Finite-temperature violation of the anomalous transverse Wiedemann-Franz law

TL;DR

This study investigates the anomalous transverse transport in Mn$_{3}$Ge across a wide temperature range, revealing a finite-temperature violation of the Wiedemann-Franz law caused by Berry curvature effects, supported by theoretical calculations.

Contribution

It demonstrates that the anomalous Lorenz ratio deviates from the Sommerfeld value at higher temperatures due to Berry curvature mismatch, providing new insights into topological transport phenomena.

Findings

Anomalous Lorenz ratio remains near the Sommerfeld value up to 100 K.

Finite-temperature violation caused by Berry curvature mismatch, not inelastic scattering.

Experimental verification of the Bridgman relation supports the results.

Abstract

The Wiedemann-Franz (WF) law links the ratio of electronic charge and heat conductivity to fundamental constants. It has been tested in numerous solids, but the extent of its relevance to the anomalous transverse transport, which represents the topological nature of the wave function, remains an open question. Here we present a study of anomalous transverse response in the noncollinear antiferromagnet Mn$_{3}$Ge extended from room temperature down to sub-Kelvin temperature and find that the anomalous Lorenz ratio remains close to the Sommerfeld value up to 100 K, but not above. The finite-temperature violation of the WF correlation is caused by a mismatch between the thermal and electrical summations of the Berry curvature, rather than the inelastic scattering as observed in ordinary metals. This interpretation is backed by our theoretical calculations, which reveals a competition between the temperature and the Berry curvature distribution. The accuracy of the experiment is supported by the verification of the Bridgman relation between the anomalous Ettingshausen and Nernst effects. Our results identify the anomalous Lorenz ratio as an extremely sensitive probe of Berry spectrum near the chemical potential.

Equations

$$L_{ij}^A=\frac{{\kappa }_{ij}^A}{T{\sigma }_{ij}^A}.$$

$$L_0=\frac{{\pi }^2}{3}{\left(\frac{k_B}{e}\right)}^2.$$

$${ϵ}_{zx}^A=\frac{T{S}_{zx}^A}{{\kappa }_{xx}}.$$

$$F_n({ϵ}_k)=({ϵ}_k-\mu {)}^n\frac{\partial f({ϵ}_k)}{\partial {ϵ}_k}.$$

$${\sigma }_{zx}^A(\mu )$$

$${\kappa }_{zx}^A(\mu )$$

$${\tilde{\sigma }}_{zx}(\xi )$$

$${\Omega }_{zx}^n(\text{k})$$

Full text

Introduction

The Berry curvature of electrons can give rise to the anomalous Hall effect (AHE)Nagaosa2010 ; Xiao2010 . This happens if the host solid lacks time-reversal symmetry, which impedes cancellation after integration over the whole Fermi surface. Explored less frequentlyLee2004 ; Miyasato2007 ; Onose2008 , the thermoelectric and thermal counterparts of the AHE (the anomalous Nernst and anomalous Righi-Leduc effects) arise also by the same fictitious magnetic fieldXiao2006 ; Onoda2008 ; Shiomi . How do the magnitudes of these anomalous off-diagonal coefficients correlate with each other? Do the established correlations between the ordinary transport coefficients hold? Satisfactory answers to these questions are still missing. A semiclassical formulation of anomalous Hall effectSinitsyn2008 is laborious, because the concept of Berry connection (or the ‘anomalous velocity’ Karplus1954 ) is based on off-diagonal matrix elements linking adjacent Bloch functions and not wave packets of semi-classical transport theorySinitsyn2008 . This makes any intuitive picture of how Berry curvature combined to a longitudinal thermal gradient can produce a transverse electric field (an anomalous Nernst response)Xiao2006 ; Mccormick or a transverse thermal gradient (an anomalous thermal Hall response)Murakami ; Qin even more challenging.

