PHYSICAL REVIEW X 9, 021053 (2019) Featured in Physics

Hear the Sound of Weyl

Zhida Song1,2 and Xi Dai1,* 1Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

(Received 4 February 2019; revised manuscript received 10 April 2019; published 17 June 2019)

Quasiparticles and collective modes are two fundamental aspects that characterize quantum matter in addition to its ground-state features. For example, the low-energy physics for Fermi-liquid phase in He-III is featured not only by fermionic near the chemical potential but also by fruitful collective modes in the long-wave limit, including several different sound waves that can propagate through it under different circumstances. On the other hand, it is very difficult for sound waves to be carried by electron liquid in ordinary due to the fact that long-range Coulomb interaction among electrons will generate a gap for ordinary electron density oscillation and thus prohibits the propagation of sound waves through it. In the present paper, we propose a unique type of acoustic collective mode in Weyl semimetals under magnetic field called chiral zero sound. Chiral zero sound can be stabilized under the so-called “chiral limit,” where the intravalley scattering time is much shorter than the intervalley one and propagates only along an external magnetic field for Weyl semimetals with multiple pairs of Weyl points. The sound velocity of chiral zero sound is proportional to the field strength in the weak field limit, whereas it oscillates dramatically in the strong field limit, generating an entirely new mechanism for quantum oscillations through the dynamics of neutral bosonic excitation, which may manifest itself in the measurements under magnetic field.

DOI: 10.1103/PhysRevX.9.021053 Subject Areas:

I. INTRODUCTION on the surface and the negative magnetoresistance [19–23] caused by the [1,24–27]. Topological semimetals are unique metallic systems with So far, the Weyl semimetal is considered a new topo- a vanishing at the [1–11]. logical state in condensed matter physics only because of Among different topological semimetals, the Weyl semi- its unique dynamics, which manifests itself in [2–5] is the most robust one because it requires no various transport experiments [19–23]. On the other hand, particular crystalline to protect it. The low-energy the unique collective modes are other types of features that quasiparticle structure of a Weyl semimetal usually contains characterize a new which is yet to be several pairs of Weyl points (WPs), isolated crossing points revealed for Weyl semimetal systems [18,28–36]. The most in 3D momentum space formed by energy bands without common collective mode in a liquid system is sound, which degeneracy. Near each WP, the surrounding quasiparticles usually requires collisions to propagate. For a neutral Fermi can be well described by the proposed by liquid such as He-III [37–40], ordinary sound can exist only Weyl 90 years ago in the context of particle physics [12].The when ωτ ≪ 1, where τ is the lifetime of the quasiparticles. WP provides not only the linear energy dispersion around it, For a clean system, the low-energy quasiparticle lifetime but more importantly, the “monopole” structure in the Berry approaches infinity with reducing , which pro- curvature, which makes the dynamics of these Weyl quasi- hibits the existence of normal sound modes at low enough particles completely different from free electrons in ordinary temperature when ωτ ≫ 1. However, there is a completely metals or and leads to many exotic properties different type of sound that emerges in the above “colli- of the Weyl semimetal, i.e., the Fermi-arc behavior [5,13–18] sionless region” called zero sound, which is purely gen- erated by the quantum-mechanical many-body dynamics *[email protected] under the clean limit [37,41–44]. In a typical Fermi-liquid system, zero sound can be simply viewed as the deforma- Published by the American Physical Society under the terms of tion of the Fermi surface that oscillates and propagates in the Creative Commons Attribution 4.0 International license. the system with the “restoration force” provided by the Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, residual interaction among the quasiparticles around the and DOI. Fermi surface. Like other types of elementary excitations in

2160-3308=19=9(2)=021053(22) 021053-1 Published by the American Physical Society ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019) condensed matter, the form functions of zero sound modes cancel out the net charge oscillation exactly, as illustrated carry irreducible representations (irreps) of the symmetry schematically in Fig. 1(d) for two pairs of WPs. Since these group of the particular system. For an electron liquid in a antiphase modes do not cause any net charge current, the normal metal, the density oscillation corresponding to the collective oscillations of the valley charge and valley trivial representation is always governed by the long-range current will be completely decoupled from the plasmon Coulomb interaction and becomes the well-known plasmon modes, and their dispersion relation remains gapless and excitation with a finite gap in the long-wave limit. Thus, zero linear in the long-wave limit, which is called “chiral zero sound modes can exist only in high multipolar channels sound” (CZS) in this paper. As we discuss in more detail ascribing to the nontrivial representation of the symmetry below, the CZS modes carry the nontrivial irreps of the group, within which the residual interactions among the corresponding little group, with which we can figure out quasiparticles are positive definite. The above condition how many CZS modes can exist with the magnetic field requires a strong and anisotropic residual interaction in being applied in some particular crystal directions. , which is difficult to be realized in normal metals. In order to clearly describe the physical process in Weyl One of the exotic phenomena of a Weyl semimetal is the semimetal systems, we divide the charge current contributed – (CME) [24,45 48], where each valley by the νth WP valley jν into two parts: the “anomalous a will contribute a charge current under the external magnetic current” jν caused by the change of the valley particle number “ ” n field. The anomalous current contributed by the CME through the CME and the “normal current” jν caused by the from a single WP valley with positive (negative) is deformation of the Fermi surface in the νth valley. For the always parallel (antiparallel) to the field direction with its general situation, the two types of currents are coupled amplitude being proportional to the particle number of that together and contribute jointly to both the CP and CME particular valley. To be specific, the anomalous current con- modes. However, in the present paper, we consider a specific a tributed by the νth valley through CME is jν ¼ eBχν= limit where only the anomalous current can survive, and both 4π2 μ − μ χ μ ð Þð ν Þ, where u and ν denote the chirality and the the CP and CZS are purely contributed by the CME. Such a imbalanced chemical potential of the νth WP, respectively, μ limit was proposed previously by Son and Yamamoto [24] is the chemical potential at equilibrium, e ¼ ∓jej is the requiring the intravalley relaxation time to be much shorter charge of the electronlike (holelike) quasiparticle, and B is than the intervalley one, which guarantees that the intravalley the magnetic field. The above CME immediately causes an relaxation process is fast enough so that any deformation interesting consequence: The particle number imbalance of the Fermi surface from its equilibrium shape can be among different valleys will induce particle transport and neglected. In the following, we call this limit the “chiral thus make it possible to form a coherent oscillation of the limit” and mainly discuss the physics of the CZS under it. valley particle numbers over space and time, which is a completely new type of collective mode induced by the CME. II. BOLTZMANN'S EQUATION METHOD On the other hand, the most common collective modes in a charged Fermi-liquid system are , and for a Let us first introduce the Boltzmann equation in the Weyl semimetal under a magnetic field, they are such chiral limit. The Boltzmann method is valid only in the semiclassical limit, where ω τ ≪ 1, ω ≪ μ, and μτ ≫ 1. collective modes where the oscillations of the valley pffiffiffiffiffiffi B B particle numbers cannot cancel each other and generate Here, ωB ¼ vF eB is the magnetic frequency, vF is the net-charge-density oscillation in real space [28–36]. Since Fermi velocity, and τ is the quasiparticle lifetime. (In this these modes are coupled to the CME current, the plasmon paper, we set ℏ ¼ 1 and the energy of WP as 0.) In the frequencies significantly depend on the magnetic field [36]. semiclassical limit, the level smearing caused by the finite Following Ref. [36], in this paper we call them “chiral quasiparticle lifetime is much larger than the Landau-level plasmons” (CPs). In general, each of the CP modes form a splitting, but it is much smaller than the chemical potential; trivial (identity) irrep of the symmetry group. As we discuss hence, the Landau-level quantization can be ignored, and in detail below, among all the CPs, there are only two the Fermi surface remains well defined. Therefore, in the branches that are fully gapped (with opposite frequencies), semiclassical limit, the collective dynamics of a Fermi- whereas the other branches are gapless. For the simplest liquid system can be described by the quasiparticle dis- Weyl semimetal with only a single pair of WPs, the little tribution function nνðk; r;tÞ through the following group at finite wave vector q contains only an identity Boltzmann equation, where ν is the valley index, k is operator under magnetic filed, indicating that all the the momentum, and r is the position of the quasiparticle, electronic collective modes propagating with wave vector q will generally cause net-charge-density oscillation and dδnνðk; r;tÞ _ ¼ð−r_ν · ∂r − kν · ∂kÞδnνðk; r;tÞ thus belong to different branches of the CP modes. dt The situation becomes completely different for a Weyl þ S½δnνðk; r;tÞ; ð1Þ semimetal with multiple pairs of WPs. Now we can have collective “breathing modes” of Fermi surfaces in different where the first term and second term on the rhs describe WP valleys so that they oscillate in an antiphase way and the drifting motion and the scattering process, respectively.

021053-2 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019)

(a) q·r- t = 0 q·r- t = π/2 q·r- t =π q·r- t = 3π/2

+,1 -,2

Mx -,1 +,2 M (b)y (c)

jCME(r,t) n(r,t) q·r- t q·r- t π 2π π 2π

(d) q·r- t = 0 q·r- t =π/2 q·r- t = π q·r- t = 3π/2

+,1 -,2

-,1 +,2

(e) (f) jCME(r,t) n(r,t) q·r- t q·r- t π 2π π 2π

FIG. 1. Chiral zero sound and chiral plasmon modes in the minimal model with four Weyl points. The symmetry group of the model is C2v consisting of a twofold rotation axis in the z direction and two mirror planes in the x-z plane and y-z plane, respectively. In (a) and (d), dashed lines represent the two mirrors, the colored disks represent the Fermi surfaces around the WPs, and the dashed gray circles represent the Fermi surfaces in equilibrium. Here, the four WPs are labeled by ðs; aÞ, with s ¼the chirality and a ¼ 1, 2 the subvalley index. (a) The volume of Fermi surfaces as functions of space and time in the chiral plasmon mode, where ðs; 1Þ and ðs; 2Þ are always in phase, making the mode even under C2 rotation. (b),(c) The chiral magnetic currents and quasiparticle densities as functions of space and time in the chiral plasmon mode. Here, the red and blue lines represent the contributions from the ð−; 1Þ [ð−; 2Þ] and ðþ; 1Þ [ðþ; 2Þ] Fermi surfaces, respectively, and the black lines represent the net current and density. (d) The volume of Fermi surfaces in the chiral zero sound mode, where ðs; 1Þ and ðs; 2Þ are always out of phase, making the mode odd under C2 rotation. (e),(f) The chiral magnetic currents and quasiparticle densities as functions of space and time in the chiral zero sound mode. The contributions from the ð−; 1Þ, ðþ; 1Þ, ð−; 2Þ, and ðþ; 2Þ Fermi surfaces are represented by the red , blue solid, red dashed, and blue dashed lines, respectively. In the chiral zero sound mode, both the net current and the net density vanish.

