Phononic-magnetic dichotomy of the thermal Hall effect in the Kitaev-Heisenberg candidate material Na$_2$Co$_2$TeO$_6$

Majorana fermions as emergent excitations of the Kitaev quantum spin liquid ground state constitute a promising concept in fault tolerant quantum computation. Experimentally, the recently reported topological half-quantized thermal Hall effect in the Kitaev material $\alpha$-RuCl$_3$ seems to confirm the Majorana nature of the material's magnetic excitations. It has been argued, however, that the thermal Hall signal in $\alpha$-RuCl$_3$ rather stems from phonons or topological magnons than from Majorana fermions. Here we investigate the thermal Hall effect of the closely related Kitaev quantum material Na$_2$Co$_2$TeO$_6$, and we show that the thermal Hall signal emerges from at least two components, phonons and magnetic excitations. This dichotomy results from our discovery that the transversal heat conductivity $\kappa_{xy}$ carries clear signatures of the phononic $\kappa_{xx}$, but changes sign upon entering the low-temperature, magnetically ordered phase. We systematically resolve the two components by considering the detailed temperature and field dependence of both $\kappa_{xy}$ and $\kappa_{xx}$. Our results demonstrate that uncovering a genuinely quantized magnetic thermal Hall effect in a Kitaev topological quantum spin liquid requires to disentangle phonon vs. magnetic contributions where the latter include potentially fractionalized excitations such as the expected Majorana fermions.

that uncovering a genuinely quantized magnetic thermal Hall effect in a Kitaev topological quantum spin liquid requires to disentangle phonon vs. magnetic contributions where the latter include potentially fractionalized excitations such as the expected Majorana fermions.
The strongly frustrated Kitaev model describes spin-1/2 degrees of freedom on a honeycomb lattice with bond-dependent interactions [1].This exactly solvable spin model attracts strong attention in the community because it exhibits a quantum spin liquid (QSL) ground state with peculiar spin excitations which fractionalize into localized Z 2 gauge fluxes (also called visons) and itinerant Majorana fermions [1][2][3] which might become exploitable for quantum memories protected from decoherence [1].Particular interest in this regard has been generated by the expectation that an external magnetic field renders the ground state a topological quantum spin liquid with chiral Majorana edge currents and a field-induced bulk gap.These edge currents are expected to give rise to a thermal Hall signal in experiments [1] with a half-quantized transversal thermal conductivity κ xy /T .For a few years, the compound α-RuCl 3 has been considered a prime candidate material [4], for which many experimental probes hint towards compatibility with the characteristics of Kitaev's model [5].Indeed, this compound exhibits a sizeable thermal Hall effect [6][7][8][9][10][11]. Interestingly, for certain ranges of temperature and magnetic field, a plateau has been reported in κ xy /T at a value which corresponds to exactly 1/2 of the quantum of the twodimensional thermal Hall conductance K QH /T = π 2 k 2 B 3h [7,9,11,12], a result which could be recognized as the key fingerprint of the topologically protected Majorana edge currents.
However, there are concerns about the uniqueness of this finding, and topological magnons [13] as well as chiral phonons [8,10] have been suggested as alternative origins of the thermal Hall effect in α-RuCl 3 .
Recently, Na 2 Co 2 TeO 6 has been proposed as a novel possible materialization of Kitaev physics in a honeycomb lattice [14,15], originating from the expectation of a bond dependent Ising-like interaction between the pseudospin-1/2 of the d 7 Co 2+ ions in high-spin 2g e 2 g configuration [14,16].The material orders antiferromagnetically below T N ≈ 27 K, and a zigzag [17] as well as a triple-q [18][19][20] ground state have been proposed following neutron scattering data.The magnetically ordered ground state implies the significance of Heisenberg and/or off-diagonal interactions additional to the expected primary Kitaev coupling [21,22].An in-plane magnetic field has been found to suppress the magnetic order at µ 0 H > µ 0 H c ≈ 10 T, in favour of strong low-energy spin fluctuations at fields near H c which are compatible with a field-induced QSL [23].Altogether, these features bear an astonishing similarity with the magnetic phase diagram of α-RuCl 3 [23].Here, we investigate the longitudinal and the transversal thermal conductivity (the thermal Hall effect), i.e. κ xx and κ xy , with the thermal current in the honeycomb plane parallel to the a axis and a magnetic field perpendicular to the plane (see Fig. 1a and methods).

