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 applications1,2,3. Particularly, in nonmagnetic crystals with broken inversion symmetry, an applied magnetic field could trigger a nonreciprocal magneto-resistance linear in the B field4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. The corresponding nonreciprocal magneto-transport (NRMT) response current can be expressed as j = χE2B, with χ 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 crystals27,28,29,30,31.

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 σxx, which is proportional to the scattering time τ. Till now, several mechanisms for NRMT in nonmagnetic materials were proposed. For example, a Berry connection polarizability mechanism independent of σxx (τ) was revealed for NRMT Hall response24,32. For longitudinal response, χ involves scattering and may originate from chiral scatterer1, magnetic self-field1, Zeeman-coupling induced Fermi surface deformation5,33, energy relaxation34, impurity resonant states35, chiral anomaly36 and Berry curvature37 mechanisms in Weyl semimetals, and etc.38,39,40. 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 τ.

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 superlattices41, leading to strong frequency doubling and energy harvesting42,43, 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.

Fig. 1: Schematics of LSK.
Fig. 1: Schematics of LSK.The alternative text for this image may have been generated using AI.
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a Schematic of actions of Lorentz force and skew scattering on electron motion. b Quantum geometric character of skew scattering process, as exemplified by a Wilson loop involving three states on Fermi surface. The corresponding skew scattering rate is proportional to the Berry curvature flux through the loop. \(| {u}_{k}\rangle\) denote the eigenstates.

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∑lflvl, 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 ∝ E2B. 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 ∝ E2B. (Hence, in calculating the current, it is sufficient to take vl as the unperturbed band velocity.) This f LSK can be obtained from the Boltzmann kinetic equation:

$$({\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}+{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}){f}_{l}=({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}+{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}){f}_{l},$$
(1)

where hat denotes linear operators, \({\widehat{{{\mathcal{D}}}}}_{{{\bf{E}}}}=-\frac{e}{\hslash }{{\bf{E}}}\cdot {\partial }_{{{\bf{k}}}}\) and \({\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}=-\frac{e}{\hslash }({{{\bf{v}}}}_{l}\times {{\bf{B}}})\cdot {\partial }_{{{\bf{k}}}}\) give the electric force and Lorentz force driving terms, \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\) and \({\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\) 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 \({\omega }_{{l}^{{\prime} }l}^{a}\) is related to the Wilson loop connecting the three involved electronic states \(l,\,{l}^{{\prime} }\), and l″ (schematics in Fig. 1b): \(W(l,{l}^{{\prime} },{l}^{{\prime\prime} })=\left\langle {u}_{l}| | {u}_{{l}^{{\prime} }}\right\rangle \left\langle {u}_{{l}^{{\prime} }}| | {u}_{{l}^{{\prime\prime} }}\right\rangle \left\langle {u}_{{l}^{{\prime\prime} }}| | {u}_{l}\right\rangle,\) 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

$${{\rm{Im}}}\,W(l,{l}^{{\prime} },{l}^{{\prime\prime} })\approx \frac{1}{2}({{{\bf{k}}}}^{{\prime\prime} }-{{{\bf{k}}}}^{{\prime} })\times ({{{\bf{k}}}}^{{\prime} }-{{\bf{k}}})\cdot {{{\boldsymbol{\Omega }}}}_{l},$$
(2)

which is proportional to the Berry curvature Ω. It follows that for an isotropic model with smooth disorder potential, \({\omega }_{{l}^{{\prime} }l}^{a}\propto {{{\bf{k}}}}^{{\prime} }\cdot ({{\bf{k}}}\times {{{\boldsymbol{\Omega }}}}_{l})\), explicitly showing the skewness of scattering, i.e., an incident electron with momentum k tends to be scattered to the transverse direction k × Ωl. 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 \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}} \sim 1/\tau\) and \({\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}} \sim 1/{\tau }_{{{\rm{sk}}}}\), where the time scale of skew scattering τsk should be much larger than τ41,42,46,47.