(An explicit derivation of this equation is given in where ϵνðkÞ is the quasiparticle energy, vνðkÞ¼∂kϵνðkÞ _ γ k B 1 − B Appendix A). The time derivatives r_ν and kν are given is the quasiparticle velocity, and νð ; Þ¼ e · by the equations of motion of the quasiparticles. In the ΩνðkÞ is the phase-space volume correction due to the presence of external field and Berry’s curvature ΩνðkÞ, presence of the Berry curvature [51]. We emphasize that they can be written as [49,50] ϵνðkÞ is not the bare band energy but the renormalized quasiparticle energy due to the presence of the collective _ ϵ k γνðkÞkν ¼ −∂rϵν þ evνðkÞ × B þ eð∂rϵν · BÞΩðkÞ; ð2Þ mode. To obtain νð Þ, we first write the total energy in terms of nνðk; r;tÞ, γνðkÞr_ν ¼ vνðkÞþ∂rϵν ×ΩνðkÞ−e½ΩνðkÞ·vνðkÞB; ð3Þ

Z Z ZZ X 3k 2 0 3 d 0 3 3 0 e 0 E ðtÞ¼E þ d r γνðk; BÞϵνðkÞδnνðk; r;tÞþ d rd r δnðr;tÞδnðr ;tÞ total total 3 ϵ r − r0 ν ð2πÞ 0j j Z Z Z X d3k d3k0 3r γ k B γ k0 B δ k r δ k0 r þ d 2π 3 νð ; Þ 2π 3 ν0 ð ; Þfν;ν0 nνð ; ;tÞ nν0 ð ; ;tÞ; ð4Þ νν0 ð Þ ð Þ

021053-3 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019) where the second and third terms are the long-range the wave vector and frequency of the corresponding Coulomb interaction and the residual short-range inter- collective mode, respectively. By substituting this trial action between the quasiparticles, respectively [52]. solution and Eqs. (2) and (3) into Eq. (1), we obtain the 0 Here, ϵνðkÞ is the bare energy dispersion for the quasi- following dynamic equation 0 particle, δnνðk; r;tÞ¼nðk; r;tÞ − nF½ϵνðkÞ − μ is the   δ r q B χ PdeviationR from Fermi-Dirac distribution, and nð ;tÞ¼ ω i η eð · Þ ν η 3 3 þ ν ¼ 2 ν ν ½ðd kÞ=ð2πÞ γνðk; BÞnνðk; r;tÞ is the net charge τ 4π βνðBÞ v   r X 2 density at position . In general, the short-range interaction eðq · BÞχν e 0 þ fν ν0 þ ην0 ; ð6Þ matrix fν;ν in the above equation should have the full 4π2 ; ϵ q2 0 ν0 0 momentum dependence and be written as fν;ν0 ðk; k Þ [52]. However, here we consider the case where the Fermi where βν B is the bare compressibility of the νth valley, surfaces are small enough such that the k dependence in ð Þ 0 χν ¼1 is the chirality, and ην ¼ βνðBÞcν is the imbal- fν ν0 ðk; k Þ can be omitted. Then, the renormalized quasi- ; anced quasiparticle particle number (per unit volume) particle energy is given by the functional derivative of the for the νth valley. At zero temperature, the bare compress- total energy as ϵνðk; r;tÞ¼δE =δnνðk; r;tÞ. An elabo- total ibility is nothing but the density of states at the Fermi level. rate study of collective modes in Weyl systems with only β 0 ∼ μ2 3 one pair of WPs using the Boltzmann equation can be In the semiclassical region, there are νð Þ =vF and β B − β 0 ∼ ω2 3 ω ≪ μ found in Ref. [31]. νð Þ νð Þ B=vF. In the semiclassical limit B , we have βμðBÞ ≈ βμð0Þ. Equation (6) is the key equation of this paper, which III. THE CHIRAL LIMIT directly leads to both CP and CZS solutions. We put the To introduce the chiral limit, we decompose δnðk; r;tÞ rigorous derivation in Appendix B and give only a brief into two parts: the part that keeps the quasiparticle number introduction here in the main text. We can interpret Eq. (6) f in each valley unchanged δnνðk; r;tÞ and the part that as the continuity equation for the quasiparticle number in a the νth valley under the chiral limit, i.e., ∂ ην ∇ · jν 0, changes the valley quasiparticle numbers δnνðk; r;tÞ. In the t þ ¼ a f where jν is the CME current contributed by the νth valley. following, we refer to δnνðk; r;tÞ as the Fermi-surface For simplicity, here we set τv ¼ ∞. i∂tην gives the lhs and degrees of freedom (d.o.f.) and δnνðk; r;tÞ as the valley a −i∇ · jν gives the rhs of Eq. (6). In the chiral limit, each of d.o.f. Since the intravalley scattering preserves the quasi- the Weyl valleys can be described by the Fermi-Dirac δ k r particle number in each valley, nνð ; ;tÞ can be relaxed distribution functions with time- and valley-dependent only through the intervalley scattering. On the other hand, chemical potential μν. Then, the CME current for the δf k r a a nνð ; ;tÞ can be relaxed through both the inter- and νth valley jν can be simply written as jν ¼ eBχν= 2 intravalley scattering processes. Therefore, the relaxation ð4π Þðμν − μÞ, where μ is the chemical potential in equi- a time of δnνðk; r;tÞ is always longer than the relation time librium. The above anomalous current jν is contributed by f of δnνðk; r;tÞ. We can approximate the scattering term as two effects: the change of quasiparticle number and the ν f modification of the averaged quasiparticle energy in the th δnνðr; k;tÞ δnνðr; k;tÞ valley due to the interaction, which correspond to the two S½δnνðr; k;tÞ ¼ − − : ð5Þ τ0 τv terms on the rhs of Eq. (6), respectively. In the above analysis, for simplicity, we always neglect As we prove in Appendix E, for the simplest case where k both the inter- and intravalley scattering cross sections the dependence in the form of residual interaction among are constants (without k dependence), Eq. (5) is almost the quasiparticles, which is a good approximation as long as all the FSs in such Weyl semimetal systems are small exact, and the valley d.o.f. have the form δnνðk; r;tÞ ∝ δ r δ ϵ0 k − μ k enough. To generalize our discussion, in Appendix F we nνð ;tÞ ½ νð Þ . Such a -independent scattering prove that even if we keep k dependent, the valley d.o.f. cross section is a good approximation for a small Fermi δnνðkÞ are still well defined and free of intravalley surface. Now we argue that in the chiral limit where scattering, but their form will be modified. Furthermore, τ0 ≪ τ , the Fermi-surface degrees and the valley degrees v under the chiral limit, the dynamic equation is still given are decoupled, and the collective modes are purely con- τ by Eq. (6), except that fν;ν0 has to be understood as the tributed by the valley degrees. To zeroth order of 0, 0 “k-averaged” interaction obtained from fν ν0 k; k . Please δf k r ; ð Þ nonzero nνð ; ;tÞ will be relaxed to zero in an infinitely see Appendix F for more details. short time; hence, the Fermi-surface degrees are always in To understand more about the chiral limit, we need to δf k r 0 equilibrium, i.e., nνð ; ;tÞ¼ . Therefore, to obtain the find the upper bound of τ0 below which the zeroth-order dynamic equation in the chiral limit, we can simply assume discussion is valid. In Appendix E, we deal with the effect δnνðk; r;tÞ¼δnνðk; r;tÞ. Here we take the trial solution as of finite τ0 in the standard second-order perturbation theory. iqð·r−ωtÞ 0 δnνðk; r;tÞ¼cνe δ½ϵνðkÞ − μ, where q and ω are Here we describe only the main conclusion: Finite τ0

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X0 introduces an effective damping term of approximately jG0j jG0jjG ðBÞ ∩ Gνj 2 2 N ðBÞ¼ − c ; ð9Þ τ0v q for the collective modes. In order to stabilize the CZS B F ν jGνj jGcð ÞjjGνj collective modes, the Hermitian part of Eq. (6) must be larger than the non-Hermitian part, or, equivalently, the where the summation of ν will be carried out over all eigenfrequency should be much larger than the damping inequivalent WPs. (Two WPs are equivalent if they are rate. Since the gapped CPs are coupled to the Coulomb related by some symmetry operation.) G0 is the maximal interaction, which dominates Eq. (6) in the long-wave (magnetic) point group of the (magnetic) , Gν is limit, the conditions for the gapped CPs to be stable the subgroup of G0 that leaves the νth WP invariant, and jGj 1 τ ≪ 3 q B ϵ q2 τ 2 q2 ≪ are (i) = v f½e ð · Þ=ð 0 Þg and (ii) 0vF is the number of elements in G. Here we take the Weyl 3 q B ϵ q2 f½e ð · Þ=ð 0 Þg. These two conditions are automati- semimetal TaAs [5] in space group I41md (#109) as an cally satisfied in the long-wave limit, and hence, the gapped example to show the usage of Eqs. (8) and (9). Since TaAs CPs are always stable against τ0. On the other hand, since is time-reversal symmetric and the maximal point group the CZSs and gapless CPs are decoupled from the Coulomb of I41md is C4v, we obtain G0 ¼ C4v þ TC4v, where T interaction, as we show in the model below and prove represents the time reversal. Totally, there are 24 different generically in Appendix B, the conditions for CZSs and WPs in TaAs, which can be divided into two classes: eight 1 gapless CPs to be stable are (i) ≪ feðq · BÞ½1 þ βðBÞf= WPs located at the k 0 plane and 16 WPs located off the τv z ¼ β B τ 2 q2 ≪ q B 1 β B β B k ¼ 0 plane. The WPs within the same class can be related ½ ð Þg and (ii) 0vF feð · Þ½ þ ð Þf=½ ð Þg. z These two conditions can be satisfied at some q only if by operations in G0 and are considered to be equivalent   2 2 2 4 2 2 from a symmetry point of view. The corresponding little τ0 e B ½1 þ βðBÞf ω μ f ≪ ∼ B 1 : groups that leave the WPs unchanged are G1 ¼fEg and τ 2 β2 B μ4 þ 2π2 3 ð7Þ v vF ð Þ vF G2 ¼fE; TC2g, respectively. Therefore, from Eq. (6), there are 24 independent variables in total leading to the For simplicity, here we assume isotropic Fermi surfaces same number of independent modes. Assuming the mag- such that βðBÞ¼½μ2=ð2π2v3 Þ. Thus, the upper bound of F netic field is applied along the C4 rotation axis, we obtain τ0 below which Eq. (6) is valid is given by Eq. (7). G ðBÞ¼C4, and hence, N ¼ 6 and N ¼ 18. Now let us analyze the (magnetic) point symmetry group c CP CZS As we discuss above, in the semiclassical region we of Eq. (6). Since the wave vector q enters Eq. (6) only 2 2 always have βνðBÞ − βνð0Þ ∼ ðω =μ Þβνð0Þ, which is through the q · B term, the symmetry group of Eq. (6) is B derived in detail in Appendix B. Thus, to the leading- much higher than the little group at q. In fact, all the point order effect of the magnetic field, we can omit the B group operations or combinations of point group operations dependence in βνðBÞ. Then, Eq. (6) is in first order of B, and the time reversal that preserve q · B, B, and fν;ν0 will q B and the corresponding symmetry group becomes higher keep Eq. (6) invariant. We emphasize that · is invariant than G ðBÞ. This higher symmetry group denoted as G ð0Þ under proper rotations and time reversal, but it changes c c B consists of all the proper rotations, time reversal (if sign under inversion, and transforms as a vector under present), and time reversal followed by proper rotations proper rotations, keeps invariant under the inversion, but (if present) in the original group. Thus, Gcð0Þ is nothing but changes sign under time reversal. Therefore, only two types the chiral subgroup of the little group at q ¼ 0. Therefore, of operations can leave Eq. (6) invariant: proper rotations under semiclassical approximation, the number of CPs and B with an axis parallel to and time reversal followed by CZSs [Eqs. (8) and (9)] should be calculated with G 0 B cð Þ twofold proper rotations with an axis perpendicular to . instead of G ðBÞ. In this paper, we denote the group consisting of these c symmetry operations as G B , which is either a magnetic cð Þ IV. MINIMAL MODEL FOR CHIRAL ZERO point group or a point group, depending on whether or SOUND not it contains combinations of a point group and the time-reversal operations. The solutions of Eq. (6) form the At last, we consider a model Weyl semimetal system representations for the group GcðBÞ, which can be divided with only two pairs of WPs with point group symmetry C2v, into two categories: the trivial and nontrivial irreps. It is as illustrated schematically in Fig. 1. For convenience, we then easy to see that the CP solutions belong to the trivial split the valley index ν into a chirality index s ¼1 and a irreps and the CZS solutions belong to the nontrivial ones. subvalley index a ¼ 1, 2. Under the C2 rotation, the ðs; 1Þ To be specific, as we prove in Appendix B, the multiplicity WP and the ðs; 2Þ WP transform to each other; under the of the trivial irrep or the number of CPs is given by Mx mirror, the ðþ;aÞ WP and the ð−;aÞ WP transform to each other. Thus, the representation matrices formed by ην X0 0 jG0jjGcðBÞ ∩ Gνj τx σ N ðBÞ¼ ; ð8Þ can be written as Dsa;s0a0 ðC2Þ¼ a;a0 s;s0 and Dsa;s0a0 ðMxÞ¼ CP B 0 ν jGcð ÞjjGνj τ σx τx;y;z σx;y;z a;a0 s;s0 , where and are Pauli matrices in the and the multiplicity of the nontrivial irreps or the number of chirality space and subvalley space, respectively, and τ0 and CZSs is given by σ0 are two-by-two identity matrices. In the following, we

