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

https://doi.org/10.1038/s41586-019-1630-4 J. Gooth1*, B. Bradlyn2, S. Honnali1, C. Schindler1, N. Kumar1, J. Noky1, Y. Qi1, C. Shekhar1, Y. Sun1, Z. Wang3,4, B. A. Bernevig5,6,7 & C. Felser1 Received: 14 November 2018

Accepted: 19 July 2019 An insulator is a correlated topological phase, which is predicted to arise from Published online: 7 October 2019 the formation of a charge-density wave in a Weyl semimetal1,2—that is, a material in There are amendments to this paper which electrons behave as massless chiral . The accompanying sliding mode in the charge-density-wave phase—the phason—is an axion3,4 and is expected to cause anomalous magnetoelectric transport efects. 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 felds. The positive contribution to the magnetoconductance originates from the anomalous axionic contribution of the to the phason current, and is locked to the parallel alignment of the electric and magnetic felds. By rotating the magnetic feld, 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 fnd experimental evidence for in strongly correlated topological condensed matter systems, which have so far been elusive in any other context.

Axions are elementary particles that have long been known in quantum dispersion relation (Fig. 1a). The Weyl nodes always occur in pairs of field theory3,4, but have not yet been observed in nature. However, it opposite ‘handedness’ or . At low energies and in the absence has been recently understood that axions can emerge as collective of interactions, the chirality is a conserved quantum number and the electronic excitations in certain crystals, the so-called axion insulators5. two chiral populations do not mix. Parallel electric and magnetic fields Despite being fully gapped to single-particle excitations in the bulk (E‖B), however, enable a steady flow of between the and at the surface, an axion insulator is characterized by a quantum left- and right-handed nodes25. This induced breaking of the chiral mechanical equation of motion, which includes a topological θE·B is a macroscopic manifestation of a quantum anomaly in term, where E and B are the electric and magnetic fields inside the relativistic field theory and gives rise to a positive longitudinal mag- insulator and θ plays the role of the dynamical axion field. Physically, netoconductance in these systems. While the single-particle states the average value of θ is determined by the microscopic details of the in Weyl systems have been well studied experimentally, many-body band structure of the system, and gives rise to unusual magneto-electric interaction effects remain largely unexplored. By turning on strong response properties, such as quantum anomalous Hall conductivi- interactions, a CDW that links the two Weyl nodes and gaps the Weyl ties6–9, the quantized circular photo-galvanic6,10,11 effect and the chiral fermions can be induced in Weyl semimetals. magnetic effect6,12–14. The prospect of realizing an axion insulator has A CDW is the energetically preferred ground state of the strongly inspired much theoretical and experimental work. Very recently, sig- coupled electron–phonon system in certain quasi-one-dimensional natures of a dynamic axion field have been found on the surface of conductors at low temperatures26. It is characterized by a gap in the magnetically doped thin films15–17. However, the single-particle excitation spectrum and by a gapless collective mode axionic in these systems—the axionic polariton5—has so formed by electron–hole pairs. The density of the electrons and the far eluded experimental detection. Alternatively, axion insulators have position of the lattice atoms are periodically modulated with a period been predicted to arise in Weyl semimetals that are unstable towards larger than the original lattice constant (Fig. 1a). The phase ϕ of this the formation of a charge-density wave (CDW)1,2,18–21. condensate is of fundamental importance for experiments: its time

In their parent state, Weyl semimetals are materials in which low- derivative is related to the electric current density (jcdw ≈ ∂ϕ/∂t) carried energy electronic quasiparticles behave as chiral relativistic fermions by the collective mode upon application of an electric field26. Owing to without rest mass, known as Weyl fermions22–24. Weyl fermions exist at its relation to the phase, this current-carrying collective mode is called isolated crossing points of the conductance and valence bands—the so- the phason. In most cases, the wavevectors of the CDW are incommen- called Weyl nodes—and their energy can be approximated with a linear surate with the lattice, and the CDW is pinned to impurities. Therefore,

1Max Planck Institute for Chemical Physics of Solids, Dresden, Germany. 2Department of Physics and Institute for Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, USA. 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, China. 4University of Chinese Academy of Sciences, Beijing, China. 5Department of Physics, Princeton University, Princeton, NJ, USA. 6Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, Berlin, Germany. 7Max Planck Institute of Microstructure Physics, Halle, Germany. *e-mail: [email protected]

Nature | Vol 575 | 14 November 2019 | 315 Article

2.0 a –F +F c T E E c F ΔE F E E 8 kq kq

k k x x 1.5 CDW n n 6

a /k ) x x π q –1 T 3

) 1.0 0

4 )]/d(10 b 0 log( U / av 0.1 d[log( U / 0.5 2 W2 0.0 E W1 F E (eV)

0.0 0 –0.1

k 246810 12 14 x T–1 (×103 K–1)

Fig. 1 | Charge-density wave in the Weyl semimetal (TaSe4)2I. a, Periodic ρ, normalized to the ρ0 = ρ(300 K) (left axis, black) and its logarithmic modulations of the carrier density n in real space x create a gap in the Weyl derivative (right axis, red) as a function of T−1 , where T is the temperature. cones of chirality +χ and −χ in the energy–momentum (E–kx) space at the Fermi av denotes the average value of the logarithmic derivative below the transition energy EF. The lattice parameter a is modulated by π/kq, where kq is the distance temperature Tc. of the Weyl nodes. b, Band structure of (TaSe4)2I. c, Electrical resistivity