Temperature dependence of κ xx and κ xy
Fig. 1b shows a comparison of κ xx at zero magnetic field, which is very similar to our previous findings [23], and at µ 0 H = 16 T with H c. Apparently, apart from a slight reduction of about 11%, the overall curve shape is preserved even at this relatively large field.Only a weak change is also observed concerning the onset temperature of magnetic ordering which we infer from the cusp in dκ xx /dT shown in the inset of Fig. 1b as T N ≈ 25 K, which is only slightly lower than in zero field [15,23].Note that an in-plane magnetic field has been reported to have a much stronger impact: κ xx becomes strongly enhanced while the magnetic order is suppressed for µ 0 H 10 T [23].We first focus on the latter, paramagnetic regime.Here the transversal heat conductivity very well follows an exponential scaling κ xy /T ∼ exp(−T /T 0 ) which recently has been proposed as a generic feature for the temperature dependence of the thermal Hall effect of charge neutral excitations [24].According to the modeling the observation of this scaling suggests a nontrivial topology of the heat carrying quasiparticles with a finite and essentially temperature independent effective Berry curvature density.Unfortunately, the exponential scaling is expected to universally hold for all types of quasiparticles and thus it does not allow to draw clear-cut conclusions about the type of the excitation causing the thermal Hall effect.However, a direct comparison of −κ xy /T with κ xx /T (see Fig. 3) strikingly yields a perfect match of both curves for T > T N , i.e., in the whole paramagnetic regime.The only difference is here a temperature independent offset since κ xy /T → 0 at high temperature whereas κ xx /T approaches a finite value.Thus, in view of the almost perfect matching temperature dependencies of κ xx /T and κ xy /T and the fact that the heat carrying quasiparticles contributing to the longitudinal κ xx are phonons (see Ref. [23] and discussion further below), it is tempting to interpret this observation as evidence for a phononic origin of the thermal Hall for T T N .Indeed, this conclusion seems to be corroborated by findings for α-RuCl 3 where the similarities between κ xx /T and κ xy /T have been interpreted as significant evidence for a phononic origin of the thermal Hall effect [10].Furthermore, the recent demonstrations of phonon thermal Hall effect in several magnetic insulators supports the notion that similar temperature dependencies of κ xy /T and a phononic κ xx /T are a general fingerprint of phononic thermal Hall effect [25][26][27].
However, the low-T behavior of κ xy in the magnetically ordered regime is difficult to be explained by a sole phononic contribution to κ xy .As can be inferred clearly from the data in Fig.On the other hand, if one assumes that the sign of the phonon thermal Hall effect is always negative, the only possibility is that a second positive contribution to κ xy rapidly gains importance, causing the sign change of κ xy in the magnetically ordered phase.In this case, this second contribution must be due to transport by magnetic degrees of freedom.
Among these, within this paper we do not distinguish further since magnetic fluctuations remain strong even at low temperature [23] and hence both magnons as well as possible fractionalized excitations such as Majorana fermions and visons a priori can not be excluded.

