Large oscillatory thermal hall effect in kagome metals

The thermal Hall effect has been a powerful technique to probe the exotic nature of correlated quantum matter^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16}. The thermal Hall effect is the thermal analog of the electrical Hall effect, which detects the transverse temperature gradient in the presence of longitudinal heat current and perpendicular magnetic field. The thermal Hall effect measured the Bogoliubov quasiparticles directly in the high-temperature superconducting cuprates^2,3,4,5 and Fe-based superconductors⁶. It measured the magnon excitations in the quantum magnets^7,8,9,10 and resolved the Majorana quantizations of the edge state in the Kitaev spin liquid candidate α-RuCl₃^11,12. On the other hand, recent progress points to the phonon origin of the thermal Hall effect in some correlated materials. For example, the Mott insulator state of cuprates is reported to establish a giant thermal Hall effect^13,14. The latest report reveals the large phonon thermal Hall effect in quantum paraelectric material SrTiO₃¹⁵.

The thermal Hall effect is much more universal than the electrical Hall effect. It is the consequence of the chirality of the carriers, whether they are fermions or bosons. The observation of these unconventional thermal Hall effects has provided many fresh insights to the exotic nature of correlated quantum matter. However, the debate has always been whether the origins of these reported effects are due to fermions or bosons^{16,17,18,19,20}. Thus, a question needs to be answered on how to confirm whether a thermal Hall signal is a fermionic or bosonic response. For example, the thermal Hall signal in α-RuCl₃ is debated to come from either phonons, magnons, or the proposed edge state fermions^{1,10,11,12,21}. One remarkable progress is the observation of QOs in the thermal conductivity of α-RuCl₃¹, which reveals the fermionic nature of the spin liquid phase because QOs are the result of Landau Level quantization for Fermi surfaces. This observation suggests a crucial potential path to separate the fermionic and bosonic thermal Hall effects by resolving the QOs. The temperature variance of the oscillatory component will further answer if the original theory based on Fermi Liquid theory for QOs in magnetization and electrical resistivity can describe the pattern of QOs in the thermal transport¹.

However, even the thermal Hall effects are most often negligibly small in real materials. The observation of QOs in the magnetothermal effect is only limited to the two elemental metals, aluminum and zinc^22,23,24,25. For correlated metals, the report of transverse magnetothermal QOs is entirely missing. In this report, we chose the recently discovered Kagome metal CsV₃Sb₅ as a platform^26,27, in which the unusual electronic structure has already led to the demonstration of large anomalous Hall, Nernst, and thermal Hall effects^28,29,30,31, the electronic nematicity^32,33,34,35, and the pairing-density-wave phenomena in the superconducting and possible pseudogap state^36,37. We present the first observation of the QOs in the thermal Hall effect in quantum-correlated materials. In CsV₃Sb₅, we were able to determine the Wiedemann–Franz (WF) ratio between the thermal Hall and electrical Hall QOs. At the ground state, the ability of charged quasiparticles to transport heat and charge is governed by the universal WF law. Violations of the WF law are typically an indication of unconventional quasiparticle dynamics, such as inelastic scattering, semimetal physics, or new phases of matter^{38,39,40,41,42,43,44}. Indeed in CsV₃Sb₅, we found the low-temperature oscillation amplitude of thermal hall conductivity is enhanced by a factor of 2.5 compared with that in electrical Hall conductivity multiplied by the Sommerfeld value L₀ and the absolute temperature T, which cannot be explained by the conventional WF ratio. This strong violation of the oscillation WF law challenges the fundamental concept of Landau quasiparticles and is suggestive of an exotic correlated quantum phase.

