Introduction

No matter how exotic, a metal follows the linear current (I)-voltage (V) relationship, i.e., Ohm’s law. Ohm’s law serves as a universal law of metal and holds because all metals possess the Fermi energy, dominating any other energy scales1,2. Recently, Ohm’s law was challenged in the Weyl metallic Bi1-xSbx (x = 3–4 %) alloy and ZrTe5, which evokes that the nontrivial band topology of a Weyl metallic state leads to electrical responses beyond the linear response theory3,4. The origin underlying this violation of Ohm’s law is the topology-induced nonlinear current, proportional to E3. This violation of Ohm’s law is a unique example in which the nonlinearity is unveiled in the response function of a metal, particularly in the weak-field regime.

Nonlinearity can also manifest in dynamic responses. In a weak field regime, a conventional metal is a linear system, meaning that its electromagnetic response is linearly proportional to the external perturbation. This linearity implies that the frequency of the time-dependent response should be the same as that of the perturbation. For example, if a sinusoidal current is applied to a conventional metal, the electrical field will change sinusoidally with the same frequency. However, this is not true for exotic metals with nonlinear characteristics5,6,7. The response of such an exotic metal can have the fundamental harmonic component with the same frequency as the perturbation and other harmonic components, even time-independent ones. These nonlinear responses are known as rectification and high harmonic generation. Rectification converts alternating current (AC), which periodically reverses direction, to direct current (DC), which flows in one direction. Rectification is known to require inversion symmetry breaking by definition8. High harmonic generation is a process where a sinusoidal electric field with frequency nω is generated when a sinusoidal electric field with frequency ω drives a system9. Here, n is an integer greater than 1. To date, a bulk metal that exhibits both rectification and high harmonic generation has not been discovered if the bulk material respects inversion symmetry. While bulk rectification has been rarely observed in semiconductors without inversion symmetry8,10,11,12,13,14, high harmonic generation usually occurs under intense electromagnetic fields in an insulator with large anharmonicity15,16,17,18, which allows for simultaneous absorption of multiple photons with the same frequency by bound electrons. In contrast, rectification in phononic systems, such as granular crystals, arises from mechanical asymmetries19. While mesoscopic rectification mechanisms rely on quantum coherence and asymmetric scattering, which are absent in macroscopic systems. These differences highlight the need to explore alternative mechanisms for rectification beyond structural asymmetry and quantum coherence.

One potential candidate of the nonlinear metal is the Weyl metallic Bi1-xSbx (x = 3–4%) alloy20,21, which occurs in the parallel electric field E and magnetic field B. The Weyl state is characterized by the separation of Weyl nodes under these fields, leading to a chiral anomaly that fundamentally alters the electromagnetic response. This anomaly manifests as a nonlinear current-voltage (I-V) curve, experimentally confirmed in previous studies22,23. The violation of Ohm’s law in this system is due to its nontrivial band topology3,4,24,25, which modifies the law of electromagnetism. Coulomb’s law and Faraday’s law of induction are revised with the θ term representing the distance between two nodal points in the band structure. As a result, the electromagnetic wave equations become nonlinear26,27,28. Interestingly, the equations governing axion electrodynamics, a consequence of the chiral anomaly, are mathematically similar to the Navier-Stokes equations for fluid flow. Actually, some similarity exists between the hydrodynamics of fluids and electromagnetic responses of the Weyl metal29,30,31,32,33. As introduced in the hydrodynamics, Reynolds number Re can be defined within the Weyl metal’s electromagnetism framework. See the supplementary text [Section 9]. Re can help predict “flow patterns” of the electric field; at low Re, flows are dominated by laminar, and at high Re, the flow tends to be turbulent. In the Weyl metal’s electromagnetism framework, we obtain \({{\mathrm{Re}}}=Ew(4\alpha g\mu )\), where E is the magnitude of the electric field, w is the width of the sample, α is the fine structure constant, g is the Lande g factor, and μ is the permeability. We emphasize that the Re does not depend on electrical conductivity. This implies that such nonlinear phenomena may be ubiquitous in Weyl metals, where the nonlinear term arises due to the chiral anomaly. In the case of Bi1-xSbx (x = 3–4%), Re is estimated to be quite large, larger than 2000 in the calculation based on the values of the material parameters and sample dimensions. This estimation suggests the possible existence of nonlinear electromagnetic responses in Bi1-xSbx (x = 3–4%). In this system, the high effective Reynolds number can result in “turbulent” electric field patterns, similar to the complex, non-uniform patterns observed in turbulent fluids. This analogy implies that, just as turbulence breaks symmetries in fluids, applied currents in our system might lead to symmetry breaking away from equilibrium.

The nonlinear current caused by the chiral anomaly and proportional to E3, was measured experimentally and derived from the Boltzmann equation4,34,35. Introducing this anomaly-driven nonlinear current into the Maxwell equations, we obtain the so-called van der Pol nonlinear differential equation36,37,38, which describes the dynamic evolution of observed electric fields as a response to a sinusoidal current in Bi1-xSbx (x = 3–4%). The chiral anomaly enters through the conduction term Janom ∝E3 (or more explicitly, σ3E3), where σ3 includes material-specific parameters and anomaly-related factors, embedding its contribution within the nonlinear damping term of the equation. One notable feature of this equation is the appearance of a single characteristic spatial length scale l, which is not present in conventional metals. This harmonic potential term, associated with this length scale l and commonly found in insulators, is crucial in predicting the nonequilibrium steady state where externally driving electric currents make an inhomogeneous pattern. As the van der Pol equation is nonlinear, we anticipate several different bifurcation phenomena, including symmetry-breaking bifurcation, period-doubling bifurcation, and more, before the system eventually reaches a chaotic state25,26,27.

