Experimental Progress in Thermal Hall Conductivity Research on Strongly Correlated Electronic Systems

XU Hao ,CHENG Shu-fan ,BAO Song ,WEN Jin-sheng ,2

1.National Laboratory of Solid State Microstructures & Department of Physics,Nanjing University,Nanjing 210093,China

2.Collaborative Innovation Center of Advanced Microstructures,Nanjing University,Nanjing 210093,China

Abstract:Thermal Hall effect(THE)is to describe the phenomenon where heat carriers are deflected by an external magnetic field applied perpendicular to the heat flow,and thus the carriers gain transverse velocity,leading to a finite temperature gradient on the two sides orthogonal to the heat flow and field.THE is predicted to occur in systems with nontrivial Berry curvatures and thus can reveal topological properties,similar to the electrical Hall effect.However,THE is not limited to charge excitations as in the electrical Hall effect,but rather,to all kinds of excitations that are able to conduct heat,making it possible to explore the exotic properties in strongly correlated electronic systems,which are typically insulators.Therefore,THE is more universal than the electrical form and has become a powerful probe in detecting charge-neutral excitations,such as phonons and magnons.Moreover,there are some sources such as chiral phonons,which are beyond a simple nontrivial-Berry-curvature scenario,that can also give rise to THE;examining THE wherein will shed light on the complex microscopic mechanism hidden in materials.Despite these,heat signals are much weaker than electrical ones.Especially for measurements of the thermal Hall conductivity,it is often needed to collect weak signals on top of a large background.This makes measuring the THE challenging—but thanks to the sustained efforts of the community,this field is developing rapidly in recent years,with many interesting results on the measurements of the thermal Hall conductivity.In this review article,we try to summarize some of these exciting accomplishments,point out remaining outstanding issues,and suggest possible future directions.

Key words:thermal Hall effect;topology;quantum spin liquid;multiferoics;pseudogap phase

CONTENTS

I.Introduction 160

A.Background 161

1.Fourier’s Law 161

2.Onsager’s Rule 161

B.Measurements 161

II.THE in Strongly Correlated Electronic Materials 162

A.Magnon Hall Effect Induced by DMI 162

1.Lu2V2O7163

2.Cu(1,3-bdc) 164

3.Perovskite Oxides 165

B.Phonon Hall Effect 167

C.Materials with Controversial Origins of THE 168

1.α-RuCl3168

2.Multiferroics(Fe1-xZnx)2Mo3O8174

3.Cuprate High-Temperature Superconductors 175

III.Conclusion and Outlook 176

Acknowledgments 179

References 179

I.INTRODUCTION

The ordinary Hall effect in conductors is caused by the transverse Lorentz force acting on free electrons when a perpendicular magnetic field is applied.The magnetic field and the Lorentz force play an essential role here to deflect the electrons.However,with the discovery and explanation of quantum Hall effect[1–3]and fractional quantum Hall effect[4–6],the nonzero Berry curvature Ω has been considered as the root cause of the transverse velocity and the magnetic field is just one of the approaches to generate the transverse component,which is especially true in real crystalline materials when carriers are treated as quasiparticles.In the topological band theory[7–13],quasiparticles can gain transverse velocity from nonzero Berry curvature,highly similar to the electrons gaining transverse velocity due to the Lorentz force.The overall effect,the Hall conductivity,is determined by the integral of the Berry curvature of the energy band over the occupied Brillouin zone.In some ferromagnetic materials,with the help of spontaneous magnetization,the Berry curvature can be nonzero even when a perpendicular magnetic field is absent,and the Hall conductivity can be nonzero(anomalous Hall effect[14,15])or even quantized(quantized anomalous Hall effect[16])without a magnetic field,suggesting again the close relationship between the Berry curvature and the Hall conductivity.Importantly,the Berry curvature is intricate to the wavefunctions of the energy bands,nothing to do with the constituent quasiparticles.Thus it is plausible that spins,as well as charge-neutral excitations such as magnons and phonons,can also result in the Hall effect as long as their energy bands are with nontrivial Berry curvatures[17–20].For spins,there are spin Hall effect[21–23]and quantum spin Hall effect[9,24–26].For magnons and phonons,since they are both heat carriers,such a process can result in a“thermal Hall effect”(THE),which describes the generation of a finite temperature gradient on the two sides of the material along the direction orthogonal to the applied heat current and magnetic field,analogous to the electrical Hall effect.THE is originally considered to occur in conductors where electrons are dominant heat carriers.In that case,THE is simply caused by Lorentz force,just the same as the Hall effect.It is a repetitive phenomenon and the explanation is a little trivial.Involved with topological band structure,however,THE is predicted to occur even in insulators and that is the attractive and interesting point.Therefore,in our discussions,“THE”is actually referred to that in insulators.Note that although the magnetic field does not generate the Lorentz force for THE,a field perpendicular to the heat current is still needed,in analogy to the electrical Hall effect.This will be discussed in detail later.Likewise,an anomalous THE version can also occur,which does not require the application of an external field[27].As such,THE greatly expands the range of transport measurements from electrical to thermal form,which is in principle applicable to any quasiparticle carrying thermal energy.It is particularly helpful in unveiling the exotic properties of strongly correlated electronic systems,which are mostly Mott insulators with frozen charge degrees of freedom,difficult for electrical transports to access[28,29].