021053-5 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019) omit the matrix subscripts for brevity. Without loss of the same phase, while the quasiparticle densities with the generality, we choose the form of residual interaction as opposite oscillate with opposite phases. Since the 0 0 x 0 0 x x x f ¼ f0τ σ þ f1τ σ þ f2τ σ þ f3τ σ , where we set CME current from the νth valley jν is proportional to χνην, f0 ≥ f1 ðf2 − f3Þ and f0 ≥−f1 ðf2 þ f3Þ to ensure a net current oscillation will be generated by the CP mode, that the interaction is positive semidefinite. The magnetic which couples to the long-range Coulomb interaction and field is applied in the z direction. Applying the representa- leads to a finite plasmon frequency in the long-wavelength tion matrices to Eq. (6), one can easily verify that the C2 limit. In contrast, in the CZS mode the valley densities with symmetry is kept but the Mx symmetry is broken. Thus, the the same chirality oscillate with opposite phases, leading to solutions will form the irreps of C2. By diagonalizing the exact cancellation of CME currents from differen Eq. (6), we obtain two branches of CPs valleys. Therefore, the CZS mode will be completely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decoupled from the charge dynamics and can keep its i e q B acoustic nature in the long-wavelength limit. ωð1;2Þ q ð z Þ ζ2 q − ζ2 q ð Þþ ¼ 2 0ð Þ 1ð Þ; ð10Þ It is insightful to compare the possibility to have zero τv 4π βðBÞ sound modes in ordinary metals and Weyl semimetals 2 2 where ζ0ðqÞ¼1þβðBÞ½f0 þf1 þ2e =ðϵ0q Þ and ζ1ðqÞ¼ under magnetic field. The collective modes for the former 2 2 βðBÞ½f2 þ f3 þ 2e =ðϵ0q Þ, and two branches of CZSs metals have been discussed in detail in Ref. [43]. Using the qffiffiffiffiffiffiffiffiffiffiffiffiffiffi description developed above for an ordinary metal, all the i e q B collective modes can be derived from the dynamics of ωð3;4Þ q ð z Þ ξ2 − ξ2 ð Þþ ¼ 2 0 1; ð11Þ δf r k τv 4π βðBÞ Fermi-surface d.o.f. nνð ; ;tÞ, which describe the small deviation of the quasiparticle occupation at the Fermi where ξ0 ¼ 1 þ βðBÞðf0 − f1Þ and ξ1 ¼ βðBÞðf2 − f3Þ.In surface. For a system with approximately the sphere 2 2 2 the long-wave limit, we have ζ0ðqÞ − ζ1ðqÞ ≈ ½ð4e Þ= symmetry, it can be expanded using the sphere harmonics 2 θ ϕ ðϵ0q Þ½1 þ βðBÞðf0 þ f1 − f2 − f3Þ, so the CP modes YLMð k; kÞ. Therefore, the longitudinal mode is formed by are gapped and the plasmon frequency is approximately the proper linear combination of the sphere harmonics with sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ 0 and becomes the plasmon mode. The transverse 2 1 β − − modes are described by the sphere harmonics with m ≠ 0. e ðqzBÞ þ ðBÞðf0 þ f1 f2 f3Þ 2 : ð12Þ Among them, the channel with m ¼1 will be absorbed 2π q β B ϵ0 j j ð Þ into the Maxwell equation to describe the possible elec- trical magnetic wave, which contains no solution for On the other hand, the CZS modes have linear dispersions frequency below the plasmon edge. Therefore, the only along the magnetic field direction with the sound velocity qffiffiffiffiffiffiffiffiffiffiffiffiffiffi possible channels to have zero sound modes in an ordinary eB 2 2 metal system are the channels with jmj ≥ 2, provided that cðBÞ¼ ξ0 − ξ1: ð13Þ 4π2βðBÞ the effective residual interaction in these channels is Here we give a rough estimation of the sound velocity for a positive definite to survive the Landau damping. These typical Weyl semimetal system. For simplicity, we set conditions are difficult to fulfill and so is zero sound in ordinary metal. Therefore, for Weyl semimetals under f 0, B 10 T, μ 30 meV, v 2 eV Åℏ−1; then we ¼ ¼ ¼ F ¼ magnetic field, the CME provides a unique mechanism ≈ 0 34 ℏ−1 ≈ 5 104 obtain cðBÞ . eV Å × m=s. to stabilize the CZS with any form of residual interaction The eigenvectors of the two CP modes are that does not cause instability. At least in the chiral limit, the dynamics of CZS involves only the anomalous current ηð1;2Þ λ q −1 λ q −1 T ¼½ 1;2ð Þ; ; 1;2ð Þ; ; ð14Þ but not the normal current, and hence, it is free of Landau pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 damping. where λ1;2ðqÞ¼½ζ0ðqÞ ζ0ðqÞ − ζ1ðqÞ=ζ1ðqÞ, and the eigenvectors of the two CZS modes are V. THERMAL PROPERTIES OF CHIRAL ZERO ð3;4Þ T SOUND η ¼½λ3;4; −1; −λ3;4; 1 ; ð15Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ξ ξ2 − ξ2 ξ The existence of the CP and CZS in the chiral limit leads where 3;4 ¼ð 0 0 1Þ= 1. In the above expres- to several interesting physical phenomena under the exter- η sions, the bases of the vector are ordered as nal magnetic field. Here we introduce two of them. The first 1 − 1 2 − 2 ηð1;2Þ ðs; aÞ¼ðþ; Þ, ð ; Þ, ðþ; Þ, ð ; Þ. are invariant one is the CZS contribution to the specific heat. The CZS ð3;4Þ under C2 and hence form the trivial irrep, whereas η will modes can be viewed as a set of 1D collective modes change sign under C2 and hence form the nontrivial irrep. dispersing only along the magnetic field. As we derive The CP mode ηð1Þ and the CZS mode ηð3Þ are schematically in Appendix G, the specific heat contributed by the CZS κ ≈ 2 Λ2 6 ≪ Θ plotted in Figs. 1(a) and 1(d), respectively. We can find is ðB; TÞ kBT =½ cðBÞ for temperature T CZS, Θ Λ clearly from Fig. 1 that the CP is such a mode that the where CZS ¼ cðBÞ =kB is the corresponding Debye tem- quasiparticle densities with the same chirality oscillate with perature for the CZS, cðBÞ is the sound velocity [Eq. (13)],

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(a) (b)

) except that the field dependence of the compressibility is 3 0.03 T=10 CZS

B modified. As we calculate in Appendix I, the compress- k ( ibility in the strong field can be expressed as β B 0.02 ð Þ¼ βð0ÞðBÞþβð1ÞðBÞþ, where the βðl≥1ÞðBÞ terms oscillate