Here, we present a study of correlations between the anomalous off-diagonal transport coefficients of a magnetic solid, with a focus on the relation between anomalous electrical, $\sigma^{A}_{ij}$, and thermal, $\kappa^{A}_{ij}$, Hall conductivities. The anomalous Lorenz ratio is defined as:

[TABLE]

We track $L^{A}_{ij}$ in order to compare it with the Sommerfeld value:

[TABLE]

We find that over a wide temperature range ($0.5$ K $<T<100$ K), $L^{A}_{ij}$ and $L_{0}$ remain close to each other, but a deviation starts above 100 K. We argue that this observation implies a hitherto unnoticed mechanism for finite-temperature violation of the Wiedemann-Franz (WF) law and points to a small (10 meV) energy scale in the Berry spectrum of Mn3Ge that is absent in Mn3Sn. This experimental observation is backed by theoretical calculations identifying the source of the Berry curvature in this family. By directly measuring the anomalous Ettingshausen and anomalous Nernst effects, we verify the validity of the Bridgman relation connecting the two transverse thermoelectric coefficients to each other. This confirms that the Bridgman relation, a consequence of Onsager reciprocity and based on thermodynamics of irreversible processesCallen , holds regardless of microscopic details. Finally, we quantify the anomalous transverse thermoelectric response $\alpha^{A}_{ij}$ and find that the ratio of $\alpha^{A}_{ij}/\sigma^{A}_{ij}$ tends to saturate towards a value close to $k_{B}/e$ in the high-temperature limit.

Following theoretical propositionsChen2014 ; Kubler2014 , a large AHE was found in Mn${}_{3}X$ ($X=$ Sn, Ge) family of non-collinear antiferromagnets Nakatsuji2015 ; Nayak2016 ; Kiyohara2016 below a remarkably high Néel temperatureZimmer1973 ; Tomiyoshi1982 ; Tomiyoshi1982b . These newcomers to the emerging field of antiferromagnetic spintronicsSmejkal2018 , present a distinct profile of the Hall resistivity in which the extraction of the anomalous Hall conductivity becomes straightforward. An anomalous thermoelectric (Nernst)Ikhlas2017 ; Li2017 and Righi-LeducLi2017 , counterparts of AHE, were also observed in Mn3Sn. In the case of Mn3Sn the triangular order is destroyed at finite temperatureNakatsuji2015 ; Ikhlas2017 ; Li2017 . This is not the case of Mn3Ge where the fate of these signals can be followed down to sub-Kelvin temperatures.

Basic properties

The room-temperature field-dependence of the three transport properties in Mn3Ge is shown in Fig. 1. Like in Mn3Sn, a hysteretic jump is triggered at a well-defined magnetic field, marking the nucleation of domains of opposite polarity induced by magnetic field Li2018 . The large jump, the small magnetic field required for inverting polarity and the weakness of the ordinary Hall response lead to step-like profiles contrasting with other topological solids exhibiting AHE Liang2017 ; Liang2018 ; Shekhar2018 . A step-like profile of anomalous transverse response (for other varietiesSakai2018 ; Liu2018 ) makes the extraction of the anomalous component straightforward. Panels of Fig. 1 show the measured Hall resistivity, Nernst signal and thermal Hall resistivity, which were used to extract electric, thermoelectric and thermal Hall conductivities.

Figure 2 presents a number of basic properties of the system under study. The spin textureZimmer1973 ; Tomiyoshi1982 ; Tomiyoshi1982 ; Nakatsuji2015 is shown in Fig. 2a. This magnetic order is stabilized thanks to the combination of Heisenberg and Dzyaloshinskii-Moriya interactionsLiu2017 . As seen in Fig. 2b, which shows the magnetization, it emerges below $T_{N}=370$ K. The small residual ferromagnetism has been attributed to the residual magnetic moment of octupole clusters of Mn atomsSuzuki2017 in this pseudo-Kagome lattice.