There are various models to generate nontrivial Berry curvatures that can be responsible for THE.For example,models with spin-spin interaction–Dzyaloshinskii-Moriya interaction[28,30–34](DMI),spinlattice coupling[35–37]or simply ionic bonding[38]are suggested to explain or predict THE.Apart from these,other factors beyond Berry curvatures can also lead to a finite or even huge THE,such as skew scattering[39,40],where scattering is much enhanced by antiferrodistortive transition,and chiral phonons as suggested in cuprates[41].As a consequence,THE opens a door for exploring materials concerning magnetic interactions and spin-phonon interactions.It also provides a probe to detect exotic charge-neutral excitations like Majorana fermions and to reveal the mystery microscopic mechanism through which we can further understand the nature of various unusual properties in materials.

Despite the many advantages of THE,measurements of the thermal Hall conductivity have been challenging due to the weak THE.It is required substantial efforts to enhance the weak THE response while maintaining a low background level.Due to the weak signals,many extrinsic factors such as the sample quality,contacts,and environmental noise can have signif-icant impacts on the measurement results.Nevertheless,a lot of fascinating experimental progress on THE in strongly correlated electronic systems has been made in recent years,providing many insights into the quantum physics underlying these materials.This review aims at reflecting on some of the interesting developments of the THE experiments.

The rest of the paper is organized as follows.In Sec.I.,we introduce the background knowledge including the law of heat conduction(also known as Fourier’s Law),Onsager’s rule,and some measurement methods.In Sec.II.,we discuss thermal Hall transport observed in DMI systems,pure phonon systems,and in the materials such asα-RuCl3,multiferroics,and superconductors,where the origin of the thermal Hall conductivity is still under debate.In Sec.III.,we make a brief conclusion of the review and provide some perspectives for future research.

A.Background

1.Fourier’s Law

Analogous to the electrical case which is defined by Ohm’s Law,thermal conductivity is defined by Fourier’s Law.

In Ohm’s Law,j,σ,andErepresent electrical current density,conductivity,and electric field,respectively,while in Fourier’s Law,q,κ,and?Tare the heat flux,thermal conductivity,and temperature gradient,respectively.Generally,σandκare matrices because of anisotropy.The inverse matrix ofκis defined as the thermal resistivity matrix,ρ.If we only consider two dimensions:

whereκxyis the thermal Hall conductivity.

2.Onsager’s Rule

Onsager’s Rule[42,43]reveals the microscopic timereversal symmetry.As shown in Eq.2,even though we only consider two dimensions,we have four independent coefficients unsettled.Onsager’s Rule points out that when choosing the correct thermodynamic forceXand fluxJ[43],the linear response coefficients confirmLij(H)=Lji(-H).That means in zero fields,the coefficients must be symmetric withLij=Ljiwhile the antisymmetric terms can occur only when a magnetic field is applied.For heat flowq,the corresponding thermodynamic force could be?Tor.The former is usually taken for convenience.

B.Measurements

Fig.1 shows the schematic of the measuring apparatus,where there is a heater providing heat flow and a heat bath which is usually Copper or LiF in order to avoid the thermal Hall background,for the thermal Hall conductivity in these two materials is substantially zero.Applying a perpendicular magnetic fieldH,the misalignment of the thermometersTBandTCcan be eliminated by using

when the temperature gradientΔTcaused by the misalignment is field symmetric.When the longitudinal signal is hysteretic,we need to use

instead[44],whereΔ*TmeansHchanges in an opposite direction compared toΔT.For example,ifΔTis measured withHsweeping from negative to positive,thenΔ*Tshould be sweeping from positive to negative.

In most cases,when applying a heat flow inxaxis(qy=0),|?xT|ought to be much lager than|?yT|.Additionally,in order to obtainκxxandκxysimultaneously in one measurement,we assume thatκxx=κyy,andκxy=-κyx.Thus the thermal conductivity can be calculated using Eqs.5 and 6,deduced from Eq.2:

and

Fig.1.Schematic of the apparatus for the thermal Hall conductivity measurements.TA,TB,and TC denote the temperature read from the corresponding thermometers.q and H are the heat current and magnetic field applied,respectively.

We note that in the formulas above,it is assumed that the thermal conductivity is isotropic in the plane.In the case where there is an in-plane anisotropy but the anisotropy is not so strong,likeκyy≈ακxx(αis close to 1)[45],we can also get similar results withκxx=-qx/?xT,andκxy=-ακ2xx?yT/qx.It should be noted thatκxx≈-qx/?xTworks for most cases as long as|?xT|?|?yT|andκxyis not extremely larger thanκxx,analogous to the Ohm’s LawRxx=Ux/Ix.

The constraintκxy=-κyxis typically used for three reasons.First,note that the off-diagonal termκxycan be decomposed into two parts,the symmetric partSxyand the antisymmetric partAxy,where

For two-dimensional materials(κzi=κiz=0,i=x,y,z),Eqs.5 and 6 and the measuring methods are undoubtedly correct,but for the three-dimensional materials the difference between the two cases needs to be considered.The numerical difference must be small in order to make sure Eqs.5 and 6 still work.For instance,when only consideringκij=-κji,

and therefore high-symmetry axis is preferred to be chosen as thex/y/z-axis so that the diagonal terms are much larger than the off-diagonal terms,making the differenceΔnegligible.After discussing some basic knowledge about thermal Hall conductivity,we will next show some measurement results of THE on some strongly correlated electronic systems.

II.THE IN STRONGLY CORRELATED ELECTRONIC MATERIALS

A.Magnon Hall Effect Induced by DMI

Dzyaloshinskii-Moriya interaction(DMI)(Eq.8)is a type of antisymmetric spin coupling firstly sug-gested by Dzyaloshinskii purely from symmetry perspectives in order to explain weak ferromagnetism of mainly antiferromagnetic crystals,such asα-Fe2O3[30].Moriya further suggested the explicit form of the“DM vector”Dij(Eq.9)and Moriya’s Law[31].Later on,Fert and Levy[32]suggested the Ruderman-Kittel-Kasuya-Yosida(RKKY)interaction of the DM type(Eq.10).Moriya’s Law and Eq.10 both point out the direction of the DM vector.