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 E2B. Let us analyze how the E and B field factors bring in the scaling parameters. First of all, the scaling parameter τ comes from the out-of-equilibrium distribution δf, arising from the combined action of scattering and E-field driving. To have the highest scaling power in τ, one should look for skew scattering: an E-field factor gives a δf ∝ τ for conventional scattering; but with skew scattering, it leads to a contribution ∝ τ2/τsk, which is the highest, a fact well known from previous studies on anomalous Hall effect and nonlinear Hall effects41,42,47,48,49. 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 states50, then it cannot bring additional τ 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 \({({\tau }^{2}/{\tau }_{{{\rm{sk}}}})}^{2}\). (ii) An even higher contribution occurs if B enters via Lorentz force. Combined with a factor of E, they together bring a τ2 factor, which just corresponds to the ordinary Hall conductivity ∝ τ2 51. Then, combined with one skew scattering for the remaining E factor, we have a result ∝ τ4/τsk, 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 ∝ EBτ2 and the skew scattering ∝ Eτ2/τsk, leading to an NRMT response scaled as E2Bτ4/τsk.

The above analysis clarifies why LSK has a higher τ (σxx) 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 approximations44,49,52 (Methods) and expand the distribution function as

$$f={f}^{0}+{\sum }_{i,j}\left[{f}^{(i,j)}+{f}_{B}^{(i,j)}\right].$$
(3)

Here, f 0 is the equilibrium Fermi-Dirac distribution. In the off-equilibrium part, we explicitly separate out the components fB 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 ∝ E iV−j. In this notation, we have \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}={\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}^{(0,-2)}\) and\({\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}={\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}^{(0,-3)}\). Figure 2 illustrates the diagrammatical way to construct each f (i, j). The rules are: (1) Each node is a component of distribution function, and the construction starts from f 0; (2) An arrow with label O pointing from node A to B means f B has a contribution from f A acted by the operator \(\widehat{O}\), and the degrees of E, B, V must be balanced between f B and \(\widehat{O}{f}^{A}\); (3) Here, we have three types of arrow labels with the correspondence: \(\,{{\rm{E}}}\,\to -\tau {\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\), \(\,{{\rm{L}}}\,\to -\tau {\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\), and \({{\rm{sk}}}\to \tau {\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\). (4) The component at a node is obtained by summing all contributions associated with arrows pointing to it. In addition, there is no arrow from f 0 with L or sk label, since they cannot produce nonequilibrium distribution without E. Following these rules, one can readily obtain any desired component f (i, j). For example, according to rule (2) and rule (3), \({f}^{(1,1)}=\tau {\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}^{(1,2)}=-{\tau }^{2}{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{0}\), which nicely recovers the result from detailed derivations (see Methods).

Fig. 2: Diagrammatic approach to solve the kinetic equation. Each node denotes a component f (i, j), and each arrow denotes an operation (rules are described in the text).
Fig. 2: Diagrammatic approach to solve the kinetic equation. Each node denotes a component f (i, j), and each arrow denotes an operation (rules are described in the text).The alternative text for this image may have been generated using AI.
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Nodes in each column share the same dependence on τ. From left to right, the components are ∝ τ0, τ1, τ2, τ3 and τ4, respectively. The components relevant to LSK are highlighted.

Our target is to solve f LSK, which is at the second order of electric field and involves one Lorentz force action (\({\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\)) and one skew scattering (\({\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\)). According to the diagrammatic approach, we identify it as \({f}_{B}^{(2,5)}\) in Fig. 2. Its expression can thus be read off from the diagram as

$$\begin{array}{l}{f}^{{{\rm{LSK}}}}={f}_{B}^{(2,5)}=-{\tau }^{4}\left[{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\{{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}},{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\}+{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\{{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}},{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\}+{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\{{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}},{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\}\right]{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{0}.\end{array}$$
(4)

Here, {. , . } is the anticommutator of two operators. Combined with the band velocity, it gives the LSK response current in NRMT: \({{{\bf{j}}}}^{{{\rm{LSK}}}}=-e{\sum }_{l}\,{f}_{l}^{{{\rm{LSK}}}}{{{\bf{v}}}}_{l}\), from which the response tensor χLSK can be extracted (\(\,\,{j}_{a}^{{{\rm{LSK}}}}={\chi }_{abcd}^{{{\rm{LSK}}}}{E}_{b}{E}_{c}{B}_{d}\), where summation over repeated Cartesian indices is implied).