T8 0.01 units) (arb. as the lth harmonics of 1=B. Under finite temperature,

th T=0.1 zz CZS the ratio between the first and zeroth components is Specific heat 0101 2 304 5 15 20 approximately T S /(eB) μ   / CZS ex ð1Þ expð−π 2 Þ β ðBÞ ω ω τ0 S νðμÞ π ≈ B B ex; − 0 μ cos ; ð16Þ βð Þ μ 2π2 kBT 4 FIG. 2. (a) The specific heat (per unit volume) in the four-WPs ðBÞ sinchð ω2 Þ eB model is plotted as a function of temperature. The specific heat is B 3 x − −x 2 μ plotted in the unit of kBΛ , where kB is Boltzmann’s constant, and where sinchðxÞ¼ðe e Þ=ð xÞ, and Sexð Þ is the area Λ is the cutoff in the momentum integral. The temperature T is enclosed by the extreme circle (perpendicular to B) on the plotted in units of Debye temperature for the CZS mode, Fermi surface. Here we assume eB > 0 and μ > 0. Because Θ Λ CZS ¼ cðBÞ =kB, where cðBÞ is the speed of the CZS. of Eqs. (6) and (13), the oscillation in βðBÞ will lead to the (b) The thermal conductivity in the four-WPs model is plotted oscillation in the sound velocity of the CZS. Substituting μ as a function of magnetic field. Here, Sexð Þ is the area enclosed Eq. (16) for Eq. (13), we obtain the first-order oscillation of by the extreme circle on the Fermi surface that is perpendicular to ω 0 2μ the sound velocity as the magnetic field. The parameters are set as B ¼ . ,   1 2τ 0 001μ Θ 10 ð1Þ kBT ¼ =ð 0Þ¼ . , and T= CZS ¼ and 0.1 for the ξ0 β ðBÞ c B ≈ cð0Þ B 1 − ; 17 blue line and red line, respectively. ð Þ ð Þ 2 2 ð0Þ ð Þ ξ0 − ξ1 β ðBÞ where cð0ÞðBÞ is the nonoscillating component of the sound Λ is the momentum space cutoff, and k is the Boltzmann B velocity. As both the specific heat and thermal conductivity constant, while in the high-temperature region (T ≫ Θ ), CZS are functions of the sound velocity, the oscillation in the specific heat is κ B; T ≈ k Λ3= 3π2 . To be specific, ð Þ B ð Þ the velocity leads to oscillations in the specific heat and in Fig. 2(a) we plot the specific heat as a function of thermal conductivity as well. As an example, in Fig. 2(b) temperature using some typical parameters for the Weyl we plot the thermal conductivity as a function of magnetic semimetal systems. Although such a temperature depend- field. In normal metals, the thermal conductivity is mainly ence is similar to the quasiparticle contribution to the contributed by electrons and acoustic . The specific heat, the two can be distinguished from each part couples only indirectly to the magnetic field and other by their different field dependence. Another unusual usually does not change much with the field. Therefore, property caused by the CZS is the thermal conductivity. the part that oscillates with the field is mainly contributed Since the CZS disperses only along the field direction, the by the free electrons in the normal metal, which satisfies the thermal current carried by the CZS modes can flow only Wiedemann-Frantz law. As we introduce above, for the along this direction. As a result, the thermal conductivity Weyl semimetals in the chiral limit, since the CZS can tensor contributed by the CZS modes has only one nonzero propagate only along the magnetic field, the thermal entry. As we derive in Appendix G, if the magnetic field conductivity along the field will be contributed by both is applied along the z direction, the thermal conductivity the CZS and free electrons leading to the dramatic violation σth δ δ τ 2 κ τ is given by ij ¼ i;z j;z sðTÞc ðBÞ ðB; TÞ, where sðTÞ is of the Wiedemann-Frantz law, which is absent for thermal the relaxation time for the CZS excitations. In the weak conductivity along the perpendicular direction. Early theo- τ ω ≪ 1 ≪ Θ field and low-temperature region ( 0 B , T CZS), retical studies of the electronic contribution to the thermal th as κðB; TÞ ∝ T=cðBÞ and cðBÞ ∝ B, we obtain σzz ∝ TB. conductivity in Weyl semimetal without considering the τ ω ≪ 1 2 In the weak field and high-temperature region ( 0 B , CZS modes obtain the B dependence for the thermal ≫ Θ κ∼ σth ∝ 2 T CZS), as const, we obtain zz B . conductivity under magnetic field [53], which is quite In order to discuss the specific heat and thermal different from the contribution from the CZS introduced conductivity in the strong field region (τ0ωB ≳ 1), we need above. Such a field-dependent violation of the Wiedemann- to rederive the dynamic equation under the strong field, Frantz law has already been seen in the thermal conduc- where the electronic states are already Landau levels. In tivity measurement of TaAs under a magnetic field, this case, since the compressibility oscillates with the field, indicating the possible contribution from the CZS [54]. as a consequence, the velocity of the CZS as well as the We note that for realistic systems which are not deeply in thermal conductivity, in general, should also oscillate with the chiral limit, the CZS will also acquire nonzero velocity the field. Here we focus only on the case ωB ≪ μ so that along the transverse direction of the magnetic field as well, there are still a large number of Landau levels below the which is caused by the accompanying normal current chemical potential. As we introduce in Appendix H, it turns during the oscillation. Therefore, the CZS or the gapless out that the dynamic equation has the same form of Eq. (6), CP can also contribute to the thermal conductivity along the

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_ transverse direction, but the effect should be much k ¼ −∂rϵðk; r;tÞþer_ × B; ðA1Þ less by orders of magnitude than that of the longitudinal _ direction. r_ ¼ ∂kϵðk; r;tÞ − k × ΩðkÞ; ðA2Þ

VI. DISCUSSION AND SUMMARY where e ¼ −jej (jej) is the electronlike (holelike) quasi- particle, and The above-mentioned quantum oscillations in specific heat and thermal conductivity can be viewed as strong ΩðkÞ¼−ih∂kuðkÞj × j∂kuðkÞi ðA3Þ evidence for the existence of the CZS but still indirect. is the Berry curvature. The decoupled equations are It will be more convincing if we can also have direct ways _ to measure it. In this regard, the direct ultrasonic meas- γðk; BÞk ¼ −∂rϵðk; r;tÞþe∂kϵðk; r;tÞ × B urement of these materials under magnetic field and low ∂ ϵ k r B Ω k temperature may be difficult but worth trying. Another þ e½ r ð ; ;tÞ · ð Þ; ðA4Þ possible experiment is inelastic neutron-scattering spec- γ k B r_ ∂ ϵ k r ∂ ϵ k r Ω trum. Although the corresponding scattering cross section ð ; Þ ¼ k ð ; ;tÞþ r ð ; ;tÞ × ðkÞ for electrons may be very small, the existence of the CZS − e½∂kϵðk; r;tÞ · ΩðkÞB; ðA5Þ can still be inferred from the spectrum of certain phonon modes, which have the same symmetry representation as where the CZS and can hybridize with it when they intersect γðk; BÞ¼1 − eB · ΩðkÞðA6Þ each other at some particular wave vector to form the “ mode”. is the phase-space measure. Now we denote the distribution In summary, we propose that an exotic collective mode, function over phase space as ρðk; r;tÞ, and due to particle the chiral zero sound, can exist in a Weyl semimetal under number conservation, we have magnetic field with the chiral limit, where the intervalley _ _ 3 3 scattering time is much longer than the intravalley one. ρðk þ dtk; r þ dtr_;tþ dtÞð1 þ dt∂k · k þ dt∂r · r_Þd kd r The CZS can propagate along the external magnetic field ¼ ρðk;r;tÞd3kd3r; ðA7Þ with its velocity being proportional to the field strength in the weak field limit and oscillating in the strong field. The and hence, CZS can lead to several interesting phenomena, among ∂ which the giant quantum oscillation in thermal conductivity 0 ρ k r ∂ k_ k_ ∂ ∂ r_ r_ ∂ ¼ ∂ ð ; ;tÞþ½ð k · Þþ · k þ r · þ · r is the most striking and can be viewed as “smoking gun” t evidence for the existence of it. × ρðk; r;tÞ ∂ _ ¼ ρðk; r;tÞþ∂k · ½kρðk; r;tÞ þ ∂r · ½r_ρðk; r;tÞ: ACKNOWLEDGMENTS ∂t Z. S. and X. D. acknowledge financial support from ðA8Þ the Hong Kong Research Grants Council (Project Here we neglect the scattering term in Boltzmann’s No. GRF16300918). Z. S. also acknowledges the Depart- equation. ment of Energy Grant No. desc-0016239, the National From now on, we assume there are a few valleys and Science Foundation EAGER Grant No. DMR 1643312, label the quantities in different valleys with a subscript ν. Simons Investigator Grants No. 404513, No. ONR For each valley, we introduce a weighted distribution N00014-14-1-0330, and No. NSF-MRSECDMR DMR function nνðk; r;tÞ¼ρνðk; r;tÞ=γðkÞ, then the multivalley 1420541, the Packard Foundation Grant No. 2016- Boltzmann equation is given by 65128, and the Schmidt Fund for Development of Majorana Fermions funded by the Eric and Wendy ∂ 0 ¼ γνðk; BÞ nνðk; r;tÞþf−∂rϵνðk; r;tÞ Schmidt Transformative Technology Fund. ∂t þ e∂kϵνðk; r;tÞ × B þ e½∂rϵνðk; r;tÞ · BΩνðkÞg APPENDIX A: BOLTZMANN’S EQUATION · ∂knνðk; r;tÞþf∂kϵνðk; r;tÞþ∂rϵνðk; r;tÞ AND COLLECTIVE MODES × ΩνðkÞ − e½∂kϵνðk; r;tÞ · ΩνðkÞBg · ∂rnνðk; r;tÞ; Let us first derive the Boltzmann equation, whichp appliesffiffiffiffiffiffi ðA9Þ when the Landau-level splitting, i.e., ωB ¼ vF eB,is smaller than the imaginary part of the quasiparticle self- where, again, the scattering is neglected. In deriving energy and the chemical potential μ. The semiclassical Eq. (A9), we make use of the relations ∂r ·½γνðk;BÞr_ν¼0 _ equations of motion of the Weyl are [49,50] and ∂k · ½γνðk; BÞkν¼0. Because of Eqs. (A4) and (A5),

021053-8 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019) these two relations are satisfied as long as (i) k is not at the In the presence of the collective mode, the single-particle Weyl point, where the semiclassical method does not apply, energy ϵνðk; r;tÞ should be determined self-consistently. and (ii) ∂r ·∂kϵðk;r;tÞ¼0, which is automatically satisfied With quasiparticle excitation, the total energy is a func- in our approximation for quasiparticle energy [Eq. (A13)]. tional of the distribution function [52]

Z Z ZZ X 3k 1 1 0 3 d 0 3 3 0 0 E t E d r γν k; B ϵν k δnν k; r;t d rd r δn r;t δn r ;t totalð Þ¼ total þ 3 ð Þ ð Þ ð Þþ2 ϵ r − r0 ð Þ ð Þ ν ð2πÞ 0j j Z Z Z 1 X 3k 3k0 3 d d 0 0 d r γν k; B γν0 k ; B fν ν0 δnν k; r;t δnν0 k ; r;t ; þ 2 2π 3 ð Þ 2π 3 ð Þ ; ð Þ ð Þ ðA10Þ νν0 ð Þ ð Þ where the second and third terms denote, respectively, the where φ is the scalar potential determined by the Poisson long-range Coulomb and residual short-range interaction equation among the quasiparticles around the WPs. Here, e 0 −∂2φ r δ r δnνðk; r;tÞ¼nνðk; r;tÞ − nF½ϵνðkÞ − μðA11Þ r ð ;tÞ¼ nð ;tÞ: ðA14Þ ϵ0 is the deviation of distribution from equilibrium, nFðϵÞ¼ 1=f1 þ exp½−ðϵÞ=ðkBTÞg is the Fermi-Dirac distribution, Now we assume the deviation from equilibrium takes the Z form of the plane wave X d3k δn r;t γν k; B δnν k; r;t ð Þ¼ 2π 3 ð Þ ð ÞðA12Þ ν ð Þ iðq·r−ωtÞ δnνðk; r;tÞ¼δnνðkÞe : ðA15Þ is the charge density at position r and time t, and fν;ν0 is a real matrix due to the Hermitian condition of the Hamil- Following this definition, we can rewrite the quasiparticle tonian. The k dependence of fν;ν0 is neglected since we energy as consider the case where the Fermi surfaces are very small Z   compared to the Brillouin zone. The quasiparticle energy X 3 0 2 0 d k e can then be derived as the functional derivation of the total ϵν k; r;t ϵν k fν ν0 ð Þ¼ ð Þþ 2π 3 ; þ ϵ q2 energy ν0 ð Þ 0 Z γ k0 B δ k0 iðq·r−ωtÞ X 3k0 × ν0 ð ; Þ nν0 ð Þe : ðA16Þ 0 d 0 0 ϵν k;r;t ϵν k fν ν0 γν0 k ;B δnν0 k ;r;t ð Þ¼ ð Þþ 2π 3 ; ð Þ ð Þ ν0 ð Þ The equation of motion to first order of δnνðkÞ is φ r þ e ð ;tÞ; ðA13Þ given by

