# Axionic charge-density wave in the Weyl semimetal (TaSe4)2I

### Subjects

An Author Correction to this article was published on 15 May 2020

## Abstract

An axion insulator is a correlated topological phase, which is predicted to arise from the formation of a charge-density wave in a Weyl semimetal1,2—that is, a material in which electrons behave as massless chiral fermions. The accompanying sliding mode in the charge-density-wave phase—the phason—is an axion3,4 and is expected to cause anomalous magnetoelectric transport effects. However, this axionic charge-density wave has not yet been experimentally detected. Here we report the observation of a large positive contribution to the magnetoconductance in the sliding mode of the charge-density-wave Weyl semimetal (TaSe4)2I for collinear electric and magnetic fields. The positive contribution to the magnetoconductance originates from the anomalous axionic contribution of the chiral anomaly to the phason current, and is locked to the parallel alignment of the electric and magnetic fields. By rotating the magnetic field, we show that the angular dependence of the magnetoconductance is consistent with the anomalous transport of an axionic charge-density wave. Our results show that it is possible to find experimental evidence for axions in strongly correlated topological condensed matter systems, which have so far been elusive in any other context.

## Main

Axions are elementary particles that have long been known in quantum field theory3,4, but have not yet been observed in nature. However, it has been recently understood that axions can emerge as collective electronic excitations in certain crystals, the so-called axion insulators5. Despite being fully gapped to single-particle excitations in the bulk and at the surface, an axion insulator is characterized by a quantum mechanical equation of motion, which includes a topological θE·B term, where E and B are the electric and magnetic fields inside the insulator and θ plays the role of the dynamical axion field. Physically, the average value of θ is determined by the microscopic details of the band structure of the system, and gives rise to unusual magneto-electric response properties, such as quantum anomalous Hall conductivities6,7,8,9, the quantized circular photo-galvanic6,10,11 effect and the chiral magnetic effect6,12,13,14. The prospect of realizing an axion insulator has inspired much theoretical and experimental work. Very recently, signatures of a dynamic axion field have been found on the surface of magnetically doped topological insulator thin films15,16,17. However, the axionic quasiparticle in these systems—the axionic polariton5—has so far eluded experimental detection. Alternatively, axion insulators have been predicted to arise in Weyl semimetals that are unstable towards the formation of a charge-density wave (CDW)1,2,18,19,20,21.

In their parent state, Weyl semimetals are materials in which low-energy electronic quasiparticles behave as chiral relativistic fermions without rest mass, known as Weyl fermions22,23,24. Weyl fermions exist at isolated crossing points of the conductance and valence bands—the so-called Weyl nodes—and their energy can be approximated with a linear dispersion relation (Fig. 1a). The Weyl nodes always occur in pairs of opposite ‘handedness’ or chirality. At low energies and in the absence of interactions, the chirality is a conserved quantum number and the two chiral populations do not mix. Parallel electric and magnetic fields (EB), however, enable a steady flow of quasiparticles between the left- and right-handed nodes25. This induced breaking of the chiral symmetry is a macroscopic manifestation of a quantum anomaly in relativistic field theory and gives rise to a positive longitudinal magnetoconductance in these systems. While the single-particle states in Weyl systems have been well studied experimentally, many-body interaction effects remain largely unexplored. By turning on strong interactions, a CDW that links the two Weyl nodes and gaps the Weyl fermions can be induced in Weyl semimetals.

A CDW is the energetically preferred ground state of the strongly coupled electron–phonon system in certain quasi-one-dimensional conductors at low temperatures26. It is characterized by a gap in the single-particle excitation spectrum and by a gapless collective mode formed by electron–hole pairs. The density of the electrons and the position of the lattice atoms are periodically modulated with a period larger than the original lattice constant (Fig. 1a). The phase ϕ of this condensate is of fundamental importance for experiments: its time derivative is related to the electric current density (jcdw ≈ ∂ϕ/∂t) carried by the collective mode upon application of an electric field26. Owing to its relation to the phase, this current-carrying collective mode is called the phason. In most cases, the wavevectors of the CDW are incommensurate with the lattice, and the CDW is pinned to impurities. Therefore, only upon applying a certain threshold electric field Eth (above which the electric force overcomes the pinning forces) is the CDW depinned and free to ‘slide’ over the lattice27,28, thereby contributing to the electrical conduction. The resulting conduction behaviour is strongly nonlinear and the electrical resistivity drops with increasing electric field.

In a Weyl semimetal, the CDW can couple electrons and holes with different chirality—that is, it generates a complex mass in the Dirac equation, which mixes the two chiral populations of the system even in zero magnetic field1,2,18,19,20. The resulting gapped state is the axion insulator and the phase of the CDW is identified as the axion field (ϕθ). The phason is the axionic mode of this system and its dynamics is therefore described by the topological θE·B term. Compared with its high-energy version, this condensed matter realization of the axion has the advantage of being accessible in magnetoelectric transport experiments.

The pairing of electrons and holes of opposite chirality and the corresponding gapping of the Weyl cones induced by the CDW imply that the chiral anomaly is irrelevant for the single-particle magnetoelectric transport in the axion insulator state. We find, however, that the axionic phason mode leads to a positive longitudinal contribution to the magnetoconductance. This connection can be understood through a calculation based on a phenomenological effective-field theory for electrical conductivity (Methods). Applying EB to the Weyl–CDW system leads to an extra anomalous contribution from the chiral anomaly to the phason current, which can be identified as a signature of the axionic character of the CDW. Aligned E and B fields generate an additional collective chiral flow of charges with a rate that is proportional to E·B. The calculation yields a positive magnetoconductance contribution of σCDW,xx(B) = c1 + c2aχB + c3 (aχB)2 in the axion insulator state, where c1, c2 and c3 are material-specific parameters, aχ = 1/(4π2) is the chiral anomaly coefficient and B denotes the component of B that is parallel to E. We note that, owing to the threshold voltage caused by the impurity pinning in real material systems, c1, c2 and c3—and therefore σCDW,xx(B)—can in general depend on V (Methods). In the low-field limit, where several Landau levels are occupied, the conductance is given by the quadratic term, recovering the characteristic, chiral-anomaly-induced, positive longitudinal magnetoconductance of a non-interacting Weyl semimetal. In particular, in this limit we obtain a Bcos2ϕ dependence of the magnetoconductance on the angle ϕ between E and B. In the high-field limit, the magnetoconductance behaves linearly with the magnetic field, similarly to the ultra-quantum limit of a Weyl semimetal. This is the fingerprint of the axion, which allows us to probe the presence of the axionic phason mode through its effect on the electrical transport in a Weyl–CDW condensed matter system.