From Eq. (4), the scaling behavior f LSK, χ LSK ∝ τ4/τsk is consistent with our previous analysis. However, to discuss the scaling of χ LSK with σxx, we have to distinguish two regimes. In the low temperature regime where impurity scattering dominates, σxx is usually varied by fabricating samples with varying impurity density ni. For example, in Refs. 53,54, this is done by making metal films with different thicknesses such that the effective ni from surface scattering is varied. Since both τ and τsk are ∝ 1/ni, we expect for such cases, \({\chi }^{{{\rm{LSK}}}}\propto {\sigma }_{xx}^{3}\). On the other hand, at elevated temperatures where phonon scattering is substantial, τ (and σxx) is usually varied by temperature, due to phonon scattering. Meanwhile, phonons do not contribute to skew scattering53,55,56, so τsk is still from impurity scattering. Therefore, one should observe \({\chi }^{{{\rm{LSK}}}}\propto {\sigma }_{xx}^{4}\). These scaling behaviors are distinct from all previous mechanisms for NRMT.

In the analysis above, we have focused only on the LSK contribution. Other contributions to NRMT can be formulated in a similar way. In Supplementary Note 3, we list the results and explicitly demonstrate that LSK is the leading contribution for high mobility systems.

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 τ = ± labels two Dirac valleys connected by time reversal operation \({{\mathcal{T}}}\), and σ’s are Pauli matrices. This model describes the topological surface states of SnTe57 and Pb1−xSnxTe(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 χLSK components for both longitudinal and transverse NRMT can reach a similar order of magnitude, with (Supplementary Note 2)

$$\left|{\chi }^{{{\rm{LSK}}}}\right| \sim \frac{{e}^{4}w}{{\pi }^{2}{\hslash }^{5}D}\frac{{\tau }^{4}}{{\tau }_{{{\rm{sk}}}}},$$
(6)

where D is the density of states. The results from numerical calculations (using parameters of SnTe57,59) 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 (χ Z)33, which are found to be much smaller than the LSK mechanism.

Fig. 3: LSK in surface states of TCI.
Fig. 3: LSK in surface states of TCI.The alternative text for this image may have been generated using AI.
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a Dispersion of a 2D gapped Dirac valley in (5). b Calculated LSK nonlinear conductivities versus chemical potential for this model. For comparison, the dashed lines (χ Z's) show the contribution from nonlinear Drude mechanism with Fermi surface deformation by Zeeman coupling to magnetic moment. Here, we take parameters relevant to SnTe, with vx/ℏ = vy/ℏ = 4 × 105m/s, Δ = 10meV, w/v = 0.1, ni = 1010 cm−2, and averaged disorder strength V0 = 10−13 eV ⋅ cm2.

The nonreciprocity is often characterized by the coefficient η = δσ/σ = − δρ/ρ, measuring the change in conductivity (resistivity) when the current direction is reversed. Here, we find η from LSK can reach  ~ 20% under B = 1 T and E = 104 V/m. This value is orders of magnitude larger than several previous results of NRMT in 2D electron gas under similar field strengths7,8,10. Another figure of merit is the nonreciprocal coefficient γ = − η/IB, where I is the driving current2. For a sample width of 1 μm, we estimate γ here can reach  ~ 105 A−1T−1, which is very large, considering that most reported γ values are below 103 A−1T−1 1,4,5,7,8,9,10.

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 conditions60. In a Weyl semimetal, the low-energy physics is from states around Weyl points60. A generic model for a Weyl point can be written as

$$H=w{k}_{z}+v{{\bf{k}}}\cdot {{\boldsymbol{\sigma }}},$$
(7)

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

We perform numerical calculation for this Weyl model using parameters typical of Weyl semimetal materials (such as TaP family61). Figure 4a illustrates the obtained f LSK distribution. For bulk materials, one usually characterizes NRMT using an intrinsic coefficient \(\gamma {\prime}=\gamma A=-\chi /{\sigma }_{xx}^{2}\), where A is the cross sectional area of the sample5,27,36. Figure 4b shows the result. One finds that \(\gamma {\prime}\) can reach 3 × 10−6 m2A−1T−1 for μ = 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 proposed36. This demonstrates LSK could dominate the NRMT response in Weyl semimetals.