A carrier density of $n=3.1\times 10^{22}$ cm*-3* is extracted from the magnitude of the ordinary Hall numbersupplement in agreement with a previous reportKiyohara2016 . The electrical resistivity shows little variation with temperature (Fig. 2c). and its magnitude of 150 $\mu\Omega$cm implies a mean free path as short as 0.9 nm, compatible with the fact that Mn3X crystals are not stoichiometricTomiyoshi1982 ; Kiyohara2016 . In our crystals, we found the Mn:Ge ratio ranges from 3.32:1 to 3.35:1supplement . Since one tenth of Ge sites are occupied by Mn atoms, the average distance between these defects is $\sqrt[3]{10}$ times the average lattice parameter ($a=0.53$ and $c=0.43$ nm) and comparable to the short electronic mean free path.

The Seebeck coefficient (Fig. 2d) has a nonmonotonic temperature dependence with a peak around 60 K, and a sign change above 200 K, and a large low-temperature slope indicative of electronic correlations. The $T$-linear electronic specific heat (Fig. 2e), is as large as $\gamma=24.3$ mJ mol*-1K-2*, thirty times larger than copper and five times larger than ironKittel . Assuming a single spherical Fermi surface corresponding to the known carrier density, such a $\gamma$, implies an effective mass as large as $m^{*}=14.5\,m_{e}$, which should not be taken literally given that the system is multiband. The slope of the Seebeck coefficient at low-temperature ($S/T\simeq-0.2~{}\mu$VK*-2*) correlates with $\gamma$ yielding $q=\frac{SN_{Av}e}{T\gamma}\simeq 1$, as observed in other correlated systemsBehnia2004 (Fig. 2f). On the other hand, the low temperature resistivity presents a weak upturn impeding the detection of any $T$-square term and the Wilson ratio (of the specific heat and magnetic susceptibility Coleman2007 ) implies that mobile electrons do not play any significant role in the magnetic responsesupplement .

The anomalous transverse WF law

For each temperature, we measured $\sigma^{A}_{zx}$ and $\kappa^{A}_{zx}$, identified as jumps in $\sigma_{zx}(B)$ and $\kappa_{zx}(B)$. This led to the determination of $L^{A}_{zx}$ at each temperature and a comparison with $L_{0}$ to check the WF law.

Our main finding is presented in Fig. 3. Below 100 K, the anomalous Lorenz ratio, $L^{A}_{zx}$ was found to be flat with a magnitude slightly larger than the Sommerfeld value, equal to it within the experimental margin. The results were reproduced by two different measuring methods (with resistive sensors and thermocouplessupplement ) and in two different samples. In one of them, we checked the persistent validity of this equality below 1 K down to 0.3 K. As seen also in the figure, $L^{A}_{zx}$ begins a steady downward deviation from ${L_{0}}$ above 100 K. Interestingly, $\sigma^{A}_{zx}$ begins a steady decrease itself above 100 K. The temperature dependence of $\sigma^{A}_{zx}$ is similar to what was reported in previous reportsNayak2016 ; Kiyohara2016 and its zero-temperature magnitude (which depends on stoichiometryKiyohara2016 ) is in agreement with was reported for a similar Mn content Kiyohara2016 ; supplement . As seen in Fig. 3d, the anomalous Lorenz ratio in Mn3Ge and in Mn3Sn behave very differently, in spite of the fact that resistivity in both shows only a slight change with temperature (Fig. 3e), in contrast to elemental ferromagnets.

The Bridgman relation

Several previous reports of the violation of WF law have been refuted afterwards. One may therefore wonder if our data can be validated by an independent criteria. The answer is affirmative. Their validity is supported by the verification of the Kelvin relation (for normal longitudinal transport coefficients) and the Bridgman relation (for anomalous transverse coefficients). Their violation counter the thermodynamics of irreversible processes.

