Chiral Anomalies Induced Transport in Weyl Metals in Quantizing Magnetic ﬁeld

PHYSICAL REVIEW RESEARCH 2, 033511 (2020)

Chiral anomalies induced transport in Weyl metals in quantizing magnetic ﬁeld

Kamal Das,* Sahil Kumar Singh,† and Amit Agarwal ‡ Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

(Received 24 March 2020; accepted 10 August 2020; published 29 September 2020)

Weyl metals host relativistic chiral quasiparticles, which display quantum anomalies in the presence of external electromagnetic fields. Here, we study the manifestations of chiral anomalies in the longitudinal and planar magnetotransport coefficients of Weyl metals, in the presence of a quantizing magnetic field. We present a general framework for calculating all the transport coefficients in the regime where multiple Landau levels are occupied. We explicitly show that all the longitudinal and planar transport coefficients show quantum oscillations which are periodic in 1/B. Our calculations recover the quadratic-B dependence in the semiclassical regime and predict a linear-B dependence in the ultraquantum limit for all the transport coefficients.

DOI: 10.1103/PhysRevResearch.2.033511

I. INTRODUCTION with the magnetic field gives rise to charge and energy imbalance between the opposite chirality Weyl fermions [29]. Weyl metals (WMs) host massless chiral relativistic quasi- However, the bulk of the theoretical work till date has particles which show very interesting and novel phenomena been focused on the semiclassical transport regime which [1–15]. One of the most interesting aspects of massless rel- displays a quadratic-B dependence of all transport coefficients ativistic chiral fluids, in quantum field theory is the break- [6,8,34,41–48] or only in the ultraquantum regime [18](see down of the chiral gauge symmetry in the presence of an Fig. 1). Recently, a unified framework for calculating the elec- external electromagnetic field [16–18]. This results in the trical magnetoconductivity in all the distinct transport regimes chiral anomaly (CA) which manifests in the nonconservation have been developed in Refs. [49–51]. So it is natural to of chiral charge [16–19]. A condensed matter realization of ask what happens to the other magnetotransport coefficients. this was first explored in the lattice theory of Weyl fermions Can all the distinct transport coefficients be explored within a by Nielsen and Ninomiya in 1983 [18]. They predicted that unified framework? the CA will give rise to a positive longitudinal magneto- In this paper, we attempt to answer these and other related conductivity, which is linear in the magnetic field strength questions. Here, we generalize the framework of Refs. [50,51] (B), for ultrahigh magnetic field in the diffusive limit. With for calculating all the magnetotransport coefficients in the the recent realizations of WM [20], there have been sev- regime where multiple Landau levels are occupied and con- eral experiments which report positive magnetoconductivity nect them to the different CAs. We explicitly show that all the or negative magnetoresistance and attribute it to the CA longitudinal and planar transport coefficients show quantum [1–4,21–24]. oscillations which are periodic in 1/B. Our calculations re- Relativistic chiral fluids in a gravitational field also display cover the quadratic-B dependence in the semiclassical regime the mixed chiral-gravitational anomaly, which results in non- and predict a linear-B dependence in the ultraquantum limit conservation of the chiral energy [25,26]. This manifests in for all the magnetotransport coefficients. The rest of the paper the magnetothermal experiments in the form of positive mag- is organized as follows: In Sec. II, we discuss the Landau netothermopower and positive magnetothermal conductivity quantization in WMs, and in Sec. III, we connect the anoma- [26–29], both of which have also been observed in recent lous magnetotransport in WMs to different CAs. In Sec. IV, experiments [30,31]. In addition to their manifestations in we present the general formalism of calculating all the CA- longitudinal magnetotransport, CAs have also been shown to induced magnetotransport coefficients. In Sec. V, we calculate give rise to the planar Hall effects in all transport coefficients the longitudinal magnetotransport coefficients, followed by [29,32–40]. Recently, we predicted another anomaly, the ther- the exploration of the magnetotransport coefficients in the mal chiral anomaly in which a temperature gradient collinear planar Hall setup in Sec. VI. We discuss the experimental pos- sibilities in Sec. VII and summarize our results in Sec. VIII. *[email protected] † [email protected] II. LANDAU QUANTIZATION IN WEYL METAL ‡[email protected] In the presence of a magnetic field, the Hamiltonian of a Published by the American Physical Society under the terms of the single Weyl cone (of chirality s), after Peierls substitution, is Creative Commons Attribution 4.0 International license. Further given by distribution of this work must maintain attribution to the author(s) s and the published article’s title, journal citation, and DOI. Hˆ = svF σ · (pˆ + eA). (1)

2643-1564/2020/2(3)/033511(11) 033511-1 Published by the American Physical Society DAS, SINGH, AND AGARWAL PHYSICAL REVIEW RESEARCH 2, 033511 (2020)