Recently, a positive magnetoconductance has been observed29 in the CDW phase of the predicted Weyl semimetal Y2Ir2O7 because of a reduced de-pinning threshold voltage in longitudinal magnetic fields. However, the phason current was found to be independent of B. To obtain evidence for the axionic phason, it is therefore desirable to go beyond these experiments.

In our study, we use the quasi-one-dimensional material (TaSe4)2I (Extended Data Fig. 1a), which has recently been shown to be a Weyl semimetal30 in its non-distorted structure (Fig. 1b). Its Fermi surface is entirely derived from Weyl cones with 24 pairs of Weyl nodes of opposite chirality close to the intrinsic Fermi level, EF (10 meV < E − EF < 15 meV). At temperatures below the transition temperature of Tc = 263 K, (TaSe4)2I forms an incommensurate CDW phase, undergoing a Peierls-like transition31, which is accompanied by the opening of a gap of approximately 260 meV to single-particle excitations. In a recent study, the momentum (k) space positions of the electronic susceptibility peaks, calculated from the Fermi pockets of the Weyl points, and the k = (222) satellite reflections experimentally observed in XRD measurements of the CDW phase of (TaSe4)2I have been found to match, indicating that the CDW nests the Weyl nodes and therefore breaks the chiral symmetry30. In addition, (TaSe4)2I has been previously shown to exhibit nonlinear electrical-transport characteristics32,33 in zero magnetic field upon crossing a threshold voltage Vth, which are associated with a current-carrying sliding phason. Therefore, this material is a strong candidate for an axion insulator and can be used for the experimental observation of the axionic phason.

Crystals of (TaSe4)2I grow in millimetre-long needles with aspect ratios of approximately 1:10, reflecting the one-dimensional crystalline anisotropy of the material (Methods and Extended Data Fig. 1). We measured the electrical transport of five (TaSe4)2I samples (A, B, C, D, E), with the electrical current I applied along the c axis of the crystals (Methods). In all samples, the low-bias four-terminal electrical resistivity ρ of the single-quasiparticle excitation was measured as a function of temperature T (Extended Data Fig. 1). The bias current-dependent transport properties were then measured for samples A, B and E. All investigated samples show similar electrical-transport properties. At 300 K, the ρ value of the crystals is around ρ0 = ρ(300 Κ) = 1,500 mΩ cm (±400 mΩ cm) with an electron density of n(300 Κ) = (1.7 ± 0.1) × 1021 cm−3 and a Hall mobility of μ(300 Κ) = (2.3 ± 0.1) cm2 V−1 s−1 (Methods and Extended Data Fig. 2), which are in good agreement with the literature32. Here, ρ0 is the average value of all samples investigated and the uncertainty is the standard deviation. The uncertainties given for n and μ originate from the errors of the Hall fits. From further analysis of the Hall measurements (Methods), we find that the Fermi level in our samples is located precisely in the middle of the CDW gap and therefore at the position of the initial Weyl cones in (TaSe4)2I (Extended Data Fig. 4). The data presented in the main text were obtained from measurements on sample A.

We first characterized the single-particle transport at zero magnetic field (|B| = 0 T). Figure 1c (left axis) shows the ρ/ρ0 ratio of the single-particle state on a logarithmic scale as a function of T−1. For all samples investigated, we observe an increasing ρ with decreasing T, consistent with the ρT dependence of (TaSe4)2I reported in the literature32,33. The plot of the logarithmic derivative (Fig. 1c, right axis) versus the inverse temperature, T−1, exhibits a well pronounced peak at around the expected CDW transition temperature, Tc = 263 K. The saturation of the logarithmic derivative at a constant value av above Tc−1 demonstrates that ρ/ρ0 follows the thermal activation law ρ/ρ0 ≈ exp[ΔE/(kBT)], allowing us to deduce the size of the single-particle gap ΔE in units of the Boltzmann constant, kB. By averaging over the saturated range, we obtain a value of av = 0.72 ± 0.04 and thus ΔE = (259 ± 14) meV, which is in excellent agreement with the literature31,32.

Next, we investigate the motion of the CDW at zero magnetic field in the pinning potential created by impurities. Figure 2a–c shows the variation of the measured voltage V as a function of the applied d.c. current I at four selected temperatures. Consistent with a sliding phason mode, nonlinearity in the VI curves appears below Tc at high bias currents, that is, high voltages (Extended Data Fig. 5). Accordingly, the VI curves can be well represented by a simple phenomenological model based on Bardeen’s tunnelling theory, as suggested by previous electrical-transport experiments on CDWs (Methods, Fig. 2d, Extended Data Fig. 6). To gain further insight into the collective charge transport mechanism, we calculate the differential electrical resistance dV/dI from the VI curves and plot it versus the corresponding V for various values of T (Fig. 2e). For all temperatures investigated, dV/dI is constant at low V. However, below Tc, dV/dI decreases above a threshold voltage Vth, indicating the onset of the collective phason current26. Vth is determined as the onset of the deviation from the zero baseline of the second derivatives d2V/dI2 (see, for example, Extended Data Fig. 5). The magnitude of the corresponding Eth = Vth/L (L is the distance between the voltage probes) and its temperature dependence (Fig. 2f) are in agreement with the literature32. Further analysis suggests that the nonlinear VI characteristics are not the result of local Joule heating or a consequence of contact effects (Methods, Extended Data Fig. 6). These observations provide evidence for a propagating, current-carrying CDW state.

