Wiedemann–Franz behavior at the Weyl points in compressively strained HgTe

Domain: physics
Published: 2025-11-29
Editorial outcome: major_revision

Abstract
Weyl semimetals, with their unique electronic band structure, have drawn significant interest for their potential to explore quantum anomalies in condensed matter systems. In this study, we investigate the large positive magneto-thermal conductance associated with the gravitational anomaly – one of the predicted anomalies – for a Weyl semimetal based on a compressively strained HgTe layer. We clearly identify the Weyl regime in our device and accurately extract the thermal conductance by performing thermometry measurements at liquid helium temperatures using fully electronic methods. We observe the anticipated increase in thermal conductance, and it perfectly matches the electrical conductance according to the Wiedemann–Franz law. This finding indicates that, despite the unique electronic spectrum of Weyl semimetals, the mechanism governing heat transport in this system is the same as that for electrical transport, with no additional violations of conservation laws.

Introduction
The ability of electrons to carry heat provides a unique approach for exploring mesoscopic systems. Hence, significant interest has been recently attracted to the electron thermal transport in Weyl semimetals, characterized by having their electronic excitations governed by the Weyl equations1. These equations feature a linear band dispersion with relativistic electrodynamic properties, reminiscent of the ones encountered in particle physics2. This has led to Weyl semimetals being proposed as a condensed matter platform for studying phenomena analogous to those predicted in high-energy physics, such as quantum anomalies. In particular, for Weyl semimetals an increase of electrical conductance with magnetic field has been connected to the chiral anomaly3,4, while a positive magneto-thermal conductance has been predicted5,6 and attributed to the gravitational anomaly leading to the violation of separate conservation laws for energy-momentum tensors within chiral Weyl cones.

However, several challenges complicate the observation of these anomalies. One significant obstacle is that the Fermi level of available materials often lies far from the Weyl point, making it difficult to attribute positive magneto-conductance exclusively to the chiral anomaly, as it can also arise from various other mechanisms7,8. Observing the gravitational anomaly is even more challenging because heat, unlike charge, is not conserved and can easily dissipate from the electron subsystem into the crystal lattice9. Earlier experiments10–12 have been carried out at relatively high lattice temperatures, where the electron transport is significantly influenced by interaction with phonons, leading to phonon drag and rapid electron-phonon relaxation. Furthermore, exploring thermoelectric phenomena requires using the correct set of constitutive equations13,14 that reflect the experimental observables (voltage and heat flow). In contrast, the sets employed in refs. 10 and 11 do not meet this criterion. The issue is particularly evident in ref. 11, where the equations chosen lead to an incorrect sign of thermopower correction to the heat conductance. The constitutive equations for thermoelectric coefficients are often derived without directly referencing experimentally observable quantities, notably as in ref. 15, which is widely regarded as a primary reference in this field. However, it is essential to formulate these coefficients using directly measurable quantities to accurately capture their physical meaning. An experiment cannot measure short-circuited thermal conductance, and in an open circuit, there is always a mixture of the thermopower to the thermal conductance14.

Compressively strained (≈0.3%) HgTe is a well-established Dirac semimetal16, in which bulk inversion asymmetry leads to a splitting of the Dirac point into four Weyl points16,17. In zero magnetic field, this splitting is small, resulting in transport behavior characteristic of a Dirac semimetal. An in-plane magnetic field breaks the time-reversal symmetry and induces a larger separation of the Weyl nodes in momentum space18–20, similar to how Dirac nodes split into Weyl nodes in other non-magnetic Dirac/Weyl semimetals, such as CdAs21 and NaBi22. This leads to transport signatures typical of Weyl semimetals16,21,22.

We previously have studied the chiral anomaly, a negative magnetoresistance at the Weyl points in a compressively strained 66 nm thick HgTe layer16, grown by molecular beam epitaxy (MBE)23. These layers have a low intrinsic carrier density and the Fermi level can be tuned (by means of a gate voltage) from the valence band into the conduction band, enabling direct access to the Weyl points, massive and massless (i.e., topological) Volkov–Pankratov surface states, and bulk bands within the same device. In the present paper, we use this ability to unequivocally identify the Weyl regime and address the gravitational anomaly via thermal conductance measurements. These transport phenomena are fundamentally due to the linear dispersion near the Weyl points, as illustrated by the band structure in Fig. 1 for compressively strained HgTe (calculated using k⋅p theory24).

In Fig. 1b, we highlight the surface states (shown in maroon) that emanate from the Weyl nodes. The presence of topological surface states near the band crossings is a common property of Dirac and Weyl semimetals25–27. Only when the band crossings happen at high symmetry points of the Brillouin zone the surface states are absent. In the presence of a perpendicular magnetic field, these surface states can contribute to quantum oscillations and even give rise to quantum Hall plateaus16, which complicates an interpretation of magnetoresistance features as signatures of Weyl fermions, using the methods proposed in refs. 28, 29.

To detect the electronic thermal conductance exclusively and avoid a strong contribution of a phonon drag component, we aim for thermal separation between the electron and phonons subsystems and cool the device to a lattice temperature of 1.33 K. At this temperature, electrons primarily transfer heat to the lattice by emitting acoustic phonons, leading to a much lower electron-phonon relaxation compared to higher temperatures. Additionally, we utilize all-electronic methods of temperature measurement (via electronic noise)30 and heating (using a heater with hot electron diffusion)31,32.

As expected for a Weyl semimetal, we observe that the thermal conductance increases with the applied in-plane magnetic field. When we compare the thermal and electrical conductance, we find them matching according to the fundamental Wiedemann–Franz law for all relevant Fermi level positions. This indicates that the gravitational anomaly does not manifest in the heat transport in this system. We discuss the implications of our result for the observation of quantum anomalies in Weyl semimetals.

Results
For our experiment, we utilize a device shaped into a symmetric H-bar structure33 (see inset of Fig. 2a). The two 40 μm long and 4 μm wide channels are used as heater and detector. In the center, the channels are connected by a 4.5 μm wide and 6 μm long island of which the transport properties are probed. Both channels and the island are equipped with separate electrostatic gates, allowing for independent control of the carrier density. Unless specified otherwise, we gate the heater and the detector channels into the conduction band, while the gate voltage at the island Vg,i is varied to study transport at different positions of the Fermi level.

Figure 2a shows the zero-field longitudinal resistance, Rxx (B=0) of the island at lattice temperature T0 ≈ 1.33 K, as the Fermi level is tuned from the conduction band to the valence band. Weyl points are located near the resistance maximum of the gate sweep, corresponding to the charge neutrality point in the band structure. We characterize the island resistance Rxx (B) by measuring it as a function of in-plane magnetic field B (applied parallel to the direction of heat transport in the island) at different island gate voltages Vg,i using lock-in measurement techniques in Fig. 2b. As shown in Fig. 2b, we observe a negative magnetoresistance below 5 T16 for a wide range of Vg,i. The maximum effect is observed around Vg,i = −0.29 V, where Rxx (0) ≈ 2 kΩ also peaks as a function of Vg,i (see Fig. 2a). The negative magnetoresistance occurs exclusively in this field orientation; in all other orientations, we observe a positive magnetoresistance—strong when the magnetic field is perpendicular to the sample plane, and weak when the field lies in-plane but is oriented perpendicular to the current direction (see ref. 16 and Supplementary Note 1).

As discussed extensively in ref. 16, the observed negative magnetoresistance is attributed to the linear band crossings around the Weyl points (i.e., the chiral anomaly), while the small resistance increase up to 0.5 T is related to weak antilocalization.

To study the thermal transport in our Weyl material, we use current heating, running a DC current Iheat through the heater channel, which increases the electron temperature at its center to Thot, above the lattice temperature T0. This temperature increase results in a heat flow, with rate Q, through the island into the detector channel. This heat flow subsequently raises the electron temperature at the detector side, Tcold.

To determine Thot (and Tcold), we measure the power spectral density of the voltage fluctuations Sv of the detector channel using a conventional technique with resonant coupling to a cryogenic amplifier30. By carefully designing the electrical circuit we ensure that no other parts of the sample contribute to Sv, also excluding the potential influence of non-equilibrium noise due to the temperature difference34,35. This allows the extraction of an average electronic temperature of the detector Tdet via the Johnson-Nyquist noise expression Sv = 4kB Tdet Rch36, where Rch is the resistance of the channel (Fig. 3a). Using the symmetry of the H-bar, we can obtain the average heater temperature Thot without needing to install and calibrate a second cryogenic amplifier by changing the configuration and passing the current through the detector channel instead (Fig. 3a), while ensuring that Rch is the same for heater and detector channels.

Assuming linear response, the electron thermal conductance K increases linearly with temperature as K=κT, where κ is the temperature-independent thermal conductance coefficient. The heat transport equation through the island can then be written as (this relation is valid for both coherent and incoherent conductors)37

Q=κ=2(Thot2 − Tcold2), (1)

where Thot and Tcold are the electron temperatures at the entrance and exit of the island, respectively. Since the dominant mechanism of heat loss in the channels is simply through electron diffusion to the leads, the temperatures inside the channels are not uniform but rather develop profiles towards the leads. Thot and Tcold are thus not identical to the measured average temperatures Tdet and Theat, respectively. However, they can be accurately determined from these measurements by taking the temperature profile into account. Meanwhile, the heat flow Q is inferred from the rise in Tdet and is calibrated using the Joule power required to bring the heater to the same temperature (see Methods for details on both procedures).

Guided by Eq. (1), we plot the measured values of Q against Thot2 − Tcold2 (Fig. 3b). The observed linear relationship confirms that the thermal conductance scales linearly with temperature. For further analysis, we focus on the thermal conductance coefficient κ=K/T.

In Fig. 3c, the extracted κ is shown for different positions of the island Fermi level from n-type to p-type through the Weyl point. We observe the largest κ in the n-region (for gate voltage Vg,i = 0.2 V), where it exhibits a slight monotonous increase with magnetic field. Closer to the Weyl point (Vg,i = −0.2 V and Vg,i = −0.29 V), κ is more than 6 times smaller and non-monotonous, following 1/Rxx in its behavior. Perhaps surprisingly, when moving into the p-region (Vg,i = −0.4 V and Vg,i = −0.5 V) the thermal conductance looks nearly the same as at the Weyl point. We will argue below, that this is simply due to the enhanced hole-phonon scattering due to the large effective mass of the holes in the valence band of HgTe.

In a non-interacting fermion system, charge and heat transport are interconnected, as described by the fundamental Wiedemann-Franz law, κT=GL0T13. This law relates the electrical and thermal conductance through the Lorenz number L0, which for electrons with charge e is given by L0 = (π2/3)(kB/e)2 ≈ 2.44 × 10−8 WΩK−2. To check if the Wiedemann-Franz law is obeyed for our Weyl semimetal, in Fig. 3d we show the Lorenz ratio L=κ/G with G=1/Rxx taken from the corresponding traces in Fig. 2b. For any Vg,i we observe that the variations in magneto-thermal conductance match the variations in magneto-electrical conductance, leading to an almost constant L across the entire studied field range. Importantly, L ≈ L0 both at the Weyl point and in the n-conducting region. The slight increase of L above L0 at Vg,i = 0.2 V is well understood since at this gate voltage the measured island resistance is overestimated due to the comparable sheet resistivities of island and channels.

At the same time, we observe L to be significantly lower than L0 in the p-regime at Vg,i = −0.4 V and Vg,i = −0.5 V. We show in Supplementary Note 4, that in the p-regime the heat loss to the phonons in the island is almost two times larger than both for n-regime and close to the Weyl point. This accounts for extra heat loss and reduces the measured L.

According to Onsager’s generalized transport equations, the presence of thermopower S modifies the thermal conductance in a circuit with no net electric current so that13,14,38

κ=κ0 − S2G: (2)

Here, κ0 is the intrinsic thermal conductance associated with short-circuited configuration. We note that the effect described by Eq. (2) does not alter the mechanism of heat transport by individual particles but rather describes the macroscopic effect of the presence of a thermoelectric voltage on heat transport, so that the Wiedemann-Franz law should hold for κ0. For a proper analysis of κ, the contribution S2G has to be accounted for.

To address this, we measure the development of thermopower across the island with gate voltage using a second harmonic technique. The sign-changing behavior in Fig. 4a is in good agreement with the theoretical prediction of the Seebeck coefficient S = −(π2/3)(kB2T/e)(d ln G/dε)ε=εF, when the energy is taken at the Fermi level εF. The observed deviation in magnitude of the experimental value for S is not surprising, as the exact expression for S requires separately accounting for the contributions from all bands38.

Figure 4b depicts the variation of thermopower with the in-plane magnetic field B at Vg,i = −0.29 V. Despite a slight increase with B, the contribution S2G remains at least three orders of magnitudes smaller than the measured thermal conductance coefficient κ, which thus does not require extra correction.