Indeed, when the amplitude of the driving sinusoidal current increases across a critical amplitude Ic, we uncover the change of electric-field dynamics from a relatively simple steady-state with electrical high harmonic generation to a dynamic phase with rectification in addition to the electrical high harmonic generation. Surprisingly, this steady state above Ic exhibits not only symmetry-allowed odd harmonic signals but also symmetry-forbidden rectification, implying the dynamic breaking of inversion symmetry. In the language of complex nonlinear systems, this dynamic phase transition is a symmetry-breaking bifurcation39,40,41, a precursor to a period-doubling bifurcation that leads to chaos42. In a symmetry-breaking bifurcation, the orbit can spontaneously become asymmetric even though the potential energy is symmetric. Our observed nonlinear dynamic phase corresponds to an intermediate symmetry-breaking bifurcated state above the critical current Ic.

Bi1-xSbx crystallizes in a rhombohedral lattice with space group \({{{\rm{R}}}}\bar{3}m\) and belongs to a centrosymmetric structure. Bi1-xSbx, at the topological critical point (xc ≈ 3–4%) between a band “insulator” (x < xc) and a topological insulator (x > xc) is an archetypal three-dimensional (3D) Dirac metal with linear band dispersion and topological Dirac nodes20,25,43,44,45,46. Nontrivial band topology is revealed in this system when time-reversal symmetry is broken. Upon applying an external magnetic field B, finely tuned Bi1-xSbx (x ≈ xc) undergoes a topological phase transition into a Weyl metal with a fundamental change of the band topology20,21,25,43,47. The Dirac bands are split into pairs of the Weyl bands with opposite monopole charges (chirality) along the direction of B. The Weyl metallic state in Bi1-xSbx (x ≈ xc) is robust against the residual energy gap \({E}_{g}\) because a large Lande g factor of ~100 (18, 22) ensures crossing of the Zeeman-split bands even at low B if \(g{\mu }_{B}B\, > \,{E}_{g}\)(4), where \({\mu }_{B}\) is the Bohr magneton. A Weyl metallic state realized with parallel E and B features the negative longitudinal magnetoresistance (LMR) and the violation of Ohm’s law, stemming from the chiral anomaly3,4,23,43,48,49,50,51. Owing to the easy manipulation of the splitting direction of the Weyl nodes by the external B, a transport experiment sensitive to that direction could be performed more readily in Bi1-xSbx (x ≈ xc) than in any other Weyl metals with broken inversion symmetry50,52,53 in which the splitting direction is fixed by symmetry. In Weyl semimetals without inversion symmetry, the nonlinear Hall effect arises due to the interplay between tilted Weyl cones and the anomalous velocity, both influenced by the chiral anomaly. The tilted cones, which describe the energy dispersion near the Weyl nodes, lead to a modified Hall effect distinct from the conventional one seen in other systems. However, it is important to explore the potential emergence of phenomena when inversion symmetry is preserved, as this could lead to unique transport effects that differ from those in Weyl semimetals with broken inversion symmetry or tilted Weyl cones.

Results and discussion

In the previous study, the I-V characteristic curves for E // B were nonlinear, while they were linear in other E and B configurations, and the nonlinearity was more pronounced with increasing B3,4. The nonlinear conductivity Δσ evaluated from the I-V curves was proportional to E2. This nonlinearity was attributed to the higher-order current component resulting from the chiral anomaly. In a steady state with parallel E and B, a finite chiral chemical potential \({\mu }_{5}\) develops4. This is due to the nonconservation of the particle number in a given chirality. The conservation law is recovered when two opposite chiralities are simultaneously considered. The particle number conservation across a pair of the Weyl bands with different chirality creates a quantum mechanical correlation between charge carriers in different Weyl bands (Fig. 1a). As the Boltzmann transport theory predicts, such correlation with the chiral chemical potential gives rise to higher-order current components.

Fig. 1: Signature of nonlinear transport phenomena in direct current (DC) measurement for longitudinal magnetoresistance configuration.
figure 1

a Schematic diagram for measuring longitudinal magnetotransport and charge pumping effect between two Weyl bands with opposite chirality for B // E, showing development of Fermi energy imbalance. b Magnetic field dependence of longitudinal resistivity \({\rho }_{{{{\rm{LMR}}}}}(B)\) was measured in the Bi1-xSbx (x~3–4 %) single crystal at various temperatures, revealing negative longitudinal magnetoresistance. c The nonlinearity of current density J against electric field E was measured at T = 1.7 K and various magnetic fields B. d The nonlinearity of J against E was determined by subtracting a linear slope in different magnetic fields. e The nonlinear conductance curves \(\Delta \sigma\) show a quadratic dependence on electric field, with greater nonlinearity at higher B. f The scaling curves of the \(\Delta \sigma (E,B)\) data taken at T = 1.7 K collapse into a single universal curve and are found to be well described by the relationship \({\Delta }{\sigma }_{L}={b}_{2}(B)\cdot {E}^{2}\) with \({b}_{2}(B)={C}^{(2)}{B}^{2}+{C}^{(1)}{B}^{4}\). This result is in good agreement with the Boltzmann transport theory previously reported4,35.

