TNDD response from these localized modes.Firstly, we comment on the meaning of “localizedmodes” discussed here. In an isolated localized phaseof matter, like a many-body localized phase(see Ref. for reviews), thermalization breaks down and the mean-ing of a temperature-gradient is unclear. We are
NOT discussing the TNDD response in this situation.In realistic quantum materials, the magnetic localizedmodes are coupled with a thermal bath (e.g., phononthermal bath) and a local temperature is well defined.To facilitate the discussion, one may consider a systemwith a U (1) spin rotation symmetry in order to sharplydefine a magnetic localized mode. In addition, we assumea finite mobility gap ∆ of the U (1) charge, and magneticlocalized excitations may exist below ∆. Assuming l m being the mean-free path for mobile magnetic excitations,practically the localized magnetic modes may fall intotwo regimes according to the localization length ξ : (1): ξ ≪ l m . This is the more common situationrealized in practical materials. Here the localized mag-netic modes may be extrinsic magnetic impurity atoms,or may form at crystalline defects. They may also format the centers of the vortices of valence bond solid (VBS)order . Typically the localization length ξ of these mag-netic modes is of the same order as the lattice spacing a ,while l m ≫ a in a reasonably clean Mott insulator.It is difficult to model a magnetic localized mode with ξ ∼ a since lattice scale details cannot be neglected. In-stead, we consider the following situation a ≪ ξ ≪ l m sothat a low energy effective description is still valid. As acrude model for such magnetic localized modes, one mayconsider a quantum dot of size ξ in the presence of a tem-perature gradient; for instance, the left (right) edge of thequantum dot is in contact with a heat reservior at tem-perature T L ( T R ). The modes in the quantum dot aretravelling ballistically since ξ ≪ l m . Consequently theright-mover (left-mover) in the quantum dot is at tem-perature T L ( T R ). Such a nonequilibrium ensemble isquantitatively comparable with a large (energy-)diffusivesystem in the presence of the same temperature gradientbut with l m ∼ ξ (for example, see Eq.(C7)). Namely,in the present situation, ξ replaces the role of l m in ourestimate Eq.(3). we conclude that the dimensionless ra-tion T N DD ( ω ) contributed by such localized modes isreduced by a factor of ∼ ξ/l m . (2): ξ ≫ l m . In this situation, the system hosts would-be mobile modes. These modes scatter with disordermultiple times before eventually become localized. Forinstance, Anderson weak-localization in two spatial di-mensions happens with ξ parametrically larger than l m .It is instructive to consider a system size L satisfying ξ > L > l m . For such a system size the localizationphysics is not present yet. Because photon absorptionis still a local process, we expect that the contributionto the TNDD response from such localized modes to becomparable with that from mobile modes.In summary, the contribution to TNDD response fromlocalized modes in the regime ξ ≪ l m can be safely ne-glected. In the opposite regime ξ ≫ l m , the localizedmodes still contribute to TNDD significantly. Neverthe-less, the localized modes in the latter regime are would-be extended (mobile) states in the absence of disorder. Appendix B: Spin-orbit coupling and the estimate ofTNDD response
From the discussion in the main text and Eq.(1), up tomatrix element effects, the TNDD spectroscopy directlyprobes the joint density of states
