Lorentz skew scattering and giant nonreciprocal magneto-transport

Domain: chemistry
Published: 2026-03-07
Editorial outcome: minor_revision

Abstract
Despite extensive research, the primary mechanism governing nonlinear (nonreciprocal) magneto-transport in clean samples remains unknown. This theoretical gap has impeded the development of design principles for efficient nonreciprocal devices. Here, we unveil a hitherto unexplored effect in nonreciprocal magneto-transport from cooperative action of Lorentz force and skew scattering. The significance of this Lorentz skew scattering mechanism lies in that it dominates both longitudinal and transverse responses in highly conductive systems, and it exhibits a scaling behavior distinct from all known mechanisms. At low temperature, it shows a cubic scaling in linear conductivity, whereas the scaling becomes quartic at elevated temperature when phonon scattering kicks in. Applying our developed microscopic theory to surface transport in topological crystalline insulator SnTe and bulk transport in Weyl semimetals leads to significant results, suggesting a new route to achieve giant transport nonreciprocity in high-mobility materials with topological band features.

Introduction
Nonreciprocal transport phenomena have received significant attention, as they manifest intriguing physics of electronic quantum geometry and form the basis for rectification and diode applications [1–3]. Particularly, in nonmagnetic crystals with broken inversion symmetry, an applied magnetic field could trigger a nonreciprocal magneto-resistance linear in the B field [4–26]. The corresponding nonreciprocal magneto-transport (NRMT) response current can be expressed as j = \chi E^2 B, with \chi denoting the response tensor. This phenomenon was first studied in chiral crystals (known as electrical magnetochiral anisotropy) [1] and recently actively explored also in various achiral crystals [27–31].

In experiment, to understand microscopic mechanisms of a transport phenomenon, a common practice is to perform a scaling analysis, i.e., to analyze how the response coefficient varies as a function of the linear conductivity \sigma_{xx}, which is proportional to the scattering time \tau. Till now, several mechanisms for NRMT in nonmagnetic materials were proposed. For example, a Berry connection polarizability mechanism independent of \sigma_{xx} (\tau) was revealed for NRMT Hall response [24, 32]. For longitudinal response, \chi involves scattering and may originate from chiral scatterer [1], magnetic self-field [1], Zeeman-coupling induced Fermi surface deformation [5, 33], energy relaxation [34], impurity resonant states [35], chiral anomaly [36] and Berry curvature [37] mechanisms in Weyl semimetals, and etc [38–40]. It is noted that these mechanisms (except the Berry connection polarizability mechanism) typically exhibit a \chi \propto \sigma_{xx}^2 scaling. Consequently, a natural question arises: Is there a mechanism in NRMT that displays distinct scaling behavior? Moreover, what mechanism yields the highest power in the scaling relation? The answer holds unique significance for NRMT, as such a contribution would dominate the response in clean samples with large \tau.

The above mentioned fundamental gap in our understanding of NRMT has hindered the discovery of design principles for efficient, low-power nonreciprocal devices. While skew scattering dominates B-field-free nonlinear transport in highly conductive materials like graphene superlattices [41], leading to strong frequency doubling and energy harvesting [42, 43], the primary NRMT mechanism in such systems remains elusive. Resolving this could not only reveal a hidden field-induced nonreciprocal transport mechanism but also unlock novel pathways to unprecedentedly significant nonreciprocity and rectification capabilities.

In this work, we resolve the above issues by unveiling a new mechanism for NRMT — Lorentz skew scattering (LSK), which is resulted from the cooperative action of skew scattering (a quantum effect of scattering which induces trajectory skewness) and Lorentz force by magnetic field, as sketched in Fig. 1. This mechanism does not require spin-orbit coupling, and it manifests Berry curvature on Fermi surface. Importantly, we show that LSK is the leading contribution with the highest degree in the scaling relation for high mobility systems. Specifically, at low temperatures when impurity scattering dominates, it gives \chi \propto \sigma_{xx}^3 scaling; whereas at elevated temperatures when phonon scattering becomes substantial, the LSK contribution would scale as \sigma_{xx}^4. Because of its distinct scaling and quantum geometric character, it should be dominating in highly conductive samples and strongly enhanced by topological band features around Fermi level. We demonstrate our theory by studying surface transport in topological crystalline insulator SnTe and bulk transport in Weyl semimetals. The estimated LSK contribution can be orders of magnitude larger than previously studied mechanisms. As the NRMT in most reported works is rather weak, our finding offers a new insight for amplifying this nonreciprocal effect, which is promising for low-dissipative rectification applications.

Results
Geometric and scaling characters of LSK
We consider a diffusive system under weak applied E and B fields in the semiclassical regime. The electric current is generally expressed as j = -e \sum_l f_l v_l, where -e is the electron charge, l = (n,k) is a collective index labeling a Bloch state, f is the distribution function, and v is the electron velocity. To study NRMT response, we focus only on the part of the current \propto E^2 B. For our proposed LSK mechanism, B field enters via Lorentz force, while skew scattering enters via the collision integral. They together generate an out-of-equilibrium distribution f_{LSK} \propto E^2 B. (Hence, in calculating the current, it is sufficient to take v as the unperturbed band velocity.) This f_{LSK} can be obtained from the Boltzmann kinetic equation:

(\hat{D}_E + \hat{D}_L)f = (\hat{I}_c + \hat{I}_{sk})f, (1)

where hat denotes linear operators, \hat{D}_E = -eE \cdot \partial_k and \hat{D}_L = -e(v_l \times B) \cdot \partial_k give the electric force and Lorentz force driving terms, \hat{I}_c and \hat{I}_{sk} correspond to the conventional collision integral and the skew-scattering collision integral, respectively (see Methods) [44].

The leading contribution to skew scattering is from third-order on-shell scattering processes. Assuming weak spin-independent disorders, the skew scattering rate \omega_{l'l}^a is related to the Wilson loop connecting the three involved electronic states l, l', and l'' (schematics in Fig. 1b): W(l,l',l'') = \langle u_l | u_{l'} \rangle \langle u_{l'} | u_{l''} \rangle \langle u_{l''} | u_l \rangle, where |u_l \rangle is the periodic part of the Bloch state. This quantity is associated with the Pancharatnam-Berry phase arg(W) [45]. For an infinitesimal Wilson loop in k space, one finds

Im W(l,l',l'') \approx \frac{1}{2} (k'' - k') \times (k' - k) \cdot \Omega_l, (2)

which is proportional to the Berry curvature \Omega. It follows that for an isotropic model with smooth disorder potential, \omega_{l'l}^a \propto k' \cdot (k \times \Omega_l), explicitly showing the skewness of scattering, i.e., an incident electron with momentum k tends to be scattered to the transverse direction k \times \Omega_l. Since the Lorentz force also deflects electrons, their cooperative action should affects both longitudinal and transverse current flows. And since for transport, scattering events occur mainly around Fermi level, one expects that skew scattering and hence LSK mechanism would be enhanced if there is substantial Berry curvature distribution on Fermi surface.

It should be noted that our main aim is to reveal the NRMT mechanism with the highest scaling power. For this purpose, detailed treatment of collision integrals is not needed; and a scaling analysis suffices, e.g., it is convenient to take \hat{I}_c \sim 1/\tau and \hat{I}_{sk} \sim 1/\tau_{sk}, where the time scale of skew scattering \tau_{sk} should be much larger than \tau [41, 42, 46, 47].

Without performing detailed analysis on the kinetic equation, one may actually understand the scaling character of LSK and show that it is the dominant contribution in an intuitive way. The NRMT response is proportional to E^2 B. Let us analyze how the E and B field factors bring in the scaling parameters. First of all, the scaling parameter \tau comes from the out-of-equilibrium distribution \delta f, arising from the combined action of scattering and E-field driving. To have the highest scaling power in \tau, one should look for skew scattering: an E-field factor gives a \delta f \propto \tau for conventional scattering; but with skew scattering, it leads to a contribution \propto \tau^2/\tau_{sk}, which is the highest, a fact well known from previous studies on anomalous Hall effect and nonlinear Hall effects [41, 42, 47–49]. Hence, the mechanism we are seeking must involve skew scattering.

Next, let us consider how the B-field factor enters. There are two cases. (i) If B field enters via correction of band structure or correction to density of states [50], then it cannot bring additional \tau factors, since by itself a B field cannot drive a non-equilibrium. In such a case, the maximal scaling power is for each E factor associated with skew scattering, leading to a result scaled as (\tau^2/\tau_{sk})^2. (ii) An even higher contribution occurs if B enters via Lorentz force. Combined with a factor of E, they together bring a \tau^2 factor, which just corresponds to the ordinary Hall conductivity \propto \tau^2 [51]. Then, combined with one skew scattering for the remaining E factor, we have a result \propto \tau^4/\tau_{sk}, higher than other contributions. This is the LSK contribution we are looking for.

In the LSK process, as illustrated in Fig. 1a, the magnetic field bends the electron trajectory via Lorentz force, leading to a transverse motion, and then skew scattering further deflects this trajectory. Therefore, the LSK scaling may be intuitively argued as a composition of the usual Hall response \propto EB\tau^2 and the skew scattering \propto E\tau^2/\tau_{sk}, leading to an NRMT response scaled as E^2 B \tau^4/\tau_{sk}.

The above analysis clarifies why LSK has a higher \tau (\sigma_{xx}) scaling power than other mechanisms. In Supplementary Note 3, we further present a systematic analysis on all NRMT contributions, which indeed confirms this conclusion. This result indicates LSK should dominate NRMT in highly conductive materials. Combined with the revealed connection between skew scattering and Berry curvature, it offers a new strategy to achieve giant NRMT response.

Diagrammatic approach to LSK
To derive the formula for LSK contribution from Eq. (1), we use the method of successive approximations [44, 49, 52] (Methods) and expand the distribution function as f = f_0 + \sum_{i,j} [f^{(i,j)} + f_B^{(i,j)}]. (3)

Here, f_0 is the equilibrium Fermi-Dirac distribution. In the off-equilibrium part, we explicitly separate out the components f_B which are linear in B, and to keep track of the degrees in E field and scattering potential V, we use the notation Q^{(i,j)} to indicate a quantity \propto E^i V^{-j}. In the diagrammatic approach, we have \hat{I}_c = \hat{I}_c^{(0,-2)} and \hat{I}_{sk} = \hat{I}_{sk}^{(0,-3)}. The diagrammatic approach, we identify it according to Fig. 2. Its expression can thus be read off from the diagram as

f_{LSK} = f_B^{(2,5)} = -\tau^4 \hat{D}_E \{\hat{D}_L, \hat{I}_{sk}\} + \hat{D}_L \{\hat{I}_{sk}, \hat{D}_E\} + \hat{I}_{sk} \{\hat{D}_E, \hat{D}_L\} \hat{D}_E f_0. (4)

Here, {.,.} is the anticommutator of two operators. Combined with the band velocity, it gives the LSK response current in NRMT: j_{LSK} = -e \sum_l f_{LSK} v_l, from which the response tensor \chi_{LSK} can be extracted (j_{LSK} = \chi_{LSK}^{abcd} E_b E_c B_d, where summation over repeated Cartesian indices is implied).