In contrast, all the n 1 LLs are achiral (support both right and left movers), and they play an important role in quantum oscillations. Each of the LLs is highly degenerate and the degeneracy = / π 2 is specified by D 1 (2 lB ). The density of states (DOS) of the LL spectrum is shown in Fig. 2(c). The group velocity of the quasiparticles in these LLs is given by s ∂ −svF n = 0, vs = n = (3) nz ∂ 2 /s . h¯ kz h¯vF kz n n 1 FIG. 1. (a) The three different magnetotransport regimes. In Here, s → s (s → s ) for the positive (negative) energy the semiclassical regime (purple area), multiple Landau level are n n+ n n− branches. The carriers of the lowest LLs have a constant occupied, but the impurity scattering or thermal smearing makes ω < /τ velocity, and are either left movers (for s = 1) or right movers them indistinguishable (¯h c kBT orh ¯ ). In the quantum oscil- =− lation regime (blue), multiple Landau level are occupied and they (for s 1). Next, we discuss the origin of quantum anoma- are distinguishable, while in the ultraquantum regime (green), only lies from the lowest chiral LLs and explore their manifestation in electric and thermal magnetotransport experiments in WM the lowest Landau level is occupied (¯hωc >μ). (b) The magnetic field dependence of the magnetoconductance in the three transport [25,27,29,41,52]. regimes for a WM. III. EQUILIBRIUM CURRENTS AND COEFFICIENTS OF CHIRAL ANOMALIES Here, −e is the electronic charge, vF is the Fermi velocity, σ = (σx,σy,σz ) is a vector composed of the three Pauli One simple way to understand the origin of CAs in matrices, and A is the vector potential corresponding to the WM is to calculate the equilibrium (no externally applied magnetic field B = ∇ × A. Considering the magnetic field bias voltage or temperature gradient) current, in the pres- along the z direction and using the Landau gauge with A = ence of a magnetic field. The equilibrium charge and en- (−By, 0, 0), it is straightforward to calculate the energy spec- ergy current for each Weyl node is given by { js , js }= e,eq E,eq trum of the Hamiltonian in Eq. (1). The Landau level (LL) dkz s {− ,s} s s D n 2π vnz e n fn . Here, fn is the Fermi-Dirac distri- energy spectrum is given by β s −μ s ≡ / + ( n ) bution function for the nth LL: fn 1 [1 e ] with −sh¯v k n = 0, β ≡ 1/k T , and μ denotes the chemical potential. These s F z B ± = (2) s n ± 2 + ω 2 . expressions for the current are valid for both the n+ and the (¯hvF kz ) 2n(¯h c ) n 1 s μ n− branches, which are chosen depending on being positive = Here, n denotes the LL index, kz is the z component of the and negative, respectively. For all the LLs with n 0, the crystal momentum, and we have√ defined the cyclotron fre- integrand is an odd function of kz, and thus their contribution quency, ωc = vF /lB with lB = h¯/(e|B|) being the magnetic to the total charge and energy current is identically zero. Thus, length scale. In the rest of the paper, we will use B =|B|.The only the n = 0 LL can contribute to the equilibrium charge energy spectrum of Eq. (2) is shown in Figs. 2(a) and 2(b).We and energy current. To evaluate this contribution, we will con- emphasize that the lowest LL are chiral in nature, i.e., right sider the chemical potential to be positive (μ>0). Evaluating (left) movers for negative (positive) chirality node, and will these expressions using integration by parts [28,29], we obtain play a crucial role in the CAs [18] discussed in this paper. s =− μCs + Cs , je, e kBT B (4) eq 0 1 Cs Cs s 2 0 s 2 2 2 jE, = μ + μk T C + k T B. (5) eq 2 B 1 B 2 s Here, the coefficients Cν for ν ={0, 1, 2} are given by s − μ ν ∂ s s e dkz s n fn Cν = v − . (6) 2πh¯ 2π nz k T ∂s n B n μ> s → s For our choice of 0, we have n n+. A brief derivation of Eqs. (4) and (5) is sketched in Appendix A. We emphasize here that Eqs. (4)–(6) are exact expressions and these do not require any approximation. In Eq. (6), only the chiral lowest = Cs ∝ FIG. 2. The Landau level energy spectrum in a WM for (a) pos- LLs (n 0) contribute to the sum and we have ν s,the itive and (b) negative chirality Weyl node, respectively. The lowest chirality of the Weyl node. Thus, the existence of the chiral LLs disperse linearly and are chiral in nature. Here, k˜z ≡ h¯vF kz/μ. LLs is an essential ingredient to obtain the nonzero charge and (c) The corresponding density of states (DOS) of the LL spectrum. energy currents for each node even in equilibrium. These chi- The DOS is scaled by D/(hvF ). The DOS for the lowest (n = 0) ral currents are in turn related to the CAs in WMs [27,29,52]. LL is constant, and it increases as the energy moves away from the We had earlier derived equations similar to Eqs. (4) and (5) Weyl nodes. The oscillation in the DOS will also manifest in all the in the semiclassical regime, where the Berry curvature of the transport coefficients. Weyl nodes played an important role [29].