only upon applying a certain threshold electric field Eth (above which the pinning in real material systems, c1, c2 and c3—and therefore σCDW,xx(B‖)— electric force overcomes the pinning forces) is the CDW depinned and can in general depend on V (Methods). In the low-field limit, where free to ‘slide’ over the lattice27,28, thereby contributing to the electrical several Landau levels are occupied, the conductance is given by the conduction. The resulting conduction behaviour is strongly nonlinear quadratic term, recovering the characteristic, chiral-anomaly-induced, and the electrical resistivity drops with increasing electric field. positive longitudinal magnetoconductance of a non-interacting Weyl 2 In a Weyl semimetal, the CDW can couple electrons and holes with semimetal. In particular, in this limit we obtain a B‖cos ϕ dependence of different chirality—that is, it generates a complex mass in the Dirac the magnetoconductance on the angle ϕ between E and B. In the high- equation, which mixes the two chiral populations of the system even field limit, the magnetoconductance behaves linearly with the magnetic in zero magnetic field1,2,18–20. The resulting gapped state is the axion field, similarly to the ultra-quantum limit of a Weyl semimetal. This is insulator and the phase of the CDW is identified as the axion field the fingerprint of the axion, which allows us to probe the presence of (ϕ = θ). The phason is the axionic mode of this system and its dynamics the axionic phason mode through its effect on the electrical transport is therefore described by the topological θE·B term. Compared with in a Weyl–CDW condensed matter system. its high-energy version, this condensed matter realization of the axion Recently, a positive magnetoconductance has been observed29 in has the advantage of being accessible in magnetoelectric transport the CDW phase of the predicted Weyl semimetal Y2Ir2O7 because of a experiments. reduced de-pinning threshold voltage in longitudinal magnetic fields. The pairing of electrons and holes of opposite chirality and the cor- However, the phason current was found to be independent of B. To responding gapping of the Weyl cones induced by the CDW imply that obtain evidence for the axionic phason, it is therefore desirable to go the chiral anomaly is irrelevant for the single-particle magnetoelectric beyond these experiments. transport in the axion insulator state. We find, however, that the axionic In our study, we use the quasi-one-dimensional material (TaSe4)2I phason mode leads to a positive longitudinal contribution to the mag- (Extended Data Fig. 1a), which has recently been shown to be a Weyl netoconductance. This connection can be understood through a calcu- semimetal30 in its non-distorted structure (Fig. 1b). Its Fermi surface is lation based on a phenomenological effective-field theory for electrical entirely derived from Weyl cones with 24 pairs of Weyl nodes of opposite conductivity (Methods). Applying E‖B to the Weyl–CDW system leads chirality close to the intrinsic Fermi level, EF (10 meV < E − EF < 15 meV). At to an extra anomalous contribution from the chiral anomaly to the temperatures below the transition temperature of Tc = 263 K, (TaSe4)2I phason current, which can be identified as a signature of the axionic forms an incommensurate CDW phase, undergoing a Peierls-like transi- character of the CDW. Aligned E and B fields generate an additional tion31, which is accompanied by the opening of a gap of approximately collective chiral flow of charges with a rate that is proportional to E·B. 260 meV to single-particle excitations. In a recent study, the momentum The calculation yields a positive magnetoconductance contribution (k) space positions of the electronic susceptibility peaks, calculated 2 of σCDW,xx(B‖) = c1 + c2aχB‖ + c3 (aχB‖) in the axion insulator state, where from the Fermi pockets of the Weyl points, and the k = (222) satellite 2 c1, c2 and c3 are material-specific parameters, aχ = 1/(4π ) is the chiral reflections experimentally observed in XRD measurements of the CDW anomaly coefficient and B‖ denotes the component of B that is parallel phase of (TaSe4)2I have been found to match, indicating that the CDW to E. We note that, owing to the threshold voltage caused by the impurity nests the Weyl nodes and therefore breaks the chiral symmetry30. In

316 | Nature | Vol 575 | 14 November 2019 aed Data 80 K 20 2 Bardeen Ohmic 0 105 V ( V)

–20 80 K 1 Vth –2 02 I (mA) 104 I (mA)

b

5 3 0 10 0102030 0 V (V) V (V) d V /d I ( Ω ) f 2 –5 130 K 10

–20 020 80 K I (mA) 105 K ) 10 130 K c 2 –1 101 155 K 180 K

(V cm 205 K th 0 E 230 K V (V ) 100 255 K 1 280 K 180 K 305 K –2 –100 0 100 100 150 200 250 10–3 10–2 10–1 100 101 102 I (mA) T (K) V (V)

Fig. 2 | Propagation of the charge-density wave in (TaSe4)2I. a–d, Voltage– resistance dV/dI as a function of V at various temperatures T. f, Threshold current (V–I) characteristics at 80 K (a), 130 K (b), 180 K (c) and 80 K (d) without electric field Eth = Vth/L as a function of temperature T. L is the distance between magnetic field. In d, light dotted lines are fits to Bardeen’s tunnelling theory the voltage probes. The error bars denote the variation between the positive

(V > Vth), and green dotted lines denote linear Ohmic fits (V < Vth). e, Differential and negative threshold voltage, Vth.

addition, (TaSe4)2I has been previously shown to exhibit nonlinear for an axion insulator and can be used for the experimental observation electrical-transport characteristics32,33 in zero magnetic field upon of the axionic phason. crossing a threshold voltage Vth, which are associated with a current- Crystals of (TaSe4)2I grow in millimetre-long needles with aspect carrying sliding phason. Therefore, this material is a strong candidate ratios of approximately 1:10, reflecting the one-dimensional crystalline

a b Vth c d B⊥I B⊥I B⊥IB80 K ⊥I 30 80 K 80 K 105 0.0 0.0 0 T 0 T 9 V 12 V 15 V 3 T 6 T 1.5 T 1.5 T 18 V 21 V 24 V 27 V 9 T Δ (d I /d V ) (mS) Δ (d I /d V ) (mS) 0 3.0 T 3.0 T –0.1 –0.1 V (V) –9 –6 –3 0369 80 120 160 4 4.5 T 4.5 T d V /d I ( Ω ) 10 B (T) T (K) 6.0 T 6.0 T 7.5 T 7.5 T –30 g h 9.0 T 9.0 T 103 B||I 80 K B||I 3 T –2 02 0102030 0.6 I (mA) V (V) 6 T 9 V 9 T 0.5 12 V 0.5 V e f th 15 V B||I B||I 0.4 18 V 30 80 K 80 K 21 V 105 0.3 24 V 0 T 0 T