From Eq. (4), the scaling behavior f_{LSK}, \chi_{LSK} \propto \tau^4/\tau_{sk} is consistent with our previous analysis. However, to discuss the scaling of \chi_{LSK} with \sigma_{xx}, we have to distinguish two regimes. In the low temperature regime where impurity scattering dominates, \sigma_{xx} is usually varied by fabricating samples with varying impurity density n_i. For example, in Refs. [53, 54], this is done by making metal films with different thicknesses such that the effective n_i from surface scattering is varied. Since both \tau and \tau_{sk} are \propto 1/n_i, we expect for such cases, \chi_{LSK} \propto \sigma_{xx}^3. On the other hand, at elevated temperatures where phonon scattering is substantial, \tau (and \sigma_{xx}) is usually varied by temperature, due to phonon scattering. Meanwhile, phonons do not contribute to skew scattering [53, 55, 56], so \tau_{sk} is still from impurity scattering. Therefore, one should observe \chi_{LSK} \propto \sigma_{xx}^4. These scaling behaviors are distinct from all previous mechanisms for NRMT.

Giant LSK response in Dirac surface states
We first apply our theory to the 2D Dirac model:

H = \tau_w k_y + v_x k_x \sigma_y - \tau v_y k_y \sigma_x + \Delta \sigma_z, (5)

where \tau = \pm labels two Dirac valleys connected by time reversal operation T, and \sigma's are Pauli matrices. This model describes the topological surface states of SnTe [57] and Pb_{1-x}Sn_xTe(Se) [58] at low temperatures. The spectrum for one valley is shown in Fig. 3a.

To have Lorentz force, we take B field to be in the z direction. By Eq. (4), near the bottom of the upper Dirac band, we estimate that the \chi_{LSK} components for both longitudinal and transverse NRMT can reach a similar order of magnitude, with (Supplementary Note 2)

|\chi_{LSK}| \sim \frac{e^4 w \tau^4}{\pi^2 \hbar^5 D \tau_{sk}}, (6)

where D is the density of states. The results from numerical calculations (using parameters of SnTe [57, 59]) are plotted in Fig. 3b, which exhibit a peak near band bottom, because of the sizable Berry curvature in this region. In the figure, for comparison, we also plot the results from the nonlinear Drude mechanism with Fermi surface deformation by Zeeman coupling to electronic magnetic moment (\chi_Z) [33], which are found to be much smaller than the LSK mechanism.

The nonreciprocity is often characterized by the coefficient \eta = \delta \sigma / \sigma = -\delta \rho / \rho, measuring the change in conductivity (resistivity) when the current direction is reversed. Here, we find \eta from LSK can reach \sim 20\% under B = 1 T and E = 10^4 V/m. This value is orders of magnitude larger than several previous results of NRMT in 2D electron gas under similar field strengths [7, 8, 10]. Another figure of merit is the nonreciprocal coefficient \gamma = -\eta / IB, where I is the driving current [2]. For a sample width of 1 \mu m, we estimate \gamma here can reach \sim 10^5 A^{-1}T^{-1}, which is very large, considering that most reported \gamma values are below 10^3 A^{-1}T^{-1} [1, 4, 5, 7–10].

Giant LSK nonreciprocity in Weyl semimetal
We have shown that to have pronounced LSK response, the system should have high mobility and large Berry curvature on Fermi surface. Weyl semimetals satisfy these conditions [60]. In a Weyl semimetal, the low-energy physics is from states around Weyl points [60]. A generic model for a Weyl point can be written as

H = w k_z + v k \cdot \sigma, (7)

which acts as a monopole for Berry curvature field. Since LSK contribution is T-even, a pair of Weyl points connected by T give the same contribution.

We perform numerical calculation for this Weyl model using parameters typical of Weyl semimetal materials (such as TaP family [61]). Figure 4a illustrates the obtained f_{LSK} distribution. For bulk materials, one usually characterizes NRMT using an intrinsic coefficient \gamma' = \gamma A = -\chi / \sigma_{xx}^2, where A is the cross sectional area of the sample [5, 27, 36]. Figure 4b shows the result. One finds that \gamma' can reach 3 \times 10^{-6} m^2 A^{-1} T^{-1} for \mu = 5 meV above Weyl point. Such LSK contribution is at least an order of magnitude larger than the chiral anomaly contribution and other mechanisms previously proposed [36]. This demonstrates LSK could dominate the NRMT response in Weyl semimetals.

Methods
Skew scattering in kinetic equation
For a homogeneous system, the Boltzmann kinetic equation at steady state reads:

\dot{k} \cdot \frac{\partial f}{\partial k} = \hat{I}\{f\}. (8)

Under applied uniform electric and magnetic fields, \dot{k}, the force on the electron wave packet, contains two parts \hat{D}_E and \hat{D}_L, corresponding respectively to the electric force and the Lorentz force [73]. For the LSK contribution to NRMT, the magnetic field enters via the Lorentz force \hat{D}_L, and other B-related terms in the kinetic equation are not relevant.

The skew scattering effect is contained in the collision integral \hat{I}. The collision integral is \hat{I}\{f_l\} = - \sum_{l'} (\omega_{l'l} f_l - \omega_{ll'} f_{l'}), where \omega_{l'l} is the scattering rate from l to l'. The scattering rate is determined quantum mechanically by the wave functions of the wave packets, their energies, and details of the scattering processes involved. It can always be decomposed into symmetric and anti-symmetric parts: \omega_{l'l}^s = (\omega_{l'l} + \omega_{ll'})/2 and \omega_{l'l}^a = (\omega_{l'l} - \omega_{ll'})/2. The two parts are associated respectively with the conventional collision integral \hat{I}_c f_l = - \sum_{l'} \omega_{l'l}^s (f_l - f_{l'}) and the skew-scattering collision integral \hat{I}_{sk} f_l = - \sum_{l'} \omega_{l'l}^a (f_l + f_{l'}) [44]. Here, we are considering the weak disorder regime. Then, the symmetric part \omega^s is dominated by processes of second order in scattering potential V, i.e., \hat{I}_c \propto V^2. Meanwhile, the leading order (skew scattering) process for the anti-symmetric part \omega^a is of third order. For example, for quasi-elastic scattering with spin-independent scattering potential, we have

\omega_{l'l}^a \approx \frac{4\pi^2}{\hbar} n_i \sum_{k''} \langle V_{kk''} V_{k''k'} V_{k'k} \rangle \delta(\epsilon_l - \epsilon_{l'}) \delta(\epsilon_l - \epsilon_{l''}) \times Im W(l,l',l''), (9)

where n_i is disorder density, V_{kk'} is the Fourier component of disorder potential, \epsilon_l is the wave packet energy, and W(l,l',l'') is the Wilson loop discussed in the main text.

Method to analyze the kinetic equation
The kinetic equation can be solved by the method of successive approximation which collects terms at each order of fields and scattering strength. In the main text, substituting the distribution function (3) into the Boltzmann kinetic equation (1) and collecting terms at each order of fields and scattering strength, we obtain a set of coupled linear equations.

For example, for terms linear in E (i=1), we have

\hat{I}_c f^{(1,2)} = \hat{D}_E f_0, \hat{I}_c f_B^{(1,4)} = \hat{D}_L f^{(1,2)}, (10)

and the remaining equations share common forms of

\hat{I}_c f^{(1,j)} = -\hat{I}_{sk} f^{(1,j+1)} (j < 2),
\hat{I}_c f_B^{(1,j)} = \hat{D}_E f^{(1,j-2)} - \hat{I}_{sk} f_B^{(1,j+1)} (j < 4). (11)

Note that for a fixed i in f^{(i,j)} and f_B^{(i,j)}, j has an upper bound, given by the conventional Drude-Boltzmann theory [51, 52]. For f^{(1,j)} and f_B^{(1,j)}, the highest value is j=2 and j=4, respectively.

Similarly, one can write down the equations at E^2 (i=2) order. These equations include

\hat{I}_c f^{(2,4)} = \hat{D}_E f^{(1,2)}, \hat{I}_c f_B^{(2,6)} = \hat{D}_E f_B^{(1,4)} + \hat{D}_L f^{(2,4)}, (12)

and the remaining equations share the common forms of

\hat{I}_c f^{(2,j)} = \hat{D}_E f^{(1,j-2)} - \hat{I}_{sk} f^{(2,j+1)} (j < 4),
\hat{I}_c f_B^{(2,j)} = \hat{D}_E f_B^{(1,j-2)} + \hat{D}_L f^{(2,j-2)} - \hat{I}_{sk} f_B^{(2,j+1)} (j < 6). (13)

Note that in our notation, one has \hat{I}_c = \hat{I}_c^{(0,-2)} and \hat{I}_{sk} = \hat{I}_{sk}^{(0,-3)}. One can check that in each equation, the (E, V^{-1}, B) order is balanced on the two sides.