We now test the axionic nature of the phason. For this purpose, we measure the nonlinear VI characteristics at fixed T, but now in the presence of a background magnetic field. We apply a magnetic field between −9 T and +9 T in steps of 1 T, oriented perpendicular and in parallel to the direction of the current flow. All experimental observations are symmetric in B (compare Fig. 3 and Extended Data Fig. 7). As shown in Fig. 3a, b, no B modulation of the VI characteristics and dV/dI can be seen within the measurement error of 0.1% for B perpendicular to I (EB) across the whole magnetic-field and temperature ranges investigated. To enable a direct comparison to the theory, we calculate the differential magnetoconductance Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T. No Δ(dI/dV)B is observed (Fig. 3c, d) in the perpendicular-field configuration. However, when the magnetic field is aligned with I (EB) (Fig. 3e, f)), the VI characteristics and dV/dI display a strong dependence on the magnitude of the magnetic field |B| above Vth and at low temperatures. Comparing the electrical currents at fixed T, B and V, we find a strong enhancement of I with increasing |B|. Because Vth itself is independent of |B| in any direction up to ±9 T (Fig. 3b, f), the observed B dependence seems to originate from a field-dependent contribution to the phason current. As shown in Fig. 3g, h, we consistently observe a large positive Δ(dI/dV)B for EB and T ≤ 155 K (see Extended Data Figs. 7, 8). The profile of this curve can be well described by a quadratic function below 7.5 T, according to the prediction for an axionic phason. To demonstrate the quadratic behaviour at low magnetic fields, we perform second-order polynomial fits, $${d}_{1}{B}_{\parallel }+{d}_{2}{B}_{\parallel }^{2}$$ (d1 and d2 are free parameters) to the experimental data and find that $${d}_{1}{B}_{\parallel }\ll {d}_{2}{B}_{\parallel }^{2}$$ (Extended Data Fig. 9). By comparison to our theoretical transport model (Methods), the dominant quadratic behaviour for all B < 7.5 T investigated is consistent with the estimated Fermi level in our samples near the position of the initial (undistorted) Weyl nodes. Above 7.5 T, Δ(dI/dV)B deviates from the quadratic function, which is not yet understood. We note that Δ(dI/dV)B in our experiment depends on V (Extended Data Fig. 9). As explained above, we attribute this observation to the impurity pinning of the CDW in our sample. The associated pinning potential induced strongly non-Ohmic IV characteristics, which are not included in our theoretical calculation.

To test the angular variation of the positive Δ(dI/dV)B, we perform VI measurements at 80 K and fixed 9 T for various angles φ (Fig. 4a). In accordance with the theoretical predictions for the axionic phason current, we find that Δ(dI/dV)φ is described by a cos2φ dependence. The observed pattern of angular dependence and the consistent functional dependence of the longitudinal Δ(dI/dV)B on B are the fundamental signatures of the axionic phason and the associated chiral anomaly. This supports the identification of (TaSe4)2I as an axion insulator.

In conclusion, our measurements reveal a positive longitudinal magnetoconductance in the sliding mode of the quasi-one-dimensional CDW–Weyl semimetal (TaSe4)2I, a signature that is linked to the presence of an axionic phason. We theoretically and experimentally find a $${B}_{\parallel }^{2}$$ dependence of the longitudinal magnetoconductance and a cosine-squared dependence on the relative orientation of I and B. Our results show that it is possible to find experimental evidence for axions, which is particularly elusive in other systems, in strongly correlated topological condensed matter systems.

## Methods

### Phenomenological theory of the magnetotransport in an axionic CDW system

In this work, we model the magnetotransport in a CDW–Weyl system (that is, an axion insulator)1 using an effective-field theory. For simplicity, we start with a quasi-one-dimensional Weyl semimetal with exactly two Weyl nodes—although the real structure of (TaSe4)2I is much more complicated. We assume that the Fermi surface consists of disconnected pockets surrounding these two Weyl nodes. Furthermore, we assume that the CDW ordering wavevector is commensurate with the Weyl node separation kq and use natural units. To simplify the discussion, we neglect the effects of impurity pinning. We then discuss the effects of impurity pinning by examining the modifications to the B = 0 conductivity.

### Effective-field theory for the semi-classical limit, where many Landau levels are occupied

Our strategy is to introduce an order parameter

$$|\varDelta |{{\rm{e}}}^{i\varphi }=\langle {c}_{-({{\bf{k}}}_{{\bf{q}}}/2)+{\rm{\delta }}{\bf{k}}}^{\dagger }{c}_{({{\bf{k}}}_{{\bf{q}}}/2)+{\rm{\delta }}{\bf{k}}}\rangle$$
(1)

for the CDW order by decoupling the electron–phonon interaction Hamiltonian. Here $${c}_{({{\bf{k}}}_{{\bf{q}}}/2)+{\rm{\delta }}{\bf{k}}}$$ is an operator that annihilates an electron displaced by wavevector δk from the Weyl node centred at ±kq/2. By integrating out the electrons and neglecting the fluctuations in the gapped amplitude mode |Δ|, we arrive at an effective-field theory for the phason mode ϕ of the CDW. Let us confine ourselves to zero temperature. By neglecting the chiral charge of the Weyl nodes (we will add it later), we find from Anderson34 the action (working in units where ħ = 1)

$$\begin{array}{ll}{S}_{0} & \,=\frac{{m}_{{\rm{F}}}}{2}N(0)\int \left[-\frac{1}{2}{\left(\frac{{\rm{\partial }}\phi }{{\rm{\partial }}t}\right)}^{2}+\frac{1}{2}{c}_{\perp }{({\hat{{\bf{k}}}}_{{\bf{q}}}\times {\rm{\nabla }}\phi )}^{2}+\frac{1}{2}c{({\hat{{\bf{k}}}}_{{\bf{q}}}\cdot {\rm{\nabla }}\phi )}^{2}\right.\\ & -\left.2{e}^{\ast }\frac{{v}_{{\rm{F}}}}{{m}_{{\rm{F}}}}{\hat{{\bf{k}}}}_{{\bf{q}}}\cdot {\bf{E}}\phi \right]{{\rm{d}}}^{3}x{\rm{d}}t\end{array}$$
(2)

Here we have defined

$${m}_{{\rm{F}}}=4\frac{{|\varDelta |}^{2}}{\lambda {\omega }_{{{\bf{k}}}_{{\bf{q}}}}^{2}}$$
(3)

with λ the electron–phonon coupling, and $${\omega }_{{{\bf{k}}}_{{\bf{q}}}}$$the phonon energy at wavevector kq. N(0) is the density of states at the Fermi level, vF is the Fermi velocity in the direction of the CDW wavevector, $${\hat{{\bf{k}}}}_{{\bf{q}}}$$, and e* is the screened electron charge. The velocities c and c correspond to the transverse and longitudinal (phason) velocities, respectively. The final term in (2) corresponds to the sliding (Froelich) mode, excited by an electric field aligned with the CDW ordering vector.