Methods
Material and fabrication
The device is fabricated on a 72nm thick HgTe layer sandwiched between two 15.2nm top and bottom (Hg,Cd)Te layers on a GaAs substrate, grown by MBE. A CdTe/ZnTe virtual substrate applies a 0.28% compressive strain, lifting the Γ8 band degeneracy at the Γ point of the Brillouin zone to form Weyl points. The presence of the 0.28% compressive strain was confirmed by high-resolution X-ray diffraction (HRXRD) measurements, as discussed in detail in ref. 16.

The H-bar device consists of two channels connected by a central island. The channels are 40 μm long and 4 μm wide, while the island measures 6 μm in length and 4.5 μm in width. The device is patterned using electron beam and optical lithography in conjunction with wet etching. Ohmic contacts to the device are fabricated by depositing AuGe and Au layers, each with a thickness of 80nm. The top gate electrodes are composed of 5nm Ti and 100nm Au, with a 15nm HfO2 gate insulator, grown via atomic layer deposition, separating them from the mesa. The wide Ohmic contacts, fabricated using conventional lithography techniques, ensure homogeneous current injection and thereby eliminate current-jetting effects42, while the electrostatic gate enables tuning of the Fermi level in the thin layer to the Weyl point.

The carrier density and mobility were determined from Hall measurements performed on a gated Hall bar device (dimensions: 1200 μm × 200 μm) fabricated from the same wafer (Supplementary Note 8). Within the applied gate voltage range, the carrier density can be tuned from n ≈ 3.2 × 1011 cm−2 at Vg,i = 0.1 V to n ≈ −2.2 × 1011 cm−2 at Vg,i = −0.5 V. According to our band structure calculations, the Fermi level remains within a few meV of the Weyl points throughout this range. From the zero-field conductance, we estimate the mobility in the n-type conduction (Vg,i = 0.1 V) regime to be μ ≈ 60 × 103 cm2/(Vs). The mobility decreases as the Fermi level approaches the Weyl point, reaching μ ≈ 22.8 × 103 cm2/(Vs) at Vg,i = −0.2 V.

Transport measurements
The device is cooled in exchange gas thermally connected to a pumped 4He cryostat, resulting in base temperature T0 ≈ 1.33 K. The copper chip carrier is attached to a copper sample receptacle, resulting in a large effective cooling area and stable phonon temperature. In close proximity of the sample, a calibrated resistance thermometer is attached, allowing to measure the lattice temperature.

The device resistance is measured using a standard lock-in technique, wherein an alternating current with a frequency of ≈ 3 Hz and an amplitude of less than 60 μV is applied across the device and a 100 kΩ series resistor, across which the current is measured. The resistance of the island is measured using a four-terminal configuration, with the legs of the H-bars serving as the current and voltage probes. Magnetic field dependent properties are measured by applying a magnetic field at a sweeping rate of 0.1 T/min, aligned with the direction of heat and charge flow in the island and within the device plane.

Noisethermometry
We measure the power spectral density Sv from the voltage fluctuations picked up from the detector channel using a home-built low-temperature amplifier (LTA) with a voltage gain ALT ≈ 5 (the gain is determined by calibration of the measurement circuit using its thermal noise, the long-term stability is δALT/ALT < 1%), connected to one side of the detector channel through a coaxial cable. The amplifier is located outside of the sample chamber, submerged in a pumped liquid helium bath, away from the magnetic field. An air-core superconducting inductor is connected in parallel to the amplifier input, resulting in a resonance around fres in a 30 kHz bandwidth (Δf) defined with a Gaussian filter. We convert VSA to spectral power density of the voltage fluctuations Sv using Sv = VSA2 / (Δf ALT2 ART2).

When the heating current is applied to the side opposite the LTA connection, the electronic noise is converted to the average detector temperature using the Johnson-Nyquist relation Sv = 4kB Tdet Rch36. Similarly, when the heating current is applied on the same side as the LTA connection, we extract the average heater temperature using: Sv = 4kB Theat Rch. In this latter case, the extraction is reliable, as the resistance of the heater channel remains almost constant with heating current (see Supplementary Note 3).

The heat transport regime of heater and detector
Accurate extraction of the thermal conductance requires a clear understanding of the heat transport regime in the heater and detector channels. We identify this regime by analyzing the average heater temperature Theat as a function of the heating current Iheat in a broad range, extending up to Theat ≈ 12 K. The observed behavior reflects the underlying electron diffusion and electron-phonon relaxation mechanisms43–46.

Without allowing for phonon emission by hot electrons, one expects Theat = T0 [1 + (ν+ν−1) arctan(ν)]1/2 45 with the dimensionless parameter ν = 3eVbias / (2πkB T0), where Vbias = Iheat Rch is the voltage bias on the channel. This formula describes a transition from quadratic heating with Theat ≈ T0 [1 + ν2/2] at eVbias ≪ kBT0 to a linear temperature increase with Theat ≈ (3/4) eVbias / (2kB) when eVbias ≫ kBT0. In the range up to T ~ 3 K, we rather observe a slower growth of Theat (Fig. 3a). For higher temperatures, we observe a characteristic bending (Supplementary Note 4), indicating a phonon contribution to the heat relaxation in channels.

We identify the phonon relaxation law by matching the experiment with a finite element simulation based on the heat transport equation (see Supplementary Note 4). We find the electron-phonon relaxation per area in a form qph = Σph (T3 − T03) with coefficient Σph ≈ 0.19 Wm−2K−3. The power 3 corresponds to the relaxation via emission of polar acoustic phonons at low temperature47. A relaxation rate following the same dependence has been observed previously for narrow HgTe quantum wells48.

Determination of local temperatures Thot and Tcold
Johnson-Nyquist thermometry provides a measure of the average electron temperatures Theat and Tdet. However, Eq. (1) involves local temperatures, which differ due to spatial temperature profiles along the channels.

Assuming electronic diffusion is the dominant heat transport mechanism, the heat equation can be solved analytically for a uniform strip. This yields

Thot = T0 [1 + (Theat/T0)2 − 1]1/2

Tcold = T0 [1 + (Tdet/T0)2 − 1]1/2

The difference between these expressions arises from the distinct heating geometries: in the heater channel heat is generated uniformly along its length, whereas in the detector channel it is injected locally at the center.

The analysis of heat transport in the channels reveals that electron-phonon relaxation plays a significant role, thereby modifying the spatial temperature profiles and potentially affecting the result of Eq. (3). To account for this, we determine the actual local temperatures Thot and Tcold through detailed numerical modeling, which incorporates both electron-phonon coupling and the full device geometry (see Supplementary Note 6). We find that in the heater channel, the relation Thot(Theat) still closely follows the prediction of Eq. (3), while in the detector channel, the local temperature satisfies Tcold = T0 + (Tdet − T0) / [1 + 2.63 (Tdet/T0 − 1)] within the relevant range.

Evaluation of the heat flow rate Q
The average temperature increase of the detector channel, Tdet reflects the heat flow Q entering the detector through the island. To determine Q(T0) for a given detector temperature Tdet = T0, we identify the heater current I0 that brings the heater channel to the same average temperature, such that Theat(I0) = T0. The corresponding Joule power in the heater, PJ = (I0)2 Rch, serves as a reference for calibrating the detector response, as the channels are identical in their thermal transport properties (see Supplementary Note 5). However, Q and PJ are not identical, as the power required to reach a given temperature depends on whether heat is introduced uniformly or locally within the channel.

In the limit of small temperature increase above the base temperature T0, and assuming purely electronic diffusive heat transport without phonon losses for a uniform strip, the average temperatures are given by49

Theat = T0 + (PJ Rch) / (12 L0 T0), Tdet = T0 + (QJ Rch) / (8 L0 T0)

The difference between these expressions stems from the fact that the heater is uniformly heated by Joule dissipation, while the detector experiences localized heating at its center due to the incoming heat flow. Matching Theat = Tdet = T0 yields Q = αQ PJ with αQ = 2/3. The prefactor accounts for different spatial profiles of heating. For greater accuracy, we refine αQ using a numerical model that incorporates the actual device geometry and electron-phonon relaxation. The resulting calibration factor αQ lies in the range 0.613 ≤ αQ ≤ 0.625, with weak temperature dependence (see Supplementary Note 6).

Using the above, our calibration procedure is:
1. We measure Theat as a function of the heater current.
2. To find the heat flow for some detector temperature Tdet = T0 we interpolate Theat to find I0 where Theat(I0) = T0.
3. We calculate the heat flow Q(T0) = αQ (I0)2 Rch.

This method provides a robust and self-calibrated value of Q, where the influence of device geometry and phonon losses are accounted for through αQ. As a result, uncertainties in these effects contribute only as second-order corrections to the final value of Q.