These equations allow us to sequentially solve f^{(i,j)} and f_B^{(i,j)} at each order and analyze their scaling behavior. For instance, f^{(1,2)} = \hat{I}_c^{-1} \hat{D}_E f_0 \sim \tau \hat{D}_E f_0 is the familiar one responsible for Drude conductivity, f_B^{(1,4)} = \hat{I}_c^{-1} \hat{D}_L f^{(1,2)} \sim \tau^2 \hat{D}_L \hat{D}_E f_0, and so on. Moreover, the structure of these equations enables a systematic diagrammatic approach, as we explained in the main text, which offers an efficient method to tackle the kinetic equation for nonlinear transport.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors investigate the mechanism of nonreciprocal magneto-transport (NRMT), which is characterized by a linear dependence on the magnetic field. They focus on the combination of the Lorentz force and skew scattering, examining the relaxation-time dependence of the resulting transport as well as its manifestation in topological crystalline insulator and SnTe.\nNonreciprocal transport is a kind of nonlinear transport phenomenon and remains an essential and actively studied topic. In particular, the magnetic field response of such effects is still not fully understood. This work aligns with those efforts, and I appreciate the authors’ focus on extrinsic effects arising from scattering processes. However, I find that the theoretical novelty of the study is limited, and several aspects of the formulation appear questionable. Moreover, the claimed “new mechanism” of “Lorentz skew scattering” does not convincingly establish itself as a distinct concept. Based on these points, I do not consider the manuscript suitable for publication in Nature Communications. My detailed comments are as follows:\n- While the Introduction emphasizes the importance of quantum effects, the analysis is based on semiclassical equations (the Boltzmann equation). Furthermore, as discussed below, the collision term is not treated in a quantum-mechanical manner. It is therefore questionable whether the effect described can rightly be called a “quantum effect in scattering.”\n- Given the central role of the scattering mechanism in the proposed NRMT, it is unclear how the assumption of a constant relaxation time of “conventional collision” can be justified. A proper microscopic analysis and evaluation of the collision integral in the Boltzmann equation is necessary.\n- In the presence of a magnetic field, it is well known that modifications to the density of states proportional to Ω · B should be included in the Boltzmann formalism. However, these corrections do not appear to be taken into account in the present treatment.\n- In the microscopic theory of the extrinsic anomalous Hall effect, contributions such as skew scattering naturally emerge as vertex corrections to the self-energy. In the present treatment of the collision term, it is unclear whether the relation between the self-energy and the vertex correction (i.e., the Ward identity) is respected.\n- The authors refer to their approach as a “diagrammatic approach,” but in fact, what they employ is a standard solution of the Boltzmann equation under the constant relaxation time approximation. Moreover, they selectively retain only the terms that exhibit a specific relaxation-time dependence. Within the same level of approximation, other contributions should also be carefully evaluated.", "reviewer_2": "I have carefully read and reviewed the manuscript “Lorentz Skew Scattering and Giant Nonreciprocal Magneto-Transport” by Cong Xiao et al. I do find the central idea—achieving large nonreciprocity by combining skew scattering with the Lorentz force under a magnetic field—interesting. I also find the paper itself to be mostly well-written and well-structured. However, I cannot recommend publication in Nature Communications. I think that the scope of interest is too narrow for a broad readership that is important criteria for Nature Communications. The paper positions the proposed LSK mechanism as a new route to large nonreciprocal transport in noncentrosymmetric systems. However, the motivation for a broad audience is not fully convincing: among many extrinsic and intrinsic mechanisms already known, this adds one more effect rather than unifying or significantly extending the field. Nevertheless, I do believe that the work will be valuable to specialists in nonreciprocal transport and is suitable for a more focused physics journal.\n1) I am a bit confused about the role of spin-orbit coupling (SOC). The manuscript claims that the present LSK mechanism does not require SOC. But skew scattering fundamentally arises from asymmetric scattering mediated by SOC. Furthermore, in Berry-curvature-driven effect, sizable SOC is usually needed to open band gaps at Fermi surface and enhance the Berry curvature. Can the authors clarify if and how their mechanism operates in the strict absence of any SOC? If SOC is indeed essential, the text should be revised accordingly.\n2) In Eq. (5), the mass term (Δ) explicitly originates from SOC and opens a gap, yet in Eq. (6) it vanishes. If skew scattering (or Berry curvature?) relies on Δ, one would expect the nonreciprocal signal to vanish as Δ→0. Does the nonreciprocity go to zero without a mass gap? Also, can the authors clarify whether the LSK mechanism originates from skew scattering (extrinsic) or Berry curvature (intrinsic)?\n3) While it is plausible that skew scattering induces a transverse velocity and the Lorentz force converts part of this into a longitudinal component, it is not immediately obvious why this combination yields a second order (nonreciprocal) response. Can you provide an intuitive physical picture or simple toy model illustrating how the two processes together produce a nonlinear effect.\n4) In the Introduction, the authors list various mechanisms for nonreciprocal transport. It would help readers to classify these into intrinsic versus extrinsic effects, especially for longitudinal nonreciprocity.\n5) Recently, nonreciprocal transports have been very extensively investigated in superconductivity. In particular, very recent work has shown giant nonreciprocal transport in an organic superconductor with structural chirality [Sato, Phys. Rev. Res. 7, 023056 (2025)]. Although superconductivity is beyond the current scope, it may be fair to shortly acknowledge these developments and briefly contrast them with the present mechanism.\n6) I am just wondering that if LSK-induced nonreciprocity is as large as claimed, why has it not yet been observed in candidate materials such as graphene, BiTeBr, or Te thin flakes? Are there any experimental challenges (e.g., sample quality, difficulty in tuning carrier number, or competing mechanisms)?", "reviewer_3": "The manuscript by Xiao et al. presents a novel mechanism, termed Lorentz skew scattering (LSK), for nonreciprocal magneto-transport (NRMT) in highly conductive systems. The authors demonstrate that the cooperative action of Lorentz force and skew scattering leads to distinct scaling behaviors in the response tensor, with potential dominance in clean samples and topological materials. The work is theoretically rigorous and supported by numerical calculations, offering significant insights into the design principles for nonreciprocal devices. The findings are timely and could have broad implications for both fundamental research and applications in rectification and diode technologies. However, several aspects of the manuscript require clarification and improvement to meet the high standards of Nature Communications.\n1. While the manuscript provides a thorough theoretical framework for LSK, the physical intuition behind the mechanism could be more clearly explained. A more intuitive description of how the interplay between the Lorentz force and skew scattering generates the observed scaling behaviors (e.g., cubic vs. quartic) would improve accessibility for a broader audience. The inclusion of a simplified schematic or analogy to illustrate this cooperative effect is recommended.\n2. The theoretical predictions are compelling, but the manuscript would benefit from a more direct comparison with existing experimental data. For instance, are there experimental results in the literature that align with the predicted $\\chi\\propto\\sigma_{xx}^{3}$ or $\\sigma_{xx}^{4}$ scaling? If not, could the authors suggest specific experiments or materials where such scaling might be observed?\n3. The manuscript highlights the dominance of LSK in highly conductive systems but does not adequately address potential limitations or competing mechanisms in real materials. For example, impurity resonance states, which can also enhance bilinear magnetoconductivity [as reported in Physical Review B 109, 155154 (2024)]), could influence the observability of LSK. Additionally, as proposed in Ref. [38] [Phys. Rev. B 108, L241104 (2023)], when the Fermi energy is near the edge of the energy gap, where $\\epsilon_{mn}$ is small, a nonlinear conductivity $\\propto\\tau^{3}\\sim\\Gamma^{-3}$ can arise independently of the LSK mechanism. How can the LSK be distinguished from these competing mechanisms? A discussion of these factors would provide a more balanced perspective.\n4. The diagrammatic approach (Fig. 2) is innovative but somewhat dense. A step-by-step explanation of how the diagrams translate to the mathematical expressions (e.g., Eq. 4) would aid comprehension. For example, results like $f_{B}^{\\left( 1,6\\right) }$ are not presented. Could the authors briefly explain why these terms are omitted from the diagram? Are they excluded because they are deemed unimportant? Additionally, Fig. 2 includes many terms, yet the manuscript only discusses the $f_{B}^{\\left( 2,5\\right) }$ term as the LSK mechanism, stating that higher-order LSK mechanisms can be neglected due to their relatively small magnitude. It is necessary to provide a more detailed rationale for why the other terms are not discussed.\n5. The use of the Boltzmann equation method to discuss nonlinear responses is a widely employed approach. Well-studied effects such as the Berry curvature dipole, quantum metric dipole, and second-order nonlinear Drude term can all be derived from this method. Among these, the $f^{\\left( 2,4\\right) }$ in Fig. 2 corresponds precisely to the second-order nonlinear Drude term. However, I wonder why intrinsic contributions, such as the quantum metric dipole term or other terms, are not visible in the diagram. What specific aspects of the Boltzmann equation formulation account for this absence?\nOverall, the manuscript presents a significant theoretical advancement in the field of nonreciprocal transport, with potential applications in device engineering. However, the clarity of the mechanism and its experimental relevance need further elaboration. Once these issues are resolved, the manuscript will be suitable for publication in Nature Communications.", "reviewer_4": "Referee Report: \"Lorentz Skew Scattering and Giant Nonreciprocal Magneto-Transport\"\nRecommendation: Accept after Minor Revisions\nOverall Assessment\nThe manuscript presents a compelling theoretical investigation of nonreciprocal magneto-transport in non-centrosymmetric systems, with a focus on the role of Lorentz-force-induced skew scattering. The study addresses a significant gap in understanding nonlinear Hall effects and offers quantitative predictions of \"giant\" nonreciprocal transport coefficients. The work is timely, analytically rigorous, and conceptually novel. While minor revisions are suggested below, the core contributions are sound and merit publication.\nMajor Strengths\n1. Novel Mechanism: The identification of Lorentz skew scattering as a distinct source of nonreciprocal transport (beyond Berry curvature or spin-orbit-related skew scattering) is an important theoretical advance.\n2. Quantitative Impact: The prediction of nonreciprocal magneto-transport response coefficient scaling as $\\sigma_{xx}$ (or $\\tau)$ highlights the potential for giant effects in clean systems, offering testable experimental signatures.\n3. Analytical Clarity: The derivations and illustrations are presented clearly and logically (e.g., diagrammatic approach in FIG 2).\nMinor Revisions Required\n1. During calculating the current, you use the unperturbed band velocity. In principle, external electric field will induce additional anomalous velocity via the Berry curvature [D. Xiao, M.