0 ¼ −γνðk; BÞωδnνðkÞ  Z    X d3k0 e2 q v k − q B v k Ω k δ k δ μ − ϵ0 k γ k0 B δ k0 þf · νð Þ e · ½ νð Þ · νð Þg nνð Þþ T½ νð Þ 2π 3 fν;ν0 þ ϵ q2 ν0 ð ; Þ nν0 ð Þ ν0 ð Þ 0

− ie½vνðkÞ × B · ∂kδnνðkÞ; ðA17Þ

0 where vνðkÞ¼∂kϵνðkÞ and δTðϵÞ¼−∂ϵnFðϵÞ. For convenience, we replace the 3D variable k in Eq. (A17) with an energy ϵ and a 2D wave vector σ on the energy surface. The integration over k in the νth valley can be rewritten as Z Z∞ Z 1 d3k ¼ dϵ d2σ ; ðA18Þ ϵ jvνðϵ; σÞj 0 R where ϵ means σ takes a value on the 2D surface with fixed energy ϵ. Apparently, the solution of Eq. (A17) takes the form

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δnνðkÞ¼−∂ϵnFðϵ − μÞδnνðσÞ; ðA19Þ where σ takes a value on the Fermi surface. Integrating the energy, Eq. (A17) becomes

γνðσ; BÞ 0 ¼ − ωδnνðσÞ jvνðσÞj  Z    X 2σ0 2 γ σ0 B d e ν0 ð ; Þ 0 þfq · vˆνðσÞ − eq · B½vˆνðσÞ · ΩνðσÞg × δnνðσÞþ fν ν0 þ δnν0 ðσ Þ 2π 3 ; ϵ q2 v σ0 ν0 ð Þ 0 j ν0 ð Þj

− ie½vˆνðσÞ × B · ∂kδnνðσÞ; ðA20Þ

Z 2 d σ γν σ; B dρν where vˆ νðσÞ¼vνðσÞ=jvνðσÞj and β B ð Þ νð Þ¼ 3 ¼ ðB4Þ ð2πÞ jvνðσÞj dμ ∂ X ∂ ∂ v k ∂ σ k ¼ νð Þ ∂ϵ þ k i ∂σ : ðA21Þ is the compressibility of the νth valley, and ρν is total i¼1;2 i particle density of the νth valley. Then, Eq. (A20) can be rewritten as In Eq. (A20), all the quantities are defined on the Fermi      surfaces, so we omit the energy dependence of these γνðσ; BÞ i i f quantities; e.g., vνðσÞ is shorthand for vνðμ; σÞ. ω þ δnν þ ω þ δnνðσÞ jvνðσÞj τv τ0

¼þfq · vˆνðσÞ − eq · B½vˆνðσÞ · ΩνðσÞg APPENDIX B: THE CHIRAL LIMIT     X 2 δ δ e We can decompose n into two parts: the first part n δnν σ fν ν0 βν0 B δnν0 × ð Þþ ; þ ϵ q2 ð Þ changes particle numbers in different valleys, and the ν0 0 δf second part n preserves the particle number in each valley − ie½vˆνðσÞ × B · ∂kδnνðσÞ: ðB5Þ but deform the shape of Fermi surface in each valley. We refer to δn as the valley d.o.f. and δfn as the Fermi-surface In the following, we study the physics in zeroth order of d.o.f. In the general case, these two d.o.f. are strongly τ0 and leave the discussion on the finite τ0 effect for coupled. However, as we argue below, in the chiral limit, Appendix E. To zeroth order of τ0, the Fermi-surface d.o.f. f the dynamics of these two d.o.f. is decoupled. In the are always in thermal equilibrium, i.e., δnνðσÞ¼0, and any presence of a scattering term, δn in general damps with deviation from equilibrium will be immediately killed by time, but the valley degrees and the Fermi-surface degrees the strong scattering. By integrating σ into Eq. (B5), we get can have different relaxation times. We denote the relax- a generalized eigenvalue equation f ation time of δn as τ0, whereas the relaxation time of δn is   τ i e q · B v. Then the time derivative term in Eq. (A20) should be ω χ η ð Þ η þ ν ν ¼ 2 ν replaced by τ 4π βνðBÞ     v   X 2 i i f e e ωδnν σ → ω δnν σ ω δnν σ : B1 q B fν ν0 ην0 : ð Þ þ τ ð Þþ þ τ ð Þ ð Þ þ 4π2 ð · Þ ; þ ϵ q2 v 0 ν0 0 τ The chiral limit refers to the case where 0 is much smaller ðB6Þ τv, i.e., τ 0 ≪ 1 Here, χν ¼1 is the chirality of the νth valley, and ην ¼ τ : ðB2Þ v βνðBÞδnν is the disequilibrium quasiparticle number in the This limit can be achieved when the intravalley scat- νth valley. In deriving Eq. (B6), we apply tering is much stronger than the intervalley scattering. In Z Z 2 Appendix E, we discuss the relaxation times contributed by d σvˆνðσÞ · ΩνðσÞ¼ dS · ΩνðσÞ¼−2πχν: ðB7Þ impurity scattering. In the simple case in Appendix E, δnν is defined as Z Now let us discuss the symmetry of Eq. (B6). 1 2σ γ σ B Apparently, Eq. (B6) has a higher symmetry than the little δ d νð ; Þ δ σ nν ¼ 3 nνð Þ; ðB3Þ group of q: It contains all the symmetries that preserve the βνðBÞ ð2πÞ jvνðσÞj chiralities of WPs and the direction of magnetic field. The where direction of q is irrelevant to the symmetry. This is because

021053-10 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019) in the chiral limit, the electric field proportional to q enters and the equation only through the q · B term and thus couples q X0 B ∩ only to the chiral d.o.f. Therefore, finite breaks only the B jG0j − jG0jjGcð Þ Gνj NCZSð Þ¼ B ; ðB13Þ symmetries changing the chiralities. In the following, we ν jGνj jGcð ÞjjGνj denote the symmetry group of Eq. (B6) as GcðBÞ.We emphasize that some antiunitary symmetry, like time respectively. Here, ν sums over all inequivalent WPs, and Gν reversal followed by a crystalline symmetry, can also keep is the little group of the νth WP. the chiralities and the magnetic field invariant. And, since fν;ν0 is a real matrix, these antiunitary symmetries act on Eq. (B6) as unitary operators. The explicit representation APPENDIX C: CZS matrix of all these symmetries is given in Eq. (B9). η B PIf ν is not a trivial irrep of Gcð Þ, then there must be It should be noticed that, to leading order of magnetic ν ην ¼ 0, implying that it does not cause any charge- β B β 0 field, i.e., setting νð Þ¼ νð Þ, Eq. (B6) even has a density oscillation. Thus, for nontrivial irreps, we can omit B symmetry higher than Gcð Þ: The magnetic field enters the Coulomb term, and the corresponding modes are the q B the equation only through term · ; thus, the direction CZSs. Now let us solve the equation of motion for the CZS. of magnetic field becomes irrelevant to the symmetry. Notice that the matrix on the rhs of Eq. (B6) is real and 0 We denote this higher symmetry group as Gcð Þ, which symmetric, so we diagonalize it as consists of all the symmetries of the zero field system that do not change chirality. 1 1 X fν;ν0 þ δν;ν0 ¼ Oν;aλaOν0;a; ðC1Þ The solutions of Eq. (B6) must form irreps of GcðBÞ.As β B β¯ B νð Þ ð Þ a we show in the next two sections, the trivial irreps of GcðBÞ ¯ always couple to the charge-density oscillation, and thus, where βðBÞ is the averaged βνðBÞ, O is an orthogonal we call these modes forming trivial irreps CPs. As we matrix, and the λa’s are dimensionless numbers. Applying prove, only two of the CPs are gapped, whereas other CPs the transformation are gapless in the long-wave limit. On the other hand, all X the nontrivial irreps are decoupled from density oscillation, 0 ην ¼ Oν;aηa; ðC2Þ so we call them the CZSs. Now let us calculate the number a of trivial irreps in the solution to Eq. (B6). We first consider a set of symmetry-related WPs in the inner Brillouin zone, we can rewrite Eq. (B6) as   and one of them has the little group GW. We denote the X i 0 e 0 maximal (magnetic) point group of the space group as G0, ω Oν χνOν 0 η 0 q · B λ η : C3 þ τ ;a ;a a ¼ 4π2 ð Þ a a ð Þ then each symmetry-related WP can be represented by a v ν;a0 coset representative of G0=GW, Applying the transformation G0 ¼ h1G þ h2G þ: ðB8Þ W W X 1 1 pffiffiffiffiffi Ξ ffiffiffiffiffi χ ffiffiffiffiffiffi η00 λ η0 The representation formed by the valley degrees is given by a;a0 ¼ p Oν;a νOν;a0 p ; a ¼ a a; ðC4Þ ν λa λa0  1 0 ∈ if gh hGW; we get a regular eigenvalue problem ∀ g ∈ G0;D0 ðgÞ¼ ðB9Þ h;h 0 else;  eðq · BÞ Ξη00 ¼ η00; ðC5Þ jG0j=jGWj;g∈ GW; 2 ¯ i 4π βðBÞðω þ τ Þ TrDðgÞ¼ ðB10Þ v 0;g∉ GW: where Ξ has the symmetry of G ðBÞ. The dispersion of the B c Now we reduce D to irreps of Gcð Þ. The number of trivial CZS is given by irreps is given by i eðq · BÞ 1 X B ∩ ω q ;n1; …;N B : jG0jjGcð Þ GWj CZS;nð Þþτ ¼ 4π2β¯ B ξ ¼ CZSð Þ B TrDðgÞ¼ B : ðB11Þ v ð Þ n jGcð Þj ∈ B jGcð ÞjjGWj g Gcð Þ ðC6Þ Therefore, for a system with a few sets of nonequivalent ξ Ξ WPs, the number of CP modes and CZS modes are given by Here, n is the nth eigenvalue of . It should be noticed that Ξ is real and symmetric (such ξ ’ λ ’ X0 B ∩ that the n s are real) only if all a s are positive. Thus, the B jG0jjGcð Þ Gνj NCPð Þ¼ B ðB12Þ number of CZS modes is given by Eq. (B13) only if ν jGcð ÞjjGνj Eq. (C1) is positive definite. Otherwise, only irreps where

021053-11 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019) all λa’s are positive correspond to physically observable and thus, the Coulomb term must be considered. However, 2 2 modes. The irreps having negative λa’s in general have as ½e =ðϵ0q Þ is a rank-one operator, in the long-wave complex ξ and so are not stable. limit, there should be only one channel that responds to Coulomb interaction. To separate this channel, we define APPENDIX D: CP the projection operator Pν ν0 ¼ð1=N Þ, where N is the P ; W W number of WPs, and divide the terms on the rhs of Eq. (B6) For the trivial irreps, in general we have ν ην ≠ 0. Therefore, the trivial irreps contribute to density oscillation, into four components:

           e2 1 e2 1 1 2 þ f þ ¼ P 2 þ f þ P þ Q f þ Q ϵ0q βðBÞ ϵ0q βðBÞ βðBÞ       1 1 þ P f þ Q þ Q f þ P: ðD1Þ βðBÞ βðBÞ