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formalism, the nonequilibrium distribution function (NDF), s gn, within the relaxation time approximation, satisfies the following equation [29,50]: gs − gs gs − f s ∂ s + ˙ s · ∇ s + s · ∇ s =− n ¯n − ¯n n . t gn kn kgn r˙n rgn (8) τ τv s Here,g ¯n is the local equilibrium distribution function consid- ered to be the Fermi function for the nth LL with a node- dependent chemical potential μs and temperature T s. The first term in the right-hand side of Eq. (8) represents the relaxation of the NDF to the local equilibrium through the intranode scattering rate 1/τ. The intranode scattering does not alter the number of carriers in the respective node, and its impact FIG. 3. The temperature dependence (x = βμ) of the dimen- is similar to that in other metals as well. In contrast, the sionless functions associated with the CAs. Note that in x →∞ → μ> second term represents the internode scattering with a relax- or T 0 for a finite 0 limit, these functions are constant as /τ shown in the orange shaded region. In this region, the coefficient of ation rate of 1 v, which attempts to undo the impact of the chiral imbalance [Eqs. (9) and (10)] and restore a steady-state thermal chiral anomaly F1 → 0. However, it has a finite value in the finite x regime. For this paper, we will mostly focus on the regime carrier distribution function. In a typical WM with broken μ>k T or x > 1, which is shaded in blue, and on the μ k T time-reversal symmetry, the Weyl nodes are separated in the B B μ regime, shaded in orange. momentum space. If we are in a regime of small so that the Fermi wave vector is smaller than the separation of the Weyl nodes, then we have τ τ [8]. Cs Cs Cs v The quantities 0, 2, and 1 are known as the coefficient The idea of a steady state in the presence of CAs and of the chiral (or axial) anomaly [41], the coefficient of the internode scattering becomes more evident from the continu- mixed chiral (or axial) gravitational anomaly [25,27,52] and ity equations of particle number and heat density for electric the coefficient of the thermal chiral (or axial) anomaly [29], field and temperature gradient applied along the direction of Cs respectively. We emphasize that the coefficient 1 vanishes magnetic field. Integrating Eq. (8) over all the states in a single βμ →∞ in the limit and has been relatively unexplored. cone, we obtain the particle number conservation equation Evaluating Eq. (6), we obtain (within the linear response) to be ⎧ F βμ ν = , (7a) ⎨⎪ 0( ) 0 ∂N s N s − N s e + ∇ · s + Cs =− 0 . Cs =− r Jn eEB 0 (9) ν s F1(βμ) ν = 1, (7b) ∂t τ 4π 2h¯2 ⎩⎪ v F2(βμ) ν = 2. (7c) ∇ · s = Cs ∇ Here, r Jn kB 1B T is the divergence of the particle {N s, N s}= dkz { s, s } current. The quantities 0 D n 2π fn gn are F = 1 the total particle number density in each Weyl cone before Here, we have defined 0(x) 1+e−x for the chiral anomaly, F (x) = x + ln(1 + e−x ) for the thermal chiral anomaly, and after applying external fields, respectively. In Eq. (9), the 1 1+ex s s 2 2 terms with C EB and C B∇T represent CA-induced flow of and F (x) = π + 2Li (−e−x ) − 2x ln(1 + e−x ) − x for 0 1 2 3 2 1+ex particle [29]. the mixed chiral-gravitational anomaly. The x dependence of Following a similar procedure, we construct the continuity these functions is explicitly shown in Fig. 3.Inthex →∞ s ≡ s − μ equation for heat density. Multiplying Eq. (8)byñ n limit, these functions are constant [see Eq. (20)], with F2 F → and integrating over all the states of a given Weyl node, we being the largest of the three and 1 0. However, the obtain thermal chiral anomaly coefficient F1 has a finite value for ∂Qs Qs − Qs finite x. + ∇ · s + Cs =− 0 . r JQ eEB 1kBT (10) In equilibrium, the total charge and energy current from ∂t τv all the Weyl nodes in a WM adds up to zero, as opposite ∇ · s = 2Cs ∇ chirality nodes always appear in pairs in a WM. However, the Here, r JQ kB 2TB T is the divergence of the heat cur- + − + chiral charge current ( j − j ) and energy current ( j − {Qs , Qs}= dkz s{ s, s } e,eq e,eq E,eq rent. The quantities 0 D n π ñ fn gn are the − 2 jE,eq) are nonzero even in equilibrium. More interestingly, heat density in each cone, before and after applying external Cs Cs ∇ in the presence of an external electric field or a temperature fields, respectively. In Eq. (10), EB 1 and 2B T represent gradient, this leads to charge and energy imbalance between the CA-induced flow of the heat density [29]. In constructing pair of opposite chirality Weyl nodes. Below, we explore the Eqs. (9) and (10), we have used the fact that the intranode consequence of this in magnetotransport experiments. scattering does not change the number of carrier and energy of each Weyl node. To evaluate the charge and heat currents in this steady IV. NONEQUILIBRIUM CURRENTS AND CHIRAL state, we now calculate the NDF from Eq. (8). Within the ANOMALIES linear response regime, it is reasonable to expect that the An applied electric field or temperature gradient drives change in chemical potential and temperature in each cone the system out of equilibrium. In the Boltzmann transport is small: δμs ≡ μs − μ<μand δT s ≡ T s − T < T . Then to