-C. Chang & Q. Niu,“Berry phase effects on electronic properties”, Rev. Mod. Phys. 82, 1959 (2010)]. Can you explain a bit in which parameter regime we can sately use the unperturbed band velocity or show that there exists no nonreciprocal magneto-transport from the anomalous veloctity? A qualitative discussion will be useful for theoretical and experimental pespectives.\n2.In Figs. 3(a) and 4(a), what is the exact values of kx and kz (such as in the unit of lattice constant)? Also, briefly explain the parameters in Hamiltonina (5), expecially the $w$.\n3. Equation (4) shows the expansion of nonequilirium distribution with $\\tau$, which, in principle, requires small $\\tau$. However, this work considers a “clean” sample, implying that $\\tau$ can be sizable. In FIG. 3, the disorder strength, parameterized by multiplying impurity density $n_iV_0=1$meV, is much smaller than the gap $\\Delta=10$ meV (and $k_Fv_{x,y}$). $\\tau$, propotional to $n_iV^2_0$, is indeed large in this sample. A qualitative/quatitative discussion about how to balance “clean” sample and small $\\tau$ will be helpful for readers.\nFinal Recommendation\nThis manuscript makes a valuable contribution to the theory of nonlinear transport. The proposed revisions are straightforward and will enhance clarity, impact, and accessibility without requiring major rework. I recommend acceptance after the authors address the minor points above. The work will interest researchers in spintronics, topological matter, and nonlinear transport."}, "author_response": "Reply to the Reviewers\nReviewer #1 (Remarks to the Author):\nThe authors investigate the mechanism of nonreciprocal magneto-transport (NRMT), which is\ncharacterized by a linear dependence on the magnetic field. They focus on the combination of the\nLorentz force and skew scattering, examining the relaxation-time dependence of the resulting transport\nas well as its manifestation in topological crystalline insulator and SnTe.\nNonreciprocal transport is a kind of nonlinear transport phenomenon and remains an essential and\nactively studied topic. In particular, the magnetic field response of such effects is still not fully\nunderstood. This work aligns with those efforts, and I appreciate the authors’ focus on extrinsic effects\narising from scattering processes. However, I find that the theoretical novelty of the study is limited,\nand several aspects of the formulation appear questionable. Moreover, the claimed “new mechanism”\nof “Lorentz skew scattering” does not convincingly establish itself as a distinct concept. Based on\nthese points, I do not consider the manuscript suitable for publication in Nature Communications. My\ndetailed comments are as follows:\nAnswer: We thank the reviewer for acknowledging the importance of the research topic of our\nmanuscript by stating that “Nonreciprocal transport is a kind of nonlinear transport phenomenon and\nremains an essential and actively studied topic. In particular, the magnetic field response of such\neffects is still not fully understood. This work aligns with those efforts, and I appreciate the authors’\nfocus on extrinsic effects arising from scattering processes”.\nWe regret that the novelty of our work was not fully appreciated by the reviewer. Below, we shall\nclarify the novelty and significance of our work to the reviewer, and elaborate why it belongs to Nature\nCommunications.\nNonreciprocal magneto-transport (NRMT) was first reported about 20 years ago and has received\ntremendous interest in recent years. Several mechanisms have been proposed for NRMT (as\nsummarized in the introduction part of our paper), and they typically give a 𝜒 ∝ 𝜎(cid:2870) scaling, where\n(cid:3051)(cid:3051)\n𝜒 is the NRMT response coefficient.\nNow the Question is: Is there any new mechanism for NRMT having a distinct scaling behavior? And\nwhat mechanism gives the highest power in the scaling relation? The answer is of unique significance,\nbecause such mechanism dominates the NRMT response in high-mobility systems. Evidently,\nknowing the answer will offer crucial insight for developing high-efficiency nonreciprocal devices.\nTill now, these fundamental questions remain unsolved.\nIt is our current work that solved these outstanding problems.\n• It predicts a new and distinct NRMT mechanism, the Lorentz skew scattering (LSK)\nmechanism, which was not known before and leads to a scaling behavior distinct from ALL\nknown mechanisms;\n• This LSK mechanism is of unique significance in that: We proved that it has the highest\nscaling power, so it dominates the NRMT signal in high-mobility systems.\n• Specifically, we showed that LSK gives a 𝜒 ∝ 𝜎(cid:2871) scaling at low temperature, and a 𝜒 ∝ 𝜎(cid:2872)\n(cid:3051)(cid:3051) (cid:3051)(cid:3051)\nscaling at elevated temperature when phonon scattering kicks in.\nThus, the novelty of our work is clear: The above conclusions are discovered here for the first time.\nThe unveiled distinct characters of LSK mechanism and its connection to band geometric properties\n(such as Berry curvature) not only provide sharp signals for analyzing experimental results but\nalso offer a new design principle to achieve unprecedentedly high NRMT in high-mobility\nsystems with topological band structures. As examples, our calculation for LSK NRMT in SnTe and\nTaP resulted huge values, which surpass all other mechanisms and other systems reported so far,\nas summarized in Table I of the main text.\nMore importantly, we are pleased to update the reviewer that our theoretical prediction has been\nconfirmed by two very recent experiments. It is also worth noting that these experiments were done\non two of the systems suggested in our paper. One is on high-mobility graphene/hBN superlattices, by\nPan He and Jian Shen’s group at Fudan University (arXiv:2511.03381). The other is on n-doped high-\nmobility BiTeBr, by Weibo Gao’s group at Nanyang Technological University (arXiv:2511.03273).\nThese works confirm the dominance of LSK mechanism in the studied high-mobility material systems.\nIn particular, in high-mobility graphene superlattices, the LSK mechanism gives rise to a record-high\ntransverse nonlinear conductivity of ~36,000μmV(cid:2879)(cid:2869)Ω(cid:2879)(cid:2869) under 𝐵 = 0.3 T, which is orders of\nmagnitude larger than all previously reported transverse nonlinear transport and NRMT. These\nexperimental findings confirm our theory and indicate our proposed LSK mechanism as an efficient\nroute to giant nonreciprocal magneto-transport, with promising potential for applications.\nIn addition, our work will also inspire new directions of research. For example, the proposed LSK may\nalso play leading roles in other nonlinear magneto-transport processes, such as nonreciprocal thermal\nand thermoelectric magneto-transport. It is anticipated to lead to significant B-field-tunable nonlinear\nNernst effect and nonlinear thermal Hall rectification in graphene superlattices.\nWe hope the above discussion has fully settled the reviewer’s concern on the novelty and significance\nof our work. In the revised manuscript, we have highlighted these points in the Introduction part. And\nwe have added a paragraph at the end (using Noted added) to mention the two very recent experiments\nwhich verified our prediction.\nIn the following detailed reply, we will fully address the reviewer’s questions and comments. We thank\nthe reviewer for the report, which have helped us improve our manuscript.\n(1) - While the Introduction emphasizes the importance of quantum effects, the analysis is based on\nsemiclassical equations (the Boltzmann equation). Furthermore, as discussed below, the collision term\nis not treated in a quantum-mechanical manner. It is therefore questionable whether the effect\ndescribed can rightly be called a “quantum effect in scattering.”\nAnswer: We thank the reviewer for the comment. First of all, we would like to remind that the\nsemiclassical theory itself is a quantum approach. It is named ‘semiclassical’ because it adopts a\ndescription of dynamics of electron wave packets in phase space; it does not mean the quantum effects\nare neglected. Actually, the semiclassical theory fully captures quantum effects and is an accurate\ntheory in the semiclassical regime. This has been demonstrated in numerous cases (see e.g., Rev. Mod.\nPhys. 82, 1959 (2010)). A notable example is the anomalous Hall effect, where the semiclassical theory\nwas shown to capture all the essential quantum effects (J. Phys.: Condens. Matter 20, 023201 (2008);\nRev. Mod. Phys. 82, 1539 (2010)).\nIn the semiclassical equations of motion, the quantum effects are encoded in the wave packet energy\nand the Berry curvature terms. In the kinetic equation (the Boltzmann equation) for wave packets, the\nquantum effects are captured in the collision terms. The quantum effect of scattering in our manuscript\nmeans clearly the skew scattering, which indeed IS a quantum mechanical scattering effect and treated\nquantum mechanically. To see this, the collision term in the kinetic equation is 𝐼(cid:4632){𝑓} =\n(cid:3039)\n−∑ (𝜔 𝑓 −𝜔 𝑓 ) and the scattering rate 𝜔 is determined quantum mechanically, by the wave\n(cid:3039)(cid:4594) (cid:3039)(cid:4594)(cid:3039) (cid:3039) (cid:3039)(cid:3039)(cid:4594) (cid:3039)(cid:4594) (cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039)\nfunctions and energies of the wave packets and by the scattering potential. For example, for quasi-\nelastic scattering processes,\n2𝜋\n𝜔 = 〈|𝑇 |(cid:2870)〉 𝛿(𝜀 −𝜀 ),\n(cid:3039)(cid:4594)(cid:3039) ℏ (cid:3039)(cid:4594)(cid:3039) (cid:3030) (cid:3039) (cid:3039)(cid:4594)\nwhere 〈…〉 denotes average over disorder configurations, and 𝑇 is the scattering T-matrix\n(cid:3030) (cid:3039)(cid:4594)(cid:3039)\ndetermined by the Lippmann-Schwinger equation\n𝑉 𝑇\n(cid:3039)(cid:4594)(cid:3039)(cid:4594)(cid:4594) (cid:3039)(cid:4594)(cid:4594)(cid:3039)\n𝑇 = 𝑉 +(cid:3533) ,\n(cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039)\n𝜀 −𝜀 +𝑖ℏ0(cid:2878)\n(cid:3039) (cid:3039)(cid:4594)(cid:4594)\n(cid:3039)(cid:4594)(cid:4594)\nwhich is fully quantum mechanical. The scattering rate can always be decomposed into symmetric and\nantisymmetric parts: 𝜔(cid:3046) = (𝜔 +𝜔 )/2 and 𝜔(cid:3028) = (𝜔 −𝜔 )/2. The skew scattering comes\n(cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:3039)(cid:4594) (cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:3039)(cid:4594)\nfrom the antisymmetric part. In the case of weak disorders, 𝜔(cid:3028) ≈ 𝜔(cid:2871)(cid:3028) is from third order scattering\n(cid:3039)(cid:4594)(cid:3039) (cid:3039)(cid:4594)(cid:3039)\nprocesses. Using the above expressions, one has\n4𝜋(cid:2870)\n𝜔(cid:2871)(cid:3028) = (cid:3533)ℑ〈𝑉 𝑉 𝑉 〉 𝛿(𝜀 −𝜀 )𝛿(𝜀 −𝜀 ).\n(cid:3039)(cid:4594)(cid:3039) ℏ (cid:3039)(cid:3039)(cid:4594) (cid:3039)(cid:4594)(cid:3039)(cid:4594)(cid:4594) (cid:3039)(cid:4594)(cid:4594)(cid:3039) (cid:3030) (cid:3039) (cid:3039)(cid:4594)(cid:4594) (cid:3039) (cid:3039)(cid:4594)\n(cid:3039)(cid:4594)(cid:4594)\nThus, the quantum effects of skew scattering are indeed captured in this treatment. These points are\nwell established in the study of anomalous Hall effect (please see, e.g., Rev. Mod. Phys. 82, 1539\n(2010)).\nIn addition, we would like to stress that the pivotal purpose of this paper is to clarify the NRMT\nmechanism with highest scaling power in the scaling relation. The highest scaling power of LSK\ncompared to other NRMT mechanisms ensures that in high mobility materials the LSK is the dominant\nmechanism and is largely enhanced, with the potential to give unprecedentedly strong NRMT. For this\npurpose, one only needs to perform scaling analysis, i.e., figure out how various quantities scale with\ncharacteristic scattering time scales. For such scaling analysis in the first part of the manuscript, it is\nNOT necessary to get into the detailed calculation of the scattering rates 𝜔 , so we did not present\n(cid:3039)(cid:4594)(cid:3039)\ntheir expressions explicitly in the main text of the original paper. Probably, this caused the reviewer’s\nmisunderstanding.\nTo clarify this point, we have added the above discussions and expressions to the Method section under\n“Skew scattering in kinetic equation”.\n(2) - Given the central role of the scattering mechanism in the proposed NRMT, it is unclear how the\nassumption of a constant relaxation time of “conventional collision” can be justified. A proper\nmicroscopic analysis and evaluation of the collision integral in the Boltzmann equation is necessary.