The magnon Hall Effect induced by DMI is closely related to the geometry and symmetry of the system.The DMI acting on the magnons or spin waves is just like a gauge field,providing a phase when magnons flow in the lattice[33].Moreover,the phase is only determined byDij·n,wherenis the orientation of the spins and the indexes refer to the sites of the magnetic ions.Thus,it is obvious that how the spins are displaced and oriented in the lattice plays an important role here.

In recent studies,it is discovered that the DMI induced thermal Hall effect appears in perovskite oxides ABO3[33])and those with kagome lattice[34,46]such as pyrochlore oxides A2B2O7[28,33].They are all ferromagnetic insulators.By the way,it can be proved that the ferromagnetic ground state is stable against the DMI as long asDij=0[33].

1.Lu2V2O7

One of the typical materials that exhibit the magnon Hall effect induced by DMI is Lu2V2O7.The first observation and the successful theoretical explanation soon attracted intense attention and set offa series of research on this subject[28].The vanadium sublattice structure can be viewed as a stacking of alternating kagome and triangular lattices along the[111]direction.With the kagome lattice,nonzero DM vectors are predicted to occur.The lattice structure and the direction of DM vector are both shown in Fig.2(a).

Theκxyof Lu2V2O7as a function of the magnetic field at different temperatures is shown in Fig.2(b).It becomes nonzero belowTc=70 K and reaches the maximum whenT=50 K.As the temperature decreases,κxygets smaller and finally disappears below 10 K.Besides these,one key feature is thatκxydecreases as the magnetic field increases after a threshold field,which is evident at 50 K and below.Since Lu2V2O7is an insulator,the heat carriers can only be phonons or spins.If phonons dominate the conduction of heat,κxyought to increase with the field because the fluctuation of spins reduces,resulting in a longer mean free path of phonons due to the weakening of scattering from spin fluctuations.The magnon scenario,where the population of magnons decreases as the magnetic field increases,appears to be more consistent with the observation of the decline of the thermal Hall conductivity.

Moreover,theoretical calculations based on this scenario,as is shown in Fig.2(c),agree with the experimental data very well.The theoretical formula[28],

is derived from the Thouless-Kohmoto-Nightingale-denNijs(TKNN)-type formula for non-interacting bosons and it is corrected to

Fig.2.(a)The crystal structure of Lu2V2O7 and the direction of the DM vector Dij on each bond of a tetrahedron[28].(b)Magnetic field variation of the thermal Hall conductivity of Lu2V2O7 at various temperatures[28].Solid lines are guides to the eye.(c)Magnetic field variation of the thermal Hall conductivity at 20 K for H‖[100][28].The red solid line indicates the magnetic field dependence given by the theory(Eq.11).(d)Magnetic field variation of thermal Hall conductivity for Lu2V2O7,Ho2V2O7,In2Mn2O7[33].Red solid lines are given by the theory(Eq.12).

considering the rotational motion of magnons[33,47].The corrected theoretical outcome is compared with the experimental data in Fig.2(d).In other similar materials,Ho2V2O7and In2Mn2O7,κxyare calculated and compared with experiments[33].The high consistency between experimental and theoretical results strongly supports the magnon scenario of THE in Lu2V2O7.The successful theoretical explanation connects the DMI and the Berry curvature,pointing out a new way to generate nontrivial Berry curvature,and all of these can be verified experimentally by THE by measuringκxy.

2.Cu(1,3-bdc)

Cu(1,3-bdc)is another material with nonzero DM vectors.In this example,we can see clearly how the magnetic field induces nontrivial Berry curvatures and consequently a finite THE.As illustrated in Fig.3(a),Cu(1,3-bdc)is a kagome ferromagnet with weak anti-ferromagnetic interlayer interaction and strong in-plane ferromagnetic interaction[34].The moments lie in the kagome plane but it can be fully polarized easily along thec-axis with a small magnetic field(μ0H≈0.05 T),as shown schematically in Fig.3(b).The DM vectors point along thec-axis and it is convenient to make spins parallel toD.

Fig.3.Magnetic structure of Cu(1,3-bdc)without(a)and with(b)a perpendicular magnetic field[34].

In Ref.[46],it is shown that the longitudinal thermal conductivityκxxof Cu(1,3-bdc)grows abruptly aroundT=Tc=1.8 K(Fig.4(a))and becomes field dependent,implying spins contribute to the thermal conductivity belowTc.By subtracting the phonon backgroundκphmeasured in the high field fromκxx,one can obtain the part contributed by spins as defined byκs≡κxx-κph.The thermal Hall conductivity is also measured.When a magnetic field parallel toc-axis is applied,the thermal Hall conductivity grows from zero and it is an odd function ofHc.It is interesting that bothκsand the thermal Hall conductivityκxyhave the activated form[Fig.4(b)],indicating both come from the same carriers–magnons[46].

Chisnell et al.later on showed inelastic neutron scattering data on the magnon bands which are helpful in understanding the thermal and thermal Hall conductivity results[34].The experimental data in zero and finite fields are shown in Fig.4(c1-c3).It is clear to see that an external field perpendicular to the kagome plane opens two gaps separating the three magnon bands(Fig.4(c2)and especially Fig.4(c3)).They also calculated the spin-wave dispersion using the Holstein-Primakofftransformation.The calculated spectra are shown in Fig.4(c4).Compared the calculations with the experimental data,it is found that gaps opened by the perpendicular magnetic field lead to nontrivial magnon bands with Chern numbers-1,0 and 1 for the lowest,middle,and highest energy bands,respectively.In addition,they compared the calculated dynamic structure factor for the spin wavesS(Q,ω)with neutron scattering data.In order to avoid nonmagnetic background,they examined the differences in densities at two different applied fields,as shown in(Fig.4(d)).It is pleasing to see that the theory and the experiment are in high agreement.