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Before investigating rectification and high harmonic generations, we reproduced the violation of Ohm’s law in our new Bi1-xSbx (x ≈ xc) crystals. As reported before3,4, the longitudinal magnetoresistivity depicted in Fig. 1b suggests that the I-V curves only exhibit nonlinear when E and B are parallel (Fig. 1c). Nonlinearity is more clearly visible in the \(\Delta\, J\) plots, in which \(\Delta \,J\) is the linear-component-subtracted current density (Fig. 1d). To confirm that these phenomena are intrinsic to the samples, we ruled out external factors, as detailed previously4. We also ensured that measurements on single crystals with homogeneous current flow excluded the possibility of current jetting, confirming that the observed features originate intrinsically from the chiral anomaly in the Weyl metallic phase and are not influenced by geometrical artifacts. The current in the E // B configuration is proportional to E3, giving rise to the nonlinear conductivity \(\Delta {\sigma }_{L}\) proportional to E2, i.e., \(\Delta {\sigma }_{L}={b}_{2}(B){E}^{2}\), where \({b}_{2}(B)\) is a B-dependent coefficient. Indeed, \(\Delta {\sigma }_{L}\) follows the relation of \(\Delta {\sigma }_{L}={b}_{2}(B){E}^{2}\) at a given B (Fig. 1e). We calculated \(\Delta {\sigma }_{L}\) from the measured I-V curves at T = 1.7 K and different B. We found that \({b}_{2}(B)\) satisfies the formula of \({C}^{(2)}{B}^{2}+{C}^{(1)}{B}^{4}\) with \({C}^{(2)}=2.7\times {10}^{4}{\Omega }^{\_1}{{{{\rm{T}}}}}^{-2}{{{{\rm{V}}}}}^{-2}{{{\rm{m}}}}\) and \({C}^{(1)}=3.0\times {10}^{2}{\Omega }^{\_1}{{{{\rm{T}}}}}^{-4}{{{{\rm{V}}}}}^{-2}{{{\rm{m}}}}\), as predicted by the Boltzmann transport theory (Fig. 1f). The additional topological current proportional to E3 is the origin of the nonlinear I-V curves and a primary factor that leads to nonlinear dynamics, which we will discuss later.

To search for dynamic phases in the current-driven state, which might result in rectification and high harmonic signals, we conducted AC electrical transport experiments (Fig. 2a). Using the lock-in technique and a DC nanovoltmeter, we measured the longitudinal Vxx in the channel for n = 1, 2, and 3 [\({V}_{xx}^{n\omega }\)] and for n = 0 [\({V}_{xx}^{DC}\)], respectively, while applying a sinusoidal current \(I(t)={{\mathrm{Re}}}[{I}_{1\omega }{e}^{i(\omega t+\theta )}]\) in different E and B configurations. By varying the amplitude I1ω of the applied current, we probed the \({V}_{xx}^{n\omega }\) signals (n = 1, 2, and 3) and \({V}_{xx}^{DC}\) at fixed T and B. We convert \({V}_{xx}^{n\omega }\) to \({E}_{xx}^{n\omega }\), the nth harmonic component of the electric field and \({V}_{xx}^{DC}\) to \({E}_{xx}^{DC}\), the DC electric field, using an electrode spacing of 2.13 mm and a cross sectional area of 2.27 × 10−6 m2 of our measured sample. We conducted the same experiments for the transverse Vxy.

Fig. 2: Non-linear responses in alternating current (AC).
figure 2

a The diagram illustrates the longitudinal configuration with parallel magnetic field B and electric field E (B // E). Sinusoidal current was applied, and the amplitude of the direct current DC component and harmonics were measured with a nanovoltmeter and a lock-in technique. b The DC electric field \({E}_{xx}^{DC}\) and harmonic electric fields \({E}_{xx}^{n\omega }\) (n = 1, 2, and 3) as a function of current density J1ω at T = 1.7 K and B = 9 T. While \({E}_{xx}^{DC}\) and \({E}_{xx}^{n\omega }\) (n = 1 and 3) are finite, \({E}_{xx}^{n\omega }\) (n = 2) is zero. In a system with broken inversion symmetry, the second harmonic signal typically coexist with the rectification signal. c, d The \({E}_{xx}^{DC}({E}_{xx}^{3\omega })\) vs. J1ω curves at different B and T = 1.7 K. e, f The \({E}_{xx}^{DC}({E}_{xx}^{3\omega })\) as a function of J1ω at different frequency at T = 1.7 K and B = 9 T. \({E}_{xx}^{DC}({E}_{xx}^{3\omega })\) is independent of the frequency of the applied current. g The field dependence of β3ω(B) [βDC(B)] extracted from the slope of the linear fitting in \({E}_{xx}^{DC}({E}_{xx}^{3\omega })\) vs. (J1ω)3. The red (blue) curve is a fit using \({{{{\rm{\beta }}}}}_{3\omega ({{{\rm{DC}}}})}={{{\rm{a}}}}\cdot {B}^{2}+{{{\rm{b}}}}\cdot {B}^{4}\), showing good agreement with the experiments. h The ratio of β3ω(B) to βDC(B) is ~2.5, regardless of the B.

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Figure 2(b) presents \({E}_{xx}^{n\omega }({J}_{\omega })\) (n = 1, 2, and 3) and \({E}_{xx}^{DC}({J}_{\omega })\) measured at B = 9 T and T = 1.7 K, where \({J}_{\omega }\) is the amplitude of the driving current density. As expected, upon the increase of \({J}_{\omega }\), \({E}_{xx}^{1\omega }\) increases linearly. Surprisingly, nonzero signals are detected for the third harmonic component \({E}_{xx}^{3\omega }({J}_{\omega })\) and DC component \({E}_{xx}^{DC}({J}_{\omega })\). \({E}_{xx}^{3\omega }\) is the third harmonic signal, which is unexpected in a linear system. On the other hand, \({E}_{xx}^{DC}\) is a DC electric component. The existence of the \({E}_{xx}^{DC}\) signal implies that the present system behaves as a (nonlinear) rectifier that converts the AC to DC signals. This is unusual because a metal rectifying AC to DC has never been reported before, as far as we know. Bulk rectification can rarely occur only in a system with broken inversion symmetry8,10,11,12, which is not the case for Bi1-xSbx (x ≈ xc). Thus, the rectification that is not allowed by symmetry cannot be understood within any existing theory.