Additionally, we find from the chiral anomaly the contribution34

$${S}_{{\rm{c}}}=\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}}N(0)\int \varphi {\bf{E}}\cdot {\bf{B}}{{\rm{d}}}^{3}x{\rm{d}}t$$
(4)

This contribution to the action emerges from the chiral anomaly in the undistorted Weyl semimetal. As shown in ref. 1, the formation of the CDW causes the amplitude of the chiral anomaly to depend on the phason field. Let us now consider the effect of a uniform external magnetic field. Because we are interested in the response of the CDW to uniform fields, we restrict ourselves to spatially homogeneous configurations ϕ(xt) = ϕ(t). By varying the action S = S0 + Sc with respect to ϕ, we find from the equations of motion that ϕ satisfies

$$\frac{\partial S}{\partial \varphi }=\frac{N(0){m}_{{\rm{F}}}}{4}\frac{{\partial }^{2}\varphi }{{\partial }^{2}t}-{\bf{E}}\cdot \left(N(0){v}_{{\rm{F}}}{e}^{\ast }\hat{{{\bf{k}}}_{{\bf{q}}}}-\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}}{\bf{B}}\right)$$
(5)

To include the effects of damping, we follow Lee et al.35 and add a phenomenological damping term to this equation. Because we are interested in the dynamics of the sliding mode above the de-pinning transition, we do not consider a phenomenological pinning term. Defining an effective damping rate Γ, we thus find

$$\frac{N(0){m}_{{\rm{F}}}}{4}\frac{{{\rm{\partial }}}^{2}\varphi }{{{\rm{\partial }}}^{2}t}+\varGamma \frac{{\rm{\partial }}\varphi }{{\rm{\partial }}t}={\bf{E}}\cdot \left(N(0){v}_{{\rm{F}}}{e}^{\ast }{\hat{{\bf{k}}}}_{{\bf{q}}}-\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}}{\bf{B}}\right)$$
(6)

We can solve this equation in the long-time limit to find the steady-state solution

$$\varphi (t)={\varphi }_{0}+\frac{t}{\varGamma }{\bf{E}}\cdot \left(N(0){v}_{{\rm{F}}}{e}^{\ast }{\hat{{\bf{k}}}}_{{\bf{q}}}-\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}}{\bf{B}}\right)$$
(7)

Next, we look at the current density

$${\bf{j}}=\frac{{\rm{\delta }}S}{{\rm{\delta }}{\bf{A}}}$$
(8)

carried by the CDW sliding mode, where A is the electromagnetic vector potential. By varying the action and restricting to position-independent configurations of ϕ, we find

$${\bf{j}}=\left(N(0){v}_{{\rm{F}}}{e}^{\ast }{{\bf{k}}}_{{\bf{q}}}-\frac{{{\rm{e}}}^{2}}{{4{\rm{\pi }}}^{2}}{\bf{B}}\right)\frac{{\rm{\partial }}\varphi }{{\rm{\partial }}t}$$
(9)

Finally, by combining equation (7) with equation (9), we find that

$${j}^{i}={\sigma }_{ij}({\bf{B}}){E}_{j}$$
(10)

with the conductivity tensor given by

$${\sigma }_{ij}({\bf{B}})=\frac{1}{\varGamma }{\left(N(0){v}_{{\rm{F}}}{e}^{\ast }{\hat{{\bf{k}}}}_{{\bf{q}}}-\frac{{{\rm{e}}}^{2}}{{4{\rm{\pi }}}^{2}}{\bf{B}}\right)}_{i}{\left(N(0){v}_{{\rm{F}}}{e}^{\ast }{\hat{{\bf{k}}}}_{{\bf{q}}}-\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}}{\bf{B}}\right)}_{j}$$
(11)

In particular, the magnetoconductivity is given by

$$\frac{{\sigma }_{ij}({\bf{B}})-{\sigma }_{ij}(0)}{{\sigma }_{ij}(0)}=-\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}N(0){v}_{{\rm{F}}}{e}^{\ast }}({\hat{k}}_{q,i}{B}_{j}+{B}_{i}{\hat{k}}_{q,j})+\frac{{{\rm{e}}}^{2}}{4{{\rm{\pi }}}^{2}{(N(0){v}_{{\rm{F}}}{e}^{\ast })}^{2}}{B}_{i}{B}_{j}$$
(12)

We note that there are two types of contributions in equation (12). The first term depends on the relative orientation of the electric and magnetic fields relative to the CDW wavevector, and originates from a mixing between the chiral-anomaly (B dependent) and conventional (B independent) contributions to the CDW dynamics in equations (7) and (9). The quadratic in B contribution to the magnetoconductivity, on the other hand, has a purely chiral-anomaly origin. Although it corresponds to current carried by the sliding mode, it originates entirely from the underlying Weyl semimetal. Hence, like in the Weyl semimetal, this contribution to the magnetoconductivity is independent of the Weyl separation (that is, CDW ordering) vector.