In this example,the magnetic field plays such a role that it no longer acts directly on the quasiparticles,providing the Lorentz force,but rather“activates”the DMI by turning the direction of spins to that of the DM vectors.It is the DMI,as a perturbation to the nextnearest Hamiltonian,that opens the gaps between three magnon bands,yielding topologically nontrivial bands.And therefore,the nontrivial Berry curvature or the possible chiral edge modes can be responsible for the magnon THE[34].

3.Perovskite Oxides

The geometry and symmetry of lattice are able to have a great impact on the Berry curvature.In ourdiscussions above,the symmetry restrains the direction of DM vectors through Moriya’s Law and thus affects the thermal Hall conductivity.In this example,the influence of geometry and symmetry on the thermal Hall conductivity can be further clarified through a comparison between two different perovskite oxides with similar lattice structures but with actually different lattice geometries.The first material is BiMnO3,the structure of which is distorted due to Bi 6s2lone pair[48],which lowers the symmetry of the whole crystal and allows a nonzero-DM vector.Another important factor is its very special orbital ordering[33].It enlarges the unit cell of BiMnO3,in which there are 16 Mn atoms.The second material is YiTiO3,which has the GdFeO3-type orthorhombic crystal structure[33].Usually there is antiferromagnetic interaction between the nearest-neighboring magnetic moments of transition metals in perovskite Mott insulators[33],while the two materials above both stabilize the ferromagnetic order by orbital ordering[33,49].

It is of great importance that in YiTiO3,the magnon Hamiltonian containing DMI is invariant under the following transformation(Eq.14),from which the Berry curvature can be proved to be zero when the magnetic field is applied alongxorz-axis.

Eq.12 shows that if the Berry curvature vanishes,the thermal Hall conductivity must be zero.And it is consistent with the experimental data,where BiMnO3has nonzero thermal Hall conductivity while YiTiO3does not,as shown in Fig.5.

Fig.5.Magnetic field variation of the thermal Hall conductivity at different temperatures of BiMnO3(a)and YiTiO3(b)[33].

B.Phonon Hall Effect

Since the phonon Hall effect was first observed in Tb3Ga5O12[50],it has not attracted much attention.One of the reasons is that phonon is charge neutral thus it cannot be directly affected by magnetic fields.Therefore,the phonon Hall effect has been considered very weak.However,it is found that in SrTiO3,a paraelectric and nonmagnetic insulator(which means phonons are the only possible carriers),theκxycan reach up to 80 mW/(K·m)[51],far exceeding the expectation.The discoveries and explanation of the enhanced phonon Hall effect have broadened the horizon of THE and shed new light on the materials with giant THE.

The experimental results of thermal and thermal Hall conductivity on SrTiO3are shown in Fig.6(a)and(b).It is found thatκxyvaries with samples butκxxandκxyalways peak at the same temperature(implying the same dominant heat carriers)andκxydecreases much faster thanκxxat both sides of the peak.In addition,it seems thatκxycan even be changed after warming up above the antiferrodistortive transition temperature and cooling back for the same sample.In order to clarify the origin of the large THE,the heat transport in KTaO3is also investigated.Compared with SrTiO3,KTaO3does not undergo an antiferrodistortive transition and its low-temperature electric permittivity is five times smaller.As is shown in Fig.6(d),the thermal Hall conductivity of KTaO3is much smaller than that of SrTiO3.Here comes a question that whether it is the antiferrodistortive transition(the domains)or the electric permittivity that accounts for the large phonon Hall effect.

Later,Chen et al.elucidated that it was the domain that resulted in the enhanced THE[40].The team first considered the intrinsic thermal Hall conductivity which,similar to the magnon Hall effect,originated from the nonzero Berry curvature and was totally determined by the Berry curvature and dispersions of the phonon bands[20].Though amplified by the electric permittivityχ≈2×104,the intrinsic thermal Hall conductivityκinxyas calculated by Eq.15 is still four orders of magnitude smaller than the observed value.

whereκHis the intrinsic thermal Hall conductivity,κLthe thermal conductivity,χthe electric permittivity,?0the permittivity of vacuum,Fthe flexoelectric coupling coefficient,Bthe magnetic field,Tthe temperature,Kthe elastic moduli,cthe speed of light,ˉhthe reduced Planck constant,vthe sound velocity and?is the phonon mean free path.

Since the intrinsic factors alone cannot explain the huge THE,the team soon resorts to extrinsic factors.They assume there exist dilute defects(in STO it couldbe the domains coming from the AFD transition)that scattered phonons strongly and Eq.15 can be modified to Eq.16,

Fig.6.κxx and κxy as a function of temperature for SrTiO3((a),(b))and KaTiO3((c),(d))[51].

whereis the extrinsic thermal Hall conductivity andAa dimensionless strength to describe how much the Berry curvature would affect the skew scattering.

After taking into account the extrinsic factors,the thermal Hall conductivity that originated from the nontrivial Berry curvature is much enhanced.The resultingκxyhas a similar magnitude as the experimental data.Furthermore,the new formula also predictsκxy∝T4sinceκxx∝T3,which explains the observation thatκxydecreases faster thanκxxon the low-temperature side.Additionally,it can also explain the absence ofκxyin KaTiO3,as there is no antiferrodistortive transition in KaTiO3,and thus no antiferrodistortive domains to scatter phonons.