To check the Bridgman relation, we directly measured both the Nernst (Fig. 4a) and the Ettingshausen (Fig. 4b) effects. The former is the transverse electric field generated by a longitudinal thermal gradient, $S_{zx}=\frac{E_{z}}{\nabla_{x}T}$ and the latter is the transverse thermal gradient produced by a longitudinal charge current, $\epsilon^{A}_{zx}=\frac{\nabla_{z}T}{J_{x}}$. The anomalous Ettingshausen effect (Fig. 4b) is easily invertible by a small magnetic field, like other anomalous transverse responses. We measured $S^{A}_{zx}$ and $\epsilon^{A}_{zx}$ at several different temperatures. Combined with longitudinal thermal conductivity data $\kappa_{xx}(T)$, this allowed us to check the Bridgman relationBridgman , which links these three quantities:

[TABLE]

As seen in Fig. 4c, the two sides of the equation remain close to each other in the whole temperature range. The Bridgman relation, derivable by a thermodynamic argumentSommerfeld is based on Onsager reciprocityCallen . Its experimental validity has been confirmed in semiconductorsDelves and in superconductors hosting mobile vorticesHuebener . While there is a previous report on simultaneous measurements of anomalous Nernst and Ettingshausen coefficientsSeki , the present study is the first experimental confirmation of the validity of Bridgman relation in the context of topological transverse response. We also verified the Kelvin relation linking the Seebeck and Peltier coefficientssupplement .

The temperature dependence of the anomalous transverse thermoelectric conductivity, $\alpha^{A}_{zx}$, extracted from anomalous Nernst coefficient, is presented in Fig.4d. As expected, it vanishes in the zero-temperature limit, but is remarkably large at room temperature. This can be seen by noting that the ratio of $\alpha^{A}_{zx}$ to $\sigma^{A}_{zx}$ is close to $k_{B}/e$ at room temperature. This contrasts with the ordinary longitudinal counterpart of this ratio which includes a $T/T_{F}$ damping factor, inversely scaling with the Fermi temperature, $T_{F}$Behnia2016 .

Origin of the finite-temperature violation

Having discussed the thermoelectric response, let us now turn back to the thermal transport. The zero-temperature validity of the WF law implies that the transverse flow of charge and entropy caused by Berry curvature conforms to a ratio of $\frac{\pi^{2}}{3}\left(\frac{k_{B}}{e}\right)^{2}$. This confirms Haldane’s argument Haldane2004 that AHE is a property of the topological quasi-particles of the Fermi surface. As argued previouslyLi2017 , only the states within the thermal window of the Fermi surface can be affected by a temperature gradient and give rise to a finite $\kappa^{A}_{zx}$. However, even in the case of ordinary longitudinal transport Ziman , the WF law ceases to be valid in presence of inelastic scattering. This is because small-angle inelastic collisions decay the momentum flow less efficiently than the energy flow both for electron-phonon Ziman and electron-electronJaoui2018 scattering. However, if inelastic scattering played a significant role, one would have not observed such a drastic difference between Mn3Sn and Mn3Ge. Moreover, since the violation of the WF law is not correlated with any significant change in the magnitude of resistivity, it cannot be attributed to the gradual emergence of an extrinsic source of AHE Shiomi . Therefore, one should look for a hitherto unidentified route to the finite-temperature violation of the anomalous transverse WF law.

In the semi-classic picture of electronic transport, charge and heat conductivity are set by the mean-free-path and the Fermi radius averaged over the whole Fermi surface with a pondering factor, which isvanhouten1992 ; Ziman ; Behnia2015 :

[TABLE]