Magnetic field dependence of κ xx and κ xy
The latter scenario of a two-component thermal Hall conductivity is strongly supported by the isothermal magnetic field dependence of κ xy as presented in Fig. 4a) at selected temperatures 8 K, 16 K, and 37 K.As can be clearly inferred from the data, κ xy (H) at all these temperatures exhibits a pronounced and mostly non-monotonic field dependence.This is quite unusual since a non-monotonic field dependence is in strong contrast to other findings on various materials including α-RuCl 3 where a quasi linear field dependence of κ xy (H) has been observed [6,8,30,[33][34][35]37].
The origin of the unusual κ xy (H) can straightforwardly be connected with the field dependence of the isothermally measured κ xx (H) as shown in Fig. 4b)-d) as we will discuss in detail further below.κ xx (H) possesses two features with opposite trends in temperature.
One is a pronounced minimum at about 4-8 T which is strongest at 8 K, very weak at 16 K and absent at 37 K.The other is a decrease at higher field with increasing field.At 8 K the decrease is weak and discernible only for µ 0 H > 14 T, but with raising the temperature it becomes increasingly important.At 16 K it begins at around 10 T, and at 37 K it dominates the data for µ 0 H 4 T. Clearly, the growing importance is consistent with the data shown in Fig. 1b.
These κ xx (H) data are characteristic of a phonon heat conductivity which is subject to strong phonon-spin scattering [38][39][40][41].In fact, κ xx has already been shown to be governed by phonons with strong phonon-spin scattering for magnetic fields applied in plane [23].In the situation which we consider here, i.e. with a magnetic field perpendicular to the magnetic planes, for the (H, T )-dependence of κ xx one should consider two qualitative phonon-spin scattering mechanisms: In the magnetically ordered state T ≤ T N one can expect that the phonon spin scattering becomes particularly important (i) near field-driven spin reorientation transitions [41] and (ii) "resonant" scattering of the phonons off collective excitations such as magnons [23,39,40].
Concerning the former, an initial version of the magnetic (H, T )-phase diagram for H||c On the other hand clear-cut information about low-energy magnon excitations in zero field is available from recent inelastic neutron scattering (INS) data [42].The lowest lying magnon mode has been reported in the energy range 1-3 meV with minima at the Γ and the M points.It is thus clear, if one considers a typical Θ D of a few hundred K, that phonon-magnon scattering involving the acoustic phonon modes must be significant already at zero field.In order to obtain further insight in the field and temperature dependence of the magnon mode and κ xx (H), we investigated the magnetic field induced energy shift of this magnon mode at the Γ-point by means of high-field electron spin resonance (ESR, see Supplementary Information).At T = 3 K, the data yield an energy gap at the Γ-point of about 219 GHz (0.91 meV) at zero field, in accordance with the zero field neutron result, and a Zeeman shift to about 950 GHz (about 3.9 meV) at 14 T (corresponding to a g factor of 3.91).
If one assumes a rigid magnon band shift in magnetic field the maximum magnon energy at 16 T amounts to about 6.8 meV corresponding to about 80 K. Standard considerations for phonon heat conductivity [43], namely with τ c a total phononic relaxation time imply phonons with an energy of about 4k B T to dominate the heat conductivity.Thus, within a simple kinematic picture for the phononmagnon scattering, for temperatures smaller than some characteristic T peak < (80 K/4) 20 K one would naturally expect a minimum in κ xx (H) because in this T -range the magnetic field should drive the 'intersection' of the phonon and magnon bands through the peak of the integrand in Eq. 1.At higher temperatures, the applied magnetic field is only sufficient to drive the band intersection towards the integrand peak without actually reaching it and therefore the data should exhibit only a suppression with increasing field.Both the minimum at low T and the increasing high-field suppression is qualitatively observed, superimposing on the effect of the phase transition described above.
The above considerations should be understood within some margin of error: The actual T peak and further details of κ xx (H, T ) certainly depend on the true Θ D , the magnon density of states and the effects of temperature and magnetic field on the magnon dispersion.
Concerning the latter, it is worth pointing out that our ESR data imply that at T = 6-8 K the magnon mode starts to increasingly soften and broaden with increasing the temperature consistent with zero magnetic field results of INS [42].The broadening gets particularly strong at T > 20 K, still below T N , possibly due to enhancement of the spin fluctuations by approaching the phase transition.It is plausible that for T > T N a qualitatively similar scenario applies for paramagnon excitations.
After having established the phonon-only nature of κ xx (H) and the connection of its anomalous field dependence to phonon-magnon scattering we turn now to κ xy (H) and directly compare the data for the longitudinal and transverse heat transport channels.Very clearly, at 8 K both curves for κ xx (H) and −κ xy (H) possess a very similar relative field dependence, i.e. a minimum at 4-6 T and a saturation at µ 0 H 12 T.At the higher temperatures, 16 K and 37 K the connection between κ xx (H) and −κ xy (H) at first glance seems not so clear as seen for 8 K.However, a closer inspection reveals that for all data the derivative ∂κ xy (H)/∂H is proportional to κ xx (H) modulo a constant offset.In order to demonstrate this revealing connection we plot in Fig. 4a) a fit according to with a 1 and a 2 as fitting constants.As can be seen in the figure, this empirical ansatz describes the data very well, apart from a slight low-field deviation at 8 K, yielding parameters a 1 < 0 and a 2 > 0 (see Supplementary Information).
Eq. 2 clearly suggests that κ xy is composed of two additive components with opposite signs.Due to its weighting with the phononic κ xx , the first term can be assigned to a phononic thermal Hall effect κ xy,ph .On the other hand, the second term is strictly linear in magnetic field, which corresponds to the usual observation for κ xy (H) for the vast majority of compounds, see above.This component must be of a different origin than κ xy,ph .The most straightforward conclusion is thus that this positive component is magnetic in nature (κ xy,mag ).Thus, the measured total thermal Hall effect should be understood as the sum of the phononic and magnetic contributions to the thermal Hall effect, i.e.It is instructive to extract the separate temperature dependences of κ xy,ph and κ xy,mag , which straightforwardly can be achieved for selected temperatures from the fits shown in Our finding of a two component thermal Hall effect, i.e. with a phononic and a magnetic contribution should be placed into the context of recent theoretical and experimental findings.First of all, our conclusion of a sizeable κ xy,ph is well compatible with several recent theoretical works for magnetic systems [44][45][46][47] where phonon-spin scattering has been identified as one important source for generating a finite phononic transverse thermal conductivity.Indeed, as demonstrated above, phonon-spin scattering is the primary scattering process in the phononic κ xx at all temperatures considered.On the other hand, a magnetic thermal Hall effect has been shown to be expected for chiral magnons in Kitaev magnets [48], in addition to the expectation of a thermal Hall effect from Majorana fermion edge currents [1,49].The growing importance of κ xy,mag observed at T < T N in our data seems indeed compatible with an additional effect due to chiral magnons.We point out that a microscopic mechanism as well as a quantitative and qualitative prediction of the phononic and magnetic thermal Hall effect in Na 2 Co 2 TeO 6 and more generally in Kitaev systems still needs to be worked out.On this general scheme, our results also provide fresh input for understanding the controversially discussed thermal Hall effect in α-RuCl 3 .The experimental significance of a phononic thermal Hall effect as shown in our data for Na 2 Co 2 TeO 6 imply that a sizeable phononic thermal Hall effect should be expected in α-RuCl 3 , too, because both systems possess a very similar phonon-spin scattering phenomenology [23,39].Therefore, our results support the recent notion of a phononic thermal Hall effect in α-RuCl 3 [10] and call for a reinvestigation of the intriguing findings for a low-temperature plateau [7,11].
More specifically, the key for uncovering a genuinely quantized magnetic thermal Hall effect should be to disentangle phonon vs. magnetic contributions to the κ xy , and to look for quantized behavior only in the magnetic part, rather than in the total κ xy , which does not deserve to be quantized with the phonon contribution.