The longitudinal thermal conductivity and transverse thermal Hall signals measured in CsV₃Sb₅

The CsV₃Sb₅ single crystal was synthesized via a self-flux growth method similar to the previous reports²⁶. The measurement was performed in the Oxford Triton200-10 Cryofree Dilution Refrigerator. The longitudinal and transverse thermal conductivity were measured using a one-heater-three-thermometers technique (Fig. 1a). The longitudinal and transverse temperature differences ΔT_x and ΔT_y were read by Lakeshore Cryotronics RX102A thermometers. The H-field was varied continuously and slowly during the measurement at a stable temperature. The results were also checked to be consistent with the stepped-field method to avoid the eddy current or other heating effects. The field dependence of the measured signal was plotted as thermal and thermal Hall resistivities to prevent the error in doing the matrix inversion. (The details of the measurement are in “Methods”).

a Sketch of the thermal Hall experiment setup. Three thermometers are connected to the sample through gold wires to measure the longitudinal and transverse temperature differences ΔT_x and ΔT_y. b The magnetic field dependence and QOs of longitudinal thermal resistivity λ_xx(H). The labeled T is the average temperature on the sample. The data were recorded continuously with the H-field swept with sufficiently slow speed. c The longitudinal thermal conductivity κ_xx as a function of measured T in different magnetic fields H. The pink and blue curves with open squares are the electronic thermal conductivity at 0 T and 13 T calculated according to the WF law, respectively. d The thermal Hall resistivity λ_xy(H) displays large quantum oscillations at selected T from 0.55 K to 4.42 K. e λ_xy(H) above 5.39 K. In the low field region below ~30 K, the S-shaped anomalous thermal Hall signal shows up in λ_xy(H). Above ~30 K, λ_xy(H) shows ordinary H-linear behavior. f The temperature dependence of the thermal Hall conductivity κ_xy/T at selected H. The green curve with the open circle is the electronic thermal Hall conductivity at 13 T multiplied by the Lorenz number \({L}_{0}\left(2.44\times 1{0}^{-8}{V}^{2}{K}^{-2}\right)\). At base temperature, the employed Hall WF ratio \(L=\frac{{\kappa }_{xy}}{{\sigma }_{xy}T}\) is ~ 2.2 × 10⁻⁸ V² K⁻².

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As shown in Fig. 1a, ΔT_x and ΔT_y were measured at fixed T with H∥z and −∇T∥x. Then, the thermal resistivity matrix λ_ij was obtained using the recorded temperature gradient −∇_xT and −∇_yT (see “Methods” for experimental setup). Figure 1b displays the longitudinal thermal resistivity λ_xx as a function of the H-field. Around the zero field, λ_xx(H) shows a sudden enhancement from the superconducting state as the sample temperature is below T_c (~2.5 K). Above ~4 T, clear oscillations emerge in λ_xx(H). The T dependence of the thermal conductivity also reveals an interesting feature. In a metal, the total thermal conductivity κ_xx is the sum of the phononic (κ_ph) and electronic (κ_e) contributions. Due to different scattering mechanisms of electron and phonon, κ_xx is dominated by the electron at low T. The Wiedemann–Franz (WF) law states that in metals the WF ratio \(L=\frac{{\kappa }_{xx}}{{\sigma }_{xx}T}\) is nearly the Lorenz number \({L}_{0}\left(2.44\times 1{0}^{-8}{V}^{2}{K}^{-2}\right)\), where σ_xx is the electrical conductivity. The electronic contribution to the thermal conductivity in CsV₃Sb₅ was calculated using L₀σ_xxT, where σ_xx is measured by using the same contacts on the sample connected to the current leads (see Supplementary Fig. 2). The value of L₀σ_xxT is then compared with the total thermal conductivity. Even down to lowest T, the total thermal conductivity is found to be greater than L₀σ_xxT (Fig. 1c). The large deviation indicates a significant contribution of phonon thermal conductivity before entering the superconducting state.

Next, we turn to examine the thermal Hall effect in CsV₃Sb₅. As shown in Fig. 1e, above 30 K, λ_xy shows the H-linear conventional-metal pattern, indicating the carrier density is dominated by one band. Below 30 K, the thermal Hall resistivity λ_xy becomes strongly nonlinear in H, accompanied by a sign change. The same temperature range also coincides with the onset temperature of electronic nematicity³⁵, and the nonlinearity in Hall and Nernst signals^28,29,31, which was attributed to the enhancement of hole mobility at low T^37,45. In the low H region, λ_xy also shows a significant anomalous thermal Hall effect, which suggests the possible time-reversal-symmetry-breaking nature of charge order⁴⁶. The large anomalous signal decreases as temperature increases but stays robustly at elevated temperatures up to 30 K (see Supplementary Fig. 8).