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scattering time τv, we obtain js −eCs −eCs Cs Cs e =−τ B2 0 1 [Ds]−1 0 1 s v Cs Cs Cs Cs jQ kBT 1 kBT 2 1 2 eE × . (14) kB∇T

FIG. 4. Chiral anomalies induced charge and energy imbalance Note that in Eq. (14), the magnetic field dependence of the in a WM in the ultraquantum limit. (a) The electrical chiral anomaly charge and heat current comes from (i) the B2 term, (ii) the induced nonconservation of chiral charge in the presence of paral- DOS which depends on B via the LL spectrum, and (iii) lel electric and magnetic fields. (b) The mixed chiral-gravitational magnetic field dependence of τ . The transport coefficients anomaly induced nonconservation of chiral energy in the presence of v can now be easily obtained from the phenomenological re- a temperature gradient parallel to the magnetic field. = σ − α ∇ lations for linear response: je,i j[ ijE j ij jT ] and = α − κ ∇ σ α α κ jQ,i j[¯ijE j ¯ij jT ]. Here, , ,¯, and ¯ denote the the lowest order in δμs and δT s, the NDF can be expressed as electrical, thermoelectric, electrothermal, and constant voltage thermal conductivity matrix, respectively. In the limiting s − μ ∂ f s βμ →∞ μ gs = f s − τvs eE + n ∇T − n case of (or kBT ), the thermal chiral anomaly n n nz T ∂s coefficient Cs → 0. In the same limit, we find Ds , Ds Ds . n 1 0 2 1 τ s − μ ∂ f s Using these, Eq. (14) can be rewritten as + 1 − δμs + n δT s − n . (11) τ ∂s ⎛ ⎞ v T n 2 Ds s 1 Cs 1 Cs Cs je Ds e 0 Ds Ds e 0kB 2 E s = τ 2⎝ 0 0 2 ⎠ . In Eq. (11), the chiral chemical potential δμ and chiral s vB Ds 2 1 Cs Cs 1 Cs −∇ s jQ T Ds Ds e kB T Ds kB T temperature δT are given by 0 2 0 2 2 2 (15) δμs Cs Cs eE From Eq. (15), we note that σ ∝ (Cs )2,¯κ ∝ (Cs )2, and α ∝ =−τ B[Ds]−1 0 1 . (12) 0 2 δ s v Cs Cs ∇ α¯ ∝ CsCs. Thus, it is reasonable to associate different trans- kB T 1 2 kB T 0 2 port coefficients with different CAs. The presence of thermal Here, we have defined the magnetic-field-dependent general- chiral anomaly coefficient in the more general Eq. (14) leads Ds Ds s 0 1 ized energy density matrix, D ≡ Ds Ds , with to a more complicated dependence of the transport coeffi- 1 2 cients on the anomaly coefficients. s − μ ν ∂ s We now explore the different regimes of magnetotransport s dkz n fn Dν = D − , (13) 2π k T ∂s depending on the strength of applied magnetic field: (i) the n B n ultraquantum regime where only the lowest (n = 0) LLs are and ν ={0, 1, 2}. occupied, (ii) the quantum oscillation regime with a few Equation (12) quantifies the chiral charge and energy im- distinguishable LLs being occupied, and (iii) the semiclassical balance in the two Weyl nodes. These imbalances are propor- regime with many but indistinguishable LLs being occupied. tional to the coefficients of CAs and inversely proportional to the generalized energy density. The former is due to the fact A. Ultraquantum regime that CAs are responsible for the charge and heat imbalances. = The latter is a consequence of the fact that a smaller DOS In the ultraquantum regime, only the lowest (n 0) LL is ω μ will lead to a larger change in δμs and δT s and vice versa. occupied and we haveh ¯ c . In this regime, the CA coef- These imbalances are schematically depicted in Fig. 4 in the ficients have already been calculated in Eq. (7). We calculate ultraquantum limit with only the lowest LLs being occupied. the finite-temperature DOS and its energy moments defined in Figure 4(a) shows the δμs induced by the electrical chiral Eq. (13)tobe anomaly and Fig. 4(b) displays the δT s generated by the ⎧ F βμ ν = , (16a) mixed chiral-gravitational anomaly. In addition to these two ⎨⎪ 0( ) 0 D anomalies, the thermal chiral anomaly has a nonzero contri- Ds = F βμ ν = , ν,0 π ⎪ 1( ) 1 (16b) bution to both the charge and energy imbalances, though this 2 h¯vF ⎩ F βμ ν = . contribution is relatively smaller [29]. 2( ) 2 (16c)

V. LONGITUDINAL MAGNETOTRANSPORT Cs Note that similar to the coefficient of thermal CA, 1, the first Ds Having obtained the NDF in the Landau quantization energy moment of the DOS, 1,0 is relatively smaller than the regime, we now proceed to calculate the magnetotrans- other two. Using these expressions in Eq. (14), we obtain the port coefficients. The steady-state nonequilibrium charge and anomaly-induced charge current to be heat current for each Weyl node is defined as { js, js }= e Q e eBτ v dkz s {− , s} s s = v F F βμ + F βμ ∇ . n D 2π vnz e ñ gn. Focusing only on the anomaly- je [e 0( )E kB 1( ) T ] (17) induced contribution which is proportional to the internode h h