Thermo voltage measurement
For a thermovoltage measurement, we pass a low-frequency alternating current through one of the channels, thereby inducing a time-dependent electron temperature difference across the island. The resulting thermal voltage develops at twice the excitation frequency and is measured using a standard lock-in technique at the second harmonic frequency. This thermal voltage is subsequently converted to the Seebeck coefficient S, utilizing the electron temperature difference between the heater and detector channels, which is determined via Johnson noise thermometry in the regime where the temperature has quadratic dependence on the heating current (close to T0).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Dear Editor,\nAfter carefully reviewing this manuscript, I am impressed by the innovative and rigorous measurement techniques employed.\nThe experimental data sufficiently support the core conclusion that \"the low-temperature thermal transport of HgTe under\nstress still conforms to the Wiedemann-Franz law.\" However, the authors further assert that the compressively strained HgTe\ntransforms into a Weyl fermion system, and that thermal transport does not violate the Wiedemann-Franz law. This claim\nlacks sufficient justification, and there are several specific issues that need clarification from the authors:\n1. Mechanism of Negative Magnetoresistance\nAs stated by the authors, negative magnetoresistance may arise from multiple mechanisms, and the positive\nmagnetoresistance observed when the magnetic field exceeds 2T does conform to weak localization characteristics.\nHowever, the key question is: how do the authors rule out that the negative magnetoresistance in the magnetic field region\nbelow 2T is not caused by weak anti-localization effects? This requires more substantial experimental evidence for\nclarification.\n2. Completeness of Gate Voltage Regulation\nThe study achieves continuous control of the Fermi surface between the valence band and the conduction band through\ngate voltage. However, the data presented indicate that when Vg = -0.29V, the Fermi surface should be precisely located at\nthe Weyl point, at which point a significant positive magnetoresistance effect should be observed under vertical magnetic\nfield conditions. The absence of this critical experimental data weakens the conclusion regarding Weyl\nsemimetals.\n3.Contradictory Phenomena in Magnetic Field Response\nAn analysis combining Figures 2b and 3d reveals that although a 2T magnetic field is a turning point for the dominance of\nchiral anomalies and weak localization, the Lorenz constant shows an unusually smooth variation within the range of 0-8T.\nDoes this suggest that the thermal response across the entire magnetic field range (both <2T and >2T) originates from trivial\nelectronic states rather than Weyl fermions?\nConsidering the innovative thermal measurement techniques in this work and the cutting-edge nature of the research topic, I\nrecommend that it is suitable for publication in Nature Communications after the above key evidence is supplemented,\nparticularly providing conclusive experimental evidence that the material system possesses Weyl semimetal characteristics.", "reviewer_2": "This paper by A. Aravindnath, et al investigates the Wiedemann-Franz behavior in a Weyl semimetal based on\ncompressively strained HgTe. By performing thermometry measurements at low temperatures using fully electronic methods,\nthe research clearly identifies the Weyl regime and extracts the thermal conductance. The observed increase in thermal\nconductance with the applied magnetic field matches the increase in electrical conductance, adhering to the Wiedemann-\nFranz law. This indicates that, despite the unique electronic structure of Weyl semimetals, the thermal transport mechanism\nremains consistent with electrical transport, without violating any conservation laws. Overall, this work is interesting and\nnovel, and I would be willing to recommend it for publication in Nature Communications after the authors address the\nfollowing concerns:\nAs we know, in thermal conductance measurements, the heat flow or heat power Q flowing through the sample is a crucial and\nintuitive quantity, especially when studying scaling laws such as the Wiedemann-Franz law. However, the analysis and\ncalculation of the heat power Q in the manuscript appear overly simplistic and cursory. I suggest that the authors provide a\nclearer and more detailed explanation of this heating power.", "reviewer_3": "This paper mainly studies the thermal transport properties of compressively strained HgTe layer. The Fermi level can be\ntuned by changing the gate voltage. The authors found that even though Weyl fermions are located at the Fermi level and\ndominate transport properties, the Wiedemann-Franz law still holds, indicating that Weyl fermions do not cause distinct\nphysics in thermal transport. In my opinion, this paper does not meet the high standards of Nature Communications, and\nother journals such as Communications Physics may be more suitable.\n(1) The innovation of the paper is not sufficient to be published in NC. I don't think the results of this article are very\ninteresting.\n(2) In the “results” section, the author spent a considerable amount of space describing how the experiment was conducted,\nwhich should be included in the “methods” section. The main text should mainly describe and discuss the experimental\nresults.\n(3) The data is not abundant and the quality is not high."}, "author_response": "We thank the reviewer for carefully evaluating the manuscript and appreciate the constructive\ncriticism. Please find our detailed response below.\n1. Mechanism of Negative Magnetoresistance\nAs stated by the authors, negative magnetoresistance may arise from multiple mechanisms, and\nthe positive magnetoresistance observed when the magnetic field exceeds 2T does conform to\nweak localization characteristics. However, the key question is: how do the authors rule out that the\nnegative magnetoresistance in the magnetic field region below 2T is not caused by weak anti-\nlocalization effects? This requires more substantial experimental evidence for clarification.\nThe authors’ reply:\nCompressively strained HgTe exhibits a pronounced negative magnetoresistance (MR) when the\nmagnetic and electric fields are aligned (i.e., B E). This negative magnetoresistance, however, is\naccompanied by positive MR features at both low and high magnetic field ranges. The positive\nmagnetoresistance at low fields (<0.5T) is indeed attributed to the weak anti-localization (WAL)\neffect, a quantum interference phenomenon. The reviewer is probably referencing the weak\nlocalization (WL), another quantum interference phenomenon, that leads to resistance decrease\nwith magnetic field and is commonly observed in combination with WAL.\nSeveral arguments allow us to clearly distinguish the observed resistance decrease at intermediate\nfields (from 0.5T up to 5T) from WL:\nFirst, due to the high mobility in our sample, the observed negative MR cannot be attributed to WL.\nInstead, at the expected very low magnetic fields, HgTe exhibits WAL (Ref. A), arising from strong\nspin–orbit coupling, on top of the negative MR background.\nSecond, WL is expected to be active when electrons can pick phase from magnetic fields. This is\ntypically stronger for the field orientations, perpendicular to the current flow, due to the larger flux\nthrough electron trajectories. In contrast, in our system negative MR is only observed for in-plane\nmagnetic field, parallel to the current flow, B I. For all other field directions, we observe positive MR\n(see the answer to the next question and newly added Supplementary Fig. 1 in the Supplementary\nInformation). Specifically, for in-plane perpendicular we observe very weak positive MR, which is\nunlikely for WL.\nThird, the observed negative MR in compressed HgTe survives at high temperatures, while WL is\nsuppressed quickly as the quantum coherence is destroyed. Below we present the data adopted\nfrom the thesis of David Mahler (https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-253982,\nFig.7.8), where the temperature dependence of magnetoresistance in compressively strained HgTe\nwas measured under B E. The measurements were performed in our group on a different device\nwith the layer stack and material properties closely matching those of the device discussed in this\nwork. On top of the original figure panel b we added a mark pointing to the region of WAL.\n[Figure Redacted]\nFrom Fig.7.8 (b), we observe that the sharp WAL feature at B=0T quickly disappears with increasing\ntemperature, confirming its origin from interference effects. In contrast, the pronounced negative\nMR at intermediate magnetic fields remains largely unchanged. This effectively rules out quantum\ninterference mechanisms such as WL as the underlying cause. A noticeable change in the negative\nmagnetoresistance is only observed at 23 K, likely due to enhanced phonon scattering at elevated\ntemperatures.\nLast, in our device we have a direct control over the position of the Fermi level in the band structure\nwith an electrostatic gate. We observe prominent negative MR only when the Fermi level is tuned\nnear the position of the Weyl points, which is verified via carrier density extracted from the out-of-\nplane magnetic field Hall measurement. This does not match the expectations for WL, where no\ndirect relation to the band structure position of Fermi level is expected. At the same time, the band\nstructure calculations based on k p theory reveal the existence of linearly dispersing Weyl points in\ncompressively strained HgTe, that should result in effect of chiral anomaly only for Fermi level close\nthe Weyl points as detailed in Ref. [16].\nThese directional and energy-specific properties of our observations support the interpretation of\nthe negative magnetoresistance as a signature of the chiral anomaly.\n2. Completeness of Gate Voltage Regulation\nThe study achieves continuous control of the Fermi surface between the valence band and the\nconduction band through gate voltage. However, the data presented indicate that when Vg = -0.29V,\nthe Fermi surface should be precisely located at the Weyl point, at which point a significant positive\nmagnetoresistance effect should be observed under vertical magnetic field conditions. The\nabsence of this critical experimental data weakens the conclusion regarding Weyl semimetals.\nThe authors’ reply:\nWe have added measurements of the magnetoresistance of our device for out-of-plane magnetic\nfield configuration to the Supplementary Materials. Indeed, for an out-of-plane field, when the\nFermi level is tuned to the Weyl points, a pronounced positive magnetoresistance is observed.\nWe also performed measurements with an in-plane magnetic field, perpendicular to the electric\nfield, which results in very weak positive magnetoresistance. These observations indicate that the\nnegative magnetoresistance is exclusive to the configuration where the magnetic field is aligned\nparallel to the electric field within the device, as expected for chiral anomaly. A comprehensive\nstudy on the angular dependence of magnetoresistance actually is available in our previous paper\non compressively strained HgTe with a comparable layer stack and electronic properties (Ref. [16]),\nproviding further insights into the role of chiral anomaly in the observed transport behavior.\n3.Contradictory Phenomena in Magnetic Field Response\nAn analysis combining Figures 2b and 3d reveals that although a 2T magnetic field is a turning point\nfor the dominance of chiral anomalies and weak localization, the Lorenz constant shows an\nunusually smooth variation within the range of 0-8T. Does this suggest that the thermal response\nacross the entire magnetic field range (both <2T and >2T) originates from trivial electronic states\nrather than Weyl fermions?\nThe authors’ reply:\nWe thank the reviewer for their useful comment, that points directly to the core message of our\nstudy. The stability of the Lorenz ratio as a function of the magnetic fields demonstrates that the\nheat is transported via the same mechanism as the charge at all fields, and that it is transported via\nfermionic quasiparticles with charge -e. So, this does not mean that the thermal transport is trivial,\non the contrary, this indicates, that if the charge transport originates from the effect of the chiral\nanomaly, so does the heat transport.\nHowever, theories of Weyl semimetals predict that their thermal conductance can be influenced by\nthe so-called gravitational anomaly. Below we summarize how our observations relate to\nexpectations across different transport regimes (we only consider those, where negative\nmagnetoresistance from chiral anomaly should be present):\n1. In an ultraclean Weyl semimetal, the chiral anomaly might affect the charge transport due\nto the anomalous charge pumping between the Weyl nodes, but less so the energy and heat\ntransport. In that case one might expect L < L0.\n2. If inelastic e-e interactions are strong and dominate over all other elastic and inelastic\nscattering processes, a hydrodynamic transport regime might be realized, where electrons\nexhibit a form of collective motion, which should result in a substantial breakdown of the\nWiedemann-Franz law, presumably with L > L0. This is the regime where the gravitational\nanomaly effects might be observed in thermal transport.\n3. Otherwise, if inelastic electron-phonon scattering dominates, which is expected for higher\ntemperatures, electrons and phonons are strongly coupled and come to the local\nequilibrium. Thermal conductance then contains electron and phonon components, and\nthe electronic part trivially follows L=L0.\n4. If elastic scattering dominates and the internode scattering rate is low, both charge and\nheat are transported via the same fermionic quasiparticles, and L=L0 is expected, even in\nthe presence of the anomaly.\nPrevious studies of thermal transport in Weyl systems were done in regime (3), due to the\nmeasurements being performed at high lattice temperatures, when electron-phonon scattering is\nstrong. In contrast, we take special care to ensure that we are not operating in this regime by\nworking at low lattice temperatures with fully electronic methods of heating and temperature\ndetection, minimizing the effects of phonon drag and lattice heating. We also directly verify that the\nelectron-phonon relaxation is small enough. However, we find, that under these conditions L=L0,\nindicating that regime (2) is not realized and rather pointing to the regime (4), which is by no means\ntrivial, but does not allow to experimentally detect the effect of gravitational anomaly.\nConsidering the innovative thermal measurement techniques in this work and the cutting-edge\nnature of the research topic, I recommend that it is suitable for publication in Nature\nCommunications after the above key evidence is supplemented, particularly providing conclusive\nexperimental evidence that the material system possesses Weyl semimetal characteristics.