Fig. 4: LSK response for the Weyl model in Eq. (7).
Fig. 4: LSK response for the Weyl model in Eq. (7).The alternative text for this image may have been generated using AI.
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a Patterns of f LSK on the Fermi surface in ky = 0 plane, for E and B fields applied in x direction and μ = 10 meV. b Calculated nonreciprocal coefficient \(\gamma {\prime}\) versus chemical potential. The inset shows the obtained current responsivity. In the calculation, we take B = 0.1T, v = 4 × 105m/s, w/v = 0.4, ni = 1015cm−3, and V0 = 10−19 eV ⋅ cm3.

Discussion

The proposed LSK mechanism for NRMT is significant because it manifests quantum geometry of band structure (Berry curvature on Fermi surface) and is dominant in highly conductive samples (possessing the highest scaling power in the Drude conductivity). The comparisons of LSK and previously reported NRMT are presented in Table 1. One sees that LSK can surpass the other known mechanisms by orders of magnitude. It should be noted that LSK does not require spin-orbit coupling, as both Lorentz force and skew scattering (and finite Berry curvature) can occur in the absence of spin-orbit coupling (see Supplementary Note 1 for a model illustration). As we noted, materials with topological band features around Fermi level, such as topological semimetals, should be suitable systems to study LSK. Recently, signals of strong skew scattering effects in nonlinear Hall measurement were reported in several systems, such as graphene superlattices42,43, BiTeBr62, and Te thin flakes63. They could be promising platforms to explore LSK response as well.

Table 1 Comparison of reported nonreciprocal coefficients γ and \(\gamma {\prime}\) with different mechanisms and materials
Full size table

The LSK mechanism is not limited to electrical transport but should also provide the leading contribution to other nonlinear magneto-transport processes, such as nonreciprocal thermal and thermoelectric magneto-transport. In particular, it may play a significant role in thermal rectification64,65, which is an important direction of research.

We have focused on NRMT in normal state. Meanwhile, nonreciprocal transport also exists for superconductors, which may arise from very different physical origins2,66,67,68. In future studies, it will be interesting to explore whether LSK of quasiparticles can also play a role in that context.

Finally, the LSK induced NRMT is suitable for rectifier or detector applications, since such devices require high mobility materials, which could reduce power consumption and heat dissipation. An important metric for rectification applications is the current responsivity \({{\mathcal{R}}}={j}_{dc}/P\), which is the ratio of the output dc current to the power dissipation P69. For the Weyl semimetal case, we estimate that \({{\mathcal{R}}}\) due to LSK may reach  ~ 66 A/mW at μ = 5 meV, for B = 0.1 T and a device size of 1 μm, as shown in the inset of Fig. 4b. This value is already orders of magnitude larger than other reported rectification mechanisms69,70. All these suggest that rectification based on LSK indeed holds potential for practical applications.

Very recently, our predicted LSK contribution was successfully observed in experiments on two of our suggested candidate material systems, i.e., n-doped high-mobility BiTeBr71 and high-mobility graphene/hBN superlattice72. These works confirm our theory and indicate LSK mechanism as an efficient route to giant nonlinear and non-reciprocal transport, with promising potential for applications.

Methods

Skew scattering in kinetic equation

For a homogeneous system, the Boltzmann kinetic equation at steady state reads:

$${{{\bf{k}}}}^{\cdot }\cdot \frac{\partial {f}_{l}}{\partial {{\bf{k}}}}=\widehat{{{\mathcal{I}}}}\left\{{f}_{l}\right\}.$$
(8)

Under applied uniform electric and magnetic fields, \({{{\bf{k}}}}^{\cdot }\), the force on the electron wave packet, contains two parts \({\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\) and \({\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\), corresponding respectively to the electric force and the Lorentz force73. For the LSK contribution to NRMT, the magnetic field enters via the Lorentz force \({\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\), and other B-related terms in the kinetic equation are not relevant.