We note that the observed non-oscillating background of λ_xy is still consistent with the WF law at very low T (Fig. 1f). Since the transverse channel is usually free of phonon contribution, the WF ratio L is directly related to the electrical and thermal relaxation time. In CsV₃Sb₅, we extract and compare the non-oscillating part \({\bar{\kappa }}_{{{{{{{\rm{xy}}}}}}}}(T)/T\) with that of \({L}_{0}{\bar{\sigma }}_{xy}\), where \({\bar{\sigma }}_{xy}\) is the non-oscillating background of electrical Hall conductivity. As T drops, the WF ratio L increases and recovers to L₀. Below ~15 K, the value of \({\bar{\kappa }}_{{{{{{{\rm{xy}}}}}}}}(T)/T\) becomes quite close to \({L}_{0}{\bar{\sigma }}_{xy}\). Below 10 K, both \({\bar{\kappa }}_{{{{{{{\rm{xy}}}}}}}}(T)/T\) and \({L}_{0}{\bar{\sigma }}_{xy}\) become nearly a constant. With further drop in T, \({\bar{\kappa }}_{{{{{{{\rm{xy}}}}}}}}(T)/T\) slightly decreases. The WF law of the transverse non-oscillating background is further supported by the data of another CsV₃Sb₅ sample, as shown in Supplementary Fig. 9.

The large QOs also appear in the thermal Hall resistivity λ_xy above ~4 T (Fig. 1d, e). Differing from the H-symmetric oscillation in λ_xx, the H-antisymmetric oscillation pattern in λ_xy is intrinsic and does not come from the longitudinal pick-up (see Supplementary Fig. 4). By performing the Fast Fourier transformation (FFT) (Fig. 2a, b), the oscillations contain multiple orbits whose frequencies are consistent with those in the Shubnikov-de Haas (SdH) oscillation⁴⁷ and the de Haas–Van Alphen (dHvA) oscillation⁴⁸. Among all these four compositions, the contribution from the δ orbit (frequency F ~87 T) dominates when H is greater than ~9 T.

a The oscillatory components \({\widetilde{\lambda }}_{xx}\) and \({\widetilde{\lambda }}_{xy}\) were extracted from λ_xx and λ_xy at selected temperatures, obtained after a fifth-order polynomial background subtraction from the raw data. A primary contribution from the delta orbit (~87 T) is marked by the grey dashed line. b Fourier transformation of the quantum oscillations at field ranged from 4 T to 13 T, showing four principal frequencies, F_α= 11 T, F_β = 25 T, F_γ= 72 T, and F_δ = 87 T. c Raw data of the quantum oscillation in thermal Hall signal at different T ranging from 3.25 K to 17.15 K. Each curve is shifted with a constant. d The oscillatory components were obtained after fifth-order polynomial background subtraction. The 75 T high-pass filter is applied to emphasize the oscillation from the delta orbit (~87 T). As the temperature goes from 3.25 K to 17.15 K at magnetic field \(H=\frac{2{\pi }^{2}{k}_{B}{m}^{*}}{1.62{\mu }_{0}\hslash e}T\), the oscillation amplitude passes through zero, accompanied by a phase reversal. e The relative phase ψ of the oscillation in the T vs. 1/H phase diagram (see Supplementary Note 7 for the definition of ψ). At low T and strong H, ψ > 1, the region is colored with orange. At high T and weak H, ψ < 1, the phase becomes opposite, and the area is colored blue. The purple dashed line shows the fitted phase-shifting boundary with function \(T=\frac{1.62\hslash e}{2{\pi }^{2}{k}_{B}{m}^{*}}{\mu }_{0}H\). f In the T vs. H phase diagram, the zero amplitude locations of the oscillations are plotted as the red circles. The zeros match the ψ = 1 boundary in (e), and the error bar is estimated from the broadening of the boundary. The straight purple dashed line is linear fitting of the boundary using the same parameter as Fig. 3c. The slope of this straight line gives the value of \(\frac{1.62\hslash e}{2{\pi }^{2}{k}_{B}{m}^{*}}\), and the cyclotron mass was determined as 0.13m_e.