\nReviewer #2 (Remarks to the Author):\nThis paper by A. Aravindnath, et al investigates the Wiedemann-Franz behavior in a Weyl semimetal\nbased on compressively strained HgTe. By performing thermometry measurements at low\ntemperatures using fully electronic methods, the research clearly identifies the Weyl regime and\nextracts the thermal conductance. The observed increase in thermal conductance with the applied\nmagnetic field matches the increase in electrical conductance, adhering to the Wiedemann-Franz\nlaw. This indicates that, despite the unique electronic structure of Weyl semimetals, the thermal\ntransport mechanism remains consistent with electrical transport, without violating any\nconservation laws. Overall, this work is interesting and novel, and I would be willing to recommend\nit for publication in Nature Communications after the authors address the following concerns:\nAs we know, in thermal conductance measurements, the heat flow or heat power Q flowing through the sample is a crucial and intuitive quantity, especially when studying scaling laws such as the Wiedemann-Franz law. However, the analysis and calculation of the heat power Q in the\nmanuscript appear overly simplistic and cursory. I suggest that the authors provide a clearer and\nmore detailed explanation of this heating power.\nThe authors’ reply:\nWe thank the reviewer for their useful comments and appreciate their evaluation of our work. In the\nupdated version of the manuscript, we reworked and expanded the discussion of the extraction of\nthe heat flux, flowing through the sample and added extra details to support our analysis. We\nadded extra subsection to the “Methods” that explains our approach to finding the heat flux.\nReviewer #3 (Remarks to the Author):\nThis paper mainly studies the thermal transport properties of compressively strained HgTe layer.\nThe Fermi level can be tuned by changing the gate voltage. The authors found that even though Weyl\nfermions are located at the Fermi level and dominate transport properties, the Wiedemann-Franz\nlaw still holds, indicating that Weyl fermions do not cause distinct physics in thermal transport. In\nmy opinion, this paper does not meet the high standards of Nature Communications, and other\njournals such as Communications Physics may be more suitable.\n(1) The innovation of the paper is not sufficient to be published in NC. I don't think the results of this\narticle are very interesting.\nThe authors’ reply:\nWe respectfully disagree with the reviewer on this point. In our view, the present work makes a\nmeaningful contribution to the study of Weyl semimetals. The question of whether the\nWiedemann–Franz law is violated—and whether hydrodynamic electron thermal conductance can\nbe detected—is central to experimental efforts aimed at observing the solid-state analogue of the\ngravitational anomaly in these systems. In fact, the original proposal that Weyl semimetals could\nserve as a platform for such studies has been a key motivation for much of the thermal transport\nresearch in this field.\nHowever, a reliable experimental confirmation of nontrivial heat transport mechanisms in Weyl\nsemimetals remains elusive. This is due in part to the challenge of reaching a regime where\nelectron–phonon scattering is weak, and in part to the intrinsic difficulty of isolating the electronic\ncontribution to thermal conductance. In this work, we present an experiment that explicitly\naddresses both issues with care—yet we still do not observe the expected thermal anomaly. We\nbelieve this result provides important insight into the nature of thermal transport in this class of\nmaterials.\n(2) In the “results” section, the author spent a considerable amount of space describing how the\nexperiment was conducted, which should be included in the “methods” section. The main text\nshould mainly describe and discuss the experimental results.\nThe authors’ reply:\nWe acknowledge that the details of Johnson-Nyquist thermometry might not be interesting for every\nreader of the manuscript. Yet in our case the choice of the method of thermal conductance\nmeasurement is strongly connected to the observed physics: we want to operate in a regime where\nthe phonon contribution is small and electron-phonon coupling is weak enough to be able to\nresolve the specific mechanisms of electronic transport. In the updated version of the manuscript,\nwe moved some less important description of the measurement into the methods section, while\nkeeping the necessary details to support this message.\n(3) The data is not abundant and the quality is not high.\nThe authors’ reply:\nWe would like to draw the reviewer’s attention to the details of our study, in particular, to the\nresearch method we used. High precision Johnson-Nyquist thermometry is a very time-consuming\nexperimental technique, requiring prolonged measurements to achieve high precision due to the\nstatistical nature of the signal. The value that one should look at to understand the final precision is\nthe relative error deltaT/T. In our case we achieved deltaT of approx. 5mK for T~1.3K and 95%\nconfidence interval. By this parameter our measurement is on par with the best practices in the\nfield.\nAdditionally, to improve this parameter by n times, the measurement time needs to be multiplied by\nn^2. This can potentially result in enormous measurement times, that don’t necessary improve\nthe analytic potential of the data. A study that results in high precision thermal transport coefficients,\nhowever, still requires several months of continuous measurements.\nThe choice of the research method is not our whim - the fragility of electronic thermal transport with\nrespect to electron-phonon interactions and the need to accurately control heat flows in the\nsystem simply does not allow the use of other approaches, which we discuss in detail in the text of\nthe article.\nGiven all this, we believe that the quantity of data and their quality are sufficient to reliably support\nthe main conclusions of our article and follow the best practices in the field.\nA) Mühlbauer, M., et al. \"One-dimensional weak antilocalization due to the Berry phase in HgTe\nwires.\" Phys. Rev. Lett. 112, 146803 – 2014."}, {"round": 2, "reviewer_comments": {"reviewer_1": "This article elaborates on the research conducted to modulate the topology of HgTe by precisely applying external stress to\nalter its crystal structure. This work represents significant advancement and importance in the field of topological materials.\nHowever, the determination of whether the modulation is successful must be approached rigorously, primarily based on the\nfollowing two criteria: 1）Whether the crystal structure has indeed changed. This serves as the foundation for verifying the\nmodulation effect. 2）Whether the key experimental evidence supporting the two types of topological properties is sufficient.\nThis is crucial for ensuring the reliability of the research conclusions.\nAs these critical conditions have not yet been fully verified, it is currently not possible to scientifically conclude whether this\nresearch has achieved its intended goals.", "reviewer_2": "All my concerns have been addressed. I recommend its publications."}, "author_response": "RESPONSE TO THE REFEREE COMMENTS\nReviewer #1 (Remarks to the Author):\nThis article elaborates on the research conducted to modulate the topology of HgTe by precisely\napplying external stress to alter its crystal structure. This work represents significant advancement\nand importance in the field of topological materials. However, the determination of whether the\nmodulation is successful must be approached rigorously, primarily based on the following two\ncriteria: 1）Whether the crystal structure has indeed changed. This serves as the foundation for\nverifying the modulation effect. 2）Whether the key experimental evidence supporting the two\ntypes of topological properties is sufficient. This is crucial for ensuring the reliability of the research\nconclusions.\nAs these critical conditions have not yet been fully verified, it is currently not possible to\nscientifically conclude whether this research has achieved its intended goals.\nThe authors’ reply:\nWe thank the reviewer again for their careful evaluation of the manuscript and positive opinion\nabout its impact on the field of topological materials.\n1) We agree that the presence of compressive strain is a crucial element for looking for Weyl\nphysics in HgTe samples. The ability to introduce the compressive strain and change crystalline\nstructure of HgTe has been proven previously in the 2019 Mahler et al. paper from our group (Ref.\n[16] in the article, 10.1103/PhysRevX.9.031034). In there, we verified by High resolution X-ray\ndiffraction (XRXRD) analysis of 66 nm and 120 nm layers grown via the same MBE technology\nas the one used in this article. The paper reports, that the 66 nm layer is fully strained, with 0.3%\ncompressive strain introduced by the lattice mismatch, while the 120 nm layer is partially relaxed,\nso that the strain is 25% less. The 72 nm layer studied in the present paper is close in thickness\nto the 66 nm layer from Ref. [16], i.e. is almost fully strained according to the lattice mismatch.\nHRXRD strain data are routinely obtained for every MBE-grown sample, including the one used\nin this study, so we exactly measured the amount of strain indicated in the manuscript (0.28%).\nWe additionally reproduce the main results of HRXRD analysis from Supplementary Material of\nRef. [16] below and additionally mention the strain measurement in the methods section.\n[Figure Redacted]\n2) Please find below the answers to individual remarks regarding the experimental evidence.\nReviewer’s response:\nThe authors mention that their samples exhibit high mobility. Under such conditions, Shubnikov–\nde Haas (SdH) oscillations are typically readily observable in magnetoresistance measurements.\nThe presence of SdH oscillations would constitute compelling evidence in identifying the Weyl\nnature of the material. Therefore, I strongly encourage the authors to include this crucial data in\nthe supplementary materials to further strengthen the manuscript.\nThe authors’ reply:\nWe thank the reviewer for inviting the discussion of quantum oscillations in Weyl semimetals.\nIndeed, the position of the intercept of SdH oscillations trend is sometimes interpreted as evidence\nfor Weyl nature of the material, following, for example, the logic discussed in Ref. [28], which is\nused for high-density bulk materials.\nHowever, this approach should be applied with extreme caution, as in many Weyl semimetals\ntopological surface states emanate from the Weyl points and provide additional\nmagnetoresistance oscillations and quantum Hall plateaus at high magnetic field. This is, for\nexample, a well-established feature of (compressively strained) HgTe, Cd As and Na Bi, which\n2 3 3\nall belong to the same class of Dirac semimetals, in which the nodes do not appear at high\nsymmetry points in the Brillouin zone. The Dirac nodes are split into Weyl nodes by BIA (bulk\ninversion asymmetry) and this splitting is obviously enhanced by the Zeeman effect when a\nmagnetic field is applied.\nOur previous study (Ref. [16]) focuses exactly on the topic of magnetoresistance and its relation\nto the surface states. For our low carrier density devices, we not only observe the SdH oscillations,\nbut also the quantum Hall effect (QH) in the transverse resistance. We analyse the transverse\nconductivity, which is a combination of longitudinal and transverse resistances:\n(𝑅𝑥𝑥) (𝑅𝑥𝑦)\n−𝑅𝑥𝑦\n𝜎𝑥𝑦 = 2 2,\nwhere W is the width and L is the length𝑅 𝑥o𝑦f +the(𝑊 H𝑅a𝑥l𝑥 b/𝐿a)r. provides means for even better\nunderstanding of the bandstructure compared to just SdH oscillations:\n𝜎𝑥𝑦\n[Figure Redacted]\nThis 2D plot from Ref. [16] shows the derivative of with gate voltage, yielding peaks at\ntransitions between the Landau levels. The unusual plateau sequence is a mixture of contributions\n𝜎𝑥𝑦\nfrom the electronic topological surface states (black) and hole-like massive surface states\n(magenta). The observed quantum oscillations can therefore not be used to extract Berry\ncurvature or confirm the Weyl band structure of the material, as they originate from the surface\nstates.\nIn this plot, U = 4 V corresponds to n=13x1011 cm-2 while U = -4 V corresponds to n=-10x1011\ngate gate\ncm-2 of hole conductance. In our new paper we operate in a smaller range around Weyl points,\nso that the carrier density varies between approximately n=3.5x1011 cm-2 and n=-2.2x1011 cm-2.\nThe difference in applied gate voltages comes from the use of a different insulator (material and\nthickness) than in Ref. [16].\nWe added the corresponding references and discussion to the Supplementary information section\n1, where the magnetoresistance data for perpendicular magnetic field is discussed.\nReviewer’s response:\nThe authors repeatedly emphasize the high mobility of their samples and the proximity of the\nFermi level to the Weyl point, yet no quantitative mobility data have been provided. In fact, such\nhigh mobility can readily give rise to the controversial current-jetting effect, which remains a\nsignificant concern in the field.\nThe authors’ reply:\nWe thank the reviewer for pointing at the mobility and Fermi level position information that was\nmissing in the manuscript. We have included the corresponding information into the manuscript\n(Methods section) and Supplementary Information. In this device we can control the total carrier\ndensity in the structure via the electrostatic gate between approximately 3.5x1011 cm-2 and -\n2.2x1011 cm-2 of surface density (As already mentioned above, close to the Weyl points the\ntransport is dominated by the surface states). According to our estimates via comparison with k\ndot p calculation in Fig.1b, these carrier densities correspond to Fermi level being moved in a ±2\nmeV range around the Weyl nodes.\nThe mobility varies with the position of the Fermi level and is different for electronic and Hall\nstates. We extract it from an analysis of the zero-field resistivity value. For a density 3.7x1011 cm-\n2 it reaches approx. 77x103 cm2/(Vs), while for the valence band at Vg = -0.5V it is approx. 16x103\ncm2/(Vs). Directly at the Weyl point the carrier density cannot be reliably extracted from the Hall\nslope, not allowing to obtain a precise mobility value.\nWe agree that current jetting can be a significant concern for bulk materials, particularly when\nsoldering the metal leads directly on the bulk crystals. In our case, this phenomenon does not\noccur, as we can utilize the semiconductor technology developed over decades for our MBE-\ngrown epilayers. The contacts are fabricated using standard lithography techniques in a Hall bar-\nlike geometry, forming low-resistance (few tens of Ohms for the device studied in this paper, much\nsmaller than the resistance of the studied structures, hundreds of Ohms, see Supplementary Fig.\n3) bond pads and avoiding any possible nonuniformity. As a result, we observe the same negative\nmagnetoresistance effect in both small structures, like the island studied in our paper and in large,\nmacroscopic Hall bars studies in Ref. [16], which are hundreds of micrometres in size.\nReviewer’s response:\nIn fact, the occurrence of weak antilocalization (WAL) and weak localization (WL) is not at all\nsurprising in HgTe systems with strong spin–orbit coupling. Therefore, we remain unconvinced\nthat the negative magnetoresistance observed below 2 T arises from the chiral anomaly, unless\nthe authors can provide direct analytical evidence from SdH oscillations.\nThe authors’ reply:\nIndeed, these phenomena are routinely observed and well-studied for these materials. We would\nlike to point the reviewers to the characteristic magnitude of WL and WAL, which stems from their\nphysical origin. Both WL and WAL represent small corrections to the diffusive transport regime\nfrom quantum interference, that results in effective conductivity change on the order of one\nconductance quantum e2/h. We observe a much stronger effect of the magnetic field, where the\nconductance in the in-plane field increases by approx. 