C.Materials with Controversial Origins of THE

As this is still the early stage on the research of THE,the origin of thermal Hall conductivity in various materials is still mysterious,or under debate.In the following,we discuss THE in quantum-spin-liquid candidateα-RuCl3,multiferroic Fe1-xZnxMo3O8,and cuprate superconductors La2-xSrxCuO4as examples.

1.α-RuCl3

Quantum spin liquid is a special state where there exists a high degree of entanglement and fractional excitations[52,53].Excitations carrying fractional quantum numbers manifest novel topological properties,and the predicted non-Abelian quasiparticles can have applications in quantum computation[54–56].Furthermore,studying quantum spin liquids may shed light on the mechanism of the high-temperature superconductivity[57].To explore the exotic properties in quantum spin liquids,people have been keeping searching for candidate materials in frustrated magnets[52],which mainly include materials with triangular lattice[58–62],kagome lattice[53,63–67],and honeycomb lattice[68–70].Among them,α-RuCl3has been studied extensively recently[71,72].

The materialα-RuCl3is quasi-two-dimensional with Ru atoms forming a honeycomb lattice and each magnetic atom showsJeff=1/2 because of spin-orbit coupling and crystal field[71,73,74].The magnetic behavior can be described by a minimal“K-?！眒odel[75],in which theΓand possibly other smaller longer-range terms stabilize a zigzag antiferromagnetic state belowTN≈7.5 K without magnetic field[76].The in-plane lattice structure is shown in Fig.7(a).Thebaxis(the armchair direction),possesses a two-fold symmetry while thea-axis does not.Fig.7(b)and(c)show different stacking forms,and particularly,dashed lines in Fig.7(b)and(c)indicate a perfect and faulty stacking,respectively.The latter is common in quasi-twodimensional materials.The stacking fault will suppress the transition signal of the as-grown single crystal atTN1and induce another transition at a higher temperatureTN2,making the curve closer to that of the powder,as shown in Fig.7(d)-(f).

In this material,there have been three different perspectives on the type of heat carriers by now.The first one is the Majorana fermions,whose antiparticles are themselves.The Majorana fermions are predicted to occur in the Kiaev quantum spin liquids and they are proposed to exist inα-RuCl3[78].Since Majorana fermions are charge-neutral particles,they can only conduct heat,with the thermal Hall conductivity having the special form:

where(dis the interlayer distance),kB,,andc=1/2 are the Boltzmann constant,reduced Planck constant,and the central charge inα-RuCl3[81,82],respectively.

It was first reported that,applying a perpendicular magnetic fieldμ0Hz=12 T,theκxychanged sign from negative to positive when the temperature rose above theTN,indicating different origins ofκxybefore and after the transition[83].The field-dependentκxyalso implies spin excitations contribute significantly in the system.Then,the plateau ofκxy/T(Eq.17)was observed with the magnetic field tilted in theacplane,and the heat flowqparallel to thea-axis[81],as shown in Fig.8.Although the plateau can only exist below 5.5 K,it strongly indicates the existence of the Majorana fermions.The phase diagram is given in Fig.9 accordingly.Bruin et al.[84]then conducted a comprehensive study of the thermal Hall conductivity and demonstrated that the approximate half-integer quantization existed in a much-extended region of the phase diagram,which proved the robustness of the quantum plateau.

Later on,Yokoi et al.present another piece of evidence trying to strengthen the Majorana point[27].First,they compare the results with fields applied alongaandb-axes atT=4.8 K.Withq‖a,the plateau occurs only when the field is applied along thea-axis with field strength exceeding the critical field that fully suppresses the magnetic order,as shown in Fig.10(a).The result is explained as follows:the absence ofκxyforH‖bwas attributed to the two-fold symmetry of theb-axis,

whereUis the rotation matrixThe plateau which preserves without a perpendicular field implies that the thermal Hall conductivity arises from a topologically non-trivial Chern insulator of Majorana fermions.

Second,they compare the results with fields applied alongaand-a-axes,but with a finite outof-plane field component,as shown in Fig.10(b).The result is indeed interesting.After the critical field,κxyin different conditions are both quantized but exhibit different signs,while below the critical field,κxyhas the same sign and is nearly the same in quantity.The former is precisely explained by the Kitaev theory,which predicts the sign changes with different directions of the magnetic field,as shown in Fig.10(c).The latter is assumed to be induced by the magnon Hall effect,which is closely related to the out-of-plane magnetic field.

The results and explanations above are to some extent plausible and convincing,but the experimental results,especially the half-quantized plateau ofκxy,are still not observed by many independent groups.That is the reason why this point of view is highly controversial.