\({J}_{\omega }\) dependences of \({E}_{xx}^{DC}\) (see Fig. 2c, e) and \({E}_{xx}^{3\omega }\) (see Fig. 2d, f) are also quite notable. At large \({J}_{1\omega }\), both \({E}_{xx}^{DC}\) and \({E}_{xx}^{3\omega }\) follow the \({J}_{\omega }^{3}\) dependence i.e., \({E}_{xx}^{DC(3\omega )}={\beta }^{DC(3\omega )}\cdot {({J}_{\omega })}^{3}\), where \({\beta }^{DC(3\omega )}\) is a proportionality coefficient. These coefficients increase with increasing B (Fig. 2d), following the formula of \({\beta }^{DC(3\omega )}(B)={a}_{2}^{DC(3\omega )}\cdot {B}^{2}+{a}_{4}^{DC(3\omega )}\cdot {B}^{4}\), where \({a}_{2}^{DC(3\omega )}\) and \({a}_{4}^{DC(3\omega )}\) are the coefficients of the quadratic and quartic terms, respectively. It is notable that \({\beta }^{DC(3\omega )}\) and \({b}_{2}(B)\) have the same B dependence. Interestingly, the ratio \({\beta }_{3\omega }/{\beta }_{DC}\) is found to be ~2.5 (Fig. 2h), irrespective of the values of the applied magnetic field B.

The \({J}_{\omega }^{3}\) relation, however, does not persist at small \({J}_{\omega }\)(Fig. 3c). Careful examinations of \({E}_{xx}^{DC}\) and \({E}_{xx}^{3\omega }\) at low \({J}_{\omega }\) reveal that the \({E}_{xx}^{DC}\) signal is zero below a critical current density \({J}_{c}\), while no such critical current density \({J}_{c}\) exists in \({E}_{xx}^{3\omega }\) as clearly seen in Fig. 3b, and Supplementary Fig. S5, and Fig. S6. The \({J}_{c}\) values of \({E}_{xx}^{DC}\) are more readily observable when \({E}_{xx}^{DC}\) is plotted as a function of \({J}_{\omega }^{3}\) (Fig. 3b). The presence of \({J}_{c}\) suggests a fundamental change of dynamics in Bi1-xSbx (x ≈ xc) when the amplitude I1ω of the sinusoidal driving current increases across the critical values. Additionally, we found that \({E}_{xx}^{DC}\) and \({E}_{xx}^{3\omega }\) do not depend on frequency in a narrow frequency range as demonstrated in Fig. 2f, g, respectively. We extended our experiments up to 2000 Hz and found no significant frequency dependence in the nonlinear response. These findings support the robustness of the observed nonlinear dynamics. We also confirm that the nonlinear I-V curves, rectification, and electrical third-harmonic generation require the condition of E // B in Bi1-xSbx (x ≈ xc) as shown in the supplementary text [Section 1]. This strongly indicates that chiral anomaly plays a significant role in the emergence of these phenomena.

Fig. 3: Phase diagram and scaling of electric field data with different current density J and magnetic field B.
figure 3

a A phase diagram has been constructed in the current density J1ω-magnetic field B plane using rectification measurements. This diagram distinguishes between two regions: region 1 and region 2. While region 1 only shows 3rd harmonic generation, region 2 displays both rectification and 3rd harmonic generation. b The scaling curves of DC electric field \({E}_{xx}^{DC}(J,B)\) data taken at T = 1.7 K are presented. All the curves at different current densities and magnetic fields collapse into a single universal curve. c The enlarged \({E}_{xx}^{DC}\) as a function of J at small J in different magnetic fields at T = 1.7 K. This data shows a clear transition from region 1, where the response is nonlinear with no rectification but has an electrical third harmonic signal, to region 2, where both occur. d The electric field \({E}_{xx}^{DC}\) and \({E}_{xx}^{3\omega }\) as a function of phase θ of the driving current at I = 13 mA shows sinusoidal variation. The phase of the driving current determines the direction of the DC electric field. Sensitivity to initial conditions is a property of a nonlinear system.

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From the \({J}_{c}\) values at different B, we can construct a \({J}_{1\omega }-B\) phase diagram at T = 1.7 K as depicted in Fig. 3a. Similarly, from the \({J}_{c}\) values at different T, we also construct a \({J}_{\omega }-T\) phase diagram at B = 9 T. See the supplementary text [Section 7]. The \({J}_{c}\) line in the \({J}_{\omega }\)-B plane divides region 2 from region 1. While the response is nonlinear without rectification in region 1, it is nonlinear with rectification in region 2. The topological-in-origin rectification is the hallmark of region 2. Interestingly, \({J}_{c}\) is reduced drastically as B increases. To understand the nature of this dynamic phase transition, we apply scaling analysis to the \({E}_{xx}^{DC}-J\) relations at different B. Similarly, we apply a scaling analysis to the \({E}_{xx}^{DC}-J\) relations at different T in the supplementary text [Section 7] (Fig. S8a). In the \({E}_{xx}^{DC}-J\) relations at different B, we look for a single universal curve that satisfies the following relation:

$${E}_{xx}^{DC}(\delta J,\delta B)\propto (\delta B)^{-\frac{d}{{\nu }_{B}}}E_{xx}^{DC}\left(\frac{\delta J}{{(\delta B)}^{\frac{{\nu }_{J}}{{\nu }_{B}}}},1\right),$$
(1)