2 2 2 2 Here, ½e =ðϵ0q Þ represents the matrix where every element is ½e =ðϵ0q Þ, f1=½βðBÞg represents the diagonal matrix T f1=½βνðBÞgδν;ν0 , and Q ¼ I − P, where I is the identity matrix. We apply an orthogonal transformation V ¼ I þ S − S where S ¼ PSQ, to remove the mixing term between the P and Q subspaces. To second order of q, we find that    ϵ 1 − 0 q2 O q4 S ¼ 2 P f þ β B Q þ ð ÞðD2Þ e NW ð Þ and            e2 1 e2 1 1 T O q2 V 2 þ f þ V ¼ P 2 þ f þ P þ Q f þ Q þ ð Þ: ðD3Þ ϵ0q βðBÞ ϵ0q βðBÞ βðBÞ P 2 Using the fact ν00 Vν00;νχν00 Vν00;ν0 ¼ χνδν;ν0 þ Oðq Þ, we can rewrite Eq. (B6) as            X 2 1 e 1 1 2 ω χνην P f P Q f Q ην0 O q : þ τ ¼ ϵ q2 þ þ β B þ þ β B þ ð Þ ðD4Þ v ν0 0 ð Þ ð Þ ν;ν0

To solve this generalized eigenvalue equation, we apply where the ξn’s are eigenvalues of Ξ. Now let us analyze the the technique used in Appendix C: diagonalizing the matrix spectrum of Ξ. For convenience, we set Oν;1 as the on the rhs and transforming the equation to a regular eigenvector of P, so the corresponding eigenvalue is eigenvalue problem. Let usP write the matrix on the right-   ¯ 2 X ¯ 1 β B λ 0 e N 1 β B hand side as f =½ ð Þg a Oν;a aOν ;a. Applying the λ W β¯ B β¯ B ð Þ δ 1 ¼ 2 ð Þþ fν;ν0 ð Þþ ν;ν0 transformation ϵ0q N βν B W ν;ν0 ð Þ X 1 1 pffiffiffiffiffiX þ Oðq2Þ; ðD8Þ 00 Ξ 0 ffiffiffiffiffi Oν χνOν 0 ffiffiffiffiffiffi ; η λ Oν ην; a;a ¼ pλ ;a ;a pλ a ¼ a ;a ν a a0 ν which is singular in the limit q → 0. Then, due to ðD5Þ Eq. (D12), the Ξ matrix takes the form   0 ζ we get a regular eigenvalue problem Ξ ¼ þ Oðq2Þ; ðD9Þ ζT Ξ0 e q · B Ξη00 ð Þ η00 ¼ 2 : ðD6Þ where 4π β¯ðBÞðω þ i Þ τv

ζ1;a ¼ Ξ1;a;a¼ 2; 3…; ðD10Þ The frequencies of the CP modes are then given by and q B ω i eð · Þ CP;n þ ¼ 2 ; ðD7Þ 0 0 τ ¯ Ξ Ξ 0 2 3… v 4π βðBÞξnðqÞ a;a0 ¼ a;a ;a;a¼ ; ðD11Þ

021053-12 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019) are submatrices of Ξ. We emphasize that for a ≥ 2, λa is not Therefore, due to Eq. (D18), n ¼ 1, 2 correspond to the q → 0 Ξ0 3 … B singular. Thus, in the limit , approaches a constant gapped CP modes, whereas n ¼ ; ;NCPð Þ correspond 0 matrix, whereas ζ1;a ∼ jqj. Therefore, by diagonalizing Ξ , to the gapless CP modes. The low-energy behavior of the we can rewrite Ξ as gapless CPs is very similar with the CZSs: Both of them 0 1 have a linear dispersion in the limit q → 0. However, a vital 0 c2jqj c3jqj difference is that gapless CPs are coupled to gapped CPs B C 0 through the c ≥3 terms in Eq. (D12), whereas the CZSs are B c2jqj ξ2 0 C a TB C 2 Ξ ¼ U B 0 CU þ Oðq Þ; ðD12Þ not. As a result, the dispersions of gapless and gapped CPs B c3jqj 0 ξ C @ 3 A form anticrossings, whereas the dispersions of CZSs and ...... gapped CPs form symmetry-protected crossings. Here we give a simplified method to calculate the gapped 0 0 CP frequency. Since the gapped CP is driven by the where the ξa’s are eigenvalues of Ξ , and U is some ξ0 Coulomb interaction, for simplicity, in this method we orthogonal matrix. Now we prove that one of the a¼2;3… is omit fν;ν0 and 1=τv. From Eq. (B6), we get zero. We denote the diagonal matrix δν;ν0 χν as [χ]. Then, the projected [χ] matrix in Q subspace is Q½χQ ¼½χ− e q B 2 X P½χ − ½χP. Apparently, ην ¼ 1 and ην ¼ χν are two zero 4π2 ð · Þ e ην ¼ · ην0 ; ðD15Þ eigenvectors of Q½χQ, wherein ην ¼ 1 is in subspace P, χ β B ω − e q B ϵ q2 ν νð Þ 4π2 ð · Þ 0 0 0 ν whereas ην ¼ χν is in subspace Q.AsΞ is equivalent to Q½χQ up to an invertible transformation, Ξ0 has one zero ξ0 eigenvalue in the Q subspace. Therefore, one of the a¼2;3… and thus, 0 is zero. Here we choose ξ2 ¼ 0. In the limit q → 0,wehave the Ξ eigenvalues as 2 X e 2 ðq · BÞβνðBÞ 1 e 4π ¼ 2 e : ðD16Þ 2 ϵ0q χνβνðBÞω − 2 ðq · BÞ ξ1ðqÞ¼−ξ2ðqÞ¼jc2jjqjþOðq Þ; ðD13Þ ν 4π

ξ q ξ0 q O q2 3 … B nð Þ¼ nð Þþ ð Þ;n¼ ; ;NCPð Þ: ðD14Þ Supposing ω is a constant in the limit q → 0,wehave

  2 X e e 2 e 3 e 4 2 ðq · BÞ ð 2 q · BÞ ð 2 q · BÞ ð 2 q · BÞ 1 e χ 4π 4π χ 4π 4π ¼ 2 ν ω þ 2 þ ν 2 3 þ 3 4 þ : ðD17Þ ϵ0q ν βνðBÞω βνðBÞω βνðBÞω

To zeroth order of q, we need keep only the first two terms Thus, the scattering term is given by in the above equation. The first term must vanish due to the Z X 2 0 0 d σ γν0 σ ; B Pno-go theorem of Weyl semimetals [55,56], which says S σ ð Þ σ σ0 ½nνð Þ ¼ 3 Wν;ν0 ð ; Þ ν χν ¼ 0. Thus, we have 2π v σ0 ν0 ð Þ j ν0 ð Þj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 1 X 1 × ½δnν0 ðσ Þ − δnνðσÞ ω q → 0 e qˆ B Z CP;1;2ð Þ¼ 2 j · j ; ðD18Þ 2 0 0 4π ϵ β B d σ γν σ ; B 0 ν νð Þ − ð Þ δ σ0 − δ σ ¼ðw0 w1Þ 3 0 ½ nνð Þ nνð Þ ð2πÞ jvνðσ Þj Z where qˆ ¼ q=jqj. X 2σ0 γ σ0 B d ν0 ð ; Þ 0 þ w1 ½δnν0 ðσ Þ − δnνðσÞ: 2π 3 v σ0 ν0 ð Þ j ν0 ð Þj APPENDIX E: FINITE INTRAVALLEY SCATTERING ðE2Þ

In this Appendix, we solve Eq. (B5) to first order of τ0 We introduce the quasiparticle d.o.f. and justify the chiral limit approximation using second- Z τ τ 1 2σ γ σ B order perturbation theory. First, let us derive 0 and v δ d νð ; Þ δ σ nν ¼ 3 nνð Þ; ðE3Þ explicitly in terms of the scattering cross section. We model βνðBÞ ð2πÞ jvνðσ; BÞj the scattering cross section as f and δnνðσÞ¼δnνðσÞ − δnν. Then, Eq. (E2) can be rewrit- σ σ0 δ 1 − δ Wν;ν0 ð ; Þ¼ ν;ν0 w0 þð ν;ν0 Þw1: ðE1Þ ten as

021053-13 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019) X S½nνðσÞ ¼ ðw0 − w1Þ½βνðBÞδnν − βνðBÞδnνðσÞ þ w1 ½βν0 ðBÞδnν0 − βν0 ðBÞδnνðσÞ 0 X ν X f f ¼ −ðw0 − w1ÞβνðBÞδnνðσÞþw1 βν0 ðBÞðδnν0 − δnνÞ − w1 βν0 ðBÞδnνðBÞ ν0 ν0  X   X  X ¼ − w0βνðBÞþw1 βν0 ðBÞ δnνðσÞ − w1 βν0 ðBÞ δnνðσÞþw1 βν0 ðBÞδnν0 : ðE4Þ ν0≠ν ν0 ν0

Defining where 1 X Z β B β B 1 d2σ 1 ¼ w0 νð Þþw1 ν0 ð Þ; ðE5Þ q v 2 q v σ 2 τ0 ν fð · F;νÞ g¼ ¯ 3 ½ · νð Þ : ðE11Þ ; ν0≠ν βðBÞ ð2πÞ jvνðσÞj 1 X β 0 τ τ ¼ w1 ν ; ðE6Þ Apparently, finite 0 introduces a non-Hermitian term in the v ν0 dynamic equation of η. This term leads to a damping rate τ 2 2 ξ we can rewrite the scattering term as proportional to 0q vF= . Therefore, for zero sound to be stable, the following relation should be satisfied: δf σ δ X S δ σ − nνð Þ − nν β B δ ½ nνð Þ ¼ τ τ þ w1 ν0 ð Þ nν0 : ðE7Þ eBq q2v2 0;ν v ν0 ≫ τ F βξ 0 ξ : ðE12Þ The first term relaxes the deformation of the Fermi surfaces that do not change the quasiparticle number in each valley, Considering the intervalley scattering, the following rela- the second term relaxes the quasiparticle number in each tion should also be satisfied: valley, and the last term is feedback from the change of the total quasiparticle number. Because the scattering eBq 1 ≫ : ðE13Þ term is elastic, the total quasiparticle number on the Fermi βξ τv surface should be a constantP underR the scattering. One can 2 3 confirm this by observing ν ½ðd σÞ=ð2πÞ ½γνðσ; BÞ= The above two inequalities are equivalent to jvνðσ; BÞjS½nνðσÞ ¼ 0. For simplicity, in the following we ν τ τ τ ξ μ2 1 ω2 neglect the dependence in 0;ν; i.e., we set 0;ν ¼ 0. ≪ q ≪ B ; τ ω2 τ μ2 ðE14Þ For simplicity, here we consider isotropic Fermi surfa- vvF B 0vF ces, where jvνðσÞj, βνðσÞ, ΩνðσÞ · vˆ νðσÞ do not depend on σ, and γνðσ; BÞ¼1. According to Eq. (B5), to leading which have solutions only if f order of τ0, we get the δnνðσÞ part as 4 τ0 ω 1 ≪ B : E15 i τ 4 ξ ð Þ ðω þ τ Þχν v μ δf σ − τ q v σ v η O τ2 nνð Þ¼ i 0½ · νð Þ e ν þ ð 0Þ: ðE8Þ 2 ðq · BÞ 4π Equation (E15) gives the upper limit of τ0, above which the CZS modes become unstable. It should be noticed that 1=ξ Now we look at the leading-order effect of τ0 on the CZS 1 β¯ B modes. For a specific branch of the CZS, Eq. (E8) gives is of the order of þ ð Þjfj. Perturbation theory for gapless CP modes is similar to χ δf σ − τ q v σ ν η O τ2 the perturbation theory for the CZS modes, and the stable nνð Þ¼ i 0½ · νð Þ ¯ ν þ ð 0Þ; ðE9Þ βðBÞξ condition of the gapless CP modes is also given by Eq. (E15). Now we consider the gapped CP modes. For where ξ is the corresponding eigenvalue of the Ξ matrix the positive branch of the gapped CP, the frequency of [Eq. (C4)]. Substituting it back into Eq. (B5) and integrat- which is denoted as ω , Eq. (E8) gives ing σ, we get CP   ω χ 2 f CP ν 2 i q v ν e q B δnν −iτ0 qˆ · vν σ ην O τ : E16 ω χ η − τ fð · F; Þ g η ð · Þ η ¼ ½ ð Þ e qˆ B þ ð 0Þ ð Þ þ ν ν ¼ i 0 ν þ 2 ν 4π2 ð · Þ τ ξ 4π βνðBÞ v   X 2 e q B e η Following the above analysis, we find this term leads þ 2 ð · Þ fν;ν0 þ 2 ν0 ; ¯ 4π ϵ q to a damping rate proportional to β B qv τ0ω = 0 0 ð Þð FÞð CPÞ ν ∼ τ ω μ2 ω2 eB ðqvFÞð 0 CPÞð = BÞ. Thus, for the gapped CP to ðE10Þ be stable, there should be