\nAnswer: We thank the reviewer for the question. As we have emphasized, the pivotal purpose of this\npaper is to clarify the mechanism with highest scaling power in the scaling relation, so that it dominates\nthe NRMT response in high-mobility systems. For this purpose, it is sufficient to figure out how\nvarious quantities scale with characteristic scattering time scales, e.g., 𝜏 and 𝜏 , which characterizes\n(cid:3046)(cid:3038)\nthe conventional and the skew scattering processes 𝐼(cid:4632) ∼ 1/𝜏, 𝐼(cid:4632) ∼ 1/𝜏 . And detailed calculation\n(cid:3030) (cid:3046)(cid:3038) (cid:3046)(cid:3038)\nof collision integral is unnecessary for this purpose. This is the standard treatment for deriving the\nscaling relation. For example, it has been successfully applied in deriving the scaling relations for\nlinear and nonlinear Hall effects (Nat. Commun. 10, 3047 (2019); PRB 111, 155127 (2025)), which\nhave been widely used for interpreting experiments (please see, e.g., Nat. Mater. 18, 324 (2019); Nat.\nNanotechnol. 17, 378 (2022)).\nTo clarify this point, we have added the following discussion in the second paragraph of the right\ncolumn on page 2:\n“It should be noted that our main aim is to reveal the NRMT mechanism with the highest scaling\npower. For this purpose, detailed treatment of collision integrals is not needed; and a scaling analysis\nsuffices, e.g., it is convenient to take …”\n(3) - In the presence of a magnetic field, it is well known that modifications to the density of states\nproportional to Ω · B should be included in the Boltzmann formalism. However, these corrections do\nnot appear to be taken into account in the present treatment.\nAnswer: We thank the reviewer for the valuable question. Indeed, the 𝛀⋅𝑩 correction to the density\nof states should be included when calculating the NRMT response. It is not discussed here. The reason\nis that: As stated, the purpose of our work is to clarify the contribution with the highest scaling power,\nand we showed this is from the LSK mechanism. The contribution from the 𝛀⋅𝑩 correction is less\nimportant compared to LSK. To see this, one simply notes that if the B field comes through the 𝛀⋅𝑩\ncorrection, it will not bring in any 𝜏 or 𝜏 scaling factors by itself. Then for NRMT (with 𝐸(cid:2870)𝐵\n(cid:3046)(cid:3038)\nresponse), each 𝐸 factor at most gives a 𝜏(cid:2870)/𝜏 scaling from skew scattering process, so the\n(cid:3046)(cid:3038)\ncontribution from 𝛀⋅𝑩 correction can at most have a scaling behavior ∝ 𝜏(cid:2872)/𝜏(cid:2870) , which is lower\n(cid:3046)(cid:3038)\nthan LSK ∝ 𝜏(cid:2872)/𝜏 (note that we always have 𝜏 > 𝜏, and 1/𝜏 goes to zero in high-mobility\n(cid:3046)(cid:3038) (cid:3046)(cid:3038) (cid:3046)(cid:3038)\nlimit).\nTo clarify this point, in the revised manuscript, we have added the following discussion in the third\nparagraph of the right column on page 2:\n“Next, let us consider how the B-field factor enters. There are two cases. (i) If B field enters via\ncorrection of band structure or correction to density of states, then it cannot bring additional τ factors,\nsince by itself a B field cannot drive a nonequilibrium. In such a case, the maximal scaling power is\nfor each E factor associated with skew scattering, leading to a result scaled as (𝜏(cid:2870)⁄𝜏 )(cid:2870). (ii) An even\n(cid:3046)(cid:3038)\nhigher contribution occurs if B enters via Lorentz force. …”\nAnd in the revised Supplementary Information, we have added a new section “Dominance of LSK\nover other NRMT contributions in highly conductive systems” (Supplementary Note 3), which\nsystematically compares the scaling behaviors of LSK and other contributions (including the one from\n𝛀⋅𝑩 correction) in detail, and confirm that LSK possesses the highest scaling power thus dominates\nthe NRMT in highly conductive materials.\n(4) - In the microscopic theory of the extrinsic anomalous Hall effect, contributions such as skew\nscattering naturally emerge as vertex corrections to the self-energy. In the present treatment of the\ncollision term, it is unclear whether the relation between the self-energy and the vertex correction (i.e.,\nthe Ward identity) is respected.\nAnswer: We thank the reviewer for the comment. Indeed, as stated by the reviewer, in a quantitative\ncalculation, one needs to consider the scattering effects in detail (like vertex corrections in Green’s\nfunction approach) and to ensure basic relations (like Ward identity). However, as we emphasized, the\ntask here is different: Our task is to perform scaling analysis to clarify the mechanism that gives the\nhighest scaling power, which ensures the dominance of this mechanism in high-mobility materials. For\nthis purpose, it is unnecessary to get into those details. And it is particularly because of this character\nthat the scaling relations are so robust and so powerful to be used for analysis of real material\nexperiments, for which it is usually impossible to have detailed information about microscopic\nscattering processes.\n(5) - The authors refer to their approach as a “diagrammatic approach,” but in fact, what they employ\nis a standard solution of the Boltzmann equation under the constant relaxation time approximation.\nMoreover, they selectively retain only the terms that exhibit a specific relaxation-time dependence.\nWithin the same level of approximation, other contributions should also be carefully evaluated.\nAnswer: We thank the reviewer for the comment and suggestion. First, as clarified in our answer to\nComment 2 above, the pivotal purpose of our work is to clarify the mechanism giving the highest\nscaling power, and for this purpose, the scaling analysis using the characteristic scattering times 𝜏 and\n𝜏 is the appropriate approach. In undertaking this task, we find it convenient to analyze and organize\n(cid:3046)(cid:3038)\nvarious contributions using the diagram shown in Fig. 2 of our manuscript. From this type of diagram,\none can systematically obtain all contributions to the nonequilibrium distribution, classified by their\ndependence on external field as well as scattering potential, so this approach is especially suitable for\nthe scaling analysis. To our best knowledge, this approach was not known before. We expect it will\nbecome a powerful tool for analyzing extrinsic contributions to nonlinear transport.\nSecond, the reviewer mentioned we “selectively retain only the terms that exhibit a specific relaxation-\ntime dependence”. We did this, because it is the main purpose of this work to clarify the contribution\nwith the highest scaling power, which dominates the NRMT response in high-mobility systems. Our\nstudy demonstrated that it is from the LSK mechanism, so we focused on it in this paper.\nThird, the reviewer suggested that we also examine other contributions. Following the reviewer’s\nsuggestion, in the revised Supplementary Information, we have added a new section “Dominance of\nLSK over other NRMT contributions in highly conductive systems”, which presents the detailed\nanalysis of scaling behavior for all other contributions to NRMT. Their scaling characters can be\nsummarized in the following Table R1:\nTable. R1: Representative mechanisms of NRMT contributions of category I and II in nonmagnetic\nmaterials, and their scaling relations with respect to relaxation time (longitudinal conductivity) in the\ncase of static impurity scattering. In NRMT contributions of category I, the magnetic field enters\nthrough classical Lorentz force effect. In category II, the magnetic field enters through quantum\nmechanical effects by Zeeman coupling to electronic magnetic moment and by coupling to Berry\ncurvature. The highest scaling power in category II is 2, whereas that in category I is 3, given by the\nLSK. Therefore, LSK possesses the highest scaling power compared to other NRMT contributions and\nhence dominates in highly conductive systems.\nTo highlight this comparison, in the main text of revised manuscript, we have added the following\ndiscussion in the first paragraph of the left column on page 3:\n“The above analysis clarifies why LSK has a higher 𝜏 (𝜎 ) scaling power than other\n(cid:3051)(cid:3051)\nmechanisms. In Supplementary Note 3, we further present a systematic scaling analysis on all NRMT\ncontributions in nonmagnetic materials, which indeed confirms this conclusion.”\nIn addition, we have also computed these contributions for the 2D Dirac model Eq. (5) in our paper.\nThe results are shown in the following Fig. R1, which is also added in Supplementary Information.\nFigure R1: Dominance of LSK over other NRMT contributions in highly conductive systems. The\ncalculation is performed for the 2D model (5) in the main text. We set the same parameters as Fig. 3\nin the main text.\nReviewer #2 (Remarks to the Author):\nI have carefully read and reviewed the manuscript “Lorentz Skew Scattering and Giant Nonreciprocal\nMagneto-Transport” by Cong Xiao et al. I do find the central idea—achieving large nonreciprocity by\ncombining skew scattering with the Lorentz force under a magnetic field—interesting. I also find the\npaper itself to be mostly well-written and well-structured. However, I cannot recommend publication\nin publication\nin Nature Communications. I think that the scope of interest is too narrow for a broad readership that\nis important criteria for Nature Communications. The paper positions the proposed LSK mechanism\nas a new route to large nonreciprocal transport in noncentrosymmetric systems. However, the\nmotivation for a broad audience is not fully convincing: among many extrinsic and intrinsic\nmechanisms already known, this adds one more effect rather than unifying or significantly extending\nthe field. Nevertheless, I do believe that the work will be valuable to specialists in nonreciprocal\ntransport and is suitable for a more focused physics journal.\nAnswer: We thank the reviewer for acknowledging “I do find the central idea—achieving large\nnonreciprocity by combining skew scattering with the Lorentz force under a magnetic field—interesting”\nand “I also find the paper itself to be mostly well-written and well-structured”.\nOn the other hand, the reviewer thought “the scope of interest is too narrow for a broad readership\nthat is important criteria for Nature Communications”, because he/she considered “among many\nextrinsic and intrinsic mechanisms already known, this adds one more effect rather than unifying or\nsignificantly extending the field”. We respectfully disagree with the reviewer’s opinion.\nBelow, we clarify the unique significance and broad interest of our work, and demonstrate why it\nbelongs to Nature Communications.\nFirst of all, nonreciprocal magneto-transport (NRMT) is itself a research topic of broad interest. It was\nfirst reported about 20 years ago [PRL 87, 236602 (2001)]. In recent years, along with the rapid\nprogress of the field of nonlinear transport, NRMT has received tremendous interest and been explored\nin various material systems [e.g., Nat. Phys. 13, 578 (2017); Nat. Phys. 14, 495 (2018); Nat. Commun.\n12, 540 (2021)]. Till now, several mechanisms have been proposed for NRMT (as summarized in the\nintroduction part of our paper), and they typically give a 𝜒 ∝ 𝜎(cid:2870) scaling, where 𝜒 is the NRMT\n(cid:3051)(cid:3051)\nresponse coefficient.\nHowever, despite extensive studies, key Questions remain: Is there any new mechanism for NRMT\nhaving a distinct scaling behavior? And what mechanism gives the highest power in the scaling relation?\nThe answer is of unique significance, because such mechanism dominates the NRMT response in high-\nmobility systems. Evidently, knowing the answer will offer crucial insight for designing material\nsystems with enhanced NRMT. As a comparison, in B-field-free transport, the understanding that skew\nscattering mechanism dominates nonlinear transport in high-mobility systems has enabled the\nachievement of record high transport signals [Nat. Nanotechnol. 17, 378 (2022); Phys. Rev. Lett. 129,\n186801 (2022)]. Therefore, the task we undertake in this work will have crucial and broad impact not\nonly for fundamental understanding of NRMT but also for potential applications in rectification and\ndiode technologies.\nSpecifically, our work answers the key outstanding questions.\n• It predicts a new and distinct NRMT mechanism - the Lorentz skew scattering (LSK)\nmechanism, which was not known before and leads to a scaling behavior distinct from ALL\nknown mechanisms.\n• This LSK mechanism is of unique significance, in that: We proved that it has the highest\nscaling power, so it dominates the NRMT signal in high-mobility systems.