where \(\delta J\) and \(\delta B\) are \(\delta J\) = J – Jc and \(\delta B\) = B – Bc, respectively. Here, d is the dimension of the system, and νB and νJ are the critical exponents. This gives a scaling relation between \({(\delta B)}^{\frac{d}{{\nu }_{B}}}{E}_{xx}^{DC}(\delta J,\delta B)\) and \({E}_{xx}^{DC}\left(\frac{\delta J}{{(\delta B)}^{\frac{{\nu }_{J}}{{\nu }_{B}}}},1\right)\) when suitable d, νB, and νJ are selected. The optimal values of \(a\equiv \frac{d}{{\nu }_{{{{\rm{B}}}}}}\) = 1.4, \(b\equiv \frac{{\nu }_{{{{\rm{J}}}}}}{{\nu }_{{{{\rm{B}}}}}}\) = 2.3, and Bc ≈ 0 T yield good scaling, as presented in (Fig. 3c). As explained in the supplementary text [Section 7], we determined these values based on the minimum standard deviations. When d is equal to 3, the values of νB and νJ are 2.1 and 4.8, respectively. The existence of the scaling function implies that the present dynamic phase transition is a critical phenomenon governed by the divergence of a correlation time, or equivalently, critical slowdown.

To describe the temporal evolution of the internal electric field in the Weyl metallic state, we incorporate a current formula that includes the higher-order term induced by the chiral anomaly into the Maxwell equation. Here, we take into account the current formula from our DC current-voltage experiments, where all the magnetic-field dependent coefficients are determined. This empirical current formula finds its justification in the Boltzmann transport theory, as previously stated. Given the external sinusoidal current drive with a frequency of ω, we consider a van der Pol nonlinear differential equation to represent the dynamics of the observed electric field along the current (z) direction:

$${\ddot{E}}_{z}+({\sigma }_{1}+3{\sigma }_{3}{E}_{z}^{2}){\dot{E}}_{z}+\frac{1}{{{l}}^{2}}{E}_{z}=\omega {J}_{\omega }\,\cos (\omega t)$$
(2)

\({\sigma }_{1}\) and \({\sigma }_{3}\) are the linear and cubic coefficients for the current equation, \(J={\sigma }_{1}{E}_{z}+{\sigma }_{3}{E}_{z}^{3}\), respectively, given by experiments (Fig. 1). A characteristic length scale \({l}\) is introduced to represent an inhomogeneous electric field pattern, which will be further elaborated below. \({J}_{\omega }\) is an amplitude of the driving current density. The inversion symmetry does not allow \({E}_{z}^{2}\) (even power) contributions to be present. Equation (2) with a negative linear damping term is known as the van der Pol equation. Here, we point out that the sign of this damping term is flipped. While the van der Pol equation qualitatively captures key nonlinear features such as symmetry-breaking bifurcations and rectification, it has limitations in fully reproducing experimental scaling exponents and describing the phase diagram quantitatively. These discrepancies arise due to simplifying assumptions, such as neglecting higher-order fluctuations and material-specific corrections, but the model remains a useful framework for understanding the nonlinear dynamics in Weyl semimetals.

To figure out essential physics encoded into this van der Pol equation, we first consider the \({\sigma }_{3}\to 0\) and \({l}\,\)\(\to \infty\) limit. This reproduces the electric-field equation for conventional metals. The \({\sigma }_{3}\to 0\) condition corresponds to the linear response regime while the \(l\to \infty\) limit considers a uniform configuration of the electric field since \({l}^{-2}{E}_{z}\) results from \(-{\vec{\nabla }}^{2}{E}_{z}\). In this respect the electric field is assumed to be inhomogeneous in the transverse direction, where the modulation length is given by \(l\). The physical origin of this characteristic length scale is the potential instability of the uniform electric field toward its inhomogeneous configuration in Weyl metals. Actually, the inhomogeneous electric-field pattern can be easily derived in the axion electrodynamics as long as the background electric field is larger than its critical value54,55,56. However, this physics has never been observed in experiments. It is quite difficult to determine the critical value of the applied current density because we do not perform self-consistent calculations for the dynamics of both Weyl electrons and electric fields. In this respect we would like to point out that the introduction of the harmonic potential term is our essential ingredient. Indeed, the absence of this insulating or mass term does not allow the rectification.

The sign-modified van der Pol Eq. (2) has an inversion symmetry under the transformation \({{{{\rm{E}}}}}_{{{{\rm{z}}}}}{\to -{{{\rm{E}}}}}_{z}\) with \({{{\rm{t}}}}\to {{{\rm{t}}}}+{{{\rm{\pi }}}}/{{{\rm{\omega }}}}\). This implies that it only allows odd harmonic solutions. However, if the external driving force \({J}_{\omega }\,\) exceeds a certain critical value \({J}_{c}\), this is no longer true. By considering a small deviation of an even harmonic component η(t) from the odd harmonic solution, one can derive a homogeneous ordinary second-order differential equation for η(t). This is known as the Whittaker-Hill differential equation57,58. The Fourier coefficients of η(t) follow a homogeneous recursion relation determined by Fourier analysis. In order to obtain a nontrivial solution in this homogeneous algebraic equation, we need the determinant in the matrix form of the recursion equation to vanish. To achieve the DC electric field in the long-time limit, we enforce a special condition for an eigenvalue or the characteristic exponent of the ansatz solution. The vanishing determinant condition, along with the characteristic exponent, leads to a phase diagram for the rectification in the two-dimensional plane of the applied magnetic field and the driving electric current density. Further analysis details are available in the supplementary text [Section 10].