30%. The presence of such a strong\nlocalization effect would imply strong disorder in the system, so that we are no longer operating\nin the diffusive transport regime. This cannot be the case because of the mobilities of our\nstructures, so WL and WAL can be reliably excluded as a possible explanation. Moreover, at the\nmobilities we have in our epilayers, any WL and WAL features are restricted to few tens mT width,\nas high mobility leads to narrow WAL signal, see also Ref. [A] below.\nReviewer’s response:\nIn fact, as clearly shown in Supplementary Fig. 1 (Theta = 90°), the observed positive\nmagnetoresistance is most likely attributable to weak antilocalization (WAL) arising from strong\nspin–orbit coupling, rather than to the positive magnetoresistance expected when the Fermi level\nis at the charge neutrality point. If the latter were the case, one would expect to see a pronounced\nparabolic increase in magnetoresistance at low fields for Theta=90°.\nThe authors’ reply:\nWe strongly disagree with the interpretation of the data in Supplementary Fig. 1 (Theta = 90°) as\nWAL. If we look at the magnitude of the observed resistance increase with magnetic field, we see\nthat it increases from 2 kΩ to 7 kΩ at 4 T. Weak antilocalization effects deliver small corrections\nto conductance of the order of e2/h (typically even less than that, see for example Ref. [A]), and\ntheir suppression cannot result in such large increase of resistance with magnetic field, even in\nmaterial with strong spin-orbit coupling.\nThe observed large magnetoresistance rather exactly matches expectations for a low-carrier-\ndensity system close to the Weyl point. As we discussed above, HgTe and other Weyl semimetals\nwith similar crystalline structure possess surface states that reside close to the Weyl point, and\nwhose contribution is observed in quantum hall and magnetoresistance close to the Weyl point.\nIn our case, at -0.29V the carrier density is so low that\nThe reviewer probably expects even larger growth, as observed in some Weyl candidates, such\nas [21]. We note that this behaviour is only characteristic for 3D carriers and thus is not applicable\nto Weyl semimetal layers that also have surface states.\nReviewer’s response:\nI also recommend that the authors include relevant references when discussing these\nphenomena, in order to support their arguments appropriately.\nThe authors’ reply:\nWe include the relevant references that were used in the main text in the discussion of thermal\ntransport phenomena:\n1. Ultraclean Weyl semimetals can be analysed based on a study by Andreev and Spivak\n[8]\n2. The hydrodynamic transport regime is theoretically investigated by Lucas et al. [6] and\nby Andreev and Spivak [8]\n3. The effects of electron-phonon coupling are discussed briefly in a review by Pekola and\nKarimi [39]\n4. The elastic scattering regime is investigated by Andreev and Spivak [8].\nReviewer’s response:\nFinally, we regret to conclude that the manuscript still falls short of the standards required for\npublication in a high-profile journal such as Nature Communications. In the supplementary\nmaterials, we observed that for θ = 90°, oscillations appear in the magnetoresistance curve at\nfields above 2 T. Could the authors clarify the origin of these oscillations? Are they attributable to\nShubnikov–de Haas (SdH) oscillations? If so, we strongly encourage the authors to extract and\npresent key parameters such as the effective mass of the charge carriers, the precise location of\nthe Fermi level, and the Berry curvature. The inclusion of such quantitative analyses would\nsignificantly enhance the credibility and impact of the manuscript.\nThe authors’ reply:\nWe respectfully disagree with the evaluation that our manuscript falls short of the standards of\nthis high-profile journal. We recognize that much of the criticism appears to stem from the absence\nof certain elements—Shubnikov–de Haas oscillations in perpendicular magnetic field and the\nextraction of Berry curvature from their phase—that are often expected in works on Weyl\nsemimetals. These signatures are indeed frequently interpreted as hallmarks of Weyl physics,\ntypically presented alongside negative magnetoresistance in in-plane field.\nWe note that this analysis is only applicable to bulk crystals with high carrier density and does not\napply to layers of low-density material, as the one we investigate. In our case, no well-developed\nSdH oscillations can be observed when the Fermi level is located at the Weyl point due to the\nexceptionally low carrier density. This precisely matches our expectations for a Weyl semimetal,\nwhile the analysis of quantum oscillations at larger densities reliably assigns them to the surface\nstates emanating from Weyl nodes, all studied in detail by Mahler et al. [16]. In low-density thin\nlayer samples, oscillations with unusual phases rather emerge from a complex interplay of Landau\nlevels, than reflecting intrinsic Berry curvature and bulk Weyl physics. Therefore, an attempt to\ninterpret such oscillations in the conventional way would be fundamentally misleading. A copy of\nthe corresponding paper has been included with our resubmission.\nOn the other hand, our devices do provide direct and reliable evidence that they are in the regime\nwhere Weyl physics is expected. By controlling the carrier density via the gate voltage, we can\nadjust the Fermi level position in the band structure with high precision. The entire range of our\nstudy spans only a few meV around the theoretically predicted Weyl point, as confirmed by our\nability to tune continuously from electron-like to hole-like conduction. We identify the Weyl point\nat the gate voltage corresponding to the maximum negative magnetoresistance, which coincides\nprecisely with the lowest carrier density. The existence of Weyl cones at this point is further\nconfirmed by our k dot p bandstructure calculations, which directly reflect the underlying\ncrystalline symmetry.\nWe thank the reviewer for drawing attention to the oscillations observed in the island\nmagnetoresistance for out-of-plane field orientation. However, these features do not appear to be\nrelated to Shubnikov–de Haas oscillations. They are only present in the small island, whereas in\nthe larger channel, measured at the same Fermi level, we observe only a single broad oscillation\nassociated with a Landau-level transition of the surface state—consistent with the behaviour\nshown in Ref. [16]. This indicates that the island oscillations most likely arise from magnetic field\nfocusing effects on ballistic carrier dynamics, rather than reflecting the underlying band structure.\nThis interpretation is fully consistent with our expectations: at such a low carrier density, well-\ndeveloped SdH oscillations cannot occur at the accessible magnetic fields.\nIn summary, all our experimental observations align with the expected behaviour of a Weyl\nsemimetal with low carrier density and significant surface-state contributions. Most importantly,\nthe pronounced negative magnetoresistance observed only when the Fermi level is tuned near\nthe Weyl point serves as the key transport signature of Weyl physics in our system. Building on\nthis regime, our study of thermal conductance achieves its central goal—probing potential\nsignatures of the gravitational anomaly in Weyl semimetals at low temperatures, where electron–\nphonon relaxation can be reliably excluded.\nReviewer #2 (Remarks to the Author):\nAll my concerns have been addressed. I recommend its publications.\nThe authors’ reply:\nWe thank the reviewer for their evaluation of the manuscript.\nA) Mühlbauer, M., et al. \"One-dimensional weak antilocalization due to the Berry phase in HgTe wires.\" Phys. Rev. Lett. 112, 146803 – 2014."}, {"round": 3, "reviewer_comments": {"reviewer_1": "Dear Editor,\nI have carefully read the authors’ responses and can see that they have addressed my questions with serious consideration.\nIn principle, I agree to accept the publication of this manuscript, mainly for two reasons: (1) the authors’ methodology and\noverall approach are highly advanced, and their experimental techniques show significant innovation; and (2) at present,\nthere is no universally accepted or entirely conclusive experimental evidence for transport signatures in topological\nsemimetals. While some theoretical papers have proposed that negative magnetoresistance and the planar Hall effect could\nserve as evidence of topological semimetals, in practice, when tested in real materials, almost none have fully satisfied\nthese criteria. This work represents an important exploration in this area and is likely to attract further attention from theorists\nand researchers with relevant expertise, stimulating additional investigation into this timely scientific problem. For these\nreasons, I support its publication.\nBest\nChen", "reviewer_2": "All my concerns have been addressed. I recommend its publications."}, "author_response": "RESPONSE TO THE REFEREE COMMENTS\nReviewer’s response:\nAs for Current-jetting effect, please refers to Nature Reviews Physics 3, 394 (2021)\nThe authors’ reply:\nWe thank the referee for drawing our attention to the review discussing current-jetting effects in\nhigh-mobility structures. In this context, we would like to highlight Phys. Rev. X 8, 031002 (2018),\nwhich is cited in the review mentioned by the referee and identifies certain high-mobility Weyl\nsemimetal candidates as systems where negative magnetoresistance can originate from current-\njetting effects. In that work, a straightforward strategy to avoid current-jetting effects was\nproposed: fabricating thin, gate-tuneable films of the studied material to enable control over the\nFermi-level position. This is precisely the approach implemented in our devices.\nImportantly, current jetting occurs in structures where small, point-like contacts, directly glued\nonto a crystal, are used to inject current, leading to inhomogeneous current flow that can be\nstrongly influenced by magnetic field [see Fig. 1c in Nature Reviews Physics 3, 394 (2021)]. In\ncontrast, our devices employ wide ohmic contacts fabricated using the standard lithography\ntechniques of semiconductor technology, which are directly attached to the HgTe mesa (see Fig.2a\nof the manuscript). These diffusive contacts ensure a uniform current distribution within the\nstructure, effectively eliminating magnetic-field–induced inhomogeneities that could give rise to\ncurrent jetting.\nFinally, our devices employ a thin 72 nm HgTe epilayer, for which the cyclotron radius at the\nmagnetic fields used in our experiment is much larger than the layer thickness. Under these\nconditions, classical ballistic effects such as current jetting are strongly suppressed.\nWe added a reference to Phys. Rev. X 8, 031002 (2018) to the Methods section in a sentence,\ndiscussing how our experimental approach allows to avoid current jetting effects.\nReviewer’s response:\nThe authors state that the 30% increase in conductance under in‑plane magnetic field is far too\nlarge to originate from weak localization (WL) or weak antilocalization (WAL), since these effects\ntypically yield corrections on the order of e2/h. However, this assertion is purely qualitative. Could\nthe authors provide quantitative analysis or experimental evidence that definitively excludes\nWL/WAL as possible contributions? In other words, how do the authors demonstrate that a 30%\nchange cannot arise—at least partially—from enhanced quantum‑interference effects or\nmultichannel localization in their system? Supporting data such as temperature dependence,\nangular dependence, or field‑width analysis would be necessary to substantiate this claim.\nThe authors’ reply:\nBelow we summarize the experimental evidence that clearly rules out quantum-interference\nmechanisms as the origin of the observed negative magnetoresistance and instead points to its\nband-structure origin:\n1. Device dimensions\nWe investigated devices with vastly different geometric sizes (island of 6x4.5 µm and a Hall\nbar of 1200x200 µm) and found that the relative magnitude of the negative magnetoresistance\nremains essentially unchanged. Moreover, the resistance does not show an exponential increase\nwith length. These observations reliably exclude multimode localization as a possible\nexplanation.\n2. Temperature dependence\nAs discussed after the first review round, our devices exhibit only a weak temperature\ndependence, while the relative drop in resistance with magnetic field remains nearly constant\nup to 23 K (see Fig. 7.8 in the thesis of David Mahler, https://nbn-\nresolving.org/urn:nbn:de:bvb:20-opus-253982). This behaviour is inconsistent with both weak\nlocalization and multimode localization, which are expected to show a much stronger\ntemperature dependence.\n3. Angular dependence\nThe angular dependence of the magnetoresistance effect has been thoroughly characterized in\nPhys. Rev. X 9, 031034 (2019) (Fig. 5) and is confirmed in our measurements (see\nSupplementary Section 1). The negative magnetoresistance appears only when the magnetic\nfield is oriented in-plane and parallel to the current direction. This is opposite to the behavior\nexpected for weak localization or multimode localization, where an out-of-plane field produces\nthe strongest effect and an in-plane field has the weakest influence due to the minimal magnetic\nflux through electron trajectories.\n4. Magnetic field width\nExperimentally, we find that the negative magnetoresistance persists up to about 5 T, beyond\nwhich a positive magnetoresistance dominates. If this width were attributed to the suppression\nof quantum interference, it would correspond to a phase-coherence breakdown field B .≈ 5 T.\nφ\nUsing B\nφ\n=ℏ/(4el\nφ\n2), this yields a phase-coherence length l\nφ\n<5 nm — an unrealistically small\nvalue incompatible with the high mobility of our structures.\nReviewer’s response:\nOverall, the authors’ explanations remain rather unconvincing. The goal of their work is clear: by\napplying external strain, they break the inversion symmetry of initially centrosymmetric HgTe,\nthereby transforming it into a Weyl semimetal. From this, they draw the following\nconclusion:“This finding indicates that, despite the unique electronic spectrum of Weyl\nsemimetals, the mechanism governing heat transport in this system is the same as that for electrical\ntransport, with no additional violations of conservation laws.”\nThis is, without question, an important and potentially far‑reaching conclusion. However, before\nreaching such a statement, the authors must first provide solid experimental demonstrations that\n(i) prior to strain the system was indeed a Dirac semimetal, supported by key evidence, and (ii)\nafter strain, the transition to a Weyl semimetal has been directly confirmed by experiment. Only\nthen can the final conclusion be considered physically convincing.\nThe authors’ reply:\nThe ability to systematically control strain in HgTe layers and thereby tune their electronic band\nstructure has been extensively demonstrated in previous studies over the past 30 years. Due to the\ninherent technological constraints of molecular beam epitaxy, truly unstrained HgTe films are\nrarely realized; most experimentally accessible samples are subject to either tensile or compressive\nstrain.\nTensile-strained HgTe layers are a well-established realization of a three-dimensional topological\ninsulator, in which the strain opens a bulk band gap and profoundly alters the transport properties.\nRepresentative studies include Phys. Rev. Lett. 106, 126803 (2011); Phys. Rev. X 4, 041045 (2014);\nNano Lett. 21, 9869–9874 (2021); and Nano Lett. 21, 5195–5200 (2021).