We now note that in the idealized case in which the unordered phase is a perfect Weyl semimetal, N(0) → 0. In this case, the second term in equation (12) dominates over the first, and we recover the characteristic quadratic positive magnetoconductivity of a Weyl material36,37,38. In particular, in this limit we also recover the characteristic cosine-squared dependence of the magnetoresistance on the angle between E and B. More generally, the quadratic term in equation (12) dominates when

$$N(0){v}_{{\rm{F}}}{e}^{\ast }\ll {{\rm{e}}}^{2}|{\bf{B}}|$$
(13)

Hence, we expect a positive quadratic magnetoconductivity in this regime.

We note that, unlike recent proposals for ‘axion insulators’ in Weyl CDW materials1,2, our system preserves time-reversal symmetry and hence has multiple pairs of Weyl fermions that are coupled in the CDW phase. This forces responses such as the anomalous Hall conductance to be zero. Nevertheless, the negative magnetoresistance due to the chiral phason channels is expected to persist in the limit of long phason lifetime (that is, weak intervalley scattering).

### Microscopic argument for the quantum limit, where only the 0-th Landau level is occupied

To gain some insight into how the sliding mode can influence the magnetoresistance in the quantum limit, we take our Weyl–CDW system and apply a large magnetic field along the $${\hat{{\bf{k}}}}_{{\bf{q}}}$$ direction. We take the magnitude of B to be large enough so that only the zero-th Landau level is occupied when electron–phonon interactions are neglected. In this case, we have a truly quasi-one-dimensional system, with conserved momentum $$Q={\bf{k}}\cdot {\hat{{\bf{k}}}}_{{\bf{q}}}$$; each state is largely degenerate, with the degeneracy given by the total flux |B|A/(2π) through the system (measured in units of the flux quantum, with A the cross-sectional area). We now reintroduce the electron–phonon interaction; this causes the formation of a CDW ground state. Because the Fermi points of the Weyl nodes disperse entirely in the lowest Landau level, the low-energy physics of this transition matches that of a true one-dimensional CDW transition, provided that we multiply all results by the Landau level degeneracy |B|A/(2π). In particular, Lee et al.28 showed that the sliding mode of the CDW contributes e2n/(m*Γ) per channel to the electrical conductivity in the $${\hat{{\bf{k}}}}_{{\bf{q}}}$$ direction, where m* = m(1 + mF) is the electron effective mass, n is the one-dimensional charge-carrier density and Γ is the phenomenological damping rate (phonon lifetime). By multiplying this by the number of channels we find in the quantum limit and for current and electric fields aligned with the magnetic field (and therefore CDW)

$${\sigma }_{ij}({\bf{B}}){\hat{{\bf{B}}}}_{i}{\hat{{\bf{B}}}}_{j}\to \frac{{{\rm{e}}}^{2}n}{{m}^{\ast }\varGamma }|{\bf{B}}|$$
(14)

which is analogous to the magnetoresistance in the ultra-quantum limit of a Weyl semimetal38.

### Comment on real material systems

Having justified the origin of the negative magnetoresistance in a simplified model, we now discuss how pinning effects will change the phenomenology. First, because CDWs are always pinned by impurities in real material systems, sliding only occurs upon reaching a certain threshold electric field, that is, the threshold voltage Vth. In addition, transport measurements always also involve single-charge carriers thermally exited above the Peierls gap ΔE, which carry electrical current in parallel to the sliding CDW mode. Therefore, the B = 0 electrical conductivity (that is, the electrical resistivity) is in general non-Ohmic and depends on the applied bias voltage V. Below the Peierls transition, the total electrical conductivity for a CDW system in zero magnetic field (σij,total) is given by Bardeen’s tunnelling theory39:

$${\sigma }_{ij,{\rm{t}}{\rm{o}}{\rm{t}}{\rm{a}}{\rm{l}}}={\sigma }_{{\rm{S}}{\rm{P}}}+{\sigma }_{{\rm{C}}{\rm{D}}{\rm{W}},0}\left(1-\frac{{V}_{{\rm{t}}{\rm{h}}}}{V}\right)\exp \left(-\frac{{V}_{0}}{V}\right)$$
(15)

Here, σSP denotes the Ohmic conductivity contribution of the thermally exited single-charge carriers, Vth is the threshold voltage and σCDW,0 and V0 are voltage-independent parameters that usually depend on the temperature T. This is derived by assuming that pinning by impurities creates a gap of Eg < ΔE to CDW sliding. Equation (15) then follows from considering Zener tunnelling of the CDW across the gap. V0 is closely related to the dynamics of the sliding CDW and can be expressed as V0 = Eg/(2ξe*), where ξ is the coherence length, as in superconductors. Because Eg and e* do not depend on T, the temperature dependence of V0 provides a measure for the temperature dependence of ξ, which generally decreases with increasing T. σCDW,0 is a measure of the number of carriers forming the CDW and has been observed to increase with increasing T. The validity of equation (15) for CDW systems has been confirmed by many experiments, by fitting the measured IV characteristics to the current Itotal(V) = ISP(V) + ICDW(V), corresponding to equation (15), with ISP(V) = GspV accounting for the single-particle contribution and ICDW(V) = I0(V − Vth)exp(−V0/V) for the sliding CDW. Gsp (the single-particle electrical conductance), I0 (corresponding to σCDW,0) and V0 are the free fitting parameters. Importantly, there is no added conductivity below Vth and hence Itotal(V) = ISP(V) for V < Vth. Vth can be determined independently, as shown in Extended Data Fig. 5. By fitting Itotal(V) to our data measured at zero magnetic field for V > Vth (Extended Data Fig. 5), we find excellent agreement with Bardeen’s tunnelling theory. The extracted value of V0 decreases and I0 increases with increasing temperature, in agreement with previous reports for other CDW systems. By fitting our measured IV characteristics with the corresponding current expression Itotal(V) = ISP(V) + ICDW(V), with ISP(V) = GspV and ICDW(V) = I0(V − VT)exp(−V0/V), using the single-particle electrical conductance Gsp and I0 and V0 as the free fitting parameters, we find excellent agreement with Bardeen’s tunnelling theory (Fig. 2d, Extended Data Fig. 4).