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We highlight the linear-B dependence of the charge current. The corresponding charge conductivity is given by 2 τ σ s = e vF v F βμ F →∞ ≈ zz π 2 2 0( ). Since 0(x ) 1, this repro- 4 h¯ lB duces the previously obtained results [18,41,49,53,54]inthe limit βμ →∞. The corresponding thermoelectric conductivity is given by τ αs =− kBe vF v F βμ zz π 2 2 1( ) which is small but nonzero for finite 4 h¯ lB βμ. Remarkably, we find that the magnetic-field-induced thermoelectric conductivity is negative, unlike its semiclassical FIG. 5. (a) The variation of the longitudinal charge conductivity counterpart [29,47] and violates the Mott relation. However, with B (for μ = 25 meV). Note the linear-B dependence in the βμ →∞ αs → in the limiting case of , zz 0, consistent with the ultraquantum regime. The inset shows the low-field behavior and the results of Refs. [27,52]. As a consequence of this vanishing α, semiclassical quadratic-B dependence is shown in orange. (b) The the magnetic-field-induced change in the Seebeck coefficient, μ dependence of the longitudinal charge conductivity (for B = 2 T). S = ασ−1, is equal to the magnetic-field-induced change in Note that the spurious divergence of the longitudinal conductivity as σ . This in turn implies that the magnetoresistance (MR) in the μ → 0 in the semiclassical limit is not there in the LL picture. Here, = × 5 / τ = −9 Seebeck coefficient is equal to the MR in the resistivity. This is we have chosen vF 2 10 m sand v 10 s. in contrast with the semiclassical regime where the MR in the Seebeck coefficient is twice that of the MR in the resistivity The highest occupied LL index can be calculated to be nc = [29,31,47]. μ2/ 2ω2 int[ (2¯h c )], and we obtain The CA-induced heat current is obtained to be ⎧ ⎪ ν = 0, (21a) τ ⎪ 0 s kBT eBvF v ⎨ = − F βμ − F βμ ∇ . 2 jQ [ e 1( )E kB 2( ) T ] (18) D 2π ν = , h h Ds = βμ 1 1 (21b) ν π ⎪3 2 h¯vF ⎪ ⎩ π 2 ν = . Similar to the charge current, the heat current also shows a 3 0 2 (21c) linear-B dependence. The validity of the Onsager’s reciprocity relation,α ¯ (B) = T α(−B) can also be confirmed. The con- ≡ + nc /λ ≡ stant voltage thermal conductivity is given by Here, we have defined 0 1 n=1 2 n and 1 − nc ω /μ 2/λ3 λ = − ω /μ 2 n=1 2n(¯h c ) n with n 1 2n(¯h c ) .Note ∝ / k2T v τ that nc int[1 B]. So the number of occupied LLs is in- κs = B F v F βμ . ¯ 2( ) (19) versely proportional to B. This combined with Eq. (21a) zz 4π 2h¯ l2 B is what leads to quantum oscillations in the longitudinal magnetoconductivity with a period proportional to 1/B.As Interestingly, for finite βμ, we find thatκ ¯ s and σ s are zz zz a consistency check, we note that Eq. (21a) obtained here, is not connected via the Wiedemann-Franz law. However, the Ds βμ →∞ identical to Eq. (4) of Ref. [50]. We find that while 0 and Wiedemann-Franz law gets restored in the limit, s π 2 D are more or less temperature independent (for μ>kBT ), κs = kB 2σ s 2 and we have ¯zz 3 ( e ) zz. This linear-B dependence of Ds βμ → 1 depends inversely on and vanishes in the limit T 0. the thermal conductivity in the ultraquantum limit has also Using Eqs. (20) and (21)inEq.(15), we calculate the charge been observed in recent experiments [52,55]. current to be e2Bτ v e 2π 2 js = v F E − k 1 ∇T . (22) B. Quantum oscillation regime e 2 B βμ 2 h 0 3 0 In this section, we will discuss the scenario when multiple / LLs are occupied. For this, the chemical potential has to be The 1 0 term in the charge conductivity originates from /Ds larger than the separation between the LLs (μ>h¯ω ). Since the 1 0 term in Eq. (15), and it gives rise to oscillations c / we want to highlight the oscillations in the magnetotransport in the longitudinal conductivity which are periodic in 1 B [49,50]. The B dependence of the longitudinal conductiv- coefficients, we work in the regime kBT h¯ωc, so that the temperature broadening does not smear out the signatures ity is displayed in Fig. 5. Additionally, we find that the of the discrete LLs. These two conditions combine to yield longitudinal thermoelectric conductivity also shows quantum μ k T and thus we focus on the limit of βμ →∞in this oscillations arising from the discreteness of the LLs. In this B / 2 section. In this limit, the coefficients of CAs are given by case, the 1 0 term in Eq. (22) governs the features of the quantum oscillations. The Onsager’s reciprocity relation, ⎧ ν = , (20a) α¯ (B) = T α(−B), holds for the longitudinal thermoelectric ⎨⎪1 0 e conductivity, even in the quantum oscillation regime. Cs =− ν s 0 ν = 1, (20b) We calculate the heat current to be 4π 2h¯2 ⎩⎪ 2 π /3 ν = 2. (20c) eBτ v π 2 2e k js = k T v F 1 E − B ∇T . (23) Q B 2 βμ 2 h 3 0 0 In contrast to the anomaly coefficients, the DOS and its Equations (22) and (23) are the main results of this paper. We energy moments get contributions from all the filled LLs. have demonstrated that the longitudinal thermal conductivity