The second perspective is that the thermal Hall effect originates from the phonon Hall effect.The main reason is that according to experimental results,the curves ofκxxandκxyare similar and can be both understood by a phonon scenario.Hentrich et al.havefound unusual behavior of the thermal transport inα-RuCl3[85].They apply an in-plane magnetic field and measureκxxin thea-bplane(κab)and alongcaxis(κc).From Fig.11(a),it seems that there are two regimes according to theBandTdependences of the thermal conductivity,divided byBc≈7.5 T.In regime I,κslightly decreases as the field increases,which is especially evident forT＜TN,indicating the suppression of the long-range magnetic order.While in regime II,κapparently increases with the field.Fig.11(a2)shows that in regime II,there is always a low-temperature dip,explained by a gap originating from the magnetic excitations.(A“dip”refers to a tendency whereκgoes down first and then increases,which is not so obvious in Fig.11(a2)for thex-axis being stretched.)The regimes are distinctive,as shown in Fig.11(c).For the zero-field curve,κdecreases as the temperature increases up toTN,which implies the suppression of the long-range magnetic order.AboveTN,κincreases rapidly.The phononic heat conduc-tivity can be estimated asκph≈cVvl,where the velocityvand the mean free pathlare nearly temperature independent and the specific heatcVgrows with temperature,and thereforeκincreases with temperature.For higher temperatures,the mean free path is reduced by phonon Umklapp processes,leading to a broad peak followed by a fast decline.For theB=16 T curve,one significant signature is thatκis much larger than that in the zero fields,and another typical feature is the low-temperature dip mentioned earlier.The two features are both explained by the phonon scattering of the magnetic excitations.In regime I,magnetic excitations are low-energy and nearly gapless,and thus phonons can be scattered strongly.In regime II,magnetic energy is lifted and a gap opens,which is marked withˉhω0.Therefore,the dip appears naturally when the energy of dominant excited phonons is aroundˉhω0,where phonons are scattered strongly by magneticexcitations.In this way,ˉhω0≈kBTcan be estimated accordingly,which is shown in Fig.11(d).Furthermore,a phenomenological model considering magnetic scattering is used to fit the experimental results,which turns out to show high consistency with the experimental data,as shown in Fig.11(b).Additionally,the curves ofκabare quite similar to those ofκc,indicating the predominant heat carriers are phonons since spindominated effects will have more distinct differences for a two-dimensional material.

Fig.7.(a)The in-plane structure of layered compound α-RuCl3 viewed perpendicular to the layers along the c*-axis.(b)ABC stacking:the out-of-plane structure viewed along the b-axis.(c)ABAB stacking:viewed in the same orientation as that in(b).(d)The heat capacity of an as-grown single crystal of α-RuCl3 from 2-20 K shows the low-temperature region exhibiting one sharp Néel transition at TN=7 K.(e)The thermal evolution of the intensity of the magnetic peak(0,1,1/3).(f)The heat capacity data on the same single crystal when subject to artificial deformation(1-3).(4)is from the powder sample[77].

Fig.8.The quantum plateau of thermal Hall conductivity observed in α-RuCl3[27].

Fig.9.The phase diagram of α-RuCl3 obtained according to the results in Fig.8[81].

Fig.10.(a)Thermal Hall conductivity κxy in the antiferromagnetic(gray shaded area)and spin liquid(yellow area)states for H‖-a-axis(red circles)and H‖b-axis(blue circles)at 4.8 K.(b)Field dependence of κxy at 4.3 K in tilted field of θ=-60°(red circles)and 60°(blue circles)away from the c-axis in the a-c plane.(c)Theoretical results for the field-angular variation of the sign(Ch)of κxy[27].

Lefran?ois et al.provided another evidence for phonon Hall effect inα-RuCl3[86].κxxandκxyof five crystals from two different sources are measured,with heat currentq‖a.As shown in Fig.12,the curves ofκxxandκxyfrom five different samples are indeed similar qualitatively and the quantities are very sample dependent.The qualitative behavior ofκxxwithout a magnetic field is consistent with those in Fig.11(c).These all indicate a phononic picture and the sample dependence is attributed to different levels of disorder.

Moreover,they argued that the magnitude of the thermal Hall angle|κxy/κxx|≈1×10-3was typical for the phonon thermal Hall effect.In this spirit,κxylarger or smaller than the half-quantized value could be well explained by the different abilities of phonons to conduct heat in different materials.In addition,they compare thermal transport with and without in-plane magnetic field,as shown in Fig.12(c)and(d)and itturns out thatκxx/Tandκxy/Tno longer increased at low temperatures,which is plausible for phonons since the in-plane magnetic field could destroy the long-range order.It is also clear to see thatκxyis much smaller than the quantized valueκHQ.Therefore,from this perspective of view,phonons play an important role in the thermal transport inα-RuCl3and the plateau observed before[27,28]could only exist only if phonons’contributions are approximately zero which is unlikely to occur.

Fig.11.(a)Magnetic field and temperature dependences of the heat conductivity of α-RuCl3(sample I).(b)κab data of sample I(solid circles)and fits(solid lines)to the Callaway model for selected magnetic fields.(c)Temperature dependence of the heat conductivity of α-RuCl3 at μ0H=0 and 16 T for κab(sample I in(1))and κc(sample II in(2)).(d)Falsecolor representation of the temperature derivative?κab/?T(sample I)together with the gap energyˉhω0/kB(solid squares)extracted from the phononic fits[85].

The third and last point of view is that there might be a Fermi surface inα-RuCl3since a“de Haas–Van Alphen-like”effect is observed[44],and thus another kind of fermion rather than the Majorana fermion may come into play.In Ref.[44],they first apply a magnetic fieldHand thermal flowqboth parallel to thea-axis and find thatκxxoscillates with a field below 4.5 K,as shown in Fig.13(a).This is more clearly shown in Fig.13(b),the oscillation starts from aboutμ0H=4 T and afterμ0Hexceeds 11 T,κxxsuddenly saturates.It seems that the state between 4 T＜μ0H＜7 T is a mixture of the zigzag antiferromagnetic state and the possible quantum spin liquid state,while the quantum spin liquid and paramagnetic states have a distinct border atμ0H=11 T.Fig.13(c)shows thatκxx,regardless of the direction of the magnetic field,is periodic with the in-plane component of the field 1/μ0Ha.The oscillation is also reproducible across samples,which enhances the reliability and implies it is intrinsic in the quantum spin liquid state.Additionally,the half-quantized plateau is not observed in Fig.13(d).