METHODS
High-quality Na 2 Co 2 TeO 6 single crystals were grown by a modified flux method [15,50].
A regular bar-shaped sample of 5.05 × 1.03 × 0.10 mm 3 was cut from an as grown crystal with the sample edges parallel to the a-, a * -, and c-axes, respectively.The cut sample was mounted in a home-built probe, employing a 6-points measurement geometry, see Fig. 1a.
In our configuration, a thermal current density j q,x was generated parallel to the a-axis (i.e. the zigzag direction of the honeycomb lattice) using a chip heater.The thereby produced longitudinal thermal gradient ∇ x T was measured using a field-calibrated differential Au/Fe-Chromel thermocouple.The transverse temperature gradient ∇ y T along the a * -direction which arises upon applying a magnetic field parallel to the c-axis, i.e. perpendicular to the honeycomb layers, was measured by a second thermocouple of the same type.
The longitudinal ∇ x T and the transversal ∇ y T were measured simultaneously.The small size of the transverse signal required the application of a large heat current to achieve a reasonable signal-to-noise ratio, resulting in temperature differences in the order of ∇ x T /T 0 = 10 % and ∇ y T /T 0 = 0.01 % compared to the thermal bath temperature T 0 .At fixed temperature the heater current was varied to prove a linear behavior in ∇ y T .To consider significant heating the sample temperature was determined by extrapolating T 0 to the position of the transverse thermocouple in the center of the sample.To eliminate longitudinal contributions to the transverse signal due to a possible misalignment of the thermocouple contacts, measurements were performed under both field polarities and longitudinal components have been eliminated by antisymmetrization of ∇ y T .
High-field high-frequency electron spin resonance (HF-ESR) experiments were carried out in a frequency range 250 − 950 GHz using a home-made multifrequency spectrometer.For the generation and detection of the microwave radiation a vector network analyzer (PNA-X from Keysight Technologies), as well as a combination of a modular Amplifier/Multiplier Chain (AMC from Virginia Diodes, Inc.) and a hot electron InSb bolometer (QMC Instruments) were employed.A single crystal of Na 2 Co 2 TeO 6 was mounted into a probe head operational in the transmission mode that was put in a 4 He variable temperature inset of a superconducting magnet system (Oxford Instruments) producing fields up to 16 T (see also Ref. [51]).