Δ (d I /d V ) (mS) 27 V 1.5 T 1.5 T Δ (d I /d V ) (mS) 0 3.0 T 3.0 T 0.2 V (V) 4 4.5 T d V /d I ( Ω ) 10 4.5 T 6.0 T 0.1 6.0 T ∝B2 7.5 T 7.5 T 0.0 –30 V = V 9.0 T 9.0 T 0.0 max 103 –2 02 0102030 –9 –6 –3 0369 80 120 160 I (mA) V (V) B (T) T(K)

Fig. 3 | Evidence for an axionic phason in (TaSe4)2I. a, b, V–I characteristic e, f, Corresponding V–I characteristics (e) and dV/dI (f) for magnetic fields B

(a) and dV/dI (b) at 80 K at various magnetic fields B perpendicular to parallel to I (B‖E). g, The longitudinal magnetoconductance Δ(dI/dV)B is well the current (I⊥B). c, d, Dependence of the differential conductance described by quadratic fits at low magnetic fields. h, Temperature dependence

Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T on the magnetic field B at 80 K and at various of Δ(dI/dV)B at the highest applied voltage, Vmax. voltages V (c) and on the temperature T for various magnetic fields (d).

Nature | Vol 575 | 14 November 2019 | 317 Article V single-particle state on a logarithmic scale as a function of T−1. For all a b th samples investigated, we observe an increasing ρ with decreasing T, B B 30 consistent with the ρ–T dependence of (TaSe ) I reported in the litera- M 105 M 4 2 ture32,33. The plot of the logarithmic derivative (Fig. 1c, right axis) versus I I 0° 0° −1 80 K 80 K the inverse temperature, T , exhibits a well pronounced peak at around 15° 15° 9 T 9 T the expected CDW transition temperature, Tc = 263 K. The saturation 0 30° 30° of the logarithmic derivative at a constant value a above T −1 demon- V (V) 104 v c

45° d V /d I ( Ω ) 45° strates that ρ/ρ0 follows the thermal activation law ρ/ρ0 ≈ exp[ΔE/(kBT)], 60° 60° allowing us to deduce the size of the single-particle gap ΔE in units of 75° 75° the Boltzmann constant, k . By averaging over the saturated range, we –30 B 90° 90° 3 obtain a value of av = 0.72 ± 0.04 and thus ΔE = (259 ± 14) meV, which is 10 31,32 –2 02 0102030 in excellent agreement with the literature . I (mA) V (V) Next, we investigate the motion of the CDW at zero magnetic field c 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. 0.6 B 80 K current I at four selected temperatures. Consistent with a sliding pha- 12 V M 9 T son mode, nonlinearity in the V–I curves appears below T at high bias 15 V I c 0.5 18 V currents, that is, high voltages (Extended Data Fig. 5). Accordingly, 21 V the V–I curves can be well represented by a simple phenomenologi- 24 V cal model based on Bardeen’s tunnelling theory, as suggested by pre- 0.4 27 V vious 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 resist- 0.3 ance dV/dI from the V–I curves and plot it versus the corresponding ∝–cos2M

Δ( d I /d V ) (mS) V for various values of T (Fig. 2e). For all temperatures investigated,

0.2 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 26 current . Vth is determined as the onset of the deviation from the zero 0.1 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 0.0 distance between the voltage probes) and its temperature depend- ence (Fig. 2f) are in agreement with the literature32. Further analysis 050 100 150 200 250 300 350 suggests that the nonlinear V−I characteristics are not the result of local M (°) Joule heating or a consequence of contact effects (Methods, Extended Fig. 4 | Angular dependence of the axial current. a, The angular dependence is Data Fig. 6). These observations provide evidence for a propagating, inferred from measurements of the V–I characteristics in a tilted B(φ) of 9 T at current-carrying CDW state. 80 K, where φ denotes the angle of B with respect to the applied current I. We now test the axionic nature of the phason. For this purpose, we meas- b, Dependence of dV/dI on the angle φ for voltages above Vth. c, Δ(dI/dV) at 9 T is ure the nonlinear V−I characteristics at fixed T, but now in the presence of plotted against φ for various voltages V. The measured data are reasonably well a background magnetic field. We apply a magnetic field between −9 T and represented by fits with cos2φ. The axial current peaks when φ → 0° and +9 T in steps of 1 T, oriented perpendicular and in parallel to the direction φ → 180°. 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 V−I characteristics and dV/dI can be seen within anisotropy of the material (Methods and Extended Data Fig. 1). We the measurement error of 0.1% for B perpendicular to I (E⊥B) across measured the electrical transport of five (TaSe4)2I samples (A, B, C, D, the whole magnetic-field and temperature ranges investigated. To E), with the electrical current I applied along the c axis of the crystals enable a direct comparison to the theory, we calculate the differential

(Methods). In all samples, the low-bias four-terminal electrical resistiv- magnetoconductance Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T. No Δ(dI/dV)B is ity ρ of the single-quasiparticle excitation was measured as a function observed (Fig. 3c, d) in the perpendicular-field configuration. However, of temperature T (Extended Data Fig. 1). The bias current-dependent when the magnetic field is aligned with I (E‖B) (Fig. 3e, f)), the V−I char- transport properties were then measured for samples A, B and E. All acteristics and dV/dI display a strong dependence on the magnitude of investigated samples show similar electrical-transport properties. At the magnetic field |B‖| above Vth and at low temperatures. Comparing

300 K, the ρ value of the crystals is around ρ0 = ρ(300 Κ) = 1,500 mΩ cm the electrical currents at fixed T, B‖ and V, we find a strong enhancement