A brief summary of the thermal transport results onα-RuCl3is presented here. The evidence of Majorana quasiparticles is the half-quantized plateau and the sign changes ofκxyexplained successfully by the Kiaev theory,but the phenomenon is difficult to be reproduced by other independent groups.The evidence of the Fermi surface is the oscillation ofκxxwith1/μ0Ha.The common feature in these results is that the phenomena are observed in a narrow temperature range.However,the evidence for phonons is gained by analyzing the similar behavior ofκxxandκxyin a large temperature range.

Fig.12.(a)Thermal conductivity of α-RuCl3 in zero magnetic field for five different samples.(b)Thermal Hall conductivity of the same five samples,measured in a magnetic field of μ0H=15 T applied normal to the honeycomb layers.(c)Thermal conductivity κxx of sample O2.(d)Same as in(c)but for κxy.The horizontal dashed line marks the quantized value(κHQ)expected for Majorana edge modes,divided by 10.Arrows mark the TN.[86].

2.Multiferroics(Fe1-xZnx)2Mo3O8

Multiferroics,where elementary excitations are strongly influenced by lattice-spin coupling,manifest novel thermal transport properties.Recently,the giant THE in insulating polar magnets(Fe1-xZnx)2Mo3O8is observed[87].By now,the giantκxyin this material has been assumed to be dominated by phonons,but extremely sensitive to the magnetic structure.The microscopic mechanism is still under investigations.

The crystal structure and magnetic phase diagram of(Fe1-xZnx)2Mo3O8are shown in Fig.14.It is clear to see in Fig.14(b)that each magnetic layer(the two Fe atoms with tetrahedral and octahedral oxygen coordinations)forms a honeycomb lattice,and the DM vectors(Eq.8)lie in thea-bplane.Therefore,as is the case in the following discussion,when applying a perpendicular field,the conventional magnon Hall effect vanishes.

The specific heat shown in Fig.15(a)is proportional toT3below 32 K both in zero and high magnetic fields,indicating phonon-like excitations at low temperatures.High similarity in quantities also implies that the magnetic field will not affect the population of the quasiparticles.The thermal conductivityκxxshows abnormal behavior(Fig.15(b)).The peak around 15 K is probably due to the decrease of the mean free path and the increase of the population.In this case,the suppression ofκxxbeforeTN≈60 K in the high field can only be explained by the suppression of the mean free path,which is contradictory to the phonon scenario.This contradiction is assumed to be caused by the strong lattice-spin interaction in multiferroics since the population of spin excitationsdecreases under a magnetic field.The thermal Hall conductivityκxy,as shown in Fig.15(c),also manifests different behaviors below and above the transition temperatureTN≈50 K.BelowTN,κxyreaches its maximum quickly and decreases slowly with field,while aboveTN,κxyincreases slowly with the field,indicating different mechanisms to generateκxyin different temperature regions.In addition,κxyis one-magnitude larger than the value ofκxy～1 mW/K m in common systems,while the thermal Hall angle is still small,with

Fig.13.(a)The emergence of oscillations in κxx(H‖a)at low temperatures.(b)The oscillations over the full field range at selected temperatures.(c)Curves of the derivative dκxx/dB versus 1/H(or 1/Ha)in arbitrary units(a.u.)for Samples 1,2 and 3(Ha=H cos θ).(d)Magnetic field variation of the thermal Hall conductivity at different temperatures of α-RuCl3[44].

|κxy/κxx|≈10-3.

3.Cuprate High-Temperature Superconductors

High-temperature superconductivity in copper oxides(“cuprates”)still fascinates physicists for the novel quantum phenomena in these materials.It is a common belief[88]that the basic difficulties are actually due to the unusual properties in the“normal”state above the superconducting transition temperature,without understanding which we cannot fully comprehend the nature of high-temperature superconductivity.A schematic phase diagram in Fig.16 demonstrates the complexities and difficulties we are confronted with.As is known,without doping,cuprates are Mott insulators for the strong electron-electron interactions and in the overdoped regime,cuprates become Fermi liquid,both of which can now be described reasonably well by existing theories.However,in the intermediate regime,for instance,the pseudogap phase,there are many unusual phenomena which are difficult to understand.Previous experimental facts and theoretical models have been discussed in existing reviews[88–93].Here,we will present the thermal Halltransport properties in cuprates[29,41].

Fig.14.Crystal structure of(Fe1-xZnx)2Mo3O8,side(a)and top(b)view.Magnetic phase diagram for undoped(c)and doped(d)crystals[87].

The thermal Hall conductivity in the Cu-O plane in the parent compound La2CuO4and its doped compounds La1.6-xNd0.4SrxCuO4(Nd-LSCO),La1.8-xEu0.2SrxCuO4(Eu-LSCO),La2-xSrxCuO4(LSCO)is explored[29],as shown in Fig.17(b).A large negativeκxyis observed in the normal state,with superconductivity suppressed by the magnetic field and it suddenly changes sign and becomes even larger when the hole doping levelp＞p*,as shown in Fig.17(c).The carrier of such a largeκxyis unsettled.First,it cannot be electrons since the parent compound,a Mott insulator can also generate large signals.Second,it cannot be magnons,asκxyremains even without static magnetism.Excluding other exotic excitations,a plausible guess is phonons.This scenario is then strongly suggested by the isotropy of the thermal Hall conductivity[41]in the pseudogap phase.However,we believe much more efforts need to be devoted in order to gain further insights into the mechanism underlying the larger thermal Hall conductivity in cuprate high-temperature superconductors.