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The temperature dependence of the amplitude of the oscillatory magnetothermal effect

We further investigate the amplitude of the oscillating components of the thermal resistivity \({\widetilde{\lambda }}_{xx}\) and the thermal Hall resistivity \({\widetilde{\lambda }}_{xy}\). The first question is what the temperature dependence of the magnetothermal oscillation amplitudes should be. According to the prediction from the Fermi liquid theory, the expected temperature dependence of the oscillation amplitude of the magnetothermal effect is different from the well-established Lifshitz–Kosevich T-dependence for magnetization and resistivity. Instead,

is replaced by \({R}_{T}^{\prime\prime}(T)\), its second derivative with respect to T. This relation is a direct result of the Legendre transformation of the M(T) relation of the LK formula⁴⁹, and the detailed derivation is in the Methods. Note that the function \({R}_{T}^{\prime\prime}(T)\) changes sign near \(\frac{2{\pi }^{2}{k}_{{{{{{{\rm{B}}}}}}}}T}{\hslash }=1.62\), a 180° phase shift of the QOs is expected to be observed in λ_xx and λ_xy at elevated temperatures. Experimentally, given the fact that the thermal Hall signal in most materials is already difficult to measure, the exact \({R}_{T}^{\prime\prime}(T)\) temperature dependence of the amplitude of the oscillatory magnetothermal effect has not yet been studied. For example, even though bismuth was one of the earliest materials discovered to exhibit QOs in magnetization, resistivity, and the giant oscillatory Nernst coefficients, neither the thermal Hall effect nor the QO in the thermal transport properties is observed in bismuth^50,51.

Figure 2a shows the raw data of the QOs in λ_xyT with a constant value shifting at different T ranging from 3.25 K to 17.15 K. Due to the multiple frequency QOs, a 75 T high-pass filter is applied to only allow through the primary oscillation from the δ orbit. The oscillation component with a single frequency is plotted in Fig. 2b. Starting from 3.25 K, the oscillation amplitude gradually vanishes at specific H as T increases. The position of these nodes where \({R}_{T}^{\prime\prime}(T)\) changes sign strictly obeys the function \(T=\frac{1.62\hslash e}{2{\pi }^{2}{k}_{B}{m}^{*}}{\mu }_{0}H\). At elevated T, the oscillation appears again, accompanied by a 180° phase flip. Figure 2c shows the 2D plot of the phase sharpness ψ of the observed oscillation. (see Supplementary Note 7 for the definition of ψ). The phase diagram is divided into two regions with different colors. At low T and strong H, ψ > 1, the region is colored orange. At high T and weak H, ψ < 1, the phase shifts to the opposite, and the area is colored light blue. At the boundary of these two regions, ψ ≈ 1, which means the oscillation amplitude is ~0 and the phase is difficult to distinguish. The boundary of the phase-shifting is fitted using a straight line in the 2D H-T plot (Fig. 2d). As the T increases, the boundary linearly shifts to higher H, exactly following the linear relationship \(T=\frac{1.62\hslash e}{2{\pi }^{2}{k}_{B}{m}^{*}}{\mu }_{0}H\) provided by our model. From the slope of this line, the cyclotron effective mass m^* is estimated to be 0.13 m_e, where m_e is the free electron mass. The effective mass revealed by the phase shift is similar to the previously reported result^31,47.