This expression for the pondering factor has been obtained for both Boltzmann Ziman and Landauervanhouten1992 formalisms with $n=0$ for charge transport, $n=1$ for thermoelectric transport and $n=2$ for thermal transport. Therefore, the main source of each transport coefficient is not located at the same place in the $k$-space. The Berry curvature summed over the Fermi sheets with these pondering factors can potentially generate a mismatch between $\sigma^{A}_{ij}$ and $\kappa^{A}_{ij}$. As the temperature raises, the summations extend over a broader interval in the $k$-space inversely proportional to the thermal de Broglie length of electrons ($\Lambda=\frac{h}{\sqrt{2\pi m^{*}k_{B}T}}$)Behnia2015 . The electrical and thermal summations of the Berry curvature depend on these two length scales as well as the fine details of the spectrum, i.e. the size of the Berry curvature sources (and sinks) as well as their relative distance to the Fermi level (See Fig. 5). One can see that the validity of the WF correlation becomes harder to satisfy with warming. The higher the temperature, the larger the $k$-space area swept by the two different pondering functions and the easier to find a mismatch between $\sigma^{A}_{ij}$ and $\kappa^{A}_{ij}$ if the Berry curvature changes significantly across $\Lambda^{-1}$. In this context, one can conceive a mismatch between thermal and electrical summations of the Berry curvature (whose typical scale in the momentum space is an order of magnitude smaller).

Theoretical calculations of the anomalous Lorenz ratio

The anomalous Hall conductivity $\sigma^{A}_{zx}$ and anomalous thermal Hall conductivity $\kappa^{A}_{zx}$ are expressed in the form of the Berry curvature $\Omega_{zx}^{n}(k)$ Vafek2001 ,

[TABLE]

Here $f(\xi-\mu)=1/(e^{\frac{\xi-\mu}{k_{B}T}}+1)$ is the Fermi-Dirac function and $\hat{v}_{x/z}$ is the velocity operator,and $\tilde{\sigma}_{zx}(\xi)$ is actually the zero-temperature anomalous Hall conductivity (divided by $\frac{e^{2}}{\hbar}$). We point out that the electric and thermal coefficients are mainly different in the pondering factors for $\tilde{\sigma}_{zx}(\xi)$. As shown in Fig. 5, the pondering factor $-\frac{\partial f(\xi-\mu)}{\partial\xi}$ in Eq. 5 is a $\delta$-function while $-(\xi-\mu)^{2}\frac{\partial f(\xi-\mu)}{\partial\xi}$ in Eq. 6 displays a double-peak profile. The double-peak distance is about $5k_{B}T$, where $T$ is the temperature. This is the essential cause to violate the anomalous WF law at finite $T$. It is clear that the anomalous Lorenz ratio $L^{A}_{zx}=\frac{\kappa^{A}_{zx}}{T\sigma^{A}_{zx}}$ remains as a constant $L_{0}=\frac{\pi^{2}}{3}(\frac{k_{B}}{e})^{2}=2.44\times 10^{-8}~{}\Omega$WK*-2* at exact zero $T$, because two pondering functions summarize at the same Fermi energy, $\mu$. At finite $T$, for example, if $\tilde{\sigma}_{zx}(\xi)$ is a constant or a linear function of $\xi$, $L^{A}_{zx}$ also remains $L_{0}$.

Here, we summarize general rules about the anomalous WF law. At zero temperature, $L^{A}_{zx}=L_{0}$ is valid, as long as the total Berry curvature is smooth to energy, which is commonly true. At finite $T$, $L^{A}_{zx}$ is close to $L_{0}$ if $\tilde{\sigma}_{zx}(\xi)$ has an approximately antisymmetric profile with respect to $\mu$ in the energy window of $\sim 5k_{B}T$, because two pondering factors contribute the same summation of the Berry curvature in this case. Otherwise, $L^{A}_{ij}$ can vary strongly from $L_{0}$. Such a violation of the anomalous WF law is caused by the distribution of the Berry curvature, instead of the extrinsic scattering.

The total Berry curvature $\tilde{\sigma}_{zx}(\xi)$ is determined by the intrinsic band structure of a give material. It is not surprising that $L^{A}_{zx}$ depends sensitively on the position of $\mu$ because $\kappa_{zx}^{A}(\mu)$ and $\sigma_{zx}^{A}(\mu)$ do. In experiment, Mn3Ge(Sn) is usually off-stoichiometric in the form of Mn3+xGe(Sn)1-x Li2017 ; Nakatsuji2015 ; Nayak2016 ; Kiyohara2016 , where there are usually excess of Mn but deficiency of Ge(Sn). Such an off-stoichiometry can shift the chemical potential up in energy compared to the charge neutral point.