\n• Specifically, we showed that LSK gives a 𝜒 ∝ 𝜎(cid:2871) scaling at low temperature, and a 𝜒 ∝ 𝜎(cid:2872)\n(cid:3051)(cid:3051) (cid:3051)(cid:3051)\nscaling at elevated temperature when phonon scattering kicks in.\nThe above conclusions are discovered here for the first time. The unveiled distinct characters of LSK\nmechanism and its connection to band geometric properties (such as Berry curvature) not only provide\nsharp signals for analyzing experimental results but also offer a new design principle to achieve\nunprecedentedly high NRMT in high-mobility systems with topological band structures. As\nexamples, our calculation for LSK NRMT in SnTe and TaP resulted huge values, which surpass all\nother mechanisms and other systems reported so far, as summarized in Table I of the main text.\nMore importantly, we are pleased to update the reviewer that our theoretical prediction has been\nconfirmed by two very recent experiments. It is also worth noting that these experiments were done\non two of the systems suggested in our paper. One is on high-mobility graphene/hBN superlattices, by\nPan He and Jian Shen’s group at Fudan University (arXiv:2511.03381). The other is on n-doped high-\nmobility BiTeBr, by Weibo Gao’s group at Nanyang Technological University (arXiv:2511.03273).\nThese works confirm the dominance of LSK mechanism in the studied high-mobility material systems.\nIn particular, in high-mobility graphene superlattices, the LSK mechanism gives rise to a record-high\ntransverse nonlinear conductivity of ~36,000μmV(cid:2879)(cid:2869)Ω(cid:2879)(cid:2869) under 𝐵 = 0.3 T, which is orders of\nmagnitude larger than ALL previously reported transverse nonlinear transport and NRMT. These\nexperimental findings confirm our theory and indicate our proposed LSK mechanism as an efficient\nroute to giant nonreciprocal magneto-transport, with promising potential for applications.\nIn addition, our work will also inspire new directions of research. For example, the proposed LSK may\nalso play leading roles in other nonlinear magneto-transport processes, such as nonreciprocal thermal\nand thermoelectric magneto-transport. It is anticipated to lead to significant B-field-tunable nonlinear\nNernst effect and nonlinear thermal Hall rectification in graphene superlattices.\nThe discussion above clearly demonstrates the unique significance and broad interest of our work. It\naddresses key outstanding questions of an effect which itself has been attracting broad interest. The\nresults have broad impact not only on fundamental knowledge but also on potential technological\napplications. In addition, its broad interest is also demonstrated by its potential in opening new\ndirections of research and by the active follow-up of experimental groups to confirm the predictions.\nBesides, the broad interest of our work was also acknowledged by the other reviewers. Particularly,\nReviewer 3 pointed out that “the manuscript presents a significant theoretical advancement in the field\nof nonreciprocal transport” and “The findings are timely and could have broad implications for both\nfundamental research and applications in rectification and diode technologies”.\nThus, we believe the discussion has fully settled the reviewer’s concern. We thank the reviewer’s\ncomments which have helped us a lot in refining the core contribution of our manuscript. In the revised\nmanuscript, we have highlighted these points in the Introduction part. And we have added a paragraph\nat the end (using Noted added) to mention the two very recent experiments which verified our\nprediction.\nThe reviewer offered several valuable comments, which we address in detail below. The manuscript is\nalso revised and improved accordingly. Please read the following.\n(1) I am a bit confused about the role of spin-orbit coupling (SOC). The manuscript claims that the\npresent LSK mechanism does not require SOC. But skew scattering fundamentally arises from\nasymmetric scattering mediated by SOC. Furthermore, in Berry-curvature-driven effect, sizable SOC\nis usually needed to open band gaps at Fermi surface and enhance the Berry curvature. Can the\nauthors clarify if and how their mechanism operates in the strict absence of any SOC? If SOC is indeed\nessential, the text should be revised accordingly.\nAnswer: We thank the reviewer for the question. The answer is: the LSK mechanism does NOT require\nSOC, as stated in our manuscript. The skew scattering process is not necessarily mediated by SOC. As\nshown in Eq. (2) of our manuscript, the skew scattering is associated with Berry curvature of electrons,\nand this Berry curvature may or may not arise from gap opening by SOC. As a well-known example,\nBerry curvature in graphene (a system with negligible SOC) can arise from gap opened by staggered\npotential rather than SOC [PRL 99, 236809 (2007)], and skew scattering also occurs. Indeed, the recent\nexperiment on graphene/hBN superlattices [Nat. Nanotechnol. 17, 378 (2022)] attributed the huge\nnonlinear transport signal to enhanced skew scattering processes associated with gap opened by\nstaggered potential from hBN substrate, not involving SOC.\nTo clarify this point, we have added the following sentence in the first paragraph of the Discussion\nsection (left column of page 5):\n“It should be noted that LSK does not require spin-orbit coupling, as both Lorentz force and skew\nscattering (and finite Berry curvature) can occur in the absence of spin-orbit coupling.”\nAnd in the revised Supplemental Information, we have added the calculation of LSK induced NRMT\nfor the graphene/hBN model in Supplementary Note 1 “LSK in gapped graphene model without spin-\norbit coupling”. The nonzero result of LSK is shown below in Fig. R2:\nFigure R2: LSK in gapped graphene model without spin-orbit coupling. a Energy spectrum of the\neffective model of graphene/hBN superlattice on one valley. b Calculated LSK conductivity 𝜒(cid:3013)(cid:3020)(cid:3012)\n(cid:3051)(cid:3051)(cid:3051)(cid:3053)\nversus chemical potential.\n(2) In Eq. (5), the mass term (𝛥) explicitly originates from SOC and opens a gap, yet in Eq. (6) it\nvanishes. If skew scattering (or Berry curvature?) relies on 𝛥, one would expect the nonreciprocal\nsignal to vanish as 𝛥 → 0. Does the nonreciprocity go to zero without a mass gap? Also, can the\nauthors clarify whether the LSK mechanism originates from skew scattering (extrinsic) or Berry\ncurvature (intrinsic)?\nAnswer: We thank the reviewer for the question. Eq. (6) gives the conductivity near the band edges as\n𝜇 ∼ Δ . Here, 𝜒(cid:3013)(cid:3020)(cid:3012) depend on Δ , but in Eq. (6), this Δ dependence is implicit. In such case, the\ndensity of state is 𝐷 ∼ Δ/2𝜋𝑣(cid:2870) and other factors in Eq. (6) are also functions of Δ. Therefore, 𝜒(cid:3013)(cid:3020)(cid:3012)\ndoes depend on Δ. And when Δ → 0, the LSK indeed vanishes, as shown in the following Fig. R3:\nFigure R3: Calculated LSK conductivity 𝜒(cid:3013)(cid:3020)(cid:3012) versus gap value in model (5) of the main text. The\n(cid:3051)(cid:3051)(cid:3051)(cid:3053)\nchemical potential is set as 𝜇 = 40 meV. The LSK conductivity vanishes when the gap is closed. When\nΔ > 𝜇, the chemical potential locates in the gap and 𝜒(cid:3013)(cid:3020)(cid:3012) also vanishes.\nIn the revised manuscript, we have added the above figure into the Supplementary Information\n(Supplementary Fig. 2).\nThe reviewer asked: “can the authors clarify whether the LSK mechanism originates from skew\nscattering (extrinsic) or Berry curvature (intrinsic)?” As shown in Eq. (2), the skew scattering and\nBerry curvature are related. We have demonstrated that skew scattering is significantly enhanced for\nelectronic states having large Berry curvatures, and this helps to enhance the LSK contribution which\ninvolves skew scattering.\n(3) While it is plausible that skew scattering induces a transverse velocity and the Lorentz force\nconverts part of this into a longitudinal component, it is not immediately obvious why this combination\nyields a second order (nonreciprocal) response. Can you provide an intuitive physical picture or simple\ntoy model illustrating how the two processes together produce a nonlinear effect.\nAnswer: We thank the reviewer for the valuable suggestion. The physical picture for LSK process is\nactually quite intuitive. The magnetic field enters through the Lorentz force, which bends a\nlongitudinally moving electron towards the transverse direction. This leads to a transverse current\nproportional to 𝐸𝐵𝜏(cid:2870) , same as the usual Hall current. Then, the skew scattering process further\ndeflects electrons in this current, i.e., converts part of the Hall current into a longitudinal flow. This\nconversion beings an additional factor of 𝐸𝜏(cid:2870)/𝜏 , as known in anomalous Hall effect. Therefore, the\n(cid:3046)(cid:3038)\ncombined LSK process gives an NRMT response scaled as 𝐸(cid:2870)𝐵𝜏(cid:2872)/𝜏 , as we discussed in the paper.\n(cid:3046)(cid:3038)\nIn the revised manuscript, we have added the above intuitive picture in the last but one paragraph of\nthe right column on page 2, as follows:\n“In the LSK process, as illustrated in Fig. 1a, the magnetic field bends the electron trajectory via\nLorentz force, leading to the usual Hall current ∝𝐸𝐵𝜏(cid:2870). Then, skew scattering further deflects this\ncurrent, with an additional factor of 𝐸𝜏(cid:2870)/𝜏 . Thus, their cooperative action gives an NRMT response\n(cid:3046)(cid:3038)\nscaled as 𝐸(cid:2870)𝐵𝜏(cid:2872)/𝜏 .”\n(cid:3046)(cid:3038)\n(4) In the Introduction, the authors list various mechanisms for nonreciprocal transport. It would help\nreaders to classify these into intrinsic versus extrinsic effects, especially for longitudinal nonreciprocity.\nAnswer: We thank the reviewer for the suggestion. In transport studies, an intrinsic effect refers to a\ncontribution completely independent of scattering and arises solely from the band structure geometry;\nwhereas an extrinsic effect arises from and depends explicitly on scattering processes. The mechanisms\nfor longitudinal NRMT reported so far, as listed in the Introduction part, are all extrinsic effects. In the\nrevised manuscript, to clarify this point, we have added the following sentence in the second paragraph\nof introduction:\n“For longitudinal response, χ involves scattering and may originate from …”\n(5) Recently, nonreciprocal transports have been very extensively investigated in superconductivity. In\nparticular, very recent work has shown giant nonreciprocal transport in an organic superconductor\nwith structural chirality [Sato, Phys. Rev. Res. 7, 023056 (2025)]. Although superconductivity is\nbeyond the current scope, it may be fair to shortly acknowledge these developments and briefly\ncontrast them with the present mechanism.\nAnswer: We thank the reviewer for the suggestion and for bringing the interesting reference Phys. Rev.\nRes. 7, 023056 (2025) into our attention. The NRMT effect in superconductors is indeed beyond the\nscope of this work, and the mechanisms in superconductors are indeed quite different from normal-\nstate systems. For example, the phenomenon observed in Phys. Rev. Res. 7, 023056 (2025) was\nattributed to the superconducting fluctuations combined with parity mixing effects.\nFollowing the reviewer’s suggestion, we have added the following discussion in the third paragraph\nof the left column on page 5:\n“We have focused on NRMT in normal state. Meanwhile, nonreciprocal transport also exists for\nsuperconductors, which may arise from very different physical origins [2,66-68]. In future studies, it\nwill be interesting to explore whether LSK of quasiparticles can also play a role in that context.”\n(6) I am just wondering that if LSK-induced nonreciprocity is as large as claimed, why has it not yet\nbeen observed in candidate materials such as graphene, BiTeBr, or Te thin flakes? Are there any\nexperimental challenges (e.g., sample quality, difficulty in tuning carrier number, or competing\nmechanisms)?\nAnswer: We thank the reviewer for the valuable question. We are pleased to update the reviewer that\nour theoretical prediction has been confirmed by two very recent experiments. It is also worth noting\nthat these experiments were done on two of the systems suggested in our paper. One is on high-mobility\ngraphene/hBN superlattices, by Pan He and Jian Shen’s group at Fudan University (arXiv:2511.03381).