\nCompressively strained layers, as investigated in Phys. Rev. X 9, 031034 (2019) and in the present\nwork, exhibit substantially different transport behavior. In contrast to the tensile case, these layers\nshow no indication of a bulk gap, and when tuned close to charge neutrality, they display a negative\nmagnetoresistance in in-plane magnetic fields. The classification of such structures as Weyl\nsemimetals is ultimately supported by these distinct transport signatures, together with detailed\nband-structure calculations incorporating the experimentally determined strain values from the\ngrown layers.\nMoreover, the authors’ replies often avoid the essential questions. For example, they argue that\nsurface‑state SdH oscillations interfere with bulk SdH signals, and therefore the usual SdH analysis\ncannot be applied. This argument is problematic. As they themselves emphasize, the strained\nsamples are Weyl semimetals with extraordinarily high mobility (77 103cm2 /Vs) and with the\nFermi energy very close to the Weyl point. Under such ideal conditions, the complete absence of\nSdH oscillations in magnetoresistance is highly counterintuitive and contradicts basic physical\nexpectations, as well as previous studies (Science 350, 413 (2015); Phys. Rev. X 5, 031023 (2015);\nNat. Phys. 14, 1125 (2018)). Of course, there are known examples, such as Co₂MnGa, which\nARPES has confirmed to be a topological semimetal yet still exhibits no SdH oscillations.\nHowever, in that material the absence of oscillations is attributed to an extremely high carrier\ndensity that produces large Landau‑level spacing, making oscillatory magnetoresistance difficult\nto observe when the Fermi surface crosses Landau levels. In contrast, the experimental conditions\nreported for strained HgTe are optimal for observing topological features—low carrier density,\nhigh mobility, and a Fermi level precisely tuned to the Weyl point—yet no direct evidence of\ntopological changes is observed. This is difficult to accept. Although the authors provide many\nexplanations and checks, these do not directly relate to or convincingly demonstrate the topological\ntransition itself.\nThe authors’ reply:\nWe would like to clarify that SdH oscillations in perpendicular magnetic field are indeed observed\nin our devices when tuned slightly away from the Weyl point, as demonstrated in Phys. Rev. X 9,\n031034 (2019) and in Supplementary Fig. 7. Our point, however, is that these oscillations arise\npredominantly from the topological surface states rather than the bulk electrons, making it\nimpossible to use them to extract the Berry phase of the bulk. Accessing SdH oscillations of the\nbulk would require suppressing the mobility of the surface states. Thus, there is no contradiction\nwith basic physical principles or previous work; rather, we exercise caution in interpreting SdH\noscillations as a definitive signature of bulk topology in this material. Directly at the Weyl point,\nthe carrier density is simply too low for a distinct SdH oscillation pattern to be observable. Again,\nthe band structure of HgTe-based structures has been explored extensively over the past 50 years,\nand the referee suggestions are at odds with a very well established chapter of semiconductor\nphysics.\nThat said, I find the experimental method used to break inversion symmetry by strain application\nboth technically precise and conceptually appealing. The quality of this experimental approach is\nimpressive, and the study could stimulate productive debate among experts on this highly relevant\nproblem. After all, research on transport signatures of the chiral anomaly in topological semimetals\nis still ongoing, and a clear consensus has not yet been reached. For this reason, while I remain\nskeptical about the claimed evidence of a Dirac–to–Weyl transition, I cautiously support\npublication of this work. Its controversial aspects are, in my view, part of what makes it interesting\nand may encourage further investigation and discussion in the field.\nThe authors’ reply:\nWe thank the referee for their thoughtful assessment of our work and for recognizing the relevance\nof both the topic and our experimental approach. We fully agree that the research field of Weyl\nsemimetals remains far from settled. One major challenge is that many of the proposed electronic-\ntransport signatures used to identify Weyl physics are not uniquely attributable to the presence of\nWeyl nodes. The unusual phase offsets observed in SdH oscillations are often attributed to a\nnontrivial Berry phase, yet in practice they may arise from the coexistence of multiple conduction\nchannels—such as topological surface states and bulk carriers—with distinct effective masses.\nLikewise, the large positive magnetoresistance observed in perpendicular magnetic fields is a well-\nknown characteristic of nearly compensated semimetals and does not, by itself, provide conclusive\nevidence for Weyl physics.\nThis leaves the observation of a negative magnetoresistance, interpreted as a manifestation of the\nchiral anomaly, as one of the few remaining transport-based indicators. However, even this\nsignature is not unique: in several high-mobility systems, similar behavior has been shown to result\nfrom classical current-jetting effects. Moreover, theoretical works have suggested that negative\nmagnetoresistance may appear in materials with any band crossings, not necessarily those hosting\nthe Weyl nodes.\nThus, our present study is aimed not only at identifying the thermal transport properties of\ncompressively strained HgTe but also exploring their potential as a probe of Weyl physics. This is\nparticularly appealing, as the observation of quantum anomalies in a semiconductor platform is\none of the major points that initially sparked interest in Weyl semimetals. We selected a material\nsystem, where the ability to adjust the band structure by strain engineering has already been\ndemonstrated, while conventional fabrication methods are easily available to create complex\nstructures with precise control of the Fermi level position. Unfortunately, our findings indicate that\nthermal transport does not exhibit distinct anomalies that could serve as a signature of Weyl points,\nthereby challenging current theories, that suggest quantum anomalies expected in particle physics\nto occur in the thermal transport in Weyl semimetals. The only anomaly our devices do demonstrate\nis the chiral anomaly in the electrical conductance, and they do so very convincingly.\nRESPONSE TO REVIEWERS' COMMENTS\nReviewer #1 (Remarks to the Author):\nDear Editor,\nAfter carefully reviewing this manuscript, I am impressed by the innovative and rigorous\nmeasurement techniques employed. The experimental data sufficiently support the core\nconclusion that \"the low-temperature thermal transport of HgTe under stress still conforms to the\nWiedemann-Franz law.\" However, the authors further assert that the compressively strained HgTe\ntransforms into a Weyl fermion system, and that thermal transport does not violate the\nWiedemann-Franz law. This claim lacks sufficient justification, and there are several specific\nissues that need clarification from the authors:\nThe authors’ reply:\nWe thank the reviewer for carefully evaluating the manuscript and appreciate the constructive\ncriticism. Please find our detailed response below.\n1. Mechanism of Negative Magnetoresistance\nAs stated by the authors, negative magnetoresistance may arise from multiple mechanisms, and\nthe positive magnetoresistance observed when the magnetic field exceeds 2T does conform to\nweak localization characteristics. However, the key question is: how do the authors rule out that\nthe negative magnetoresistance in the magnetic field region below 2T is not caused by weak anti-\nlocalization effects? This requires more substantial experimental evidence for clarification.\nThe authors’ reply:\nCompressively strained HgTe exhibits a pronounced negative magnetoresistance (MR) when the\nmagnetic and electric fields are aligned (i.e., B∥E). This negative magnetoresistance, however, is\naccompanied by positive MR features at both low and high magnetic field ranges. The positive\nmagnetoresistance at low fields (<0.5T) is indeed attributed to the weak anti-localization (WAL)\neffect, a quantum interference phenomenon. The reviewer is probably referencing the weak\nlocalization (WL), another quantum interference phenomenon, that leads to resistance decrease\nwith magnetic field and is commonly observed in combination with WAL.\nSeveral arguments allow us to clearly distinguish the observed resistance decrease at\nintermediate fields (from 0.5T up to 5T) from WL:\nFirst, due to the high mobility in our sample, the observed negative MR cannot be attributed to\nWL. Instead, at the expected very low magnetic fields, HgTe exhibits WAL (Ref. A), arising from\nstrong spin–orbit coupling, on top of the negative MR background.\nReviewer’s response:\nThe authors mention that their samples exhibit high mobility. Under such conditions, Shubnikov–\nde Haas (SdH) oscillations are typically readily observable in magnetoresistance measurements.\nThe presence of SdH oscillations would constitute compelling evidence in identifying the Weyl\nnature of the material. Therefore, I strongly encourage the authors to include this crucial data in\nthe supplementary materials to further strengthen the manuscript.\nSecond, WL is expected to be active when electrons can pick phase from magnetic fields. This is\ntypically stronger for the field orientations, perpendicular to the current flow, due to the larger flux\nthrough electron trajectories. In contrast, in our system negative MR is only observed for in-plane\nmagnetic field, parallel to the current flow, B∥I. For all other field directions, we observe positive\nMR (see the answer to the next question and newly added Supplementary Fig. 1 in the\nSupplementary Information). Specifically, for in-plane perpendicular we observe very weak positive\nMR, which is unlikely for WL.\nReviewer’s response:\nThe authors repeatedly emphasize the high mobility of their samples and the proximity of the\nFermi level to the Weyl point, yet no quantitative mobility data have been provided. In fact, such\nhigh mobility can readily give rise to the controversial current-jetting effect, which remains a\nsignificant concern in the field.\nThird, the observed negative MR in compressed HgTe survives at high temperatures, while WL is\nsuppressed quickly as the quantum coherence is destroyed. Below we present the data adopted\nfrom the thesis of David Mahler (https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-253982,\nFig.7.8), where the temperature dependence of magnetoresistance in compressively strained\nHgTe was measured under B∥E. The measurements were performed in our group on a different\ndevice with the layer stack and material properties closely matching those of the device discussed\nin this work. On top of the original figure panel b we added a mark pointing to the region of WAL.\n[Figure redacted]\nFrom Fig.7.8 (b), we observe that the sharp WAL feature at B=0T quickly disappears with\nincreasing temperature, confirming its origin from interference effects. In contrast, the pronounced\nnegative MR at intermediate magnetic fields remains largely unchanged. This effectively rules out\nquantum interference mechanisms such as WL as the underlying cause. A noticeable change in\nthe negative magnetoresistance is only observed at 23 K, likely due to enhanced phonon\nscattering at elevated temperatures.\nLast, in our device we have a direct control over the position of the Fermi level in the band structure\nwith an electrostatic gate. We observe prominent negative MR only when the Fermi level is tuned\nnear the position of the Weyl points, which is verified via carrier density extracted from the\nout-of-plane magnetic field Hall measurement. This does not match the expectations for WL, where no\ndirect relation to the band structure position of Fermi level is expected. At the same time, the band\nstructure calculations based on k⋅p theory reveal the existence of linearly dispersing Weyl points in\ncompressively strained HgTe, that should result in effect of chiral anomaly only for Fermi level close\nthe Weyl points as detailed in Ref. [16].\nThese directional and energy-specific properties of our observations support the interpretation of\nthe negative magnetoresistance as a signature of the chiral anomaly.\nReviewer’s response:\nIn fact, the occurrence of weak antilocalization (WAL) and weak localization (WL) is not at all\nsurprising in HgTe systems with strong spin–orbit coupling. Therefore, we remain unconvinced\nthat the negative magnetoresistance observed below 2 T arises from the chiral anomaly, unless\nthe authors can provide direct analytical evidence from SdH oscillations.\n2. Completeness of Gate Voltage Regulation\nThe study achieves continuous control of the Fermi surface between the valence band and the\nconduction band through gate voltage. However, the data presented indicate that when Vg = -\n0.29V, the Fermi surface should be precisely located at the Weyl point, at which point a significant\npositive magnetoresistance effect should be observed under vertical magnetic field conditions.\nThe absence of this critical experimental data weakens the conclusion regarding Weyl\nsemimetals.\nThe authors’ reply:\nWe have added measurements of the magnetoresistance of our device for out-of-plane magnetic\nfield configuration to the Supplementary Materials. Indeed, for an out-of-plane field, when the\nFermi level is tuned to the Weyl points, a pronounced positive magnetoresistance is observed.\nWe also performed measurements with an in-plane magnetic field, perpendicular to the electric\nfield, which results in very weak positive magnetoresistance. These observations indicate that the\nnegative magnetoresistance is exclusive to the configuration where the magnetic field is aligned\nparallel to the electric field within the device, as expected for chiral anomaly. A comprehensive\nstudy on the angular dependence of magnetoresistance actually is available in our previous paper\non compressively strained HgTe with a comparable layer stack and electronic properties\n(Ref. [16]), providing further insights into the role of chiral anomaly in the observed transport\nbehavior.\nReviewer’s response:\nIn fact, as clearly shown in Supplementary Fig. 1 (Theta = 90°), the observed positive\nmagnetoresistance is most likely attributable to weak antilocalization (WAL) arising from strong\nspin–orbit coupling, rather than to the positive magnetoresistance expected when the Fermi level\nis at the charge neutrality point. If the latter were the case, one would expect to see a pronounced\nparabolic increase in magnetoresistance at low fields for Theta=90°.\n3.Contradictory Phenomena in Magnetic Field Response\nAn analysis combining Figures 2b and 3d reveals that although a 2T magnetic field is a turning\npoint for the dominance of chiral anomalies and weak localization, the Lorenz constant shows an\nunusually smooth variation within the range of 0-8T. Does this suggest that the thermal response\nacross the entire magnetic field range (both <2T and >2T) originates from trivial electronic states\nrather than Weyl fermions?\nThe authors’ reply:\nWe thank the reviewer for their useful comment, that points directly to the core message of our\nstudy. The stability of the Lorenz ratio as a function of the magnetic fields demonstrates that the\nheat is transported via the same mechanism as the charge at all fields, and that it is transported\nvia fermionic quasiparticles with charge -e. So, this does not mean that the thermal transport is\ntrivial, on the contrary, this indicates, that if the charge transport originates from the effect of the\nchiral anomaly, so does the heat transport.