JDOS ( ~ ω ) of the mo-bile magnetic excitations: δ ˆ n α ( ω ) ≡ α ˆ n ( ω ) − α − ˆ n ( ω ) ∝ ~ ω · JDOS ( ~ ω ) · ∇ T · τ (B1)In order to estimate the optical absorption coeffient α ˆ n in a Mott insulator, one need to estimate the strength ofelectric polarization and the magnetic dipole moment. Itturns out that they are comparable in a typical transitionmetal Mott insulator, which is drastically different fromthe case of a band metal/insulator. In the latter casethe electric polarization carried by a typical particle-holeexcitation is ∼ e · a where e is the electron charge and a is the lattice constant, while the magnetic moment car-ried by the same excitation is of the order of a Bohrmagneton µ B . For a given electromagnetic wave, themagnetic dipole energy scale µ B · B is smaller than theelectric dipole energy scale e · a · E by roughly a factorof the fine-structure constant ∼ / ζ · e · a where the dimen-sionless factor ζ ∼ t/U ) . On the other hand, themagnetic dipole moment carried by the same excitationis still ∼ µ B . As a result, they would have comparablesizes for typical 3d transition metal Mott insulators with t/U ∼ α ˆ n ( ω ) ∼ n r ǫ c π ~ · |h f | P | i i| · ~ ω · JDOS ( ~ ω ) ∼ π n r α ζ a · ~ ω · JDOS ( ~ ω ) . (B2)where n r is the relative refractive index of the material, c is the speed of light, α is the fine structure constant ∼ / JDOS ( ~ ω ) is the joint density of statesfor the relevant excitations at photon energy ~ ω . We as-sume that the temperature is comparable with the mag-netic excitation energy scale, and we have used the typi-cal matrix element h f | P | i i ∼ ζ · e · a where a is the latticeconstant.Notice that JDOS ( ~ ω ) may be estimated as ∼ a W where a is the lattice constant and W is the band widthof the excitations. For a typical photon energy ∼ W , onefinds that ~ ω · JDOS ( ~ ω ) ∼ /a , independent of thenature of the excitations. For instance, the interband ab-sorption coefficient α ( ω ) in a band metal/insulator is typ-ically ∼ m − . The dimensionless coupling constant ζ reduces by a factor of 10 in transition metal Mott in-sultors, which gives the absorption coefficient ∼ m − ,broadly consistent with the tera-Hertz penetration depth( ∼ .The TNDD response can be similarly estimated. Wefirst consider the case of a quantum paramagnet. δ ˆ n α ( ω ) ∼ µ r ǫ c π ~ · ( ρ i − ρ f ) · · Re[ h f | P | i ih i | M | f i ] · ~ ω · JDOS ( ~ ω ) (B3)We again assume that the temperature is comparablewith the magnetic excitation energy scale, and conse-quently the effect of temperature gradient in ( ρ i − ρ f ) canbe estimated by the dimensionless factor |∇ T |· l m T where l m is the mean-free path of the magnetic excitations. Ifthe spin-orbit coupling (SOC) is strong one may estimate h f | P | i i ∼ ζea while h i | M | f i ∼ g s µ B ( g s is the g-factorthe spin magnetic moment.). Putting together we have: δ ˆ n α ( ω ) ∼ π µ r g s α ζa a · |∇ T | · l m T · ~ ω · JDOS ( ~ ω ) , if strong SOC. (B4)Here a is the Bohr radius.From Eq.(B2,B4), and a ∼ a , we can estimate thatif the spin-orbit coupling is strong and the temperatureis comparable with the magnetic excitation energy scale,the dimensionles ratio T N DD in Eq.(2)
T N DD ( ω ) ∼ α ζ |∇ T | · l m T ∼ |∇ T | · l m T , if strong SOC.(B5)Here we used the fact that for a typical transition metalMott insulator ζ ∼ − ∼ α .In the absence of the SOC, h i | M | f i = 0 because ~M = g s µ B ~S is proportional to the conserved total spin ~S (Weonly consider the spin magnetic moment. The orbitalmagnetic moment in Mott insulators is much smaller andneglected.). In the limit of a weak SOC: D/J ≪