(±400 mΩ cm) with an electron density of n(300 Κ) = (1.7 ± 0.1) × of I with increasing |B‖|. Because Vth itself is independent of |B‖| in any 1021 cm−3 and a Hall mobility of μ(300 Κ) = (2.3 ± 0.1) cm2 V−1 s−1 (Meth- direction up to ±9 T (Fig. 3b, f), the observed B dependence seems to ods and Extended Data Fig. 2), which are in good agreement with the originate from a field-dependent contribution to the phason current. 32 literature . Here, ρ0 is the average value of all samples investigated and As shown in Fig. 3g, h, we consistently observe a large positive Δ(dI/dV)B the uncertainty is the standard deviation. The uncertainties given for for E‖B and T ≤ 155 K (see Extended Data Figs. 7, 8). The profile of this n and μ originate from the errors of the Hall fits. From further analy- curve can be well described by a quadratic function below 7.5 T, accord- sis of the Hall measurements (Methods), we find that the Fermi level ing to the prediction for an axionic phason. To demonstrate the quad- in our samples is located precisely in the middle of the CDW gap and ratic behaviour at low magnetic fields, we perform second-order 2 therefore at the position of the initial Weyl cones in (TaSe4)2I (Extended polynomial fits, d12Bd+ B (d1 and d2 are free parameters) to the exper- 2 Data Fig. 4). The data presented in the main text were obtained from imental data and find that d12Bd≪ B (Extended Data Fig. 9). By com- measurements on sample A. parison to our theoretical transport model (Methods), the dominant

We first characterized the single-particle transport at zero mag- quadratic behaviour for all B‖ < 7.5 T investigated is consistent with the netic field (|B| = 0 T). Figure 1c (left axis) shows the ρ/ρ0 ratio of the estimated Fermi level in our samples near the position of the initial

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Nature | Vol 575 | 14 November 2019 | 319 Article

Methods 2  2  ∂S Nm(0) F ∂ ϕ ⁎ ^ e = 2 −⋅EkNv(0) Fe q − 2 B (5) Phenomenological theory of the magnetotransport in an ∂ϕ 4 ∂ t  4π  axionic CDW system In this work, we model the magnetotransport in a CDW–Weyl system To include the effects of damping, we follow Lee et al.35 and add a (that is, an axion insulator)1 using an effective-field theory. For simplic- phenomenological damping term to this equation. Because we are ity, we start with a quasi-one-dimensional Weyl semimetal with exactly interested in the dynamics of the sliding mode above the de-pinning two Weyl nodes—although the real structure of (TaSe4)2I is much more transition, we do not consider a phenomenological pinning term. Defin- complicated. We assume that the Fermi surface consists of discon- ing an effective damping rate Γ, we thus find nected pockets surrounding these two Weyl nodes. Furthermore, we 2  2  assume that the CDW ordering wavevector is commensurate with the Nm(0) F ∂ ϕ ∂ϕ ∗^ e 2 + Γ =⋅EkNv(0) Fe q − 2 B (6) Weyl node separation kq and use natural units. To simplify the discus- 4 ∂ t ∂t  4π  sion, we neglect the effects of impurity pinning. We then discuss the effects of impurity pinning by examining the modifications to the B = 0 We can solve this equation in the long-time limit to find the steady- conductivity. state solution

 2  Effective-field theory for the semi-classical limit, where many t ∗^ e ϕt()=+ϕ0 Ek⋅(Nv0) Fe q − 2 B (7) Landau levels are occupied Γ  4π  Our strategy is to introduce an order parameter Next, we look at the current density iϕ † Δce= c kk/2 +δ (1) −/()kkq 2+δ ()q δS j = (8) δA for the CDW order by decoupling the electron–phonon interaction Hamiltonian. Here is an operator that annihilates an electron carried by the CDW sliding mode, where A is the electromagnetic vector c(/kkq 2)+δ displaced by wavevector δk from the Weyl node centred at ±kq/2. By potential. By varying the action and restricting to position-independent integrating out the electrons and neglecting the fluctuations in the configurations of ϕ, we find gapped amplitude mode |Δ|, we arrive at an effective-field theory for  2  the phason mode ϕ of the CDW. Let us confine ourselves to zero tem- ∗ e ∂ϕ jk=(Nv0) Fe q − 2 B (9) perature. By neglecting the chiral charge of the Weyl nodes (we will  4π  ∂t add it later), we find from Anderson34 the action (working in units where ħ = 1) Finally, by combining equation (7) with equation (9), we find that