\nHowever, theories of Weyl semimetals predict that their thermal conductance can be influenced\nby the so-called gravitational anomaly. Below we summarize how our observations relate to\nexpectations across different transport regimes (we only consider those, where negative\nmagnetoresistance from chiral anomaly should be present):\n1. In an ultraclean Weyl semimetal, the chiral anomaly might affect the charge transport due\nto the anomalous charge pumping between the Weyl nodes, but less so the energy and\nheat transport. In that case one might expect L < L0.\n2. If inelastic e-e interactions are strong and dominate over all other elastic and inelastic\nscattering processes, a hydrodynamic transport regime might be realized, where electrons\nexhibit a form of collective motion, which should result in a substantial breakdown of the\nWiedemann-Franz law, presumably with L > L0. This is the regime where the gravitational\nanomaly effects might be observed in thermal transport.\n3. Otherwise, if inelastic electron-phonon scattering dominates, which is expected for higher\ntemperatures, electrons and phonons are strongly coupled and come to the local\nequilibrium. Thermal conductance then contains electron and phonon components, and\nthe electronic part trivially follows L=L0.\n4. If elastic scattering dominates and the internode scattering rate is low, both charge and\nheat are transported via the same fermionic quasiparticles, and L=L0 is expected, even in\nthe presence of the anomaly.\nPrevious studies of thermal transport in Weyl systems were done in regime (3), due to the\nmeasurements being performed at high lattice temperatures, when electron-phonon scattering is\nstrong. In contrast, we take special care to ensure that we are not operating in this regime by\nworking at low lattice temperatures with fully electronic methods of heating and temperature\ndetection, minimizing the effects of phonon drag and lattice heating. We also directly verify that the\nelectron-phonon relaxation is small enough. However, we find, that under these conditions\nL=L0, indicating that regime (2) is not realized and rather pointing to the regime (4), which is by\nno means trivial, but does not allow to experimentally detect the effect of gravitational anomaly.\nReviewer’s response:\nI also recommend that the authors include relevant references when discussing these\nphenomena, in order to support their arguments appropriately.\nConsidering the innovative thermal measurement techniques in this work and the cutting-edge\nnature of the research topic, I recommend that it is suitable for publication in Nature\nCommunications after the above key evidence is supplemented, particularly providing conclusive\nexperimental evidence that the material system possesses Weyl semimetal characteristics.\nReviewer’s response:\nFinally, we regret to conclude that the manuscript still falls short of the standards required for\npublication in a high-profile journal such as Nature Communications. In the supplementary\nmaterials, we observed that for θ = 90°, oscillations appear in the magnetoresistance curve at\nfields above 2 T. Could the authors clarify the origin of these oscillations? Are they attributable to\nShubnikov–de Haas (SdH) oscillations? If so, we strongly encourage the authors to extract and\npresent key parameters such as the effective mass of the charge carriers, the precise location of\nthe Fermi level, and the Berry curvature. The inclusion of such quantitative analyses would\nsignificantly enhance the credibility and impact of the manuscript.\nReviewer #2 (Remarks to the Author):\nThis paper by A. Aravindnath, et al investigates the Wiedemann-Franz behavior in a Weyl\nsemimetal based on compressively strained HgTe. By performing thermometry measurements at\nlow temperatures using fully electronic methods, the research clearly identifies the Weyl regime\nand extracts the thermal conductance. The observed increase in thermal conductance with the\napplied magnetic field matches the increase in electrical conductance, adhering to the\nWiedemann-Franz\nlaw. This indicates that, despite the unique electronic structure of Weyl semimetals, the thermal\ntransport mechanism remains consistent with electrical transport, without violating any\nconservation laws. Overall, this work is interesting and novel, and I would be willing to recommend\nit for publication in Nature Communications after the authors address the following concerns:\nAs we know, in thermal conductance measurements, the heat flow or heat power Q flowing\nthrough the sample is a crucial and intuitive quantity, especially when studying scaling laws such\nas the Wiedemann-Franz law. However, the analysis and calculation of the heat power Q in the\nmanuscript appear overly simplistic and cursory. I suggest that the authors provide a clearer and\nmore detailed explanation of this heating power.\nThe authors’ reply:\nWe thank the reviewer for their useful comments and appreciate their evaluation of our work. In\nthe updated version of the manuscript, we reworked and expanded the discussion of the\nextraction of the heat flux, flowing through the sample and added extra details to support our\nanalysis. We added extra subsection to the “Methods” that explains our approach to finding the\nheat flux.\nReviewer #3 (Remarks to the Author):\nThis paper mainly studies the thermal transport properties of compressively strained HgTe layer.\nThe Fermi level can be tuned by changing the gate voltage. The authors found that even though\nWeyl fermions are located at the Fermi level and dominate transport properties, the Wiedemann-\nFranz law still holds, indicating that Weyl fermions do not cause distinct physics in thermal\ntransport. In my opinion, this paper does not meet the high standards of Nature Communications,\nand other journals such as Communications Physics may be more suitable.\n(1) The innovation of the paper is not sufficient to be published in NC. I don't think the results of\nthis article are very interesting.\nThe authors’ reply:\nWe respectfully disagree with the reviewer on this point. In our view, the present work makes a\nmeaningful contribution to the study of Weyl semimetals. The question of whether the\nWiedemann–Franz law is violated—and whether hydrodynamic electron thermal conductance\ncan be detected—is central to experimental efforts aimed at observing the solid-state analogue\nof the gravitational anomaly in these systems. In fact, the original proposal that Weyl semimetals\ncould serve as a platform for such studies has been a key motivation for much of the thermal\ntransport research in this field.\nHowever, a reliable experimental confirmation of nontrivial heat transport mechanisms in Weyl\nsemimetals remains elusive. This is due in part to the challenge of reaching a regime where\nelectron–phonon scattering is weak, and in part to the intrinsic difficulty of isolating the electronic\ncontribution to thermal conductance. In this work, we present an experiment that explicitly\naddresses both issues with care—yet we still do not observe the expected thermal anomaly. We\nbelieve this result provides important insight into the nature of thermal transport in this class of\nmaterials.\n(2) In the “results” section, the author spent a considerable amount of space describing how the\nexperiment was conducted, which should be included in the “methods” section. The main text\nshould mainly describe and discuss the experimental results.\nThe authors’ reply:\nWe acknowledge that the details of Johnson-Nyquist thermometry might not be interesting for\nevery reader of the manuscript. Yet in our case the choice of the method of thermal conductance\nmeasurement is strongly connected to the observed physics: we want to operate in a regime\nwhere the phonon contribution is small and electron-phonon coupling is weak enough to be able\nto resolve the specific mechanisms of electronic transport. In the updated version of the\nmanuscript, we moved some less important description of the measurement into the methods\nsection, while keeping the necessary details to support this message.\n(3) The data is not abundant and the quality is not high.\nThe authors’ reply:\nWe would like to draw the reviewer’s attention to the details of our study, in particular, to the\nresearch method we used. High precision Johnson-Nyquist thermometry is a very time-\nconsuming experimental technique, requiring prolonged measurements to achieve high precision\ndue to the statistical nature of the signal. The value that one should look at to understand the final\nprecision is the relative error deltaT/T. In our case we achieved deltaT of approx. 5mK for T~1.3K\nand 95% confidence interval. By this parameter our measurement is on par with the best practices\nin the field.\nAdditionally, to improve this parameter by n times, the measurement time needs to be multiplied\nby n^2. This can potentially result in enormous measurement times, that don’t necessary improve\nthe analytic potential of the data. A study that results in high precision thermal transport\ncoefficients, however, still requires several months of continuous measurements.\nThe choice of the research method is not our whim - the fragility of electronic thermal transport\nwith respect to electron-phonon interactions and the need to accurately control heat flows in the\nsystem simply does not allow the use of other approaches, which we discuss in detail in the text\nof the article.\nGiven all this, we believe that the quantity of data and their quality are sufficient to reliably support\nthe main conclusions of our article and follow the best practices in the field.\nA) Mühlbauer, M., et al. \"One-dimensional weak antilocalization due to the Berry phase in HgTe\nwires.\" Phys. Rev. Lett. 112, 146803 – 2014.\nRESPONSE TO THE REFEREE COMMENTS\nReviewer #1 (Remarks to the Author):\nThis article elaborates on the research conducted to modulate the topology of HgTe by precisely\napplying external stress to alter its crystal structure. This work represents significant advancement\nand importance in the field of topological materials. However, the determination of whether the\nmodulation is successful must be approached rigorously, primarily based on the following two\ncriteria: 1）Whether the crystal structure has indeed changed. This serves as the foundation for\nverifying the modulation effect. 2）Whether the key experimental evidence supporting the two\ntypes of topological properties is sufficient. This is crucial for ensuring the reliability of the research\nconclusions.\nAs these critical conditions have not yet been fully verified, it is currently not possible to\nscientifically conclude whether this research has achieved its intended goals.\nThe authors’ reply:\nWe thank the reviewer again for their careful evaluation of the manuscript and positive opinion\nabout its impact on the field of topological materials.\n1) We agree that the presence of compressive strain is a crucial element for looking for Weyl\nphysics in HgTe samples. The ability to introduce the compressive strain and change crystalline\nstructure of HgTe has been proven previously in the 2019 Mahler et al. paper from our group (Ref.\n[16] in the article, 10.1103/PhysRevX.9.031034). In there, we verified by High resolution X-ray\ndiffraction (XRXRD) analysis of 66 nm and 120 nm layers grown via the same MBE technology\nas the one used in this article. The paper reports, that the 66 nm layer is fully strained, with 0.3%\ncompressive strain introduced by the lattice mismatch, while the 120 nm layer is partially relaxed,\nso that the strain is 25% less. The 72 nm layer studied in the present paper is close in thickness\nto the 66 nm layer from Ref. [16], i.e. is almost fully strained according to the lattice mismatch.\nHRXRD strain data are routinely obtained for every MBE-grown sample, including the one used\nin this study, so we exactly measured the amount of strain indicated in the manuscript (0.28%).\nWe additionally reproduce the main results of HRXRD analysis from Supplementary Material of\nRef. [16] below and additionally mention the strain measurement in the methods section.\n[Figure redacted]\n2) Please find below the answers to individual remarks regarding the experimental evidence.\n_______________________________________\nReviewer’s response:\nI accept this interpretation.\nReviewer’s response:\nThe authors mention that their samples exhibit high mobility. Under such conditions, Shubnikov–\nde Haas (SdH) oscillations are typically readily observable in magnetoresistance measurements.\nThe presence of SdH oscillations would constitute compelling evidence in identifying the Weyl\nnature of the material. Therefore, I strongly encourage the authors to include this crucial data in\nthe supplementary materials to further strengthen the manuscript.\nThe authors’ reply:\nWe thank the reviewer for inviting the discussion of quantum oscillations in Weyl semimetals.\nIndeed, the position of the intercept of SdH oscillations trend is sometimes interpreted as evidence\nfor Weyl nature of the material, following, for example, the logic discussed in Ref. [28], which is\nused for high-density bulk materials.\nHowever, this approach should be applied with extreme caution, as in many Weyl semimetals\ntopological surface states emanate from the Weyl points and provide additional\nmagnetoresistance oscillations and quantum Hall plateaus at high magnetic field. This is, for\nexample, a well-established feature of (compressively strained) HgTe, Cd As and Na Bi, which\n2 3 3\nall belong to the same class of Dirac semimetals, in which the nodes do not appear at high\nsymmetry points in the Brillouin zone. The Dirac nodes are split into Weyl nodes by BIA (bulk\ninversion asymmetry) and this splitting is obviously enhanced by the Zeeman effect when a\nmagnetic field is applied.\nOur previous study (Ref. [16]) focuses exactly on the topic of magnetoresistance and its relation\nto the surface states. For our low carrier density devices, we not only observe the SdH oscillations,\nbut also the quantum Hall effect (QH) in the transverse resistance. We analyse the transverse\nconductivity, which is a combination of longitudinal (𝑅 ) and transverse (𝑅 ) resistances:\n𝑥𝑥 𝑥𝑦\n−𝑅\n𝑥𝑦\n𝜎 = ,\n𝑥𝑦 𝑅2 +(𝑊𝑅 /𝐿)2\n𝑥𝑦 𝑥𝑥\nwhere W is the width and L is the length of the Hal bar. 𝜎 provides means for even better\n𝑥𝑦\nunderstanding of the bandstructure compared to just SdH oscillations:\n[Figure redacted]\nThis 2D plot from Ref. [16] shows the derivative of 𝜎 with gate voltage, yielding peaks at\n𝑥𝑦\ntransitions between the Landau levels. The unusual plateau sequence is a mixture of contributions\nfrom the electronic topological surface states (black) and hole-like massive surface states\n(magenta). The observed quantum oscillations can therefore not be used to extract Berry\ncurvature or confirm the Weyl band structure of the material, as they originate from the surface\nstates.\nIn this plot, U = 4 V corresponds to n=13x1011 cm-2 while U = -4 V corresponds to n=-10x1011\ngate gate\ncm-2 of hole conductance. In our new paper we operate in a smaller range around Weyl points,\nso that the carrier density varies between approximately n=3.5x1011 cm-2 and n=-2.2x1011 cm-2.\nThe difference in applied gate voltages comes from the use of a different insulator (material and\nthickness) than in Ref. [16].\nWe added the corresponding references and discussion to the Supplementary information section\n1, where the magnetoresistance data for perpendicular magnetic field is discussed.\n_______________________________________\nReviewer’s response:\nI accept this interpretation.