By shifting $\mu$ slightly above, we can successfully reproduce the general trend of $T$-dependence of $L^{A}_{zx}$ for both Ge and Sn compounds in Fig. 6. For Mn3Ge at $\mu=180$ meV above the charge neutral point, $L^{A}_{zx}$ drops quickly from 100 $\sim$ 250 K and goes up again after 250 K, qualitatively consistent with the experiment. The first drop is induced by dips of $\sigma^{A}_{zx}$ at $\sim$0.1 and $\sim$0.3 eV, because the $\kappa^{A}_{zx}(T)$ sums smaller Berry curvature than $\sigma^{A}_{zx}(T)$ does in this case (see Eqs.5-6). We note that the dip at $\sim$0.1 eV is induced by an anti-crossing gap while the dip at $\sim$0.3 eV is caused by a Weyl point. (See the supplementary Fig. S11) The following up-turn is related to the increase of $\sigma^{A}_{zx}$ after these dips. We also note that the theoretical violation is smaller than the experimental one. This may point to a significant role played by electronic correlations, which is neglected in the density-functional theory employed here. For Mn3Sn at $\mu=140$ meV, however, $L^{A}_{zx}$ remains close to $L_{0}$ in a large temperature range.

Different behaviors between two materials originate in their different Berry curvature $\tilde{\sigma}_{zx}(\xi)$, as shown in Fig. 6(a) and (c). For Mn3Sn, $\tilde{\sigma}_{zx}(\xi)$ is approximately anti-symmetric with respect to $\mu=140$ meV in an energy window of nearly 200 meV, inducing $L^{A}_{zx}\approx L_{0}$ up to 400 K and even above. However, $\mu$ may vary from such an ideal anti-symmetric point in different samples. This explains the observed deviation of of $L^{A}_{zx}$ in a different Mn3Sn sample, as shown in Fig.  S9. To demonstrate the sensitive role of the chemical potential, we show $L^{A}_{zx}-T$ curves for different $\mu$ in Fig. 6. Because it depends on the competition of the Berry curvature [$\tilde{\sigma}_{zx}(\xi)$] profile and the temperature ($5k_{B}T$), $L^{A}_{zx}\approx L_{0}$ at low $T$ for different $\mu$. In addition, there are interesting topological features near the chose chemical potential (see Fig. 6e-f). Along the $\Gamma-M$ line in the Brillouin zone, there are two Weyl points induced by the crossing between the lowest and second lowest conduction bands. These Weyl points were not revealed in previous study which usually focuses on the crossing between the lowest conduction band and the highest valence band. For Mn3Sn at $\mu=140$ meV, a Weyl point exists and contributes large Berry curvature to the anomalous Hall conductivity. For Mn3Ge at $\mu=180$ meV, however, the Weyl cone is strongly tilted so that negative and positive Berry curvatures nearly cancel each other near the Weyl point (see more information in the supplementary Fig. S10).

Concluding remarks

To summarize, we measured counterparts of the anomalous Hall effect associated with the flow of entropy and found that the Wiedemann-Franz law linking the magnitude of the thermal and electrical Hall effects is valid at zero-temperature, but a finite deviation emerges above 100 K. This deviation is not caused by inelastic scattering, but arises because of a mismatch in thermal and electrical summations of Berry curvature over the Fermi surface. The Bridgman relation, which links anomalous Nernst and Ettingshausen coefficients is satisfied over the whole temperature range. Finally, we observed that the room-temperature $\alpha^{A}_{zx}/\sigma^{A}_{zx}$ ratio is close to $k_{B}/e$.