2 i  jσ=(ij B)Ej (10) mF 1 ∂ϕ  1 ^ 221 ^ S = N(0)−   + cϕ(kkqq×∇ )+ cϕ( ⋅∇ ) 0 2 ∫ 2 ∂t 2 ⊥ 2    (2) with the conductivity tensor given by  ∗ vF ^ 3 −2e kEq ⋅dϕx dt  2   2  mF  1 ∗∗^ e ^ e σij()Bk= Nv(0) FeNqq− Bk  (0)veF − B (11) Γ 4π2 4π2  ij  Here we have defined In particular, the magnetoconductivity is given by Δ 2 mF =4 2 (3) B 2 λω σσij()−(ij 0) e ^ ^ k q =− kB+ Bk 2 ∗ ()qi,,jiqj σij(0) 4π Nv(0) Fe (12) with λ the electron–phonon coupling, and ω the phonon energy at 2 kq e wavevector k . N(0) is the density of states at the Fermi level, v is the + BBij q F 4π2((Nv0) e∗)2 ^ F Fermi velocity in the direction of the CDW wavevector, k q, and e* is the screened electron charge. The velocities c⊥ and c correspond to the transverse and longitudinal (phason) velocities, respectively. The final We note that there are two types of contributions in equation (12). term in (2) corresponds to the sliding (Froelich) mode, excited by an The first term depends on the relative orientation of the electric and electric field aligned with the CDW ordering vector. magnetic fields relative to the CDW wavevector, and originates from Additionally, we find from the chiral anomaly the contribution34 a mixing between the chiral-anomaly (B dependent) and conventional (B independent) contributions to the CDW dynamics in equations (7) e2 3 (4) and (9). The quadratic in B contribution to the magnetoconductivity, SNc = 2 (0)⋅∫ ϕxEBddt 4π on the other hand, has a purely chiral-anomaly origin. Although it cor- responds to current carried by the sliding mode, it originates entirely This contribution to the action emerges from the chiral anomaly from the underlying Weyl semimetal. Hence, like in the Weyl semimetal, in the undistorted Weyl semimetal. As shown in ref. 1, the formation this contribution to the magnetoconductivity is independent of the of the CDW causes the amplitude of the chiral anomaly to depend Weyl separation (that is, CDW ordering) vector. on the phason field. Let us now consider the effect of a uniform We now note that in the idealized case in which the unordered phase external magnetic field. Because we are interested in the response of is a perfect Weyl semimetal, N(0) → 0. In this case, the second term in the CDW to uniform fields, we restrict ourselves to spatially homo- equation (12) dominates over the first, and we recover the characteristic 36–38 geneous configurations ϕ(x, t) = ϕ(t). By varying the action S = S0 + Sc quadratic positive magnetoconductivity of a Weyl material . In par- with respect to ϕ, we find from the equations of motion that ϕ ticular, in this limit we also recover the characteristic cosine-squared satisfies dependence of the magnetoresistance on the angle between E and B. temperature T. This is derived by assuming that pinning by impurities

More generally, the quadratic term in equation (12) dominates when 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 ∗ 2 Nv(0)eFe ≪ B (13) 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.

Hence, we expect a positive quadratic magnetoconductivity in this Because Eg and e* do not depend on T, the temperature dependence regime. of V0 provides a measure for the temperature dependence of ξ, which

We note that, unlike recent proposals for ‘axion insulators’ in Weyl generally decreases with increasing T. σCDW,0 is a measure of the num- CDW materials1,2, our system preserves time-reversal symmetry and ber of carriers forming the CDW and has been observed to increase hence has multiple pairs of Weyl fermions that are coupled in the CDW with increasing T. The validity of equation (15) for CDW systems has phase. This forces responses such as the anomalous Hall conductance been confirmed by many experiments, by fitting the measured I–V to be zero. Nevertheless, the negative magnetoresistance due to the characteristics to the current Itotal(V) = ISP(V) + ICDW(V), corresponding chiral phason channels is expected to persist in the limit of long phason to equation (15), with ISP(V) = GspV accounting for the single-particle lifetime (that is, weak intervalley scattering). contribution and ICDW(V) = I0(V − Vth)exp(−V0/V) for the sliding CDW.

Gsp (the single-particle electrical conductance), I0 (corresponding to

Microscopic argument for the quantum limit, where only the σCDW,0) and V0 are the free fitting parameters. Importantly, there is no

0-th Landau level is occupied added conductivity below Vth and hence Itotal(V) = ISP(V) for V < Vth. Vth To gain some insight into how the sliding mode can influence the mag- can be determined independently, as shown in Extended Data Fig. 5. netoresistance in the quantum limit, we take our Weyl–CDW system By fitting Itotal(V) to our data measured at zero magnetic field for V > Vth ^ and apply a large magnetic field along the kq direction. We take the (Extended Data Fig. 5), we find excellent agreement with Bardeen’s magnitude of B to be large enough so that only the zero-th Landau level tunnelling theory. The extracted value of V0 decreases and I0 increases is occupied when electron–phonon interactions are neglected. In this with increasing temperature, in agreement with previous reports for case, we have a truly quasi-one-dimensional system, with conserved other CDW systems. By fitting our measured I–V characteristics with ^ momentum Q =⋅kkq; each state is largely degenerate, with the degen- the corresponding current expression Itotal(V) = ISP(V) + ICDW(V), with eracy given by the total flux |B|A/(2π) through the system (measured ISP(V) = GspV and ICDW(V) = I0(V − VT)exp(−V0/V), using the single-particle in units of the flux quantum, with A the cross-sectional area). We now electrical conductance Gsp and I0 and V0 as the free fitting parameters, reintroduce the electron–phonon interaction; this causes the forma- we find excellent agreement with Bardeen’s tunnelling theory (Fig. 2d, tion of a CDW ground state. Because the Fermi points of the Weyl nodes Extended Data Fig. 4). disperse entirely in the lowest Landau level, the low-energy physics of Impurity pinning in real materials, such as in our samples, has impor- this transition matches that of a true one-dimensional CDW transition, tant consequences on the experimental detection of the axionic provided that we multiply all results by the Landau level degeneracy response of CDWs. As mentioned above, the zero-field conductivity |B|A/(2π). In particular, Lee et al.28 showed that the sliding mode of the tensor will develop a nonlinear dependence on V. Additionally, the CDW contributes e2n/(m*Γ) per channel to the electrical conductivity single-particle current will always be measured in parallel to the sliding ^ in the kq direction, where m* = m(1 + mF) is the electron effective mass, CDW in experiments. By putting these facts together, we see from equa- tion (12) that the measured magnetoconductance ΔI n is the one-dimensional charge-carrier density and Γ is the phenom- Δ(VB) ∝(σBxx )−σxx(0) enological damping rate (phonon lifetime). By multiplying this by the will become non-Ohmic and will in general depend on V. Hence, the number of channels we find in the quantum limit and for current and amplitude of the characteristic cosine-squared dependence of the electric fields aligned with the magnetic field (and therefore CDW) magnetoconductivity on the angle between E and B will also become V-dependent. e2n σ ()BB^^BB→ || (14) ij ij mΓ∗