1, theTNDD response can be estimated as follows. The onlyeffect of the weak SOC is in the matrix element product: h f | P | i ih i | M | f i .For the magnetic dipole matrix element: h i | M | f i ∝ E f − E i h i | [ S, H ] | f i ∝ h f | ~D · [ S, ~S i × ~S j ] | i i . Notice thatthe operator of the commutator is a spin triplet. Thereare two possibilities: (1): the states | f i and | i i differ byspin-1 in the limit D/J →
0. For instance, | f i may bea spin triplet while | i i is a spin singlet in that limit; (2):the states | f i and | i i have the same spin in the limit D/J → h i | M | f i ∝ ( D/J ) , because the wavefunction cor-rections of | f i and | i i due to nonzero D/J need to be considered. In this situation, the electric dipole matrixelement h f | P | i i ∝ ( D/J ) since P is a spin singlet oper-ator in the limit of D/J →
0. Therefore in situation-(2)we have h f | P | i ih i | M | f i ∝ ( D/J ) .In the situation-(1), a similar consideration leads to: h i | M | f i ∝ ( D/J ) and h f | P | i i ∝ ( D/J ). So we still have h f | P | i ih i | M | f i ∝ ( D/J ) .In summary, we have the following estimate in a quan-tum paramagnet assuming the temperature is compara-ble with the magnetic excitation energy scale: T N DD ( ω ) ∼ (cid:18) DJ (cid:19) α ζ |∇ T | · l m T ∼ (cid:18) DJ (cid:19) |∇ T | · l m T , if weak SOC. (B6)Next we estimate the TNDD response in magnetic or-dered states due to magnon excitations. Even in theabsence of microscopic SOC, the (
D/J ) factor in the es-timate Eq.(B6) will be replaced by ∼ U (1) in the absence of SOC.The electric polarization operator P is expected to carryzero charge under this U (1) rotation. To have a nonzeromatrix element product h f | P | i ih i | M | f i , one must con-sider the linear-order effect of the SOC. Therefore in thiscase the ( D/J ) factor in the estimate Eq.(B6) will bereplaced by ∼ D/J . Appendix C: Details of the mean-field calculationfor TNDD
In this section we provide a detailed account of theSchwinger boson mean-field theory. The spin is repre-sented by bosonic spinons ~S i = 12 b † iα ~σ αβ b iβ , (C1)while boson number per site is subject to the constraint: b † iα b iα = κ. (C2)Although κ = 2 S for spin- S , it will be convenient toconsider κ to be a continuous parameter, taking on anynon-negative value .Considering the operator identities ~S i · ~S j = − ˆ A † ij ˆ A ij + κ and ~S i × ~S j = [ ˆ ~C † ij ˆ A ij + h.c. ], whereˆ A ij = − ˆ A ji = b iα ǫ αβ b jβ , ˆ ~C ij = ˆ ~C ji = − i b iα ( ǫ~σ ) αβ b jβ . (C3)standard mean-field decoupling of Eq.(6) leads to themean-field Hamiltonian: H MF = − J X
D/J . Under this ap-proximation we will set the parameter ( not the operator ˆ ~C ij ) ~C ij ∝ D/J to zero in Eq.(C4) below, which yieldsEq.(9) in the main text. We also focus on Sachdev’s Q = Q state, where A ij happens to have the followingspatial pattern: A ij = d ij A , and A is chosen to be real.After diagonalizing H MF in the momentum space,there are three Kramers degenerate Bogoliubov bosonbands (see Fig.2): H MF = u =1 , , X ~k,α = ↑ , ↓ E u,~k γ α † u,~k γ αu,~k . (C5)Notice that spin is not a good quantum number and ↑ , ↓ are simply labelling the two-fold Kramers degeneracy foreach band.In the presence of a temperature gradient ∇ T ( ~r ), theoccupation of Bogoliubov spinons g u,~k = h n u,~k i (where n u,~k = γ † u,~k γ u,~k ) deviates