Next, we plot the temperature variance of the oscillation amplitudes Δλ_xx and Δλ_xy in Fig. 3a. Going from the lowest temperature, the oscillation amplitudes show a concave curve. This can be understood by considering that the electron and phonon contribute to κ_xx and κ_xy additively. However, the oscillation amplitude of λ_xx and λ_xy will be influenced by the non-oscillating phonon thermal conductivity from the matrix inversion. Thus, κ_xx and κ_xy should be utilized to investigate the \({R}_{T}^{\prime\prime}(T)\) damping effect to eliminate the contribution from the phonon (see Supplementary Note 2). As shown in Fig. 3b, the purple circles and light blue squares are the calculated ratio between the oscillation amplitude of magnetothermal conductivity in various temperatures and the electrical conductivity in the lowest temperature limit. By doing the fitting for the experimental data using the \({R}_{T}^{\prime\prime}(T)\) function, the temperature smearing of the thermal conductivity QOs which follow the second derivative of R_T(T) is confirmed. Similar to the specific heat QOs in high-temperature superconductors^52,53, the observation of the \({R}_{T}^{\prime\prime}(T)\) temperature dependence and the 180° phase shift also demonstrates an essential signature of the magnetothermal QOs expected for a Fermi liquid.

a The oscillation amplitude Δλ_xxT, Δλ_xyT, Δρ_xx/L₀, Δρ_xy/L₀ as a function of temperature obtained by performing the FFT to the oscillatory components. As T increases, Δρ_xx/L₀ and Δρ_xy/L₀ damp much slower than Δλ_xxT and Δλ_xyT, indicating the magnetothermal oscillation amplitudes have a different temperature damping factor from the electrical resistivity. b To eliminate the contribution from the phonon, κ_xx and κ_xy were utilized to investigate the \({R}_{T}^{\prime\prime}(T)\) damping effect. The purple circles and light blue squares show the calculated ratio between the oscillation amplitude of the magnetothermal conductivity in various temperatures and the electrical conductivity in the zero temperature limit. The amplitudes were read directly from Supplementary Fig. 6 at 1/μ₀H ~ 0.082T⁻¹. The error bars are estimated from the broadening of the raw data due to the experimental noise. The purple and light blue solid lines are the fittings for the longitudinal and transverse thermal conductivity oscillation amplitudes using the \({R}_{T}^{\prime\prime}(T)\) function. The fitting results of the oscillation amplitude are 1.55 \({R}_{T}^{\prime\prime}(T)\) for the longitudinal thermal conductivity and 2.50 \({R}_{T}^{\prime\prime}(T)\) for the transverse thermal Hall conductivity. The experimental results showed that in the zero temperature limit, the oscillation WF ratio Δκ_xx/Δσ_xxT reaches 1.55 L₀, and Δκ_xy/Δσ_xyT reaches 2.50 L₀. The background WF ratio κ_xy/σ_xyT is plotted using the red triangles. The Lorenz number L₀ is shown as the black dashed line. c The ratio of the transverse and longitudinal thermal conductivity oscillation amplitudes for the four smallest orbits α, β, γ, and δ. The ratio is calculated using the FFT amplitude for these different orbits separately. The four different frequencies give nearly the same ratio at different T, indicating the oscillations are mainly dominated by the thermodynamic potential rather than the multiband feature. The error bars of (a) and (c) are estimated by comparing the differences of several common FFT windows.

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The strong violation of the WF law in the thermal Hall QOs of CsV₃Sb₅

The WF relation can also be checked in the oscillatory components similar to the non-oscillating background. Generally, in the limit of low temperature, strong field, and small effective mass, the phase smearing effect caused by finite temperature can be neglected. Thus, the oscillation of the same Landau level density of states at the Fermi surface has constructive interference and the oscillatory amplitudes Δκ_ij and Δσ_ij should be related by the WF relation when T → 0:

At elevated temperatures, Δκ_ij and Δσ_ij are damped by the coefficients \({R}_{T}^{\prime\prime}(T)\) and R_T(T) respectively. However, in Kagome metal CsV₃Sb₅, the WF law of the QOs is violated. Figure 3b shows the temperature dependence of the ratio between Δκ_ij(T→0) and Δσ_ij(T→0). From the WF relation, the very low T limit values for the longitudinal and transverse channels should both be L₀. In Fig. 3b, the experimental results showed that Δκ_xx(T→0)/Δσ_xx(T→0) reaches 1.55 L₀ and Δκ_xy(T→0)/Δσ_xy(T→0)T is a even larger value 2.50 L₀. By contrast, the background ratio κ_xy(T→0)/σ_xy(T→0)T has already been shown consistent with the WF law, and the ratio is L₀ in low T.