For later convenience, we use the expression of \({{{\rm{E}}}}\left({{{\rm{z}}}}\right)={by}\left(z\right)\), where \(b\,\ll \,1\) is a scaling parameter. In addition, we replace t with z = wt. Then, Eq. (2) reads \(\ddot{y}+(\alpha+3\beta {y}^{2})\,\dot{y}+\gamma y=M\,\cos (z)\), where \(\alpha=\frac{{\sigma }_{1}}{\omega }\), \(\beta=\frac{{\sigma }_{3}{b}^{2}}{\omega }\), \(\gamma={(\frac{1}{\omega l})}^{2}\), and \(M=\frac{{J}_{0}}{b\omega }\). Our theoretical phase diagram confirms that the mass term in the van der Pol equation plays a central role in the emergence of the rectification in addition to the nonlinear current damping term. Given \({{{{\rm{\sigma }}}}}_{1}\) and \({J}_{1{{{\rm{\omega }}}}}\), (Fig. 4a) shows that both \({{{{\rm{\sigma }}}}}_{3}\) and \(l\), transformed into \(\beta\) and \(\gamma\) in the theory, respectively, should be finite (on the curve) in order to have the rectification. Figure 4b, given by the vanishing determinant condition discussed above and in the supplementary text [Section 10], exhibits a three-dimensional phase diagram for the appearance of the rectification. Here, the x-axis represents the applied magnetic field, which determines the values of \({{{{\rm{\sigma }}}}}_{1}(B)\) and \({{{{\rm{\sigma }}}}}_{3}(B)\), and the y-axis expresses the applied current density \({J}_{{{{\rm{\omega }}}}}\). The z-axis denotes the characteristic length scale \(l\), translated into \({{{\rm{\gamma }}}}\) in the theory. The three parameters effectively summarize the essential physics observed in experiments. The two-dimensional surface defines the critical surface for rectification to occur. To gain further insights, we consider a horizontal cut, where the modulation length scale \(l\) in the transverse plane is fixed. The inset figure shows a two-dimensional phase diagram in the plane of the magnetic field and current density for various transverse modulation scales, exhibiting a reasonable match with the experimental phase diagram (Fig. 3a).

Fig. 4: Theoretical phase diagram and the DC electric-field component as a function of the current density.
figure 4

a The theoretical conditions for rectification in Weyl metals are that the β and γ parameters must be finite on the curve, given the linear conductivity σ1 and applied current density J0. b The rectification region is shown in the three-dimensional phase diagram as a function of the magnetic field, current density, and characteristic length scale. The inset figure shows a two-dimensional phase diagram in the magnetic field and current density plane with different transverse modulation scales. c Using the variational iteration method, an approximate DC electric field component is obtained from the van der Pol equation. This component depends on the applied current density and external magnetic fields. d The theoretical phase diagram for rectification is shown. The red points on the diagram represent the critical current density determined using the variational iteration method, while the blue curve shows the solution to the vanishing determinant condition in the linearized homogeneous recursion equation. Both methods agree well. e The scaling analysis for 8 different theoretical curves. f Parametric three-dimensional plots in the y(z) and y′(z) plane in a given J0 and B = 3 T. Here, the critical current density is Jc = 4343 (A/m2). (i) In the region where J is smaller than Jc, the thin tube region that consists of symmetric orbits of the right-hand side is expanded by a factor of 103. (ii) In the region where J is greater than Jc, strong nonlinearity with a complex orbit occurs.

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Unfortunately, we could not determine the DC electric field from the previous linear analysis, where all the coefficients of higher even harmonics can be expressed by the DC coefficient from the linearized homogeneous equation. The nonlinear term beyond the Whittaker-Hill differential equation inevitably determines the DC coefficient of the system in the steady state, which was not considered in the previous determinant analysis. This difficulty can be overcome because there is a nice variational iteration formula, which takes into account the previously neglected nonlinear term and thus, determines the DC component in an iterative way. Based on this protocol, we obtained an approximate formula of the DC electric-field component from the van der Pol equation directly and explicitly. See the supplementary text [Section 10] for the details. Figure 4(c) displays the DC electric-field component as a function of the applied current density \({J}_{1{{{\rm{\omega }}}}}\) in given external magnetic fields. Compared to the measurements given in Figs. 2b, c, e, and 3c, it seems that there exist slight mismatches in the \({J}_{1{{{\rm{\omega }}}}}\) dependence in the vicinity of the critical current density. Actually, we could not find such a nice scaling theory shown in Fig. 3b from our theoretical solutions. This point will be discussed in more details below. Here, we emphasize that the DC component of the electric field is an approximate solution given by the first iteration only. However, we point out that this approximate solution may be regarded as a reasonable answer, justified by (Fig. 4d). In Fig. 4d, the blue curve denotes the critical line for the rectification given by the vanishing determinant condition, which corresponds to the horizontal cut in Fig. 4b. On the other hand, the red dots are obtained by the condition that the DC electric-field component vanishes, given by the variational iteration formula. These two different methods show a reasonable match.

We also confirm that the amplitudes of even harmonics are vanishingly small compared to the DC amplitude, where all the even harmonics components are expressed by the DC or zeroth component in the linearized homogeneous recursion formula. See the supplementary text [Section 10] for more details. This result is consistent with the experimental observation that the even harmonics are undetected within the resolution of the experiments.