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CA-induced transport coefficients [29]. For closely spaced LLs such that μ h¯ωc, we can replace the discrete sum nc → nc over LLs by an integral: 0 0 dn. In this limit, we 2 2 obtain 0 ≈ 2(μ/h¯ωc ) and 1 ≈ 2(μ/h¯ωc ) . Using these expressions, it is straightforward to calculate ⎧ ν = , (25a) ⎪1 0 ⎨⎪ μ2 π 2 s 1 2 ν = , D ≈ 3βμ 1 (25b) ν π 2 3 3 ⎪ 2 h¯ vF ⎪ ⎩ 2 FIG. 6. (a) The variation of the longitudinal thermal conductivity π ν = 2. (25c) with B. The linear-B dependence in the ultraquantum regime (for B > 3 12 T here) is evident. The inset shows the small B behavior and the semiclassical B2 dependence is shown by the orange line. (b) The μ These expressions are identical to the DOS derived in Ref. dependence of the longitudinal thermal conductivity. The parameters [29]. are identical to those of Fig. 5 and T = 10 K. Now, the charge current can be obtained to be e2 eB 2v2 k π 2 s = ( ) F τ − B 2 ∇ . also possess quantum oscillations with features very similar to je vvF E T (26) 8π 2h¯ μ2 e 3μβ that of the charge conductivity (both dictated by 1/ 0)[56]. The dependence of the thermal conductivity on the magnetic The charge conductivity is identical to the previously reported field and the chemical potential is shown in Fig. 6. results [29,41,47] which show quadratic-B and positive mag- Remarkably, we find that the period of oscillations of netoconductivity. This semiclassical quadratic-B dependence the longitudinal transport coefficients σ andκ ¯, defined in is a well-established signature of the CA in the low-field Eqs. (22) and (23), satisfies the Onsager’s quantization rule limit, and it has also been experimentally verified [21–23]. for Shubnikov–de Haas oscillations [57] in the transverse The thermoelectric conductivity is also consistent with the conductivity. However, this is not surprising, considering the previously obtained semiclassical results [29,47], and with the fact that for both of these the origin lies in the DOS of the experimental observations in Dirac metals and/or semimetals LLs. Onsager’s quantization rule for the Shubnikov–de Haas [30,31]. A similar calculation yields the heat current oscillations in the transverse conductivity states that the period 2 2 π 2 s kBT (eB) vF 2e of conductivity oscillation (in 1/B) is given by j = τvvF E − kB∇T . (27) Q 8π 2h¯ μ2 3 βμ 1 2πe 1 = . (24) We note that the thermal conductivity obtained here is iden- B h¯ Ae tical to the previous reports [29,47] and it has recently been

Here, Ae is the extremal cross section of the Fermi surface in a measured in GdPtBi [52]. The magnetic field and the Fermi plane perpendicular to the magnetic field. For our longitudinal energy dependence of charge conductivity in this regime conductivity, we find that both the charge and thermal con- are shown in Fig. 5 on top of the quantized LL results. In 2 2 ductivity vanishes at μ = 2n(¯hωc ) . This yields the period of Fig. 5(a), the three different transport regimes, with different / = 2 /μ2 B dependences, are evident. In the inset, the yellow line oscillation to be 1(1 B) 2eh¯vF . Since the Fermi sur- = π 2 / shows the semiclassical fitting. In Fig. 5(b), the Fermi energy face in an isotropic WM is spherical with Ae kF , 1(1 B) is consistent with Eq. (24). The quantum oscillation in 1/B in dependence is displayed. In Fig. 6, we have shown the same the longitudinal components of σ andκ ¯ are explicitly shown for the thermal conductivity. in Fig. 7. VI. PLANAR HALL EFFECTS C. Semiclassical regime So far we have explored the impact of CAs in the longitudi- In this section, we show that for small B when many LLs nal magnetotransport coefficients. However, it has been shown are occupied, we can recover the semiclassical results for the that the origin of planar Hall effects and anisotropic longitudinal transport coefficients in nonmagnetic materials can also be related to CAs [32–36]. Here, we explore the impact of quantized LLs, on all the planar Hall transport coefficients and explore the possibility of quantum oscillations in them. In the planar Hall setup [37–40], we measure the longitudinal and Hall transport coefficients in the plane of the E − B (or ∇T − B) fields, as shown in Fig. 8(a). Here the electric field is applied along the z direction with a planar magnetic field applied at an angle φ, so that B = B(cos φzˆ + sin φyˆ ). However, it turns out that the calculations for the case where the electric FIG. 7. The (a) charge and (b) thermal conductivity as a function field or temperature gradient is parallel or perpendicular to the of the inverse of the magnetic field. The constant period of oscil- magnetic field are relatively easier [58,59]. Thus, we perform lation in 1/B is evident. The period of oscillation is determined by our calculations in a rotated frame of reference (z -y ), so that