Impurity pinning in real materials, such as in our samples, has important consequences on the experimental detection of the axionic response of CDWs. As mentioned above, the zero-field conductivity tensor will develop a nonlinear dependence on V. Additionally, the single-particle current will always be measured in parallel to the sliding CDW in experiments. By putting these facts together, we see from equation (12) that the measured magnetoconductance $$\frac{\Delta I}{\Delta V(B)}\propto {\sigma }_{xx}(B)-{\sigma }_{xx}(0)$$ will become non-Ohmic and will in general depend on V. Hence, the amplitude of the characteristic cosine-squared dependence of the magnetoconductivity on the angle between E and B will also become V-dependent.

### (TaSe4)2I crystal structure and single-crystal growth

(TaSe4)2I is a quasi-one-dimensional material with a body-centred tetragonal lattice (Fig. 1b). The Ta atoms are surrounded by Se4 rectangles and form chains aligned along the c axis. These chains are separated by I ions. Single crystals of Ta2Se8I were obtained via a chemical vapour transport method using Ta, Se and I as starting materials40. A mixture with composition Ta2Se8I was prepared and sealed in an evacuated quartz tube. The ampoule was inserted into a furnace with a temperature gradient of 500 to 400 °C with the educts in the hot zone. After two weeks, needle-like crystals had grown in the cold zone (Extended Data Fig. 1). Energy-dispersive X-ray spectroscopy reveals a Ta:Se:I ratio of 17.75:72.95:9.3, which matches the theoretical ratio of 18.18:72.73:9.09 within the measurement precision of 0.5.

### Electrical-transport measurements

All electrical-transport measurements were performed in a variable-temperaturecryostat (Quantum Design) equipped with a 9-T magnet. To avoid contact resistance effects, only four-terminal measurements were carried out. To reduce any possible heating effects during current sweeps, the samples were covered in highly heat-conducting, but electrically insulating, epoxy32. The temperature-dependent electrical resistivity curves were measured with a standard low-frequency (f = 6 Hz) lock-in technique (Keithley SR 830), by applying a current of 100 nA across a 100-MΩ shunt resistor. The applied bias current I = 100 nA for these measurements was chosen such that the corresponding electric field E = IR/L did not exceed the threshold field Eth > E, where L is the distance between the voltage probes. The determination of Eth is explained later. The bias current-dependent transport properties were then measured on samples A, B and E because their closely adjacent electrical contacts of L < 3 mm allowed us to cross the threshold field Eth with experimentally accessible currents (IR < 60 V) in our setup. The bias current-dependent transport experiments were limited to temperatures of T ≥ 80 K because Eth increases exponentially with decreasing T. Given the minimal contact resolution of >0.5 mm in our bulk samples, no switching could be achieved in our samples below 80 K, because the corresponding threshold voltages exceeded the measurable voltage range (<60 V). The d.c. voltage–current characteristics were measured with a Keithley 6220 current source (1014 Ω output impedance) and a Keithley 2182A/E multimeter.

### First-principles calculations

For the ab initio investigations we employ density-functional calculations (DFT) as implemented in the VASP package41. We use a plane-wave basis set and the generalized-gradient approximation for a description of the exchange-correlation potential42. As a second step we create Wannier functions from the DFT states using Wannier9043 and extract the parameters for a tight-binding Hamiltonian to further analyse the band structure.

### Estimate of the Fermi level position from Hall measurements

To estimate the Fermi level position in our samples, we performed Hall measurements on sample D. Because the single-particle resistivity of all our samples was similar throughout the examined temperature range, the Hall measurements are representative of the whole set of samples presented here. The device used for the Hall measurements is shown in Extended Data Fig. 2a. Contacts 1 and 4 are used for electrical current injection, contacts 2 and 3 are used to probe the longitudinal voltage along the sample, and contacts 5 and 6 are used to measure the Hall voltage V5,6 across the sample. The magnetic field B is perpendicular to the measurement plane. The Hall resistance as a function of the magnetic field B is calculated using RH = V5,6/I at fixed temperatures T. We chose the electrical current I applied at each temperature to be small enough to probe only single-particle transport but high enough to resolve the Hall signal V5,6. In particular, we used I = 1 μA at 125 K, I = 10 μA at 155 K and 185 K, I = 100 μA at 215 K and 245 K, and I = 2 mA at 300 K. Although the data are quite noisy, we observe a strong B dependence of RH (Extended Data Fig. 2c–h). Below 125 K, the longitudinal resistance of the sample became too high to measure the Hall response. As reported previously33, the Hall resistance at low temperatures is linear with B. At 300 K, however, RH becomes nonlinear with B at high fields, which is typical for semiconductors and insulators at elevated T, because both electrons and holes become thermally activated across the gap. At low temperatures, the band closer to the Fermi level dominates. As discussed previously, even in such a one-dimensional material as (TaSe4)2I, the Hall effect is well defined because, to the first order in B, the Lorentz force does not depend on the transverse effective mass33. Recalling the standard expression for the low-field Hall conductivity (single-charge-carrier approximation), we estimate the carrier concentration n = (dRH/dB)−1(e*d)−1 as a function of T from the slope dRH/dB of the linear fits to the magnetic field-dependent RH (d = 300 mm is the height of sample D). As seen in Extended Data Fig. 3i, n decreases by orders of magnitude when decreasing T from 300 K to 125 K, consistent with the increasing longitudinal single-particle resistivity (Extended Data Fig. 2b). At the CDW transition temperature Tc a jump in n is observed, consistent with the opening of a single-particle gap. At 300 K, the estimated carrier concentration is n = (1.7 ± 0.1) × 1021 cm−3, leading to a mobility of μ = (2.3 ± 0.1) cm2 V−1 s−1, using the Drude model μ = (e*)−1 with an electrical resistivity of ρ = 1,562 μΩ cm at 300 K. Both n and μ are in good agreement with the values previously reported33 for (TaSe4)2I.

Using the effective mass m* = 0.4m0 (m0 is the free electron mass) obtained from ARPES measurements32 for T < Tc and a single-band model, we subsequently determine the Fermi level position EF in our sample with respect to the conduction band edge.