Even though all the experimental observations cannot be fully explained, some of the key features are understood based on this modified van der Pol equation. First, the third harmonic signal naturally arises in the van der Pol equation because of the nonlinear term while inter-band transitions as an optical process are not involved due to the low frequency of the driving field. Second, the rectification and the existence of the critical \({J}_{c}\) can also be explained based on the symmetry-breaking bifurcation. It has been shown that a series of bifurcations emerge in complex nonlinear systems before the system eventually reaches a chaotic state when one of the control parameters, such as the amplitude of the external driving force is varied. However, while the van der Pol model theoretically permits chaos, real-world constraints including sample heating, increased noise, and material degradation confine the system to a quasi-periodic or periodically bifurcating regime, preventing a full transition to chaos. In certain parameter ranges, a linearly damped and externally driven van der Pol oscillator responds asymmetrically even though “the potential energy” is a single symmetric well. This dynamic symmetry breaking is a precursor of the period-doubling transition toward chaos39,40,42. The symmetry-breaking bifurcation implies that the time average of \({{{{\rm{E}}}}}_{{{{\rm{z}}}}}(t)\) can yield a nonzero value. As a property of a nonlinear complex system, symmetry-breaking bifurcation is a pathway for rectification that can occur in an inversion-symmetric system. The present result is in strong contrast with a conventional mechanism that requires inversion symmetry breaking as a necessary element for rectification. See the supplementary text [Section 10] for this symmetry-breaking bifurcation phenomenon, where the evolution of the electric-field dynamics was presented in the phase space. To gain insights into the dynamics near the chaotic state, we utilized a bifurcation-like diagram to analyze the system’s evolution based on experimental parameters as shown in Fig. 4f. Our findings indicate that the parameter values leading to chaos differ significantly from those used in our experiments, which are unattainable due to factors like heating. For further details on this symmetry-breaking bifurcation phenomenon and the evolution of the electric-field dynamics, see supplementary text [Section 10].

One might criticize that the present theoretical framework does not take into account electronic screening. Given a fairly large conductivity of the sample, charge perturbations, in particular those due to underlying conductivity and applied B-field, are essentially negligible in a metal. Electronic screening refers to the modification of the dielectric function by Weyl electrons’ polarization bubbles through their charge-fluctuation effects within the Maxwell equations. This involves integrating out Weyl electrons at finite chemical potential to derive the effective dynamics of electromagnetic fields. This screening effect must be reflected in both the linear and nonlinear conductivity of Weyl metals. In this study, we used the experimental values very near the ones obtained by fitting the data for \({\sigma }_{1}=7.5\times {10}^{3}\,(1+0.026{B}^{2})\) and for \({\sigma }_{3}=2.7\times {10}^{4}\,{B}^{2}+3\times {10}^{2}\,{B}^{4}\) in SI units, respectively, and set ω to 2000 Hz for our analysis. As derived in Section 10 of the supplementary material, we would like to emphasize that the electric field equation is identical to the static form of the Navier-Stokes equation. Here, the nonlinear term originates from the axion electrodynamics. Moreover, we point out that the Reynolds number Re is given by Re = Ew(4αgμ), indicating it is independent of conductivity. This implies that nonuniform and inhomogeneous patterns of the electric field can manifest regardless of conductivity values. Consequently, even after accounting for the screening effect, these patterns can emerge in Weyl metals. The same phenomenology observed in Bi1-xSbx (x ≈ xc) was also identified in ZrTe5, suggesting the universality of these phenomena.

There are several experimental aspects that do not seem to align with the van der Pol equation. First, we could not find the scaling function measured from experiments in our theoretical analysis. It is possible that this discrepancy between our experiments and theoretical predictions may be due to the fact that certain types of fluctuations have been overlooked in the van der Pol equation. It is worth noting that the dynamic equation of the internal electric field in the presence of the chiral anomaly is similar to the Navier-Stokes equation. This equation suggests that the effective Reynolds number is high enough to allow for spatial inhomogeneities such as turbulence, indicating that inhomogeneous patterns of electric fields cannot be ignored. We may need to incorporate the effects of the spatial pattern of electric fields into the van der Pol equation more explicitly, beyond the introduction of the characteristic length scale \(l\).

Although the bulk material itself retains inversion symmetry, the spontaneous generation of an inhomogeneous electric field with a characteristic length scale of \(l\) breaks this symmetry. This spatial asymmetry is essential for the observed DC voltage generation. This phenomenon can be termed selection-rule violating nonlinear transport as a dynamical phase transition. Figure 5a illustrates an inversion-symmetry-broken current distribution, which can generate a directional flow (DC component). A promising avenue for future research would be multi-terminal measurements to probe the local electric field configuration.

Fig. 5: Transverse DC and harmonic responses for Bi1-xSbx (x=3 – 4 %) in the longitudinal configuration.
figure 5

a Schematic diagram showing a possible current flow pattern, which generates a directional flow (DC component) from an oscillating input (AC). b The DC electric field \({E}_{xy}^{DC}\) and harmonic electric fields \({E}_{xy}^{n\omega }\) (n = 1, 2 and 3) as a function of current density J1ω at T = 1.7 K and B = 9 T for the longitudinal configuration with parallel magnetic field B and electric field E (B // E). While \({E}_{xy}^{DC}\) and \({E}_{xy}^{2\omega }\) are finite, \({E}_{xy}^{\omega }\) and \({E}_{xy}^{3\omega }\) are zero. c A phase diagram constructed in the current density J1ω-magnetic field B plane using \({E}_{xx}^{DC}\), \({E}_{xy}^{DC}\) and \({E}_{xy}^{2\omega }\) measurements at different magnetic field. d, e The enlarged \({E}_{xy}^{DC}\) and \({E}_{xy}^{2\omega }\) as a function of J at small J in different magnetic fields at T = 1.7 K, respectively.

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While a direct observation of the inhomogeneous current distribution might be elusive at present, we obtained crucial supporting evidence. We present an experimental result that strengthens the reliability and significance of our study. Excitation of the sample with AC current (frequency ω) under parallel B and E fields (where Lorentz force is zero) should yield a null Hall signal. However, we detected non-zero DC and 2ω voltage signals as shown in Fig. 5b, while the ω-linear Hall signal does not exist as expected, similar to the symmetry-forbidden signals observed in the longitudinal case. Notably, these signals appear only above a critical AC current amplitude, coinciding with the critical values determined from longitudinal measurements (Fig. 5c). Figure 5d, e show enlarged \({E}_{xy}^{DC}\) and \({E}_{xy}^{2\omega }\) as a function of J at small J in different magnetic fields at T = 1.7 K, respectively. These “Hall” signals provide compelling evidence for spontaneous inversion symmetry-breaking in the current distribution as a dynamical phase transition.