from the thermal equilibriumvalue g u,~k . For simplicity, we consider the steady stateBoltzmann equation within a single relaxation-time ap-proximation: ~v u,~k · ∇ ~r g u,~k ( ~r ) = − g u,~k ( ~r ) − g u,~k ( ~r ) τ , (C6)where g u,~k ( ~r ) = e Eu,~k/kBT ( r ) +1 , ~v u,~k = ~ ∇ ~k E u,~k . To theleading order, these give δg u,~k ( ~r ) ≡ g u,~k ( ~r ) − g u,~k ( ~r ): δg u,~k ( ~r ) ≡ δg ↑ u,~k ( ~r ) = δg ↓ u,~k ( ~r ) = ∂g u,~k ( ~r ) ∂E E u,~k τ~v u,~k · ∇ TT ( ~r )(C7)Since the velocity ~v u,~k = − ~v u, − ~k , we have: δg u,~k ( ~r ) = − δg u, − ~k ( ~r ) (C8)To be concrete, we focus on the case ˆ E = ˆ x and ˆ B = ˆ z ,with the light propagating direction ˆ n = − ˆ y and thetemperature gradient ∇ T ∝ ˆ y (the η response in Eq.(5)). In order to compute the matrix elements in Eq.(1), onewrites P x and M z in terms of the Bogoliubov bosons, andselects the relevant terms: P x → v,α,w,α ′ X ~q X v,α,w,α ′ ~q γ α † v,~q γ α ′ † w, − ~q + h.c. + 1 A v,α,w,α ′ ,t,β X ~q,~p Y v,α,w,α ′ ,t,β~q,~p γ α † v,~q γ α ′ † w, − ~q γ β † t,~p γ βt,~p + h.c.,M z = − g s µ b X i b † iα σ zαβ b iβ → v,α,w,α ′ X ~q Z v,α,w,α ′ ~q γ α † v,~q γ α ′ † w, − ~q + h.c.. (C9)The objects X v,α,w,α ′ ~q , Y v,α,w,α ′ ,t,β~q,~p , Z v,α,w,α ′ ~q are deter-mined by the Bogoliubov transformation from Eq.(9) toEq.(C5).Plugging in Eq.(1), one finds δ ˆ n α ( ω ) = 8 πµ r ǫ c d Re[ I ( ω )] , (C10)where I ( ω ) = ωA v,α,w,α ′ X ~q h X v,α,w,α ′ ~q + 1 A t,β X ~p Y v,α,w,α ′ ,t,β~q,~p · g t,~p i ∗ · Z v,α,w,α ′ ~q (1 + g v,~q + g w, − ~q ) · δ ( E v,~q + E w, − ~q − ~ ω ) . (C11)Here the bosonic factor (1 + g v,~q + g w, − ~q ) is well antici-pated from the golden rule. The factor g t,~p appears be-cause of the quartic interactions in ~P in Eq.(8).It is a good moment to study the symmetry prop-erty of I ( ω ). In thermal equilibrium, it is straightfor-ward to see that the inversion symmetry alone dictates I ( ω ) = 0, while time-reversal symmetry alone allows anonzero imaginary part of I ( ω ) (giving rise to the well-known natural circular dichroism in noncentrosymmetricsystems).Next we consider the effect of nonequilibrium occupa-tion δg u,~k in Eq.(C7). Expanding Eq.(C11) gives threecontributions, I = I ( A ) + I ( B ) + I ( C ) : I ( A ) ( ω ) ∝ X ∗ · Z · ( δg v,~q + δg w, − ~q ) ,I ( B ) ( ω ) ∝ Y ∗ · Z · g t,~p · ( δg v,~q + δg w, − ~q ) ,I ( C ) ( ω ) ∝ Y ∗ · Z · δg t,~p · (1 + g v,~q + g w, − ~q ) . (C12)While the inversion symmetry allows all these contri-butions, the time-reversal symmetry only allows theirreal parts: the directional dichroism. In addition, inthe special situation that v = w , namely if the createdtwo spinons are in the same band, obviously I ( A ) ( ω ) = I ( B ) ( ω ) = 0 due to Eq.(C8) and only I ( C ) ( ω ) is nonzero. FIG. 4. The fit log( W ~q,~p / [ p y · ( ζea g s µ B )]) = log( u ) − q E ,~q − ∆ / ∆ (i.e., Eq.(C14) with ~ u = u ˆ y = u ζea g s µ B ˆ y )with only one fitting parameter u . In each case 696 datapoints with both q E ,~p − ∆ / ∆ and q E ,~q − ∆ / ∆ between0 . . ~q (but different ~p ), the visibly different data points are muchfewer. We set A = 1, and consider three cases of different SOCstrength: case-(a): D z = D p = 0 . J (and µ = − . J );case-(b): D z = D p = 0 . J (and µ = − . J ); case-(c) D z = D p = 0 . J (and µ = − . J ). Notice that for eachcase the chemical potential µ is tuned so that the spinon gap isfixed to be ∆ = 0 . J . As shown in this figure, we numericallyfind that u = 0 . u = 0 .
151 = 0 . · . u = 0 .
603 = 0 . · .