III.CONCLUSION AND OUTLOOK

We have surveyed basic knowledge and some experimental facts of THE in strongly correlated electronic systems.The central point is to determine the type of the dominant heat carrier and figure out the microscopic mechanism to generate transverse velocity.The mechanism of magnon Hall effect in DMI systems and phonon Hall effect in SrTiO3can now be almost understood.The former furnishes the information that the intrinsic topological properties–nontrivial Berry curvature can generate aκxyand the latter delivers the message that phonons are able to generate an enhanced thermal Hall signal by skew scattering,which may be ahint for the origin of the giant thermal Hall conductivity in multiferroics and pseudogap phase.In multiferroic material(Fe0.875Zn0.125)2Mo3O8,the specific heat exhibits a phonon-like behavior and is scarcely affected by the magnetic field.However,κxxis suppressed by field,inconsistent to phononic scenario.More interestingly,under the circumstance where there is no conventional magnon Hall effect,quite largeκxyis observed and it exhibits different qualitative behavior above and underTN.Since it is known that in multiferroics,spin and lattice couple strongly with each other,it is feasible to ascribe these strange behavior to the strong spin-lattice coupling.In the pseudogap phase in the cuprate high-temperature superconductors,κxy/Twill go down towards negative values at low temperatures.Two important questions should be asked:first,how could the heat carriers(probably phonons)be scattered so strongly in the pseudogap phase and second,what is the difference between phases withp＜p*andp＞p*?It seems that both questions point to the mysterious microscopic mechanism of high-temperature superconductivity and as a result,THE may provide another key window for us to understand it.In the quantum spin liquid candidate materialα-RuCl3,origins of the thermal Hall conductivity are highly controversial.More experimental facts are needed to verify different opinions.

Fig.15. The specific heat(a),thermal conductivity(b)and thermal Hall conductivity(c)of(Fe0.875Zn0.125)2Mo3O8[87].

Fig.16.The phase diagram of hole-doped cuprates[94].AF,PG,CDW,FL,and SM represent the antiferromagnetic,pseudogap,charge-density-wave,Fermi-liquid,and strangemetal phases,respectively.T*refers to the temperature of the border of the PG and SM phase.The p*is the critical concentration separating the PG and SM phase.Tc denotes the superconducting transition temperature.

Based on the existing theories and experimental facts,some other interesting theories may get involved in the THE.For example,in spirit of the enhanced

Fig.17.(a)Phase diagram of cuprates,where short-range incommensurate spin order occurs below Tm.The colored vertical strips indicate the temperature range where κxy/T decreases towards negative values at low temperatures.(b)Thermal Hall conductivity versus temperature in a field μ0H=15 T.(c)κxy across the pseudogap critical point p*[29].skew scattering,the model for resonant scattering of phonons[96]are suggested,trying to explain the large THE in La2CuO4,but materials that meet the“resonant conditions”have not yet been found.The chirality of phonons[41,97,98]is another important factor to affect the Hall response.Chiral phonons are predicted to show Valley phonon Hall effect with strain gradient[97],similar to Valley Hall effect[99].However,relevant research is still lacking.

Besides specific microscopic models,a generic behavior of thermal Hall conductivity for both fermionic and bosinic systems has been analyzed[95]with reasonable assumptions,as shown in Fig.18.It is shown that in a very large temperature range relevant to typical experimental conditions,the thermal Hall conductivity exhibits a universal scaling behavior of the exponential form(e-T/T0),indicating that there may be a universal framework in the understanding of the THE.Nevertheless,unveiling the mystery beneath the interesting experimental observations is still urgently calling for comprehensive and collaborative work from both experimental and theoretical sides.

Fig.18. Sketch of the three temperature regions for the thermal Hall conductivity:the low-temperature region with saturating(const),power-law(Tα),or activation(e-Δ/T)behaviors that reflect different spectral properties of the underlying excitations;the intermediate temperature region with a universal scaling law of the exponential form(e-T/T0)for T～D,where D represents the characteristic energy for topologically nontrivial excitations;the high-temperature limit with a T-γ scaling,where γ=1 for bosons and γ=3 for fermions.Only the former two regions are experimentally relevant.[95].

The relationship between|κxy|andκxxis shown in Fig.19.In DMI systems,such as in Lu2V2O7and Cu(1,3-bdc),|κxy|scarcely changes withκxx,which indicates thatκxyis determined by the Berry curvature and is not much influenced by the mean scattering timeτ.In the meanwhile,however,the quantum spin ice compound Tb2Ti2O7shows the opposite behavior.Additionally,κxyin Fe2Mo3O8is quite large,even one magnitude larger than those in other materials,while its thermal Hall angle is still small,with|κxy/κxx|≈10-3.This similar result is also found inα-RuCl3where different samples show differentκxxandκxybut the ratios are all in the same magnitude,1×10-3[86],which is considered as evidence for phonon Hall effect.Therefore,this graph can,to some extend,furnish some important information betweenκxxandκxyand it may also help us classify different origins of thermal Hall conductivity efficiently.

Fig.19.(a)Thermal Hall conductivity|κxy|as a function of normalized temperature T/Tc(Tc denotes the magnetic phase transition temperature).(b)|κxy|versus κxx in various insulators[87].

The exploration of THE is just in the beginning.THE provides another perspective to unveil the mysteries in strongly correlated electronic materials.The current theories which can explain some phenomena in some systems are still incomplete but are being rapidly developed.With more and more materials with excellent thermal Hall transport properties being discovered and the properties being controlled easily,applications of thermal devices should definitely take a step forward.In addition,THE will help us further understand magnons,phonons as well as their interactions,which can also promote applications in spintronic and magnonic devices.

ACKNOWLEDGMENTS

The work was supported by National Key Projects for Research and Development of China with Grant No.2021YFA1400400,the National Natural Science Foundation of China with Grants No.12225407 and 12074174,China Postdoctoral Science Foundation with Grants No.2022M711569 and 2022T150315,Jiangsu Province Excellent Postdoctoral Program with Grant No.20220ZB5,and Fundamental Research Funds for the Central Universities.