Onsager’s quantization rule deﬁned in Eq. (24). the magnetic ﬁeld lies along the z axis of the new frame, as

033511-6 CHIRAL ANOMALIES INDUCED TRANSPORT IN WEYL … PHYSICAL REVIEW RESEARCH 2, 033511 (2020)

order in E⊥)tobe ⎛ ⎞ ⎝ kz ⎠ vnz = vF sn − sδn,0 . (30) 2n + 2 ˜2 kz lB

Here, sn = sgn(n). There is also a Lorentz velocity component along the x direction (perpendicular to the B − E plane), vnx = E sin φ/B, which gives rise to the conventional Hall effect. The velocity component along the y direction is zero. Now, it is straightforward to calculate the transport coefficients in the rotated frame of reference, z -y .UsingEq.(29) to revert back to the laboratory frame, we find that the B-dependent part of the transport coefficients picks up the familiar angular dependence given by {σ ,α , α , κ }={σ ,α , α , κ } 2 φ, zz zz ¯ zz ¯zz zz zz ¯ zz ¯zz cos (31) FIG. 8. (a) A schematic of the planar Hall geometry experiments. (b) For ease of calculations, we work in the frame of reference with {σyz,αyz, α¯ yz, κ¯yz}={σ ,α , α¯ , κ¯ } sin φ cos φ. (32) an axis aligned along the magnetic field. This is rotated with respect zz zz zz zz to the laboratory frame (aligned along the electric field or tempera- In deriving Eqs. (31) and (32), we have used the fact that = = ture gradient) by an angle φ.TheB dependence of (a) the anisotropic Lyy Lyz 0. This follows from the fact that the velocity 2 longitudinal thermal conductivity (¯κzz ∝ cos φ) and (b) the planar component along the y axis is zero. This clearly estab- Righi-Leduc effect (¯κyz ∝ cos φ sin φ) for different angles between lishes that the planar Hall transport coefficients retain the B the magnetic field and the temperature gradient. The inset shows the dependence of the longitudinal transport coefficients in the contribution of the lowest LL. linear response regime. Consequently, they also display the three regimes of (i) ultraquantum transport which is linear in B, (ii) quantum oscillation transport regime, and (iii) the shown in Fig. 8(b). The coordinates in the two frames are semiclassical transport regime which is quadratic in B. related as The B dependence of the longitudinal and the planar (Righi-Leduc) thermal conductivity is shown in Figs. 8(c) φ φ z = cos sin z . and 8(d), for different orientations of the magnetic field. The − φ φ (28) y sin cos y quantum oscillations in both of these can be clearly seen. Similar quantum oscillations will also be there in all other

If the transport coefficients are denoted by Lij in this rotated transport coefficients. frame, then the transport coefficients in the laboratory frame, ={σ ,α , α , κ } Lij ij ij ¯ ij ¯ij ,aregivenby VII. DISCUSSION In this paper, we have presented all calculations within the Lzz Lzy cos φ − sin φ Lzz Lzy = constant scattering time approximation. The energy and tem- Lyz Lyy sin φ cos φ L L yz yy perature dependence of the scattering timescale can be easily cos φ sin φ × . (29) included and it does not change the qualitative features of the − sin φ cos φ discussed transport coefficients. However, we have to be more careful in analyzing the magnetic field dependence of the scat- For an electric field applied at an angle to the magnetic tering timescale. For the short-range impurity scattering, real- field, there are two different effects at play. The parallel ized via neutral defects, the scattering rate is proportional to component of the electric field (the component parallel to the the density of states [8,53,60]. Now, in the ultraquantum limit, magnetic field: E = E cos φ) makes the crystal momentum the DOS is proportional to B, and hence τv ∝ 1/B. Thus, the → + / time dependent: kz kz eE t h¯. This modifies the NDF of transport coefficients may become completely independent of the electronic states as shown in Eq. (11), with the substitution the magnetic field in this regime. For the case of multiple E → E . The perpendicular component of the electric field filled LLs, the magnetic field dependence of the scattering (E⊥ = E sin φ) modifies the LL spectrum and the associated timescale is more complicated. Additionally, the magnetic DOS, as derived in detail in Appendix B. The modified LL field dependence for other scattering mechanisms like charged spectrum gives rise to a nonlinear (in E⊥) DOS as shown in impurities and phonons among others is still an open problem. Eq. (B9). A similar approach was used in Ref. [51] to demon- For an isolated system, the chemical potential can also depend strate that the longitudinal planar conductivity has a cos6 φ on the magnetic field strength. However in our case, the device angular dependence, and the planar Hall component has a is connected to external leads (or reservoirs), which set the cos5 φ sin φ angular dependence, arising from the nonlinear chemical potential of the system [61]. terms in the DOS. However, in this work we work in the linear As shown in Fig. 1, the semiclassical or quantum transport response regime and focus on the quantum oscillation of the regimes are quantified by ωcτ or βh¯ωc.Now,ifωcτ 1 planar thermal transport coefficients. Using the LL dispersion, along with βh¯ωc 1, then the LL broadening caused either we calculate the velocity along the magnetic field (to zeroth by the impurities or by the thermal smearing is less than