(TaSe4)2I crystal structure and single-crystal growth which is analogous to the magnetoresistance in the ultra-quantum (TaSe4)2I is a quasi-one-dimensional material with a body-centred 38 limit of a Weyl semimetal . tetragonal lattice (Fig. 1b). The Ta atoms are surrounded by Se4 rectan- gles and form chains aligned along the c axis. These chains are separated − Comment on real material systems by I ions. Single crystals of Ta2Se8I were obtained via a chemical vapour Having justified the origin of the negative magnetoresistance in a transport method using Ta, Se and I as starting materials40. A mixture simplified model, we now discuss how pinning effects will change the with composition Ta2Se8I was prepared and sealed in an evacuated phenomenology. First, because CDWs are always pinned by impurities quartz tube. The ampoule was inserted into a furnace with a tempera- in real material systems, sliding only occurs upon reaching a certain ture gradient of 500 to 400 °C with the educts in the hot zone. After two threshold electric field, that is, the threshold voltage Vth. In addition, weeks, needle-like crystals had grown in the cold zone (Extended Data transport measurements always also involve single-charge carriers Fig. 1). Energy-dispersive X-ray spectroscopy reveals a Ta:Se:I ratio of thermally exited above the Peierls gap ΔE, which carry electrical cur- 17.75:72.95:9.3, which matches the theoretical ratio of 18.18:72.73:9.09 rent in parallel to the sliding CDW mode. Therefore, the B = 0 electrical within the measurement precision of 0.5. conductivity (that is, the electrical resistivity) is in general non-Ohmic and depends on the applied bias voltage V. Below the Peierls transition, Electrical-transport measurements the total electrical conductivity for a CDW system in zero magnetic All electrical-transport measurements were performed in a variable- 39 field (σij,total) is given by Bardeen’s tunnelling theory : temperaturecryostat (Quantum Design) equipped with a 9-T magnet. To avoid contact resistance effects, only four-terminal measurements  Vth   V0  were carried out. To reduce any possible heating effects during cur- σσij,total =+SP σCDW,01− exp−  (15)  V   V  rent sweeps, the samples were covered in highly heat-conducting, but electrically insulating, epoxy32. The temperature-dependent electri-

Here, σSP denotes the Ohmic conductivity contribution of the thermally cal resistivity curves were measured with a standard low-frequency exited single-charge carriers, Vth is the threshold voltage and σCDW,0 and (f = 6 Hz) lock-in technique (Keithley SR 830), by applying a current

V0 are voltage-independent parameters that usually depend on the of 100 nA across a 100-MΩ shunt resistor. The applied bias current Article I = 100 nA for these measurements was chosen such that the corre- electrical resistivity of ρ = 1,562 μΩ cm at 300 K. Both n and μ are in 33 sponding electric field E = IR/L did not exceed the threshold field Eth > E, good agreement with the values previously reported for (TaSe4)2I. where L is the distance between the voltage probes. The determina- Using the effective mass m* = 0.4m0 (m0 is the free electron mass) 32 tion of Eth is explained later. The bias current-dependent transport obtained from ARPES measurements for T < Tc and a single-band properties were then measured on samples A, B and E because their model, we subsequently determine the Fermi level position EF in our closely adjacent electrical contacts of L < 3 mm allowed us to cross the sample with respect to the conduction band edge. threshold field Eth with experimentally accessible currents (IR < 60 V) For a bandgap material, the carrier concentration is given by in our setup. The bias current-dependent transport experiments were  E  limited to temperatures of T ≥ 80 K because Eth increases exponentially F nN=ec xp  (16) with decreasing T. Given the minimal contact resolution of >0.5 mm kTB  in our bulk samples, no switching could be achieved in our samples ∗ 3/2 2πm kTB below 80 K, because the corresponding threshold voltages exceeded where Nc =2 and h is the Planck constant. By transform- ()h2 the measurable voltage range (<60 V). The d.c. voltage–current char- ing equation (16), we obtain a description for EF that only depends on 14 acteristics were measured with a Keithley 6220 current source (10 Ω T, m and n: output impedance) and a Keithley 2182A/E multimeter.  ∗ −3/2 1 2πm kTB  First-principles calculations EnF =ln   kTB (17)  2  h2   For the ab initio investigations we employ density-functional calcula-   tions (DFT) as implemented in the VASP package41. We use a plane-wave basis set and the generalized-gradient approximation for a description The result of equation (17) as a function of T is shown in Extended Data 42 of the exchange-correlation potential . As a second step we create Fig. 2j. Within the error bar, the estimated EF is independent of the tem- 43 Wannier functions from the DFT states using Wannier90 and extract perature for T < Tc, giving a mean value of EF = 129 ± 10 meV below the the parameters for a tight-binding Hamiltonian to further analyse the conduction band edge of the CDW gap. By comparing this number to band structure. 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

Estimate of the Fermi level position from Hall measurements of the CDW gap (EF = ½ΔE). As shown previously via DFT calculations To estimate the Fermi level position in our samples, we performed and ARPES measurements, the CDW gap opens symmetrically around Hall measurements on sample D. Because the single-particle resis- the crossing points of the initial electronic band structure31. Therefore, tivity of all our samples was similar throughout the examined tem- the Fermi level position is in the centre of the initial Weyl cones. perature range, the Hall measurements are representative of the whole set of samples presented here. The device used for the Hall Excluding other intrinsic and extrinsic origins for the nonlinear measurements is shown in Extended Data Fig. 2a. Contacts 1 and 4 V–I curves are used for electrical current injection, contacts 2 and 3 are used The following analysis suggests that the nonlinear V–I characteristics to probe the longitudinal voltage along the sample, and contacts 5 are not the result of local Joule heating or a consequence of contact and 6 are used to measure the Hall voltage V5,6 across the sample. effects, but originate from an electrically driven intrinsic process in