Fig. 2 Fig. 1 .Fig. 2 .
Fig.2presents our findings for the transversal heat conductivity κ xy at 16 T as a function

Fig. 4 .
Fig. 4. Field dependence of the longitudinal and transversal thermal conductivities of Na 2 Co 2 TeO 6 for selected temperatures.a) Field dependence of the transverse thermal conductivity κ xy .Open symbols represent a fit according to Eq. 2 using κ xx -data.b)-d) Field dependence of the longitudinal thermal conductivity κ xx .

[ 29 ]
reveals a practically temperature independent magnetic transition at about 8 T for T 16 K.Associated spin fluctuations are therefore a plausible cause for the dip observed in κ xx (H) at 8 and 16 K.

Fig. 5 .
Fig.5.Decomposition of κ xy /T of Na 2 Co 2 TeO 6 at 16 T in a negative phononic and a positive magnetic contribution according to fitting results of Eq. 2 compared to the experimental data (blue).Open symbols refer to fits of the field dependence whereas filled symbols represent fits to data of temperature dependent measurements.Solid lines are guides to the eye.

Fig. 4 .
Fig. 4. In addition, we use two temperature dependent data sets for both κ xx and κ xy measured at µ 0 H = 6 T and 16 T (see Supplementary Information), and Eq. 2 for extracting κ xy,ph and κ xy,mag at 16 T.The complete data set is shown in Fig. 5 in comparison with the total measured κ xy as a function of temperature.This comparison clearly reveals that for T > T N the absolute value of κ xy,ph always is somewhat larger than that of κ xy,mag , yielding the overall negative sign.However, upon the onset at T < T N , this fine balance changes and

Fig. S1 .
Fig. S1.Frequency ν versus resonance field H res dependence of the ESR signal of Na 2 Co 2 TeO 6 for the H c axis field geometry at T = 3 K (left vertical scale) together with representative HF-ESR spectra at selected frequencies (right vertical scale).The dashed line depicts the linear dependence ν = ∆ + h −1 g c µ 0 µ B H with the g factor g c = 3.91 and the zero field magnon gap ∆ = 219 GHz (0.91 meV).
Fig. S3.Temperature dependence of the thermal Hall effect of Na 2 Co 2 TeO 6 for magnetic field applied along the c-axis.Bold symbols represent the four-point average of the data (transparent symbols) and error bars account the spread before averaging.

Fig. S5 . 2 (
Fig. S4.Temperature dependence of the thermal conductivity of Na 2 Co 2 TeO 6 for magnetic field applied along the c-axis.
[25][26][27][30][31][32][33][34][35][36]ges sign at about 12 K.This sign change is a crucial information which implies either of two scenarios for the origin of the thermal Hall effect.It is thinkable, on one hand, that κ xy indeed is purely phononic as inferred above.If this were the case the mechanism causing the off-diagonal thermal response has to fundamentally change upon cooling through T N and further upon possible reorientations of the magnetic order[23]at even lower temperatures.Given the rather complicated and barely understood magnetic phase diagram of Na 2 Co 2 TeO 6[15,23,28,29], one cannot straightforwardly exclude this possibility without further information.One should note, however, that up to present all reported examples of phonon thermal Hall effect do not exhibit a sign change[25][26][27][30][31][32][33][34][35][36].Hence this scenario seems rather unlikely.
3, κ xx /T and κ xy /T exhibit distinctly different temperature dependencies at T T N .While upon cooling κ xx /T is only moderately reduced by about 30% at the lowest temperature measured,