033511-7 DAS, SINGH, AND AGARWAL PHYSICAL REVIEW RESEARCH 2, 033511 (2020) the LL separation, making them distinguishable. This is the Here, the integration limit is kz ∈ [−∞, ∞]. The contribution regime where the discreteness of the LLs manifests itself in all to both the currents from all LLs except for the n = 0level physical quantities including transport coefficients. Here, the vanishes since the integrand is an odd function of kz for n = 0. shortest scattering timescale dominates the disorder-induced A nonzero contribution arises only from the lowest LLs. broadening. This typically turns out to be the intranode scat- The n = 0 LL for each chirality is linearly dispersing and 5 tering time (τ) in WMs. For a system with vF = 2 × 10 extends to ±∞ in energy. For such systems, which have a m/s and τ ≈ 10−12 s, we have B 0.02 T for observing vacuum state of infinite negative-energy states occupied, the quantized Landau levels in transport experiments. So a value physical currents have to be calculated by subtracting the of B ≈ 1 T is very likely to show the quantum oscillation contribution of the vacuum states. This is generally done by regime. This corresponds toh ¯ωc ≈ 5 meV, which is easily defining the current operators via normal ordering. We find much larger than the thermal energy scale for temperatures that this is equivalent to adopting the integration limit of kz ∈ T < 60 K. Furthermore, the use of Boltzmann transport equa- [0, ∞)fors =−1(orkz ∈ [−∞, 0) for s = 1) in Eq. (A1). tion assumes the presence of a weak disorder strength which Performing a variable change from kz → , and doing an preserves the shape of the Fermi surface. This is specified via integration by parts, we obtain μ /τ the condition h¯ . Thus, to explore the ultraquantum ∞ ∂ s ∞ f limit, we have to be in the regimeh ¯/τ<μ

033511-8 CHIRAL ANOMALIES INDUCED TRANSPORT IN WEYL … PHYSICAL REVIEW RESEARCH 2, 033511 (2020) sn = sgn(n), the LL spectrum is obtained to be can be written as = 2 + 2Ks θ. yc lBkx lB n(kz )sinh (B8) ˜ (˜p ) = s (v p˜ )2 + 2|n|he¯ B˜v2 − sv p˜ δ , . (B4) n z n F z F F z n 0 The perpendicular electric field in this expression lifts the degeneracy of LLs, and consequently it modifies the After doing the inverse Lorentz transformation, we obtain ρ = nc 1 DOS. Using the general formula of DOS, ( ) n=0 (2π )2 δ s − dkxdkz ( n ), a small calculation yields s 1 s (k , k ) = h¯v K (k ) + h¯v k tanh θ. (B5) nc n x z η2 F n z F x λ (,+) − λ (,−) ρ() = η2ρ 2 n n − 1 . (B9) 0 eEL n=0 y η−2 = − 2 θ Here, we have defined 1 tanh and Here, we have defined 2 | | eELy s 2 n λ ,± = ± − | | ω η−3 2. K (k ) = sgn(n) + k2 − sk δ , . (B6) n( ) 2 n (¯h c ) (B10) n z 2 z z n 0 (lBη) 2 Equation (B9) is a nonlinear function of the electric field The most notable effect of the perpendicular electric field is strength. the second term in Eq. (B5). As a result of this, the carriers In the limit of eELy , expanding Eq. (B10)inpowers have finite velocity along the x axis, perpendicular to the E–B of the strength of the electric field we obtain the DOS to be 2 plane. It is straightforward to calculate ρ() = η ρ0 0. Here, ⎛ ⎞ nc 1 0 = 2 − 1. (B11) vF ⎝ kz ⎠ E − | | ω η−3 2 vnz = sn − sδn,0 ; vnx = . (B7) n=0 1 2 n (¯h c ) η2 2n + 2 B ˜2 kz lB This expression of the DOS was used in Ref. [51] to calculate the angular dependence of the planar Hall conductivity. How- We note that the x component of the velocity is simply the ever, in this paper, we will focus on the linear response regime Lorentz velocity, which is a constant and identical for all the and retain only the term independent of the electric field. To LLs. This is what gives rise to the classical Hall effect. the lowest order in E,wehaveη → 1, and Eq. (B9) reduces We emphasize that Eq. (B3) represents a harmonic oscilla- to the expression of the DOS derived in Eq. (21), which was 2 tor with center at y = (l˜B ) p˜x/h¯. In the laboratory frame, this independent of E.

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