For a bandgap material, the carrier concentration is given by

$$n={N}_{{\rm{c}}}\exp \left(\frac{{E}_{{\rm{F}}}}{{k}_{{\rm{B}}}T}\right)$$
(16)

where $${N}_{{\rm{c}}}=2{\left(\frac{2\pi {m}^{\ast }{k}_{{\rm{B}}}T}{{h}^{2}}\right)}^{3/2}$$ and h is the Planck constant. By transforming equation (16), we obtain a description for EF that only depends on T, m and n:

$${E}_{{\rm{F}}}=\mathrm{ln}\left[{\frac{1}{2}n\left(\frac{2\pi {m}^{\ast }{k}_{{\rm{B}}}T}{{h}^{2}}\right)}^{-3/2}\right]{k}_{{\rm{B}}}T$$
(17)

The result of equation (17) as a function of T is shown in Extended Data Fig. 2j. Within the error bar, the estimated EF is independent of the temperature for T < Tc, giving a mean value of EF = 129 ± 10 meV below the conduction band edge of the CDW gap. By comparing this number to the experimentally obtained CDW gap size of ΔE = 159 ± 5 meV, we find that the Fermi level in our (TaSe4)2I samples is precisely in the middle of the CDW gap (EF = ½ΔE). As shown previously via DFT calculations and ARPES measurements, the CDW gap opens symmetrically around the crossing points of the initial electronic band structure31. Therefore, the Fermi level position is in the centre of the initial Weyl cones.

### Excluding other intrinsic and extrinsic origins for the nonlinear V–I curves

The following analysis suggests that the nonlinear VI characteristics are not the result of local Joule heating or a consequence of contact effects, but originate from an electrically driven intrinsic process in the (TaSe4)2I crystals, consistent with a sliding CDW. First, we exclude thermally driven switching due to Joule heating. Thermally driven switching corresponds to a local rise of the temperature in the (TaSe4)2I sample above Tc. Such a temperature rise would be determined by the power dissipated in the sample and the electronic circuitry. In the worst-case scenario, all of the VI Joule heating power is dissipated within the (TaSe4)2I crystals. Hence, the dissipated power necessary for the switching would approach zero as T → Tc from below. However, the electrical power Pth = Vth2/[dV/dI(Vth)] obtained at Vth from our experiments (Extended Data Fig. 6a) does not approach zero as T → Tc. Instead, Pth decreases with decreasing T. This is precisely the reverse of what we would expect from thermally driven switching. Second, we rule out contact effects as the cause of the nonlinear threshold behaviour in our experiments. As an example, we show Vth as a function of the distance of the voltage probes L at 80 K in Extended Data Fig. 6b for L = 1 mm, 2 mm and 3 mm. The observed linear VthL dependence implies that the transition is driven by the electric field and not by the absolute magnitude of I, which justifies our definition of Eth = Vth/L. It also demonstrates that the transition originates from a bulk effect, because the contacts in all of our devices are of identical size; hence, switching in the contacts would have no dependence on the length. Therefore, any change in the switching properties must result from the (TaSe4)2I crystal itself.

### Excluding other origins for the positive differential magnetoconductance

Here, we address the concern that the experimentally detected positive differential magneto-conductance Δ(dI/dV)B in (TaSe4)2I could arise from an alternative origin than the axionic nature of the CDW. Several intrinsic and extrinsic origins of the positive Δ(dI/dV)B in (TaSe4)2I can be excluded from the experimental observations and analysis. (i) Inhomogeneous current distribution caused by quenched disorder and the current-jetting effect are ruled out. The elongated geometry of the samples and the silver paint contacts used for current injection, encasing the entire ends of the wires, provide homogeneous current injection through the entire bulk of the samples44,45.Theoretical simulations with similar sample geometries and even higher carrier mobility have shown44 that magnetic fields much larger than 9 T are required for the onset of current-jetting effects. To experimentally test for inhomogeneous current distribution in our wires, we use a special contact geometry on sample E (Extended Data Fig. 10). The voltage contacts on sample E were designed as point contacts, locally probing the voltage drop across two different edges of the sample. As in all other devices used in this study, the current injection contacts 1 and 6 cover the entire ends of the (TaSe4)2I crystal. We measured simultaneously the potential difference Vij between the two pairs of nearest-neighbour contacts along the current flow (V2–3 and V4–5) at T = 80 K as a function of the applied current I and B for the perpendicular (IB) and longitudinal (EB)I measurement configuration (Extended Data Fig. 10b–g, i–n) The obtained curves are nearly identical and display only small differences within the measurement error (up to 10%). The agreement across the two pairs of contacts shows that distortions of the current path are minimal and that the current distribution is uniform within the sample. This implies that the observed positive longitudinal Δ(dI/dV)B is an intrinsic electronic effect. (ii) The absence of a Δ(dI/dV)B for perpendicular B, as well as the large temperatures at which the longitudinal Δ(dI/dV)B occurs, show that neither quantum interference effects of the condensate nor of the single quasiparticle excitations play an important role. (iii) The independence of Vth from B (Fig. 3a, e) excludes any possible effects from a reduction of Vth.

## Data availability

All data generated or analysed during this study are available within the paper and its Extended Data files. Reasonable requests for further source data should be addressed to the corresponding author.

## Change history

• ### 15 May 2020

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

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## Acknowledgements

C.F. acknowledges the research grant DFG-RSF (NI616 22/1; ‘Contribution of topological states to the thermoelectric properties of Weyl semimetals’) and SFB 1143. First-principles calculations were funded by the US Department of Energy through grant number DE-SC0016239. B.A.B. acknowledges additional support from the US National Science Foundation EAGER through grant number NOA-AWD1004957, Simons Investigator grants ONR-N00014-14-1-0330 and NSF-MRSEC DMR-1420541, the Packard Foundation and the Schmidt Fund for Innovative Research. Z.W. acknowledges support from the National Thousand-Young-Talents Program, the CAS Pioneer Hundred Talents Program and the National Natural Science Foundation of China.