\nThe other is on n-doped high-mobility BiTeBr, by Weibo Gao’s group at Nanyang Technological\nUniversity (arXiv:2511.03273). These experimental findings confirm our theory and indicate our\nproposed LSK mechanism as an efficient route to giant nonreciprocal magneto-transport, with\npromising potential for applications.\nIn the revised manuscript, we have added an “Note added” to mention these two recent experimental\nworks which confirmed our theory.\nWe thank the reviewer again for valuable questions and suggestions, which have helped us a lot in\nimproving our manuscript. With our thorough reply, careful revisions, and the recent experimental\nverifications, we hope that the reviewer will now recommend the publication of our work in Nature\nCommunications.\nReviewer #3 (Remarks to the Author):\nThe manuscript by Xiao et al. presents a novel mechanism, termed Lorentz skew scattering (LSK), for\nnonreciprocal magneto-transport (NRMT) in highly conductive systems. The authors demonstrate that\nthe cooperative action of Lorentz force and skew scattering leads to distinct scaling behaviors in the\nresponse tensor, with potential dominance in clean samples and topological materials. The work is\ntheoretically rigorous and supported by numerical calculations, offering significant insights into the\ndesign principles for nonreciprocal devices. The findings are timely and could have broad implications\nfor both fundamental research and applications in rectification and diode technologies. However,\nseveral aspects of the manuscript require clarification and improvement to meet the high standards of\nNature Communications.\nAnswer: We thank the reviewer for carefully reading our manuscript and for the high evaluation on\nthe significance and broad implications of our work: “The work is theoretically rigorous and supported\nby numerical calculations, offering significant insights into the design principles for nonreciprocal\ndevices. The findings are timely and could have broad implications for both fundamental research and\napplications in rectification and diode technologies.”\nWe thank the reviewer for the valuable suggestions and comments, which have helped us a lot in\nimproving the manuscript. In the following, we fully address these suggestions and comments.\n(1) While the manuscript provides a thorough theoretical framework for LSK, the physical intuition\nbehind the mechanism could be more clearly explained. A more intuitive description of how the\ninterplay between the Lorentz force and skew scattering generates the observed scaling behaviors (e.g.,\ncubic vs. quartic) would improve accessibility for a broader audience. The inclusion of a simplified\nschematic or analogy to illustrate this cooperative effect is recommended.\nAnswer: We thank the reviewer for the valuable suggestion. The physical picture for LSK process is\nactually quite intuitive. The magnetic field enters through the Lorentz force, which bends a\nlongitudinally moving electron towards the transverse direction. This leads to a transverse current\nproportional to 𝐸𝐵𝜏(cid:2870) , same as the usual Hall current. Then, the skew scattering process deflects\nelectrons in this Hall current sideways, i.e., converts part of the Hall current into a longitudinal current\nflow. This conversion beings an additional factor of 𝐸𝜏(cid:2870)/𝜏 , as known in anomalous Hall effect.\n(cid:3046)(cid:3038)\nTherefore, the combined LSK process gives an NRMT response scaled as 𝐸(cid:2870)𝐵𝜏(cid:2872)/𝜏 , as we discussed\n(cid:3046)(cid:3038)\nin the paper.\nIn the revised manuscript, we have added the above intuitive picture in the last but one paragraph of\nthe right column on page 2, as follows:\n“In the LSK process, as illustrated in Fig. 1a, the magnetic field bends the electron trajectory via\nLorentz force, leading to the usual Hall current ∝𝐸𝐵𝜏(cid:2870). Then, skew scattering further deflects this\ncurrent, with an additional factor of 𝐸𝜏(cid:2870)/𝜏 . Thus, their cooperative action gives an NRMT response\n(cid:3046)(cid:3038)\nscaled as 𝐸(cid:2870)𝐵𝜏(cid:2872)/𝜏 .”\n(cid:3046)(cid:3038)\n(2) The theoretical predictions are compelling, but the manuscript would benefit from a more direct\ncomparison with existing experimental data. For instance, are there experimental results in the\nliterature that align with the predicted $\\chi\\propto\\sigma_{xx}^{3}$ or $\\sigma_{xx}^{4}$ scaling?\nIf not, could the authors suggest specific experiments or materials where such scaling might be\nobserved?\nAnswer: We thank the reviewer for the valuable question. We are pleased to update the reviewer that\nour predicted scaling has been confirmed by two very recent experiments on high-mobility material\nsystems. It is also worth noting that these experiments were done on two of the systems suggested in\nour paper. One is on high-mobility graphene/hBN superlattices, by Pan He and Jian Shen’s group at\nFudan University (arXiv:2511.03381). The other is on n-doped high-mobility BiTeBr, by Weibo Gao’s\ngroup at Nanyang Technological University (arXiv:2511.03273). These experimental findings confirm\nour theory and indicate our proposed LSK mechanism as an efficient route to giant nonreciprocal\nmagneto-transport, with promising potential for applications.\nIn the revised manuscript, we have added an “Note added” to mention these two recent experimental\nworks which confirmed our theory.\n(3) The manuscript highlights the dominance of LSK in highly conductive systems but does not\nadequately address potential limitations or competing mechanisms in real materials. For example,\nimpurity resonance states, which can also enhance bilinear magnetoconductivity [as reported in\nPhysical Review B 109, 155154 (2024)]), could influence the observability of LSK. Additionally, as\nproposed in Ref. [38] [Phys. Rev. B 108, L241104 (2023)], when the Fermi energy is near the edge of\nthe energy gap, where $\\epsilon_{mn}$ is small, a nonlinear conductivity\n$\\propto\\tau^{3}\\sim\\Gamma^{-3}$ can arise independently of the LSK mechanism. How can the LSK\nbe distinguished from these competing mechanisms? A discussion of these factors would provide a\nmore balanced perspective.\nAnswer: We thank the reviewer for the suggestion and for bringing the interesting reference Phys. Rev.\nB 109, 155154 (2024) into our attention. That work discussed the effects of impurity resonance states\non NRMT, which are important when the impurity scattering is strong compared to energy scale of\nband structures. The reviewer also mentioned Phys. Rev. B 108, L241104 (2023), in which the\nhighlighted contribution is important also in the strongly disordered limit where the disorder smearing\nis larger than the interband separation around Fermi level. Those cases are distinct from the context\nstudied in our work, which instead focuses on good metals (highly conductive systems), where disorder\nsmearing is much smaller than the energy scale of intrinsic band structures. For example, in gapped\ngraphene superlattices [arXiv: 2511.03381], the disorder smearing ℏ/𝜏 is only the order of 0.06 – 0.6\nmeV (𝜏 is about 1 – 10 ps), which is much smaller than the gap at the order of 10 – 20 meV. In n-\ndoped BiTeBr semiconductor [arXiv: 2511.03273], ℏ/𝜏 is about 3 – 10 meV (𝜏 is about 60 – 220 fs),\nwhich is much smaller than the relevant interband gap at the order of 300 meV. In such high-mobility\nsystems, the LSK dominated NRMT has indeed been observed by recent experiments [Refs. 71,72].\nIt is also worth noting that in our focused case of highly conductive materials, where the disorder\nsmearing is much smaller than the interband gap, the mentioned contribution in Phys. Rev. B 108,\nL241104 (2023) actually gives a 𝜏(cid:2869) scaling (Berry curvature dipole) rather than 𝜏(cid:2871) , which is\nnaturally distinguishable from the LSK contribution. More importantly, in linear order of magnetic\nfield, this contribution is time-reversal odd, thus it does not exist in nonmagnetic systems studied in\nthis paper. Therefore, that term lies outside the scope of our discussion.\nFollowing the reviewer’s suggestion, in revision, we have cited the two mentioned works (as Refs. 35\nand 39) and added the following sentence in the second paragraph of Introduction:\n“For longitudinal response, 𝜒 involves scattering and may originate from …, impurity resonant\nstates [35]”\n(4) The diagrammatic approach (Fig. 2) is innovative but somewhat dense. A step-by-step explanation\nof how the diagrams translate to the mathematical expressions (e.g., Eq. 4) would aid comprehension.\nFor example, results like $f_{B}^{\\left( 1,6\\right) }$ are not presented. Could the authors briefly\nexplain why these terms are omitted from the diagram? Are they excluded because they are deemed\nunimportant? Additionally, Fig. 2 includes many terms, yet the manuscript only discusses the\n$f_{B}^{\\left( 2,5\\right) }$ term as the LSK mechanism, stating that higher-order LSK mechanisms\ncan be neglected due to their relatively small magnitude. It is necessary to provide a more detailed\nrationale for why the other terms are not discussed.\nAnswer: We thank the reviewer for the suggestion. Following the suggestion, in the revised manuscript,\nwe have added a new section titled “Method to analyze the kinetic equation” in Methods to give step-\nby-step derivations on how our diagrammatic method works. To aid comprehension of these rules, in\nthe revised manuscript we have added a typical example illustrating how the rules work. The example\nis added in the first paragraph of the right column on page 3, and is shown as follows:\n“For example, according to rule (2) and rule (3), 𝑓((cid:2869),(cid:2869)) = 𝜏𝐼(cid:4632) 𝑓((cid:2869),(cid:2870)) = −𝜏(cid:2870)𝐼(cid:4632) 𝐷(cid:3553) 𝑓(cid:2868), which nicely\n(cid:3046)(cid:3038) (cid:3046)(cid:3038) (cid:3006)\nrecovers the result from detailed derivations (see Methods).”\n((cid:2869),(cid:2874))\nThe reviewer mentioned 𝑓 is not presented. This is because it does not exist. For the partial\n(cid:3003)\ncontributions 𝑓((cid:3036),(cid:3037)) and 𝑓 ((cid:3036),(cid:3037)) to the nonequilibrium distribution, they are solved iteratively from the\n(cid:3003)\nkinetic equation. The range of label 𝑗 (for disorder potential) depends on the value of 𝑖 (for E field\npower); it does not run up to infinity. For example, for 𝑓((cid:2869),(cid:3037)) and 𝑓 ((cid:2869),(cid:3037)) , the highest value of 𝑗 is\n(cid:3003)\n((cid:2869),(cid:2874))\n𝑗 = 2 and 𝑗 = 4, respectively. As such, 𝑓 does not exist. This is explicitly shown in the new\n(cid:3003)\nsection “Method to analyze the kinetic equation” added to the Method.\n((cid:2870),(cid:2873))\nThe reviewer asked why we only keep 𝑓 but neglect other terms in the diagram. The reason is\n(cid:3003)\nthat our main aim in this paper is to clarify the mechanism that gives the highest scaling power and\nthus dominates the NRMT response in high-mobility systems. As analyzed in the last five paragraphs\nin the section “Geometric and scaling characters of LSK” on page 2 of the main text, we already\nshowed that LSK is the mechanism we are looking for. Therefore, in the diagram, we only focus on\nthe LSK process. The LSK involves one Lorentz force action (𝐷(cid:3553) 𝐷(cid:3553) ) and one skew scattering (𝐷(cid:3553) 𝐼(cid:4632) ),\n(cid:3006) (cid:3013) (cid:3006) (cid:3046)(cid:3038)\n((cid:2870),(cid:2873))\nthus is represented by 𝑓 , as shown explicitly in the diagrammatic approach. Meanwhile, all other\n(cid:3003)\nterms are not related to LSK and are subdominant compared to LSK in high-mobility systems. Thus,\n((cid:2870),(cid:2873))\nwe only keep 𝑓 in the discussion.\n(cid:3003)\nIn addition, in the revised Supplementary Information, we have added a new section “Dominance of\nLSK over other NRMT contributions in highly conductive systems”, which presents the detailed\nanalysis of scaling behavior for all other contributions to NRMT. Systematic formulations given there\ndetail the rationale why LSK possesses the highest scaling power and is the leading contribution in\nhighly conductive materials. The added calculation results also explicitly show that the LSK\ncontribution overwhelms other contributions to NRMT in highly conductive systems (Supplementary\nFig. 3). The calculation results are shown in Fig. R1, and we paste it here for the reviewer’s\nconvenience:\nFig. R1: Dominance of LSK over other NRMT contributions in highly conductive systems. The\ncalculation is performed for the 2D model (5) in the main text. We set the same parameters as Fig.