The skew scattering effect is contained in the collision integral \(\widehat{{{\mathcal{I}}}}\). \(\widehat{{{\mathcal{I}}}}\) is a linear operator acting on the distribution function: \(\widehat{{{\mathcal{I}}}}\{\,{f}_{l}\}=-{\sum }_{l{\prime} }({\omega }_{l{\prime} l}\,{f}_{l}-{\omega }_{ll{\prime}\, }{f}_{l{\prime} })\), where \({\omega }_{l{\prime} l}\) is the scattering rate from l to \(l{\prime}\). 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: \({\omega }_{l{\prime} l}^{s}=({\omega }_{l{\prime} l}+{\omega }_{ll{\prime} })/2\) and \({\omega }_{l{\prime} l}^{a}=({\omega }_{l{\prime} l}-{\omega }_{ll{\prime} })/2\). The two parts are associated respectively with the conventional collision integral \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}_{l}=-{\sum }_{l{\prime} }{\omega }_{l{\prime} l}^{s}(\,\,{f}_{l}-{f}_{l{\prime} })\) and the skew-scattering collision integral \({\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}_{l}=-{\sum }_{l{\prime} }{\omega }_{l{\prime} l}^{a}(\,\,{f}_{l}+{f}_{l{\prime} })\)44. Here, we are considering the weak disorder regime. Then, the symmetric part ωs is dominated by processes of second order in scattering potential V, i.e., \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\propto {V}^{2}\). Meanwhile, the leading order (skew scattering) process for the anti-symmetric part ωa is of third order. For example, for quasi-elastic scattering with spin-independent scattering potential, we have

$${\omega }_{l{\prime} l}^{a}\approx \frac{4{\pi }^{2}}{\hslash }{n}_{i}{\sum }_{l{\prime\prime} }{\langle {V}_{kk{\prime\prime} }{V}_{k{\prime\prime} k{\prime} }{V}_{k{\prime} k}\rangle }_{c}\delta ({\varepsilon }_{l{\prime} }-{\varepsilon }_{l})\delta ({\varepsilon }_{l{\prime\prime} }-{\varepsilon }_{l})\\ \times {{\rm{Im}}}W(l,l{\prime},l{\prime\prime} ),$$
(9)

where ni is disorder density, \({V}_{kk{\prime} }\) is the Fourier component of disorder potential, εl is the wave packet energy, and \(W(l,l{\prime},l{\prime\prime} )\) 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

$${\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}^{(1,2)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{0},\,{\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}_{B}^{(1,4)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\,{f}^{(1,2)},$$
(10)

and the remaining equations share common forms of

$$\begin{array}{rcl}{\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}^{(1,j)}=-{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}^{(1,j+1)} & & \,(\,\,j < 2),\\ {\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}_{B}^{(1,j)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\,{f}^{(1,j-2)}-{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}_{B}^{(1,j+1)} & & \,(\,\,j < 4).\end{array}$$
(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 theory51,52. 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 E2 (i = 2) order. These equations include

$${\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}^{(2,4)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{(1,2)},\,{\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}_{B}^{(2,6)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}_{B}^{(1,4)}+{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\,{f}^{(2,4)},$$
(12)

and the remaining equations share the common forms of

$$\begin{array}{rcl}{\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}^{(2,j)} &=& {\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{(1,j-2)}-{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}^{(2,j+1)}\,(\,\,j < 4),\\ {\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}\,{f}_{B}^{(2,j)}={\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}_{B}^{(1,j-2)} &+& {\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\,{f}^{(2,j-2)}-{\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}\,{f}_{B}^{(2,j+1)}\,(\,\,j < 6).\end{array}$$
(13)

Note that in our notation, one has \({\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}={\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}^{(0,-2)}\) and \({\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}={\widehat{{{\mathcal{I}}}}}_{{{\rm{sk}}}}^{(0,-3)}\). 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, \({f}^{(1,2)}={\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}^{-1}{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{0} \sim \tau {\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}^{0}\) is the familiar one responsible for Drude conductivity, \({f}_{B}^{(1,4)}={\widehat{{{\mathcal{I}}}}}_{{{\rm{c}}}}^{-1}{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}\,{f}^{(1,2)} \sim {\tau }^{2}{\widehat{{{\mathcal{D}}}}}_{{{\rm{L}}}}{\widehat{{{\mathcal{D}}}}}_{{{\rm{E}}}}\,{f}_{0}\), 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.