## Author information

Authors

### Contributions

B.A.B., C.F. and J.G. conceived the experiment. N.K., C. Shekhar and Y.Q. synthesized the single-crystal bulk samples. J.G., S.H. and C. Schindler fabricated the electrical-transport devices. J.G. carried out the transport measurements with the help of S.S.H. and C. Schindler. J.G. and C. Shekhar analysed the data. J.N., Z.W. and Y.S. calculated the band structure. B.B. and B.A.B. provided the theoretical background of the work. All authors contributed to the interpretation of the data and to the writing of the manuscript.

### Corresponding author

Correspondence to J. Gooth.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Jian Wang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 (TaSe4)2I crystal structure, growth, device and transport characterization.

a, Crystal structure of (TaSe4)2I. b, Sketch of the growth principle. A temperature gradient (temperatures T1 > T2) is imposed on an evacuated quartz ampule, which contains (TaSe4)2I powder at T1. The evaporated (TaSe4)2I diffuses towards the area with temperature T2 and condenses into single crystals. c, Optical micrograph of the as-grown (TaSe4)2I crystals. d, Distribution of Weyl points in momentum (k) space of chirality ±χ. e, Scanning electron microscope image of a crystal. f, Typical device used for electrical-transport measurements. g, h, Single-particle resistance of samples B, C, D and E. The electrical resistance R, normalized by R0 = R(300 K) (g) and its logarithmic derivative (h) as a function of T−1, where T is the temperature. i, Single-particle gaps of all (TaSe4)2I samples investigated. The error denotes the fitting error of 1σ. The dotted line displays the mean value of all samples.

### Extended Data Fig. 2 Hall measurements and Fermi level position of sample D.

a, Device used for the Hall measurements in a magnetic field B. b, Single-particle longitudinal resistivity ρ versus temperature T. ci, Single-particle Hall resistance RH at 125 K (c), 155 K (d), 185 K (e), 215 K (f), 245 K (g) and 300 K (h). The measured data (black) are fitted linearly (red) to extract the carrier concentration n (i). The error bars denote the error of the linear fits to 1σ. j, Estimated Fermi level position below Tc. ΔE is the single-particle gap, obtained from Extended Data Fig. 1.

### Extended Data Fig. 3 V–I characteristics of sample A at zero magnetic field and selected temperatures.

aj, VI characteristics at 80 K (a), 105 K (b), 130 K (c), 155 K (d), 180 K (e), 205 K (f), 230 K (g), 255 K (h), 280 K (i) and 305 K (j). These data were used to calculate the differential resistance dV/dI curves in Fig. 2d. At temperatures below 180 K, we start to observe nonlinearity. This nonlinearity becomes even more apparent in the dV/dI curves shown in Fig. 2d, where a deviation from the linear behaviour is already seen at 230 K.

### Extended Data Fig. 4 Fitting the nonlinear V–I characteristics of sample A at zero magnetic field.

ag, VI characteristics (black line), a linear fit Isp(V) = mV (green line) and a fit with the Bardeen model6 I(V) = Isp(V) + ICDW(V) (red line), where ICDW(V) = I0(V − VT)exp(−V0/V) at 80 K (a), 105 K (b), 130 K (c), 155 K (d), 180 K (e), 205 K (f) and 230 K (g). h, i, m (h) and the threshold voltage Vth extracted from Fig. 2 were employed to extract the fit parameters. j, I0. k, V0. The error bars in hk denote the error of the fits to 1σ.

### Extended Data Fig. 5 Dependence of the switching voltage of sample A on the contact separation.

a, b, VI characteristics (a) and differential resistance dV/dI (b) at 80 K and zero magnetic field B for various contact lengths L on sample A. ce, Second derivatives d2V/dI2 used to estimate the threshold voltages Vth shown in Fig. 2g. We define Vth as the voltage at which the last data point of d2V/dI2 (marked by the vertical line) touches the zero baseline before the global minimum upon enhancing V.

### Extended Data Fig. 6 Testing the origin of the nonlinear V–I curves of sample A.

a, The increasing Joule heating power Pth = Vth2/[dV/dI(Vth)] at the threshold, as a function of increasing T. b, The linear dependence of Vth on L shown at 80 K (the line denotes a linear fit) demonstrate that the observed effects are intrinsic to the (TaSe4)2I crystals.

### Extended Data Fig. 7 Symmetry and temperature dependence of the V–I characteristics in the magnetic field of sample A.

a, b, VI characteristics (a) and differential resistance dV/dI (b) at 80 K for sample A in magnetic fields perpendicular to the applied current, but opposite to the magnetic field in Fig. 3. c, d, Corresponding VI characteristics (c) and dV/dI (d) in magnetic fields parallel to the applied current. eg, VI characteristics at 80 K (e), 105 K (f) and 130 K (g).

### Extended Data Fig. 8 Bias-dependent data of samples A and B at 105 K.

The two samples have similar contact separation. a, b, VI characteristics (a) and differential resistance dV/dI (b) at various magnetic fields B applied perpendicular to the current direction (IB) for sample A. c, Magnetic field dependence of the magnetoconductance Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T at 105 K and at various voltages V for sample A. d, e, Corresponding VI characteristics (d) and dV/dI (e) at various magnetic fields B applied perpendicular to the current direction (IB). f, Magnetic field dependence of the magnetoconductance Δ(dI/dV)B for sample B.

a, d1. b, d2.

### Extended Data Fig. 10 Homogeneity test of the current distribution on sample E.

a, First configuration. I is injected from contacts 1 and 6 and V is measured between contacts 2 and 3. be, VI characteristics (b, d) and dV/dI (c, e) at 80 K and at various magnetic fields B for IB (b, c) and for IB (d, e). f, g, Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T for IB (f) and for IB (g). h, Second configuration. V is measured between contacts 4 and 5. in, Corresponding characteristics (i, k) and dV/dI (j, l) for IB (i, j) and for IB (k, l). m, n, Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T for IB (m) and for IB (n).

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Gooth, J., Bradlyn, B., Honnali, S. et al. Axionic charge-density wave in the Weyl semimetal (TaSe4)2I. Nature 575, 315–319 (2019). https://doi.org/10.1038/s41586-019-1630-4

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