The magnetic field B is perpendicular to the measurement plane. the (TaSe4)2I crystals, consistent with a sliding CDW. First, we exclude The Hall resistance as a function of the magnetic field B is calculated thermally driven switching due to Joule heating. Thermally driven using RH = V5,6/I at fixed temperatures T. We chose the electrical cur- switching corresponds to a local rise of the temperature in the (TaSe4)2I rent I applied at each temperature to be small enough to probe only sample above Tc. Such a temperature rise would be determined by the single-particle transport but high enough to resolve the Hall signal power dissipated in the sample and the electronic circuitry. In the worst-

V5,6. In particular, we used I = 1 μA at 125 K, I = 10 μA at 155 K and case scenario, all of the V–I Joule heating power is dissipated within

185 K, I = 100 μA at 215 K and 245 K, and I = 2 mA at 300 K. Although the (TaSe4)2I crystals. Hence, the dissipated power necessary for the the data are quite noisy, we observe a strong B dependence of RH switching would approach zero as T → Tc from below. However, the elec- 2 (Extended Data Fig. 2c–h). Below 125 K, the longitudinal resistance trical power Pth = Vth /[dV/dI(Vth)] obtained at Vth from our experiments of the sample became too high to measure the Hall response. As (Extended Data Fig. 6a) does not approach zero as T → Tc. Instead, Pth reported previously33, the Hall resistance at low temperatures is decreases with decreasing T. This is precisely the reverse of what we linear with B. At 300 K, however, RH becomes nonlinear with B at would expect from thermally driven switching. Second, we rule out high fields, which is typical for semiconductors and insulators at contact effects as the cause of the nonlinear threshold behaviour in our elevated T, because both electrons and holes become thermally experiments. As an example, we show Vth as a function of the distance of activated across the gap. At low temperatures, the band closer to the voltage probes L at 80 K in Extended Data Fig. 6b for L = 1 mm, 2 mm the Fermi level dominates. As discussed previously, even in such a and 3 mm. The observed linear Vth–L dependence implies that the transi- one-dimensional material as (TaSe4)2I, the Hall effect is well defined tion is driven by the electric field and not by the absolute magnitude of because, to the first order in B, the Lorentz force does not depend I, which justifies our definition of Eth = Vth/L. It also demonstrates that on the transverse effective mass33. Recalling the standard expression the transition originates from a bulk effect, because the contacts in for the low-field Hall conductivity (single-charge-carrier approxima- all of our devices are of identical size; hence, switching in the contacts −1 −1 tion), we estimate the carrier concentration n = (dRH/dB) (e*d) as a would have no dependence on the length. Therefore, any change in function of T from the slope dRH/dB of the linear fits to the magnetic the switching properties must result from the (TaSe4)2I crystal itself. 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 Excluding other origins for the positive differential decreasing T from 300 K to 125 K, consistent with the increasing magnetoconductance longitudinal single-particle resistivity (Extended Data Fig. 2b). At the Here, we address the concern that the experimentally detected posi-

CDW transition temperature Tc a jump in n is observed, consistent tive differential magneto-conductance Δ(dI/dV)B in (TaSe4)2I could with the opening of a single-particle gap. At 300 K, the estimated arise from an alternative origin than the axionic nature of the CDW. Sev- 21 −3 carrier concentration is n = (1.7 ± 0.1) × 10 cm , leading to a mobility eral intrinsic and extrinsic origins of the positive Δ(dI/dV)B in (TaSe4)2I of μ = (2.3 ± 0.1) cm2 V−1 s−1, using the Drude model μ = (e*nρ)−1 with an can be excluded from the experimental observations and analysis. (i) Inhomogeneous current distribution caused by quenched disorder independence of Vth from B (Fig. 3a, e) excludes any possible effects and the current-jetting effect are ruled out. The elongated geometry from a reduction of Vth. of the samples and the silver paint contacts used for current injection, encasing the entire ends of the wires, provide homogeneous cur- rent injection through the entire bulk of the samples44,45.Theoretical Data availability simulations with similar sample geometries and even higher carrier All data generated or analysed during this study are available within mobility have shown44 that magnetic fields much larger than 9 T are the paper and its Extended Data files. Reasonable requests for further required for the onset of current-jetting effects. To experimentally source data should be addressed to the corresponding author. test for inhomogeneous current distribution in our wires, we use a special contact geometry on sample E (Extended Data Fig. 10). The 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. voltage contacts on sample E were designed as point contacts, locally First-principles calculations were funded by the US Department of Energy through grant probing the voltage drop across two different edges of the sample. As number DE-SC0016239. B.A.B. acknowledges additional support from the US National in all other devices used in this study, the current injection contacts 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 1 and 6 cover the entire ends of the (TaSe4)2I crystal. We measured the Schmidt Fund for Innovative Research. Z.W. acknowledges support from the National simultaneously the potential difference Vi–j between the two pairs Thousand-Young-Talents Program, the CAS Pioneer Hundred Talents Program and the National Natural Science Foundation of China. 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 per- Author contributions B.A.B., C.F. and J.G. conceived the experiment. N.K., C. Shekhar and pendicular (I⊥B) and longitudinal (E‖B)I measurement configura- Y.Q. synthesized the single-crystal bulk samples. J.G., S.H. and C. Schindler fabricated the tion (Extended Data Fig. 10b–g, i–n) The obtained curves are nearly electrical-transport devices. J.G. carried out the transport measurements with the help of identical and display only small differences within the measurement 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 error (up to 10%). The agreement across the two pairs of contacts work. All authors contributed to the interpretation of the data and to the writing of the shows that distortions of the current path are minimal and that the manuscript. current distribution is uniform within the sample. This implies that Competing interests The authors declare no competing interests. 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 Additional information Correspondence and requests for materials should be addressed to J.G. large temperatures at which the longitudinal Δ(dI/dV)B occurs, show Peer review information Nature thanks Jian Wang and the other, anonymous, reviewer(s) for that neither quantum interference effects of the condensate nor of their contribution to the peer review of this work. the single quasiparticle excitations play an important role. (iii) The Reprints and permissions information is available at http://www.nature.com/reprints. Article