99 in case-(c). Thescaling u ∝ ( D/J ) is confirmed. Focusing on the low temperature/energy TNDD spec-troscopy, one may consider the contribution v = w = t = 1 from the lowest energy band only (see Fig.2 for aplot of the band structure), and compute I ( ω ) = I ( C ) ( ω )analytically. In this case: I ( C ) ( ω ) = ωA X ~q,~p W ~q,~p · δg ,~p (1 + 2 g ,~q ) δ (2 E ,~q − ~ ω ) , where W ~q,~p ≡ X α,α ′ ,β ( Y ,α, ,α ′ , ,β~q,~p ) ∗ · Z ,α, ,α ′ ~q . (C13) W ~q,~p is a real function satisfying W ~q,~p = − W − ~q, − ~p dueto the inversion symmetry. Taylor expanding near the Γ-point, to the leading order one expects: W ~q,~p ≈ ~ u · ~p + ~ v · ~q .In fact, interestingly, we numerically found that W ~q,~p canbe well described as W ~q,~p = ( ~ u · ~p ) e − √ E ,~q − ∆ / ∆ (C14)in the momentum regime where the relativistic disper-sion Eq.(10) holds (see Fig.4 for details). We do not at-tempt to analytically justify Eq.(C14) here since it devi-ates from the main purpose of this paper. Eq.(C13,C14) then lead to: I ( C ) ( ω ) = ωA X ~p ( ~ u · ~p δg ,~p ) · X ~q e − √ E ,~q − ∆ / ∆ (1 + 2 g ,~q ) δ (2 E ,~q − ~ ω ) . (C15)Crystal symmetry and dimensional analysis show that ~ u = u ˆ y = u ζea g s µ B ˆ y , consistent with the η responsein Eq.(5). The dimensionless number u is expect to be ∼ ( D/J ) and can be determined numerically (see Fig.4for details).With Eq.(C7,C10,10,C15) the low temperature/energyTNDD response can be computed within our mean-fieldtreatment: δ ˆ y α ( ω ) = C · (cid:2) g ( ~ ω/ (cid:3) · ( k B T ) · (cid:2) G ( z ) − z · G ( z ) + (ln z ) G ( z ) (cid:3) · e − √ ( ~ ω/ − ∆ / ∆ · ~ ω · JDOS ( ~ ω ) · ∇ y T · τ · vT . (C16)This is just the Eq.(11) in the main text.We can apply the estimate in the previous section tothe present example as follows. We firstly estimate α ˆ n due to the electric dipole processes following the goldenrule: α ˆ n ( ω ) ∼ n r ǫ c π ~ ( ζea ) [1 + 2 g ( ~ ω/ ~ ω · JDOS ( ~ ω )= 16 π n r α ζ a [1 + 2 g ( ~ ω/ ~ ω · JDOS ( ~ ω ) , (C17)where n r is material’s relative refractive index. For thesituation with k B T ∼ J ∼ ~ va and ~ ω ∼ T N DD ( ω ) in Eq.(2): T N DD ( ω ) ∼ α a ζa · u ∇ y T · τ · vT , (C18)confirming the estimate Eq.(3) since u ∝ ( D/J ) .Finally, we would like to remark on the validity of themean-field treatment. Although we performed the calcu-lation within the mean-field approach, the main com-ponent of the calculation (Eq.(C12,C13) in App.C) isjustified as long as the quasiparticle description is valid.These microscopic contributions to TNDD can be writtendown phenomenologically as a low quasiparticle-densityexpansion, up to the second order ∝ g ~p · g ~q . Some othercomponents of the calculation (e.g., the matrix elementbehavior Eq.(C14) ) may receive corrections moving be-yond the mean-field approximation, but these would notchange the result of TNDD response qualitatively. P. Anderson, Materials Research Bulletin , 153 (1973). P. A. Lee, Journal of Physics: Conference Series , 012001 (2014). Y. Zhou, K. Kanoda, and T.-K. Ng,Rev. Mod. Phys. , 025003 (2017). L. Savary and L. Balents,Reports on Progress in Physics , 016502 (2016). C. Broholm, R. J. Cava, S. A. Kivelson, D. G.Nocera, M. R. Norman, and T. Senthil,Science (2020), 10.1126/science.aay0668,https://science.sciencemag.org/content/367/6475/eaay0668.full.pdf. X.