~3\nin the main text.\nTo highlight this comparison, in the main text of revised manuscript, we have added the following\ndiscussion in the first paragraph of the left column on page 3:\n“The above analysis clarifies why LSK has a higher 𝜏 (𝜎 ) scaling power than other\n(cid:3051)(cid:3051)\nmechanisms. In Supplementary Note 3, we further present a systematic scaling analysis on all NRMT\ncontributions in nonmagnetic materials, which indeed confirms this conclusion.”\nWe have also added the following discussion in the last paragraph of the right column on page 3:\n“In the analysis above, we have focused only on the LSK contribution. Other contributions to\nNRMT can be formulated in a similar way. In Supplementary Note 3, we list the results and explicitly\ndemonstrate that LSK is the leading contribution for high mobility systems.”\n(5) The use of the Boltzmann equation method to discuss nonlinear responses is a widely employed\napproach. Well-studied effects such as the Berry curvature dipole, quantum metric dipole, and second-\norder nonlinear Drude term can all be derived from this method. Among these, the\n$f^{\\left( 2,4\\right) }$ in Fig. 2 corresponds precisely to the second-order nonlinear Drude term.\nHowever, I wonder why intrinsic contributions, such as the quantum metric dipole term or other terms,\nare not visible in the diagram. What specific aspects of the Boltzmann equation formulation account\nfor this absence?\nAnswer: We thank the reviewer for this question. Figure 2 is about the nonequilibrium part of\ndistribution function, whereas the intrinsic contribution is only related to the equilibrium Fermi-Dirac\ndistribution function 𝑓 (and field corrections enters elsewhere, e.g., through the velocity), so it is not\n(cid:2868)\npresent here.\n(6) Overall, the manuscript presents a significant theoretical advancement in the field of nonreciprocal\ntransport, with potential applications in device engineering. However, the clarity of the mechanism\nand its experimental relevance need further elaboration. Once these issues are resolved, the\nmanuscript will be suitable for publication in Nature Communications.\nAnswer: We thank the reviewer for the high evaluations of our work on its novelty, significance, broad\nimpact, and potentials in applications. We also thank the reviewer for giving valuable suggestions,\nwhich help us a lot in improving our manuscript. In the revised manuscript, we have added more\ndiscussions to clarify the physical picture and the experimental relevance. As we highlighted in the\nabove reply, a good news is that our predicted LSK mechanism has been successfully observed in two\nvery recent experiments, in two of our predicted systems, i.e., in high-mobility graphene/hBN\nsuperlattices by Pan He and Jian Shen’s group at Fudan University (arXiv:2511.03381) as well as in\nn-doped high-mobility BiTeBr semiconductor by Weibo Gao’s group at Nanyang Technological\nUniversity (arXiv:2511.03273). Thus, the issues raised by the reviewer have been fully settled. We\nhope the reviewer can recommend the publication of our work in Nature Communications now.\nReply to the Reviewers\nReviewer #1 (Remarks to the Author):\nThe authors have responded to my comments sincerely and provided careful, detailed answers. They\nhave also clarified the key points of the manuscript, which helped me better understand and organize\nthe authors’ arguments. I appreciate the effort invested in preparing a thorough reply.\nNevertheless, I remain of the view that the present manuscript is not well-suited for publication in\nNature Communications. To avoid any misunderstanding, this does not imply that I question the value\nof the study itself; rather, I believe that a more specialized journal would be a more appropriate venue.\nWhile the possibility of a relaxation-time dependence—as emphasized by the authors—is potentially\nimportant in the context of nonlinear transport phenomena, the current treatment, as stated in the reply,\nis limited to a relaxation-time “scaling” argument and would require more detailed and quantitative\nanalysis. From this perspective, I consider that the discussion should be further developed in a field-\nspecific journal rather than in a broad-interest venue.\nMore technically, although the origin of skew scattering is indeed quantum mechanical, there may be\ncompetition with other quantum effects, in particular interband coherence contributions that are not\nfully captured by a Berry-curvature description. In addition, the authors argue that vertex corrections\ncan be neglected because the analysis is presently restricted to relaxation-time “scaling.” However,\nvertex corrections could in principle modify the relaxation-time dependence itself, and it may therefore\nbe risky to exclude them a priori. I recognize that incorporating such effects is highly nontrivial and\ntechnically demanding, but addressing these issues—at least in terms of a more mature and\ncomprehensive treatment—would likely be necessary for the work to reach the level of completeness\nexpected in this area.\nAnswer: We thank the reviewer for the comment.\nWe are glad to know that the reviewer acknowledged the value of our work and our effort in addressing\nhis/her previous report. The reviewer’s remaining objection is that he/she thought that our approach\n(by using kinetic equation with scaling analysis on scattering time dependence) is not as accurate\n(or complete) as compared to certain advanced approaches, like field-theoretical or Green’s function\napproaches.\nEvery approach has its pros and cons, and “what is the right approach to adopt” is determined by the\ntask to be solved. As we have elaborated in the first-round reply and in our manuscript, the task of this\nwork is to perform scaling analysis to clarify the mechanism that gives the highest scaling power for\na nonlinear transport effect. It is NOT aimed at some detailed quantitative calculations. For this purpose,\nour approach, i.e., scaling analysis based on kinetic equation and scattering time dependence, is the\nproper approach. Actually, it has been the standard one for deriving scaling relations, as successfully\ndemonstrated in many previous works on linear or nonlinear transport (e.g., Phys. Rev. Lett. 114,\n217203 (2015); Nat. Commun. 10, 3047 (2019)). We agree with the reviewer that those more advanced\napproaches may have more accurate and complete treatment, e.g., on vertex corrections. However, for\nthe task of a scaling analysis, such quantitative improvement is unnecessary.\nIt seems the reviewer dislikes the notion of relaxation time (or scattering time) and its scaling analysis.\nFor this, we point out that these notions still form the basis of experimental studies so far. For real\nmaterials, it is usually impossible to have detailed information about microscopic scattering processes.\nIn front of the complicated transport measurement data, what can we do? The experimentalists have to\nrely on the notions of scattering times and scaling relations to analyze the experimental results. And\nour treatment in this work directly conforms with the demand from experimental studies and with what\ncan be done in experiment. This is precisely the reason why our work has already attracted significant\ninterest. As we mentioned in the last reply, our prediction here has been confirmed in two independent\nexperiments (arXiv:2511.03381 by Pan He and Jian Shen’s group at Fudan University, and\narXiv:2511.03273 by Weibo Gao’s group at Nanyang Technological University).\nFinally, we emphasize that the key novelty and significance of our work lie in that we discover a new\ntransport mechanism, the Lorentz skew scattering (LSK) mechanism, for nonreciprocal magneto-\ntransport (NRMT); and we demonstrate that it has the highest scaling power, so it dominates the NRMT\nresponse in highly conductive systems. This finding unveils a crucial insight into developing high-\nefficiency nonreciprocal devices. (Note also that our proposed LSK mechanism has a clear physical\npicture, as discussed in our paper, so its existence does not depend on what theoretical approach one\nadopts for its description.) Given its significance and its impact already made in the field, we believe\nthis work meets the high standard of Nature Communications.\nReviewer #2 (Remarks to the Author):\nThe authors have adequately addressed my scientific concerns and clarified the novelty, physical\nmechanism, and experimental relevance of the proposed effect. While I initially questioned the breadth\nof interest, the identification of a dominant high-mobility regime and the recent experimental\nindications strengthen the case. That said, nonreciprocal transport itself is by now a well-established\nresearch topic, and the present work does not introduce novelty at the level of opening a new field,\nbut rather advances it by identifying a new dominant mechanism. I therefore leave the final assessment\nof suitability for Nature Communications to the editor.\nBelow I have only one small comment.\n(3) We thank the reviewer for the valuable suggestion. The physical picture for LSK process is actually\nquite intuitive. The magnetic field enters through the Lorentz force, which bends a longitudinally\nmoving electron towards the transverse direction. This leads to a transverse current proportional to\nEBtau^2, same as the usual Hall current. Then, the skew scattering process further deflects electrons\nin this current, i.e., converts part of the Hall current into a longitudinal flow. This conversion beings\nan additional factor of Etau^2/tau_sk, as known in anomalous Hall effect. Therefore, the combined\nLSK process gives an NRMT response scaled as E^2Btau^4/tau_sk, as we discussed in the paper.\n[Comment]\nThank you for giving me a nice picture. While the intuitive picture is helpful, the explanation risks\nbeing misleading, as it may suggest that a single application of the electric field Ex directly induces\nthe full LSK process. If my understanding is correct, skew scattering acts on the nonequilibrium\ndistribution rather than directly deflecting an already formed current. Clarifying this point would\navoid confusion regarding why the response is second order in E.\nAnswer: We thank the reviewer very much for his/her recommendation.\nRegarding the comment on skew scattering, we point out that skew scattering, like any other scattering\nprocesses, enters the kinetic equation from the collision integral (right hand side of kinetic equation).\nIt does not enter from the driving terms. And its consequence, like any other scattering processes, is\nmanifested as additional contributions to the nonequilibrium distribution, which in turn contributes to\nthe current flow. This theoretical formulation has nothing contradictory with the physical picture of\nskew scattering, namely, it represents the asymmetric deflection of an incoming electron towards\ntransverse directions (see e.g., Rev. Mod. Phys. 82, 1539 (2010)). To avoid possible confusion, we\nrevised the quoted sentences on page 2 as follows:\n“In the LSK process, as illustrated in Fig. 1a, the magnetic field bends the electron trajectory via\nLorentz force, leading to a transverse motion, and then skew scattering further deflects this trajectory.\nTherefore, the LSK scaling may be intuitively argued as a composition of the usual Hall response ∝\n𝐸𝐵𝜏(cid:2870) and the skew scattering ∝ 𝐸𝜏(cid:2870)/𝜏 , leading to an NRMT response scaled as 𝐸(cid:2870)𝐵𝜏(cid:2872)/𝜏 .”\n(cid:3046)(cid:3038) (cid:3046)(cid:3038)\nReviewer #3 (Remarks to the Author):\nI have reviewed the response and the revised manuscript, and I find that all concerns raised by the\nreferees have been adequately addressed. In doing so, the authors have further strengthened the\nsignificance of the work. As noted in my previous report, this work represents a significant theoretical\nadvancement in the field of nonreciprocal transport, with potential broad implications for both\nfundamental research and applications in nonreciprocal devices. Given the novelty of the idea, the\nclarity of the physical picture, and its role in bridging different frontiers of physics research, I\nrecommend that the manuscript be accepted for publication in Nature Communications.\nAnswer: We sincerely thank the reviewer for his/her high praise of the work’s significance and broad\nimpact, and for recommending its publication in Nature Communications."}]

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