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1 ω2 k k0 k τ ≪ B when the short-range interaction fν;ν0 ð ; Þ is indepen- 0 2 : ðE17Þ qvF μ dent, this effect can be neglected safely. However, when 0 fν;ν0 ðk; k Þ becomes k dependent, it is crucial to consider Therefore, the CP modes in the long-wave limit are always this effect to obtain the correct dynamic equation. stable against the intravalley scattering. In the presence of a k-dependent interaction, the renor- malized quasiparticle energy in Eq. (A16) is modified to APPENDIX F: k-DEPENDENT SCATTERING Z   AND INTERACTION X 3k0 2 0 d 0 e ϵν k ϵν k fν ν0 k; k In this section, we consider the k dependence in the ð Þ¼ ð Þþ 2π 3 ; ð Þþϵ q2 ν0 ð Þ 0 scattering cross section and residual short-range interac- γ k0 B δ k0 tion. We show that the dynamic equations (6) and (B6) are × ν0 ð ; Þ nν0 ð Þ: ðF1Þ still correct in the chiral limit, except that the parameters should be modified. Now we neglect the r and t dependence in δnνðkÞ because the scattering process has a much shorter length scale and 1. Elastic scattering conserving the timescale than the collective mode. Here we omit the plane- renormalized energy wave factor eiðq·r−ωtÞ for simplicity. Changing k to the We emphasize that it is the renormalized quasiparticle variable ϵ, σ and writing nνðkÞ as nνðϵ; σÞ¼nFðϵ − μÞþ energy, other than the bare quasiparticle energy, that is δðϵ − μÞδnνðσÞ (as we introduce in Appendix A), we can conserved in the scattering process. This effect is not write the correction to the quasiparticle energies of the considered in Appendixes A–E. As we explain below, quasiparticles on the Fermi surface as

Z   X 2σ0 2 γ k0 B 0 0 d 0 e ν0 ð ; Þ 0 ϵνðkÞ¼μ ⇒ ΔνðσÞ¼ϵνðkÞ − ϵνðkÞ¼ fν ν0 ðσ; σ Þþ δnνðσ Þ: ðF2Þ 2π 3 ; ϵ q2 v σ0 ν0 ð Þ 0 j ν0 ð Þj

We require the renormalized quasiparticle energy to be conserved in the scattering process. Thus, the scattering term is modified to ZZ X ϵ0 2σ0 γ σ0 B d d νð ; Þ 0 0 0 0 0 S½nνðkÞ ¼ Wν ν0 ðσ; σ Þδ½ϵ þ ΔνðσÞ − ϵ − Δν0 ðσ Þ½nν0 ðϵ ; σ Þ − nνðϵ; σÞ: ðF3Þ 2π 3 v σ0 ; ν0 ð Þ j νð Þj

To linear order of δnνðσÞ, we obtain

ZZ X ϵ0 2σ0 γ σ0 B d d νð ; Þ 0 0 0 0 S½nνðkÞ ¼ Wν ν0 ðσ; σ Þδðϵ − ϵ Þ½δðϵ − μÞδnν0 ðσ Þ − δðϵ − μÞδnνðσÞ 2π 3 v σ0 ; ν0 ð Þ j νð Þj ZZ X ϵ0 2σ0 γ σ0 B d d νð ; Þ 0 0 þ Wν ν0 ðσ; σ Þfn ½ϵ þ ΔνðσÞ − Δν0 ðσ Þ − μ − n ðϵ − μÞg 2π 3 v σ0 ; F F ν0 ð Þ j νð Þj Z X 2σ0 γ σ0 B d νð ; Þ 0 0 0 ¼ δðϵ − μÞ Wν ν0 ðσ; σ Þ½δnν0 ðσ ÞþΔν0 ðσ Þ − δnνðσÞ − ΔνðσÞ: ðF4Þ 2π 3 v σ0 ; ν0 ð Þ j νð Þj

2. The valley degrees of freedom First, we decompose the scattering cross section into an intravalley component and an intervalley component In Appendix B, we decompose δnνðσÞ into two parts: f the valley d.o.f. δnν σ and the Fermi-surface d.o.f. δnν σ . 0 1 ð Þ ð Þ σ σ0 δ ð Þ σ σ0 1 − δ ð Þ σ σ0 In Appendix E, we show that if the short-range interaction Wν;ν0 ð ; Þ¼ ν;ν0 Wν ð ; Þþð ν;ν0 ÞWνν0 ð ; Þ: and the scattering cross section are k independent, δnνðσÞ ðF5Þ is a constant for each valley [Eq. (E3)]. In the following, we show that with k-dependent interaction and scattering, Correspondingly, we then decompose the scattering term 0 1 the valley d.o.f. are still well defined but their form is into an intravalley term Sð Þ and an intervalley term Sð Þ. modified. Here, we are interested only in Sð0Þ,

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Z 1 2σ0 γ σ0 B ð0Þ d νð ; Þ ð0Þ 0 0 0 S δ σ ν σ σ δ 0 σ Δ 0 σ − δ σ − Δ σ ½ nνð Þ ¼ 3 0 W ð ; Þ½ nν ð Þþ ν ð Þ nνð Þ νð Þ: ðF6Þ λ ð2πÞ jvνðσ Þj

The valley d.o.f. are undamped under the intravalley scattering. The following condition is sufficient and necessary for δnνðσÞ to be undamped under arbitrary intravalley scattering Z   X 2σ0 2 γ k0 B d 0 e ν0 ð ; Þ 0 δnνðσÞþΔνðσÞ¼δnνðσÞþ fν ν0 ðσ; σ Þþ δnν0 ðσ Þ¼aν; ðF7Þ 2π 3 ; ϵ q2 v σ0 ν0 ð Þ 0 j ν0 ð Þj

where aν is some constant. It is direct to see that when f is We decompose the short-range interaction into a independent of σ, Eq. (E3) satisfies Eq. (F7). k-independent part f¯ and a k-dependent part δf, We expand the valley d.o.f. on a set of basis functions 0 ¯ 0 X fν;ν0 ðσ; σ Þ¼fν;ν0 þ δfν;ν0 ðσ; σ Þ: ðF11Þ δnνðσÞ¼ cαhναðσÞ: ðF8Þ α Then, the basis functions subject to Eq. (F9) can be solved by series expansion in order of δf. We take the trial solution For k-independent f, we can simply set hναðσÞ¼δνα, such that for arbitrary cα, Eq. (F7) is satisfied. For k-dependent X∞ ðmÞ f, we require hνα to satisfy hναðσÞ¼δνα þ hνα ; ðF12Þ Z   m¼1 X 2σ0 2 γ k0 B d 0 e ν0 ð ; Þ σ 0 σ σ m hναð Þþ 3 fν;ν ð ; Þþ 2 0 where hð Þ is in mth order of δf. Substituting Eq. (F12) for ð2πÞ ϵ0q jvν0 ðσ Þj ν0 Eq. (F9), we obtain σ0 × hν0αð Þ¼Aνα; ðF9Þ Z X 2σ0 γ σ0 B ðmþ1Þ d νð ; Þ 0 ðmÞ hνα ðσÞ¼− δfν ν0 ðσ; σ Þh 0 ; where Aνα is a matrix, such that for arbitrary cα, Eq. (F7) is 2π 3 v σ0 ; ν α ν0 ð Þ j νð Þj satisfied, and aν is given as X m ¼ 0; 1; 2…; ðF13Þ aν ¼ cαAνα: ðF10Þ ð0Þ α where hνα ¼ δνα and

Z    X 2 0 0 2 X∞ d σ γν σ ; B e δ ð Þ ¯ δ ðmÞ σ0 Aνα ¼ να þ 3 0 fν;ν0 þ 2 ν0α þ hν0α ð Þ : ðF14Þ 2π vν σ ϵ q ν0 ð Þ j ð Þj 0 m¼1 ¯ We can properly choose fν;ν0 such that Z Z  Z  2σ γ σ B 2σ0 γ σ0 B X 2σ00 γ σ00 B d νð ; Þ d νð ; Þ 0 d νð ; Þ 00 00 0 δfν ν0 ðσ; σ Þþ δfν ν00 ðσ; σ Þδfν00 ν0 ðσ ; σ Þþ ¼ 0: 2π 3 v σ 2π 3 v σ0 ; 2π 3 v σ00 ; ; ð Þ j νð Þj ð Þ j ν0 ð Þj ν00 ð Þ j ν00 ð Þj ðF15Þ

Then, due to Eqs. (F13) and (F15), there are Z 2σ γ σ B d νð ; Þ σ δ β B 3 hναð Þ¼ να νð ÞðF16Þ ð2πÞ jvνðσÞj and   2 δ ¯ e β B Aνα ¼ να þ fν;α þ 2 αð Þ: ðF17Þ ϵ0q ¯ To be specific, we can expand fν;ν0 that fulfills Eq. (F15) in orders of f as