Finite momentum instabilities in axion electrodynamics have been explored in various contexts, encompassing both weakly correlated and strongly coupled systems. For weakly correlated systems, refer to H. Ooguri and M. Oshikawa’s study59. For strongly correlated systems, see M. Baggioli, K.-Y. Kim, L. Li, and W.-J. Li’s work60. Notably, the axion term, even when static, induces mixing among the x, y, and z components of the electric field, potentially leading to a negative eigenvalue at finite momentum if the axion term reaches its critical strength. We hypothesize that applied currents may “renormalize” the axion term, as there is no known non-renormalization theorem for this axion term, especially outside equilibrium.

It would be interesting to conduct three-dimensional “hydrodynamic” simulations to explicitly demonstrate the electric field configuration, particularly the “vortex” current flow pattern. Currently, achieving this nonlinear metallic regime, which simultaneously explains both nonlinear Hall responses and nonlinear longitudinal responses, is challenging with these three-dimensional “hydrodynamic” simulations.

Our system reveals the complex behavior of dynamical systems, especially as they evolve through symmetry-breaking bifurcations. This discovery hints at the possibility of observing chaos or turbulence in solid-state systems when fields and dimensions are manipulated, phenomena usually seen in nonlinear dynamical systems. Our research opens a new chapter on electron transport, showing unprecedented behavior under highly nonlinear and nonequilibrium conditions. Additionally, finding symmetry-forbidden rectification in a macroscopically inversion-symmetric system is both scientifically exciting and promising for technology. Optoelectronic devices relying on mechanisms intimately related to nontrivial band topology are expected to occur.

We emphasize that rectification and third harmonic generation in our experiments have no correlation with heating effects or any other extrinsic factors. To rule out heating effects, we attached a thermometer to monitor the sample temperature in real-time during current sweeps and confirmed that it remained stable within experimental errors. We also measured the same sample with different contact distances and observed negligible heat effects, providing further evidence of their absence. See Supplementary text [Section 3] for details. This lack of heating is likely due to the low contact resistance of the sample. Additionally, to verify that the observed effects do not originate from the measurement instruments, we performed the same AC experiments on an as-grown intrinsic Bi2Te3 sample. In any E and B configuration, \({E}_{xx}^{DC}\) and \({E}_{xx}^{3\omega }\) were zero, confirming that the observed effects are intrinsic to Bi1-xSbx (x ~ 3–4 %).

In conclusion, in the Weyl metallic Bi1-xSbx (x ~ 3–4 %), we have observed a dynamic phase transition from a steady state with a “trivial” nonlinear response to a dynamical phase with rectification. In the dynamical phase, the rectification, which has been believed to be impossible to exist in metal with inversion symmetry, arises as a nonlinear response. As the chiral anomaly makes the differential equation describing the time evolution of the internal electric field nonlinear, the symmetry-breaking bifurcation, which is often observed in nonlinear complex systems, occurs as this dynamic phase transition. The present results will offer a solid foundation for further understanding of nonlinear electromagnetic responses in topological materials, raising fundamental questions about the possible mechanisms for rectification and high harmonic generation. A horizon will also be open on the application side by facilitating the development of optoelectronic devices relying on mechanisms intimately related to nontrivial band topology.

Methods

Sample synthesis and characterization

The single crystals of Bi1-xSbx with nominal x = 3–4 % were synthesized by the Bridgeman method3,4,18. The small pieces of crystals with rectangular shapes were cut from a large ingot. Energy dispersive x-ray spectroscopy (EDX) measurements were performed on the selected crystals. Six contacts with the Hall bar pattern were made for the electrical transport measurements on the sample with dimensions of 6.84 mm  × 2.88 mm × 0.79 mm. Six contacts with the Hall bar pattern were made for the electrical transport measurements. Along with the transverse magnetoresistance (TMR) and Hall resistance, the longitudinal magnetoresistance (LMR) was measured at T = 1.7 K up to B = ± 9 T to confirm the negative LMR. To obtain the material parameters such as carrier concentrations and mobilities, fitting based on the two-carrier model was carried out simultaneously for Hall resistance and TMR. Parameter sets that are very similar to the previous sample were acquired in the present samples.

Transport measurements

The electrical transport was performed in a cryogen-free magnet system (Cryogenic, Inc.). DC measurements were conducted using a current source (Keithley 220) and a nano voltmeter (Agilent 34420 A). The IV curves were measured by the conventional DC method using the current source and the nanovoltmeter. High harmonic and rectification signals in the longitudinal channel (Vxx) and Hall channel (Vxy) were collected using a nanovoltmeter and two lock-in amplifiers (Stanford Research Systems SR830 and NF Corporation LI 5640), respectively with excitation frequencies between 1 Hz and 2000 Hz. To supply a constant current to the samples, a voltage-to-current converter (Stanford Research Systems CS580) was used. Different frequencies were tested to investigate the frequency dependence but there is no frequency dependence in the present frequency range. The frequencies that we used satisfy \(\omega \tau \to 0\) and \(\hslash \omega\, \ll \,{E}_{F}\), where \(\tau\) is the relaxation time and \({E}_{F}\) is the Fermi energy. These conditions ensure the applicability of the semi-classical theory to the experimental results. In all the experiments, we carefully monitored the sample temperature to confirm its stability. When a sample heating is induced, sudden slope change is detected in the IV curves at high currents. We limited the currents below these values. All the presented results are, thus, free from the heating effect. We unambiguously rule out the heating effect as the origin for topological rectification and electrical high harmonic generation.