Extended Data Fig. 1 | (TaSe4)2I crystal structure, growth, device and e, Scanning electron microscope image of a crystal. f, Typical device used for transport characterization. a, Crystal structure of (TaSe4)2I. b, Sketch of the electrical-transport measurements. g, h, Single-particle resistance of samples growth principle. A temperature gradient (temperatures T1 > T2) is imposed on B, C, D and E. The electrical resistance R, normalized by R0 = R(300 K) (g) and its −1 an evacuated quartz ampule, which contains (TaSe4)2I powder at T1. The logarithmic derivative (h) as a function of T , where T is the temperature. i, evaporated (TaSe4)2I diffuses towards the area with temperature T2 and Single-particle gaps of all (TaSe4)2I samples investigated. The error denotes the condenses into single crystals. c, Optical micrograph of the as-grown (TaSe4)2I fitting error of 1σ. The dotted line displays the mean value of all samples. crystals. d, Distribution of Weyl points in momentum (k) space of chirality ±χ. Extended Data Fig. 2 | Hall measurements and Fermi level position of sample measured data (black) are fitted linearly (red) to extract the carrier D. a, Device used for the Hall measurements in a magnetic field B. b, Single- concentration n (i). The error bars denote the error of the linear fits to 1σ. particle longitudinal resistivity ρ versus temperature T. c–i, Single-particle Hall j, Estimated Fermi level position below Tc. ΔE is the single-particle gap, resistance RH at 125 K (c), 155 K (d), 185 K (e), 215 K (f), 245 K (g) and 300 K (h). The obtained from Extended Data Fig. 1. Article

Extended Data Fig. 3 | V–I characteristics of sample A at zero magnetic field temperatures below 180 K, we start to observe nonlinearity. This nonlinearity and selected temperatures. a–j, V–I characteristics at 80 K (a), 105 K (b), 130 K becomes even more apparent in the dV/dI curves shown in Fig. 2d, where a (c), 155 K (d), 180 K (e), 205 K (f), 230 K (g), 255 K (h), 280 K (i) and 305 K (j). These deviation from the linear behaviour is already seen at 230 K. data were used to calculate the differential resistance dV/dI curves in Fig. 2d. At Extended Data Fig. 4 | Fitting the nonlinear V–I characteristics of sample A at 180 K (e), 205 K (f) and 230 K (g). h, i, m (h) and the threshold voltage Vth zero magnetic field. a–g, V–I characteristics (black line), a linear fit Isp(V) = mV extracted from Fig. 2 were employed to extract the fit parameters. j, I0. k, V0. 6 (green line) and a fit with the Bardeen model I(V) = Isp(V) + ICDW(V) (red line), The error bars in h–k denote the error of the fits to 1σ. where ICDW(V) = I0(V − VT)exp(−V0/V) at 80 K (a), 105 K (b), 130 K (c), 155 K (d), Article

Extended Data Fig. 5 | Dependence of the switching voltage of sample A on voltages Vth shown in Fig. 2g. We define Vth as the voltage at which the last data the contact separation. a, b, V–I characteristics (a) and differential resistance point of d2V/dI2 (marked by the vertical line) touches the zero baseline before dV/dI (b) at 80 K and zero magnetic field B for various contact lengths L on the global minimum upon enhancing V. sample A. c–e, Second derivatives d2V/dI2 used to estimate the threshold Extended Data Fig. 6 | Testing the origin of the nonlinear V–I curves of shown at 80 K (the line denotes a linear fit) demonstrate that the observed 2 sample A. a, The increasing Joule heating power Pth = Vth /[dV/dI(Vth)] at the effects are intrinsic to the (TaSe4)2I crystals. threshold, as a function of increasing T. b, The linear dependence of Vth on L Article

Extended Data Fig. 7 | Symmetry and temperature dependence of the V–I Fig. 3. c, d, Corresponding V–I characteristics (c) and dV/dI (d) in magnetic characteristics in the magnetic field of sample A. a, b, V–I characteristics (a) fields parallel to the applied current. e–g, V–I characteristics at 80 K (e), 105 K and differential resistance dV/dI (b) at 80 K for sample A in magnetic fields (f) and 130 K (g). perpendicular to the applied current, but opposite to the magnetic field in Extended Data Fig. 8 | Bias-dependent data of samples A and B at 105 K. The 105 K and at various voltages V for sample A. d, e, Corresponding V–I two samples have similar contact separation. a, b, V–I characteristics (a) and characteristics (d) and dV/dI (e) at various magnetic fields B applied differential resistance dV/dI (b) at various magnetic fields B applied perpendicular to the current direction (I⊥B). f, Magnetic field dependence of perpendicular to the current direction (I⊥B) for sample A. c, Magnetic field the magnetoconductance Δ(dI/dV)B for sample B. dependence of the magnetoconductance Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T at Article

Extended Data Fig. 9 | Fitting parameters of the second-order polynomial fits to the experimental longitudinal Δ(dI/dV)B‖ shown in Fig. 3g.

a, d1. b, d2. Extended Data Fig. 10 | Homogeneity test of the current distribution on f, g, Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T for I⊥B (f) and for I‖B (g). h, Second sample E. a, First configuration. I is injected from contacts 1 and 6 and V is configuration. V is measured between contacts 4 and 5. i–n, Corresponding measured between contacts 2 and 3. b–e, V–I characteristics (b, d) and dV/dI characteristics (i, k) and dV/dI (j, l) for I⊥B (i, j) and for I‖B (k, l).

(c, e) at 80 K and at various magnetic fields B for I⊥B (b, c) and for I‖B (d, e). m, n, Δ(dI/dV)B = (dI/dV)B − (dI/dV)0 T for I⊥B (m) and for I‖B (n).