Generally, κ_xx and κ_xy are related to both electrical conductivity and specific heat. A question is whether the magnetothermal QOs in CsV₃Sb₅ follow the SdH or dHvA mechanism. It has been shown that with multiple electron and hole bands, the pronounced QOs of the Hall effect depend on the relative contribution of each band, respectively⁵⁴. However, in multiband Kagome metal CsV₃Sb₅, the multiband feature only has a secondary influence on the oscillatory components κ_xx and κ_xy. Figure 3c shows the ratios between Δκ_xy and Δκ_xx of the four smallest pockets in CsV₃Sb₅. For these different pockets, the ratios are all around the same value and slightly go up in the same trend when T increases, which means the QOs of κ_xx and κ_xy follow the dHvA pattern instead of the SdH pattern. This phenomenon happens when the mean free path is comparable to the sample size or domain size at low T⁵⁵. Then, both κ_xx and κ_xy are proportional to the electronic specific heat C, and their QOs are dominated by the thermodynamic potential (see Supplementary Note 5).

Our findings reveal the QOs in the thermal Hall effect in correlated quantum materials for the first time. The oscillatory thermal Hall signal shows a characteristic 180° phase flip in elevated temperatures, which is the macroscopic effect caused by microscopic quantum phase interference due to the modification of the distribution function for thermodynamic quantities⁵⁶. Meanwhile, the observed temperature variance of the oscillatory components and the phase flip at elevated temperatures is remarkably consistent with the predictions based on the Fermi liquid theory, which is a “smoking gun” for any model. Since the magnetothermal QOs originate from the Landau level density of states (DOS), the \({R}_{T}^{\prime\prime}(T)\) temperature dependence with a characteristic phase flip can be used to identify and investigate the oscillatory behavior of magnetothermal conductivity in a wide range of quantum materials such as single element metals^57,58,59, quantum spin liquid candidates α-RuCl₃¹, semimetals^60,61,62, and high-temperature superconductors^52,53.

On the other hand, The oscillatory WF ratio enables to examine the correlated materials from a different perspective. Usually, the background WF law violation was used to reveal unconventional quasiparticle dynamics. In terms of the oscillatory WF ratio, the deviation is also highly unconventional. Whereas before claiming the deviation of the oscillatory WF ratio in CsV₃Sb₅ is related to its electronic structure, it may be that one should consider that this deviation is due to phonon. Recent works suggest that the phonon drag effect may lead to enormously large QOs in the Nernst effect in Weyl semimetals⁶³ and the significantly enhanced thermal Hall effect in doped SrTiO₃⁶⁴. Yet, the QOs in the thermal Hall effect have not been reported in these materials, and the phonon drag effect was not observed in CsV₃Sb₅ from the thermoelectric measurements. Moreover, in the limit of considerably low T as in this study, the phonon-drag effect is likely to be very small. Thus, we can rule out the possibility that phonons contribute to the thermal Hall QOs.

Since κ and σ measure the dHvA and SdH QOs, respectively, the observation of the OQs WF law violation reveals different physical manifestations of dHvA and SdH QOs mechanism. Generally speaking, both dHvA and SdH effects are due to the oscillatory density of states (DOS). Nevertheless, the SdH effect has an extra term coming from the scattering between the oscillating states and the total states on the Fermi surface, and the amplitude of the oscillation component of electrical conductivity must be normalized by dividing by the non-oscillatory background^65,66,67. The combination of these two factors can lead to the deviation of the SdH QOs amplitudes from the DOS QOs amplitudes, especially when the Landau level orbits are unconventional, such as the existence of magnetic breakdown⁶⁶ or strong correlation effects^67,68. Indeed, recent studies support the magnetic breakdown effect between conventional orbits and Chern Fermi pockets^48,69 which has the same frequency as the δ orbit observed in magnetothermal QOs in this paper. Many works also observed the strongly interacting effects with multiple interrelated phases in this system^{36,45,46,70,71,72}. In this sense, the oscillatory magnetothermal effect provides a bridge for the direct comparison between the SdH and dHvA effects and sheds light on how the intricate physics in Kagome lattice leads to novel correlation effects that may contribute to the emergence of unusual density-wave order, electronic nematicity, and putative intertwined orders^{35,36,46,70,71}.