**METHODS

Sample preparation and transport measurement:** Single crystals of Mn3Ge were grown from polycrystalline samples, using Bridgman-Stockbarger technique. The raw materials, Mn (99.99% purity) and Ge (99.999% purity), were weighed and mixed in an Argon glove box with a molar ratio of 3.3:1, loaded in an alumina crucible then sealed in a vacuum quartz ampule. The mixture was heated up to 1050 oC, remained for 2 hours to ensure homogeneity of melt, then was cooled slowly down to 800 oC to obtain polycrystalline samples. The polycrystalline Mn3Ge were ground, loaded in an alumina crucible and sealed into another vacuum quartz ampule. The growth temperature was controlled at 980 oC and 800 oC for high-temperature and low-temperature end, respectively. Finally, to obtain high temperature hexagonal phase, the quartz ampule was quenched with water. The single crystals were cut by a wire saw into typical dimensions of $0.3\times 1.5\times 2$ mm3 for transport measurements. The stoichiometry was found to be Mn3.08Ge0.92 (Mn:Ge = 3.32-3.35:1) using energy dispersive X-ray spectroscopy (EDX), This is close to the ratio of the raw materials and comparable to previous reportsKiyohara2016 .

Longitudinal and Hall resistivity were measured by the standard four-probe method using a current source (Keithley 6221) with a DC-Nanovoltmeter (Keithley 2182A) in a commercial measurement system (PPMS, Quantum Design). The thermal conductivity and thermal Hall effect were performed using a heater and two pairs of thermocouples in the PPMS in a high-vacuum environmentLi2017 . For temperatures below 4.2 K, the measurements were performed in a dilution refrigerator inserted in a 14 T superconducting magnet using one-heater-three-thermometers set-up, allowing to measure longitudinal and transverse transport coefficients with the same contacts.

Theoretical calculations: The band structure was calculated with the density-functional theory in the framework of the generalized-gradient approximation Perdew1996 . The Bloch wave functions were projected to atomic-orbital-like Wannier functions Mostofi2008 . Based on the Wannier-projected tight-binding Hamiltonian, we calculated the Berry curvature and the anomalous Hall conductivity in the clean limit. More details can be found in Ref.  Zhang2017 . As shown in Fig. 6(a) and (c), the $\tilde{\sigma}_{zx}(\xi)$ data are analytically fitted so that integrals in Eqs. 5 and 6 can be evaluated accurately with dense energy steps.

Data availability: The data that support the findings of this study are available upon reasonable request.

Acknowledgements We thank Alaska Subedi for numerous discussions. This work was supported by the National Science Foundation of China (Grant No. 51861135104 and No. 11574097) and The National Key Research and Development Program of China(Grant No.2016YFA0401704). Z. Z. was supported by the 1000 Youth Talents Plan. K. B. was supported by China High-end foreign expert program and Fonds-ESPCI-Paris and by the KITP through the National Science Foundation Grant No. NSF PHY17-48958. B.Y. acknowledges the financial support by the Willner Family Leadership Institute for the Weizmann Institute of Science, the Benoziyo Endowment Fund for the Advancement of Science, Ruth and Herman Albert Scholars Program for New Scientists, and the European Research Council (ERC) under the European Union s Horizon 2020 research and innovation programme (grant agreement No. 815869). B.Y. thanks the fruitful discussion with Yang Zhang at the Max Planck Institute in Dresden.

S1 Magnitude of the anomalous Hall conductivity

The zero-temperature anomalous Hall conductivity found here (190 $\Omega^{-1}$cm*-1*) is to be compared with previous reports of 150-450 $\Omega^{-1}$cm*-1* in Ref. Nayak2016_S and 320-380 $\Omega^{-1}$cm*-1* in Ref. Kiyohara2016_S . The difference in stoichiometry between the present sample (Mn3.1Ge0.9) and the one studied in Ref. Kiyohara2016_S (Mn3.05Ge0.95) combined to the fact that the magnitude of the anomalous Hall resistivity depends on Mn contentKiyohara2016_S , provide an explanation for this apparent discrepancy. The temperature dependent anomalous Hall conductivity in different samples cut from the same batch are similar (see Fig. S1).