-G. Wen, Phys. Rev. B , 165113 (2002). X.-G. Wen, Rev. Mod. Phys. , 041004 (2017). M. P. Shores, E. A. Nytko, B. M.Bartlett, and D. G. Nocera,Journal of the American Chemical Society , 13462 (2005),pMID: 16190686, https://doi.org/10.1021/ja053891p. M. R. Norman, Rev. Mod. Phys. , 041002 (2016). Y. Kasahara, T. Ohnishi, Y. Mizukami, O. Tanaka, S. Ma,K. Sugii, N. Kurita, H. Tanaka, J. Nasu, Y. Motome,T. Shibauchi, and Y. Matsuda, Nature , 227 (2018). M. Hermele, M. P. A. Fisher, and L. Balents,Phys. Rev. B , 064404 (2004). M. J. P. Gingras and P. A. McClarty,Reports on Progress in Physics , 056501 (2014). Or time-reversal symmetry combined with a spatial trans-lation such as in an antiferromagnet. R. Fuchs, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics , 647 (1965),https://doi.org/10.1080/14786436508224252. D. Szaller, S. Bord´acs, V. Kocsis, T. R˜o om, U. Nagel, andI. K´ezsm´arki, Phys. Rev. B , 184419 (2014). In general NDD receives contributions from higher or-der multipole processes. However in the context of Mottinsulators the electric-dipole-magnetic-dipole contributionEq.1 dominates. J. Goulon, A. Rogalev, C. Goulon-Ginet, G. Benayoun,L. Paolasini, C. Brouder, C. Malgrange, and P. A. Metcalf,Phys. Rev. Lett. , 4385 (2000). M. Kubota, T. Arima, Y. Kaneko, J. P. He, X. Z. Yu, andY. Tokura, Phys. Rev. Lett. , 137401 (2004). T. Arima, Journal of Physics: Condensed Matter , 434211 (2008). I. K´ezsm´arki, N. Kida, H. Murakawa,S. Bord´acs, Y. Onose, and Y. Tokura,Phys. Rev. Lett. , 057403 (2011). Y. Takahashi, R. Shimano, Y. Kaneko, H. Murakawa, andY. Tokura, Nature Physics , 121 (2012). Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota,S. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, andY. Tokura, Nature Communications , 2391 (2013). I. Kzsmrki, D. Szaller, S. Bordcs, V. Koc-sis, Y. Tokunaga, Y. Taguchi, H. Murakawa,Y. Tokura, H. Engelkamp, T. Rm, and U. Nagel,Nature Communications , 3203 (2014). S. Toyoda, N. Abe, S. Kimura, Y. H. Matsuda,T. Nomura, A. Ikeda, S. Takeyama, and T. Arima,Phys. Rev. Lett. , 267207 (2015). Y. Tokura and N. Nagaosa,Nature Communications , 3740 (2018). P. Jung and A. Rosch, Phys. Rev. B , 245104 (2007). I. Affleck and J. B. Marston,Phys. Rev. B , 3774 (1988). M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee, N. Na-gaosa, and X.-G. Wen, Phys. Rev. B , 214437 (2004). Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen,Phys. Rev. Lett. , 117205 (2007). O. I. Motrunich, Phys. Rev. B , 045105 (2005). S.-S. Lee and P. A. Lee, Phys. Rev. Lett. , 036403 (2005). Similar to a thermal transport experiment, if the temper-ature of the system is far below the magnetic excitationenergy, a temperature gradient would not efficiently affectthe magnetic excitation distributions and would not leadto a sizable TNDD. L. N. Bulaevskii, C. D. Batista, M. V. Mostovoy, and D. I.Khomskii, Phys. Rev. B , 024402 (2008). We only consider the contribution from the spin magneticmoment in this paper. The orbital magnetic moment in aMott insulator is a spin-singlet but is much smaller thanthe spin magnetic moment, by a factor of ( t/U ) in the( t/U )-expansion. . Y. Tokiwa, T. Yamashita, D. Terazawa, K. Kimura,Y. Kasahara, T. Onishi, Y. Kato, M. Halim, P. Gegenwart,T. Shibauchi, S. Nakatsuji, E.