\nReviewer’s response:\nThe authors repeatedly emphasize the high mobility of their samples and the proximity of the\nFermi level to the Weyl point, yet no quantitative mobility data have been provided. In fact, such\nhigh mobility can readily give rise to the controversial current-jetting effect, which remains a\nsignificant concern in the field.\nThe authors’ reply:\nWe thank the reviewer for pointing at the mobility and Fermi level position information that was\nmissing in the manuscript. We have included the corresponding information into the manuscript\n(Methods section) and Supplementary Information. In this device we can control the total carrier\ndensity in the structure via the electrostatic gate between approximately 3.5x1011 cm-2 and -\n2.2x1011 cm-2 of surface density (As already mentioned above, close to the Weyl points the\ntransport is dominated by the surface states). According to our estimates via comparison with k\ndot p calculation in Fig.1b, these carrier densities correspond to Fermi level being moved in a ±2\nmeV range around the Weyl nodes.\nThe mobility varies with the position of the Fermi level and is different for electronic and Hall\nstates. We extract it from an analysis of the zero-field resistivity value. For a density 3.7x1011 cm-\n2 it reaches approx. 77x103 cm2/(Vs), while for the valence band at Vg = -0.5V it is approx. 16x103\ncm2/(Vs). Directly at the Weyl point the carrier density cannot be reliably extracted from the Hall\nslope, not allowing to obtain a precise mobility value.\nWe agree that current jetting can be a significant concern for bulk materials, particularly when\nsoldering the metal leads directly on the bulk crystals. In our case, this phenomenon does not\noccur, as we can utilize the semiconductor technology developed over decades for our MBE-\ngrown epilayers. The contacts are fabricated using standard lithography techniques in a Hall bar-\nlike geometry, forming low-resistance (few tens of Ohms for the device studied in this paper, much\nsmaller than the resistance of the studied structures, hundreds of Ohms, see Supplementary Fig.\n3) bond pads and avoiding any possible nonuniformity. As a result, we observe the same negative\nmagnetoresistance effect in both small structures, like the island studied in our paper and in large,\nmacroscopic Hall bars studies in Ref. [16], which are hundreds of micrometres in size.\n_______________________________________\nReviewer’s response:\nAs for Current-jetting effect, please refers to Nature Reviews Physics 3, 394 (2021)\nReviewer’s response:\nIn fact, the occurrence of weak antilocalization (WAL) and weak localization (WL) is not at all\nsurprising in HgTe systems with strong spin–orbit coupling. Therefore, we remain unconvinced\nthat the negative magnetoresistance observed below 2 T arises from the chiral anomaly, unless\nthe authors can provide direct analytical evidence from SdH oscillations.\nThe authors’ reply:\nIndeed, these phenomena are routinely observed and well-studied for these materials. We would\nlike to point the reviewers to the characteristic magnitude of WL and WAL, which stems from their\nphysical origin. Both WL and WAL represent small corrections to the diffusive transport regime\nfrom quantum interference, that results in effective conductivity change on the order of one\nconductance quantum e2/h. We observe a much stronger effect of the magnetic field, where the\nconductance in the in-plane field increases by approx. 30%. The presence of such a strong\nlocalization effect would imply strong disorder in the system, so that we are no longer operating\nin the diffusive transport regime. This cannot be the case because of the mobilities of our\nstructures, so WL and WAL can be reliably excluded as a possible explanation. Moreover, at the\nmobilities we have in our epilayers, any WL and WAL features are restricted to few tens mT width,\nas high mobility leads to narrow WAL signal, see also Ref. [A] below.\n_______________________________________\nReviewer’s response:\nThe authors state that the 30% increase in conductance under in‑plane magnetic field is far too\nlarge to originate from weak localization (WL) or weak antilocalization (WAL), since these effects\ntypically yield corrections on the order of e2/h. However, this assertion is purely qualitative. Could\nthe authors provide quantitative analysis or experimental evidence that definitively excludes\nWL/WAL as possible contributions? In other words, how do the authors demonstrate that a 30%\nchange cannot arise—at least partially—from enhanced quantum‑interference effects or\nmultichannel localization in their system? Supporting data such as temperature dependence,\nangular dependence, or field‑width analysis would be necessary to substantiate this claim.\nReviewer’s response:\nIn fact, as clearly shown in Supplementary Fig. 1 (Theta = 90°), the observed positive\nmagnetoresistance is most likely attributable to weak antilocalization (WAL) arising from strong\nspin–orbit coupling, rather than to the positive magnetoresistance expected when the Fermi level\nis at the charge neutrality point. If the latter were the case, one would expect to see a pronounced\nparabolic increase in magnetoresistance at low fields for Theta=90°.\nThe authors’ reply:\nWe strongly disagree with the interpretation of the data in Supplementary Fig. 1 (Theta = 90°) as\nWAL. If we look at the magnitude of the observed resistance increase with magnetic field, we see\nthat it increases from 2 kΩ to 7 kΩ at 4 T. Weak antilocalization effects deliver small corrections\nto conductance of the order of e2/h (typically even less than that, see for example Ref. [A]), and\ntheir suppression cannot result in such large increase of resistance with magnetic field, even in\nmaterial with strong spin-orbit coupling.\nThe observed large magnetoresistance rather exactly matches expectations for a low-carrier-\ndensity system close to the Weyl point. As we discussed above, HgTe and other Weyl semimetals\nwith similar crystalline structure possess surface states that reside close to the Weyl point, and\nwhose contribution is observed in quantum hall and magnetoresistance close to the Weyl point.\nIn our case, at -0.29V the carrier density is so low that\nThe reviewer probably expects even larger growth, as observed in some Weyl candidates, such\nas [21]. We note that this behaviour is only characteristic for 3D carriers and thus is not applicable\nto Weyl semimetal layers that also have surface states.\n_______________________________________\nReviewer’s response:\nI partially accept this interpretation.\nReviewer’s response:\nI also recommend that the authors include relevant references when discussing these\nphenomena, in order to support their arguments appropriately.\nThe authors’ reply:\nWe include the relevant references that were used in the main text in the discussion of thermal\ntransport phenomena:\n1. Ultraclean Weyl semimetals can be analysed based on a study by Andreev and Spivak\n[8]\n2. The hydrodynamic transport regime is theoretically investigated by Lucas et al. [6] and\nby Andreev and Spivak [8]\n3. The effects of electron-phonon coupling are discussed briefly in a review by Pekola and\nKarimi [39]\n4. The elastic scattering regime is investigated by Andreev and Spivak [8].\n_______________________________________\nReviewer’s response:\nI accept this interpretation.\nReviewer’s response:\nFinally, we regret to conclude that the manuscript still falls short of the standards required for\npublication in a high-profile journal such as Nature Communications. In the supplementary\nmaterials, we observed that for θ = 90°, oscillations appear in the magnetoresistance curve at\nfields above 2 T. Could the authors clarify the origin of these oscillations? Are they attributable to\nShubnikov–de Haas (SdH) oscillations? If so, we strongly encourage the authors to extract and\npresent key parameters such as the effective mass of the charge carriers, the precise location of\nthe Fermi level, and the Berry curvature. The inclusion of such quantitative analyses would\nsignificantly enhance the credibility and impact of the manuscript.\nThe authors’ reply:\nWe respectfully disagree with the evaluation that our manuscript falls short of the standards of\nthis high-profile journal. We recognize that much of the criticism appears to stem from the absence\nof certain elements—Shubnikov–de Haas oscillations in perpendicular magnetic field and the\nextraction of Berry curvature from their phase—that are often expected in works on Weyl\nsemimetals. These signatures are indeed frequently interpreted as hallmarks of Weyl physics,\ntypically presented alongside negative magnetoresistance in in-plane field.\nWe note that this analysis is only applicable to bulk crystals with high carrier density and does not\napply to layers of low-density material, as the one we investigate. In our case, no well-developed\nSdH oscillations can be observed when the Fermi level is located at the Weyl point due to the\nexceptionally low carrier density. This precisely matches our expectations for a Weyl semimetal,\nwhile the analysis of quantum oscillations at larger densities reliably assigns them to the surface\nstates emanating from Weyl nodes, all studied in detail by Mahler et al. [16]. In low-density thin\nlayer samples, oscillations with unusual phases rather emerge from a complex interplay of Landau\nlevels, than reflecting intrinsic Berry curvature and bulk Weyl physics. Therefore, an attempt to\ninterpret such oscillations in the conventional way would be fundamentally misleading. A copy of\nthe corresponding paper has been included with our resubmission.\nOn the other hand, our devices do provide direct and reliable evidence that they are in the regime\nwhere Weyl physics is expected. By controlling the carrier density via the gate voltage, we can\nadjust the Fermi level position in the band structure with high precision. The entire range of our\nstudy spans only a few meV around the theoretically predicted Weyl point, as confirmed by our\nability to tune continuously from electron-like to hole-like conduction. We identify the Weyl point\nat the gate voltage corresponding to the maximum negative magnetoresistance, which coincides\nprecisely with the lowest carrier density. The existence of Weyl cones at this point is further\nconfirmed by our k dot p bandstructure calculations, which directly reflect the underlying\ncrystalline symmetry.\nWe thank the reviewer for drawing attention to the oscillations observed in the island\nmagnetoresistance for out-of-plane field orientation. However, these features do not appear to be\nrelated to Shubnikov–de Haas oscillations. They are only present in the small island, whereas in\nthe larger channel, measured at the same Fermi level, we observe only a single broad oscillation\nassociated with a Landau-level transition of the surface state—consistent with the behaviour\nshown in Ref. [16]. This indicates that the island oscillations most likely arise from magnetic field\nfocusing effects on ballistic carrier dynamics, rather than reflecting the underlying band structure.\nThis interpretation is fully consistent with our expectations: at such a low carrier density, well-\ndeveloped SdH oscillations cannot occur at the accessible magnetic fields.\nIn summary, all our experimental observations align with the expected behaviour of a Weyl\nsemimetal with low carrier density and significant surface-state contributions. Most importantly,\nthe pronounced negative magnetoresistance observed only when the Fermi level is tuned near\nthe Weyl point serves as the key transport signature of Weyl physics in our system. Building on\nthis regime, our study of thermal conductance achieves its central goal—probing potential\nsignatures of the gravitational anomaly in Weyl semimetals at low temperatures, where electron–\nphonon relaxation can be reliably excluded.\n_______________________________________\nReviewer’s response:\nOverall, the authors’ explanations remain rather unconvincing. The goal of their work is clear: by applying\nexternal strain, they break the inversion symmetry of initially centrosymmetric HgTe, thereby transforming\nit into a Weyl semimetal. From this, they draw the following conclusion:“This finding indicates that, despite\nthe unique electronic spectrum of Weyl semimetals, the mechanism governing heat transport in this system\nis the same as that for electrical transport, with no additional violations of conservation laws.”\nThis is, without question, an important and potentially far‑reaching conclusion. However, before reaching\nsuch a statement, the authors must first provide solid experimental demonstrations that (i) prior to strain the\nsystem was indeed a Dirac semimetal, supported by key evidence, and (ii) after strain, the transition to a\nWeyl semimetal has been directly confirmed by experiment. Only then can the final conclusion be\nconsidered physically convincing.\nMoreover, the authors’ replies often avoid the essential questions. For example, they argue that surface‑state\nSdH oscillations interfere with bulk SdH signals, and therefore the usual SdH analysis cannot be applied.\nThis argument is problematic. As they themselves emphasize, the strained samples are Weyl semimetals\nwith extraordinarily high mobility (77×103cm2 /Vs) and with the Fermi energy very close to the Weyl point.\nUnder such ideal conditions, the complete absence of SdH oscillations in magnetoresistance is highly\ncounterintuitive and contradicts basic physical expectations, as well as previous studies\n(Science 350, 413 (2015); Phys. Rev. X 5, 031023 (2015); Nat. Phys. 14, 1125 (2018)). Of course, there are\nknown examples, such as Co₂MnGa, which ARPES has confirmed to be a topological semimetal yet still\nexhibits no SdH oscillations. However, in that material the absence of oscillations is attributed to an\nextremely high carrier density that produces large Landau‑level spacing, making oscillatory\nmagnetoresistance difficult to observe when the Fermi surface crosses Landau levels. In contrast, the\nexperimental conditions reported for strained HgTe are optimal for observing topological features—low\ncarrier density, high mobility, and a Fermi level precisely tuned to the Weyl point—yet no direct evidence\nof topological changes is observed. This is difficult to accept. Although the authors provide many\nexplanations and checks, these do not directly relate to or convincingly demonstrate the topological\ntransition itself.\nThat said, I find the experimental method used to break inversion symmetry by strain application both\ntechnically precise and conceptually appealing. The quality of this experimental approach is impressive,\nand the study could stimulate productive debate among experts on this highly relevant problem. After all,\nresearch on transport signatures of the chiral anomaly in topological semimetals is still ongoing, and a clear\nconsensus has not yet been reached. For this reason, while I remain skeptical about the claimed evidence\nof a Dirac–to–Weyl transition, I cautiously support publication of this work. Its controversial aspects are,\nin my view, part of what makes it interesting and may encourage further investigation and discussion in the\nfield.\nReviewer #2 (Remarks to the Author):\nAll my concerns have been addressed. I recommend its publications.\nThe authors’ reply:\nWe thank the reviewer for their evaluation of the manuscript.\nA) Mühlbauer, M., et al. \"One-dimensional weak antilocalization due to the Berry phase in HgTe wires.\"\nPhys. Rev. Lett. 112, 146803 – 2014."}]

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