S2 Carrier density

The field dependence of Hall resistivity along two different directions, yield the same value for $\partial\rho_{H}/\partial B$ at high magnetic field at 2 K (see Fig.S2). The slope was found to be $R_{H}=\partial\rho_{H}/\partial B=-0.02~{}\mu\Omega\,$cm/T. Using a single-band model ($n=\frac{1}{{\rm e}R_{H}}$), we found a carrier density of 3.1$\times$1022 cm*-3*, which is comparable to Ref. Nayak2016_S . A small anomalous Hall resistivity observed previouslyNayak2016_S in the $B\parallel z$ configuration is also observed in our sample.

S3 Longitudinal thermal transport

Fig. S3 shows the temperature dependence of thermal conductivity along $x$ axis, $\kappa_{xx}$, and along $z$ axis, $\kappa_{zz}$ as well as the anomalous transverse thermal conductivity, $\kappa^{A}_{zx}$. The latter was extracted using the anomalous thermal resistivity, $W^{A}_{zx}$ and the longitudinal conductivities. Fig.S4 shows Lorenz ratio $L_{xx}$ for longitudinal transport in Mn3Ge. At finite temperature, $L_{xx}/L_{0}$ exceeds unity indicating a large contribution from phonons. As expected, the ratio tends to unity in the zero temperature limit.

S4 The Kelvin Relation

The Peltier effect is the heat current produced by a charge current. The Kelvin relation relates the Peltier and Seebeck coefficients: $\Pi_{xx}=TS_{xx}$. Fig. S5b shows $\Pi_{xx}$ and $TS_{xx}$ in Mn3Ge. The ratio of $\Pi_{xx}/TS_{xx}$ is close to 1 in the whole temperature range. The Peltier coefficient $\Pi_{xx}$ and the Ettingshausen coefficient $\epsilon_{zx}$ were obtained by symmetrizing the signals obtained by injecting positive and negative currents through the sample in order to eliminate the Joule heating effect. Comparing to Ettingshausen effect, we can measure the heat absorb/release from the sample. We measured the temperature gradient along the sample to get the heat combining with the thermal conductivityGarrido_S .

S5 Thermal evolution of anomalous transverse response

Fig. S6 displays the field dependence of the three transverse coefficients at different temperatures. The temperature dependence of the anomalous transverse transport coefficients presented in main text were deduced from these curves.

S6 Specific heat CP and magnetic susceptibility $\chi$

The Fig. S7 a) and b) show the specific heat data for Mn3Ge and Mn3Sn. As seen in the inset, the electronic contribution in Mn3Sn is 19.1 mJ mol*-1K-2*. From the low-temperature magnetization curve (Fig. S7c), the magnetic susceptibility $\chi$ is deduced to be 6.2 $\times$ 10*-3* emu Oe*-1mol-1*. As seen in Fig. S8, the magnetic susceptibility is much larger than what is expected for the Pauli susceptibility of mobile electrons according to the Wilson ratio. This implies that most of the susceptibility is due to Mn spins.

S7 The dependence of Lorenz ratio in $\rm{Mn}_{3}\rm{Sn}$ on stoichiometry

The focus of this paper is the validity and/or the violation of the anomalous transverse WF law in Mn3Ge. Nevertheless, let us notice that in the case of Mn3Sn, the published data suggests that the the precise chemical composition of the compound can effect validity or verification of the WF law with little effect on the temperature dependence of resistivity. The precise location of the chemical potential depends on the the Mn. A violation of the WF law in Mn3Sn is conceivable if the chemical potential shifts away from the $\mu=140$ meV (See Fig.6).

S8 The band structure and Berry curvature