-G. Moon, and Y. Matsuda,Journal of the Physical Society of Japan , 064702 (2018),https://doi.org/10.7566/JPSJ.87.064702. G. Jackeli and G. Khaliullin,Phys. Rev. Lett. , 017205 (2009). J. c. v. Chaloupka, G. Jackeli, and G. Khaliullin,Phys. Rev. Lett. , 027204 (2010). H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, andS. E. Nagler, Nature Reviews Physics , 264 (2019). Y. Singh and P. Gegenwart,Phys. Rev. B , 064412 (2010). J. A. Sears, M. Songvilay, K. W. Plumb, J. P. Clancy,Y. Qiu, Y. Zhao, D. Parshall, and Y.-J. Kim,Phys. Rev. B , 144420 (2015). R. D. Johnson, S. C. Williams, A. A. Haghighi-rad, J. Singleton, V. Zapf, P. Manuel, I. I. Mazin,Y. Li, H. O. Jeschke, R. Valent´ı, and R. Coldea,Phys. Rev. B , 235119 (2015). H. B. Cao, A. Banerjee, J.-Q. Yan, C. A. Bridges,M. D. Lumsden, D. G. Mandrus, D. A. Ten-nant, B. C. Chakoumakos, and S. E. Nagler,Phys. Rev. B , 134423 (2016). M. Elhajal, B. Canals, and C. Lacroix,Phys. Rev. B , 014422 (2002). A. C. Potter, T. Senthil, and P. A. Lee,Phys. Rev. B , 245106 (2013). Generally the polarization operator contains spin-tripletterms similar to DM interactions. Here for simplicity weonly consider spin-singlet terms which dominate in theweak spin-orbit coupling limit. A. H. MacDonald, S. M. Girvin, and D. Yoshioka,Phys. Rev. B , 9753 (1988). ζ also receives contribution from the magneto-elastic cou-pling. For a typical transition metal Mott insulator, thiscontribution to polarization is similar in size as the contri-bution from the t/U -expansion . S. Yan, D. A. Huse, and S. R.White, Science , 1173 (2011),https://science.sciencemag.org/content/332/6034/1173.full.pdf. S. Depenbrock, I. P. McCulloch, and U. Schollw¨ock,Phys. Rev. Lett. , 067201 (2012). Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc,Phys. Rev. B , 060405 (2013). H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D.Xie, R. Z. Huang, B. Normand, and T. Xiang,Phys. Rev. Lett. , 137202 (2017). Y.-C. He, M. P. Zaletel, M. Oshikawa, and F. Pollmann,Phys. Rev. X , 031020 (2017). S. Sachdev, Phys. Rev. B , 12377 (1992). D. P. Arovas and A. Auerbach,Phys. Rev. B , 316 (1988). N. Read and S. Sachdev,Phys. Rev. Lett. , 1773 (1991). S. Sachdev and N. Read,International Journal of Modern Physics B , 219 (1991),https://doi.org/10.1142/S0217979291000158. Notice that a single spinon excitation is not gauge invariantand does not contribute to physical responses. Y.-M. Lu, Y. Ran, and P. A. Lee,Phys. Rev. B , 224413 (2011). Y.-M. Lu, G. Y. Cho, and A. Vishwanath,Phys. Rev. B , 205150 (2017). R. Nandkishore and D. A. Huse,Annual Review of Condensed Matter Physics , 15 (2015),https://doi.org/10.1146/annurev-conmatphys-031214-014726. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn,Rev. Mod. Phys. , 021001 (2019). M. Levin and T. Senthil, Phys. Rev. B , 220403 (2004). D. V. Pilon, C. H. Lui, T. H. Han, D. Shrekenhamer,A. J. Frenzel, W. J. Padilla, Y. S. Lee, and N. Gedik,Phys. Rev. Lett. , 127401 (2013). A. Little, L. Wu, P. Lampen-Kelley, A. Baner-jee, S. Patankar, D. Rees, C. A. Bridges, J.-Q.Yan, D. Mandrus, S. E. Nagler, and J. Orenstein,Phys. Rev. Lett. , 227201 (2017). F. Wang and A. Vishwanath,Phys. Rev. B , 174423 (2006). Y. Gao and D. Xiao, Phys. Rev. Lett. , 227402 (2019). O. I. Motrunich, Phys. Rev. B73