Experimental setup of CsV₃Sb₅ sample in the main text

The CsV₃Sb₅ single crystal was synthesized via a self-flux growth method similar to the previous reports²⁶. The dimension of the CsV₃Sb₅ sample in the main text is 1.8 mm × 1.6 mm × 0.06 mm. The thickness of the sample is quite homogeneous at different positions or along its entire length. The measurement was performed in the Oxford Triton200-10 Cryofree Dilution Refrigerator. The longitudinal and transverse thermal conductivities were measured using a one-heater-three-thermometers technique (Fig. 1a). One end of the sample is thermally anchored (with H74F epoxy of Epoxy Technology)) on the copper heat bath of the probe sample chamber. To measure the longitudinal and transverse temperature differences Δ T_x and Δ T_y on the sample, the 5 mil Au wires were first attached to the sample with Ag paint. The positions of the contacts are located near the center of the sample. The sizes of the contacts are smaller than 0.05 mm. Then, a 1 kOhm thin-film resistor heater and three Lakeshore Cryotronics RX102A thermometers (1 kOhm) were thermally attached to the Au wires with Ag paint. Low thermal conductivity Lakeshore Manganin wires of 1-mil diameter were connected from the contact on the heater and thermometers to the copper wires extended to the exterior. The steady heating power P applied to the warm edge of the sample is always set to make sure the temperature change on the sample is less than 10% of the environment temperature. To ensure the magnetic field is perfectly perpendicular to the sample, the sample was first put on top of two Cu blocks with the same thickness. The copper blocks were then removed after the sample was firmly anchored.

The longitudinal and transverse temperature differences ΔT_x and ΔT_y were read by Lakeshore Cryotronics RX102A thermometers. The H-field was varied continuously and slowly (0.05 T/min) during the measurement at a stable temperature. The results were also checked to be consistent with the stepped-field method to avoid the eddy current or other heating effects. The field dependence of the measured signal was plotted as thermal and thermal Hall resistivities to prevent the error in doing the matrix inversion. The thermal and thermal Hall resistivities were measured at fixed T with H∥z and −∇T∥x. The temperature stability of the heat bath is controlled to be smaller than 10⁻⁴ K using the monitoring thermometer. There is no T gradient on the heat bath since the Cu heat bath is larger than the sample. Thus, the possible error induced by the thermal Hall effect of Cu can be ignored.

We note that in CsV₃Sb₅ various experimental probes revealed potential micro-size domain at low T due to the CDW supercells^73,74,75. A pioneering Focus-Ion-Beam work⁷² milled hexagon-shaped microstructures with a size of 10 μm to search for electrical transport anisotropy. However, given the mm sizes of samples used in the thermal transport measurement, the thermal transport properties are the averages of many domains.

The differential method and thermometer calibration

The experimental data were taken in the DC static heater method. The resistance of the thermometers is quite stable and will not drift when sweeping the magnetic field. The results were checked to be consistent with the square-wave and sine-wave heating power measurements. The resistances of the thermometers R_T1, R_T2, and R_T3 were measured by the Lock-in amplifiers (SR865) at frequency f ~ 13 Hz. (T₁, T_2, and T₃ are measured at the contacts shown in Fig. 1a) To get a better signal-of-noise ratio and further reduce the temperature fluctuation of the environment, the transverse signal was measured in a differential method: the resistance difference ΔR between the R_T2 and R_T3 was detected by a Lock-in amplifier after being amplified by a differential amplifier (×100). Then, R_T3 can be obtained by calculating R_T2+ Δ R/100. During the measurement, the amplifier was placed into a box with high permeability, and the wires were wrapped together by Al foils to reduce the H-field induced noise pick-up. To convert the exact temperature from the measured resistance, all the thermometers are calibrated as a function of temperature and magnetic field⁷⁶. With the above procedures, the resolution of the transverse temperature difference was successfully reduced to the range of 10⁻⁵–10⁻⁴ K while at the heat bath temperature T = 1 K (see Supplementary Fig. 4d).