021053-16 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019)

f¯ ¼ f¯ð1Þ þ f¯ð2Þ þ; ðF18Þ where Z Z 2 2 0 0 1 1 d σ γν σ; B d σ γν σ ; B ¯ð Þ ð Þ ð Þ σ σ0 fν;ν0 ¼ 3 3 0 fν;ν0 ð ; ÞðF19Þ βνðBÞβν0 ðBÞ ð2πÞ jvνðσÞj ð2πÞ jvν0 ðσ Þj and Z Z 1 2σ γ σ B 2σ0 γ σ0 B ¯ð2Þ d νð ; Þ d νð ; Þ fν;ν0 ¼ 3 3 0 βνðBÞβν0 ðBÞ ð2πÞ jvνðσÞj ð2πÞ jvν0 ðσ Þj  Z  X 2σ00 γ σ00 B d νð ; Þ 00 ð1Þ 00 0 ð1Þ × ðfν ν00 ðσ; σ Þ − f 00 Þðfν00 ν0 ðσ ; σ Þ − f 00 0 Þ : ðF20Þ 2π 3 v σ00 ; ν;ν ; ν ;ν ν00 ð Þ j ν00 ð Þj

3. Dynamic equation

Now we study the dynamic equation of the valley d.o.f. We first look at the scattering term. Since dnνðσÞ¼ f ð0Þ ð1Þ ð0Þ ð1Þ δnνðσÞþδnνðσÞ and S ¼ S þ S , where S is contributed by intravalley scattering [Eq. (F6)] and S is contributed by intervalley scattering, the total scattering term decomposes into four terms

ð0Þ ð1Þ ð0Þ f ð1Þ f S½nνðσÞ ¼ S ½δnνðσÞ þ S ½δnνðσÞ þ S ½δnνðσÞ þ S ½δnνðσÞ: ðF21Þ

ð0Þ In the last subsection, we prove that S ½δnνðσÞ ¼ 0. Now we make a relaxation-time approximation for the other three terms

δ σ ð1Þ nνð Þ S ½δnνðσÞ ≈− ; ðF22Þ τv

δf σ ð0Þ f ð1Þ f nνð Þ S ½δnνðσÞ þ S ½δnνðσÞ ≈− : ðF23Þ τ0

ð0Þ ð1Þ In the chiral limit, we have S ≫ S and so τ0 ≪ τv. Following the derivation in Appendix A, we obtain the linearized Boltzmann equation with k-dependent short-range interaction as      γνðσ; BÞ i i f ω þ δnνðσÞþ ω þ δnνðσÞ jvνðσÞj τv τ0

¼fq · vˆνðσÞ − eq · B½vˆνðσÞ · ΩνðσÞg½δnνðσÞþΔνðσÞ − ie½vˆνðσÞ × B · ∂k½δnνðσÞþΔνðσÞ; ðF24Þ where ΔνðσÞ is defined in Eq. (F2). To zeroth order of τ0,wehave X δnνðσÞ¼δnνðσÞ¼ cαhναðσÞ; ðF25Þ α where hναðσÞ are the bases introduced in the last subsection. Because of Eq. (F7), δnνðσÞþΔνðσÞ is a constant aν, and due to Eqs. (F10) and (F17), Eq. (F24) can be written as       X 2 γνðσ; BÞ i ¯ e ω δnν σ q vˆν σ − eq B vˆν σ Ων σ cν fν ν0 βα B cν0 : v σ þ τ ð Þ¼f · ð Þ · ½ ð Þ · ð Þg þ ; þ ϵ q2 ð Þ ðF26Þ j νð Þj v ν0 0

Integrating σ on both sides of this equation and applying Eq. (F16), we obtain

021053-17 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019)       X 2 i eðq · BÞ ¯ e ω βν B cν χν cν fν ν0 βν0 B cν0 : þ τ ð Þ ¼ 4π2 þ ; þ ϵ q2 ð Þ ðF27Þ v ν0 0

We introduce the variable ην ¼ βνðBÞcν, and then we obtain     X 2 i eðq · BÞ eðq · BÞ ¯ e ω χνην ην fν ν0 ην0 ; þ τ ¼ 4π2β B þ 4π2 ; þ ϵ q2 ðF28Þ v νð Þ ν0 0 which is of the same form as Eq. (B6). ð0Þ gnðq; r;tÞ − gn ðq; r;tÞ ∂tgnðq; r;tÞ¼−cn∂zgnðq; r;tÞ − ; τsðTÞ APPENDIX G: THERMODYNAMIC PROPERTY ðG5Þ OF CHIRAL ZERO SOUND where τ ðTÞ is the relaxation time for the CZS excitations. We treat the CZS modes as bosonic quasiparticle s ω q At low temperature, the relaxation should be proportional excitations. For each branch of CZS modes nð Þ,we τ q r to v. In the presence of a temperature gradient, the first- assign a distribution function gnð ; ;tÞ, and in equilibrium order stationary solution reads it is just the Bose-Einstein distribution, i.e., 0 1 δg ðq; rÞ¼g ðq; rÞ − gð Þðq; rÞ ≈−τ ðTÞðc ∂ TÞ ð0Þ q r n n n s n z gn ð ; ;tÞ¼ ω q ; ðG1Þ nð Þ ω q 1 expð Þ − 1 nð Þ kBT × : G6 2 2 ω ðqÞ ð Þ kBT 4sinh n 2kBT where kB is the Boltzmann constant and T is the temper- “ ” ature. Here we drop the CZS subscript for brevity. In the The thermal current is given by following, we assume the magnetic field is applied along Z ω q X 3 the z direction, so the dispersion is nð Þ¼cnqz. d q th ω q δ q r First let us calculate the specific heat per unit volume jz ¼ 3 nð Þcn gnð ; Þ jqj<Λ ð2πÞ Z n Z ∂ X 3q X 3 2 d jcnqzj d q ω q 1 κ T −τ ∂ nð Þ ð Þ¼ 3 jc q j ¼ sðTÞ 3 cnðcn zTÞ 2 ω q : ∂T jqj<Λ ð2πÞ n z − 1 q Λ 2π 2 nð Þ n expð k T Þ n j j< ð Þ kBT 4sinh 2 Z  B  kBT X Λ Λ2 − 2 2 1 dqz qz cnqz G7 ¼ k ð Þ B 2π 4π 4 2 cnqz n −Λ kBT sinh 2k T X B Therefore, the thermal conductivity is κ ¼ nðTÞ; ðG2Þ X n σth δ δ τ 2κ ¼ i;z j;z sðTÞ cn nðTÞ: ðG8Þ Λ i;j   cn Z=ðkBTÞ 3 n 3 kBT dx κnðTÞ¼kBΛ cnΛ 2π −c Λ=ðk TÞ APPENDIX H: STRONG MAGNETIC FIELD n B AND FINITE TEMPERATURE ½c Λ=ðk TÞ2 − x2 x2 × n B ; G3 4π 2 x ð Þ The above derivations are based on Boltzmann’s equa- 4sinh 2 tion, which is valid only if ωBτ0 ≪ 1, ωB ≪ μ. Thus, it is where Λ ∼ 1=a0 is the cutoff of q, a0 is a lattice constant, still unknown whether the CP and CZS modes exist in the κ and n is the specific heat contributed by the nth branch of case ωBτ0 ≳ 1, ωB ≪ μ. Here we refer to this case as the the CZS. In the two limits cnΛ ≫ kBT and cnΛ ≪ kBT,we strong field case. In this case, the Landau levels are formed, have and there are many Landau levels under the chemical 8 potential. Therefore, the system should be described by < 3 k T k Λ B ;cΛ ≫ k T; distribution functions on the Landau levels. Here we B 12cnΛ n B κnðTÞ¼ ðG4Þ expand this distribution function as an equilibrium part : Λ3 1 Λ ≪ kB 6π2 ;cn kBT: and a small deviation from equilibrium

Now let us calculate the thermal conductivity. For an 0 nνðk; α;tÞ¼nT½ϵνðk; αÞ − μ inhomogeneous system, the distribution function satisfies δ ϵ0 α − μ δ α iðq·r−ωtÞ the Boltzmann equation þ T½ νðk; Þ nνð Þe : ðH1Þ

021053-18 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019)      α Here, k is the momentum along the magnetic field, is δ ϵ0 α − μ ω i δ ω i δf α ϵ0 α T½ νðk; Þ þ nν þ þ nνð Þ the Landau-level index, νðk; Þ are the Landau levels, τv τ0 0 † n0½ϵνðk; αÞ − μ¼hψ αψ kαi is the occupation number in 0 k ¼ δT½ϵνðk; αÞ − μq · vνðαÞ δ0 ϵ −∂ϵn0 ϵ     equilibrium, and ð Þ¼ ð Þ. We assume the Landau X 2 levels as [57] δ α e β δ × nνð Þþ fν;ν0 þ ϵ q2 ν0 ðBÞ nν0 : ðH9Þ ν0 0 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 <> uk þ v k þ 2eBα; α > 0; 0 We define the disequilibrium quasiparticle number in the ϵνðk; αÞ¼ χ α 0 ðH2Þ > uk þ vk; ¼ ; νth valley as ην ¼ βνðBÞδnν. Then, to zeroth order of τ0, :> pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uk − v k2 þ 2eBjαj; α < 0; integrating k and summing over α, we get   q B q B ω i χ η eð · Þ η eð · Þ where juj < jvj such that the WP is type I [11]. In the þ ν ν ¼ 2 ν þ 2 τ 4π βνðBÞ 4π presence of scattering Eq. (E1), we can write the spectrum v   X 2 function as [58] e fν ν0 ην0 ; × ; þ ϵ q2 ðH10Þ ν0 0 1 1 2τ ϵ0 k α ω =ð 0Þ A½ νð ; Þ; ¼ 0 2 2 ; ðH3Þ which has the exact form as Eq. (B6). It should be noticed π ½ϵνðk; αÞ − μ − ω þ 1=ð2τ0Þ that due to Eq. (H2), only the zeroth Landau level contributes to the integral on the rhs. One can easily verify where τ0 is the quasiparticle lifetime [Eq. (E5)]. Therefore, that the leading-order effect of τv is introducing an effective the occupation number is given by damping rate, and the stable condition for the CZS and CP modes is still given by Eqs. (E15) and (E17), respectively. Z 1 n ðϵÞ¼ dω Aðϵ; ωÞ; ðH4Þ T 1 þ exp ω kBT APPENDIX I: QUANTUM OSCILLATION IN COMPRESSIBILITY and its derivative is given by After the Landau levels are formed, the compressibility Z at finite temperature is given by Z 1=ð2kBTÞ X δTðϵÞ¼ dω ω Aðϵ; ωÞ: ðH5Þ eB dk 0 1 þ cosh βνðBÞ¼ FðαÞ;FðαÞ¼ δ0½ϵνðk; αÞ − μ: kBT 2π 2π α ðI1Þ Similar to the weak field case, we decompose δnνðαÞ as a valley degree Here we assume μ > 0. Using the Poisson equation, we Z get [42] 1 X eB dk 0 ∞ δnν ¼ δT½ϵνðk; αÞ − μδnνðαÞðH6Þ 1 1 X βνðBÞ 2π 2π β 0 − 0þ 0þ α α νðBÞ¼Fð Þ 2 Fð Þþ2 Fð Þþ Fð Þ α¼1 Z∞ and a Fermi-surface degree 1 0 − 0þ α α ¼ Fð Þ 2 Fð Þþ d Fð Þ δf α δ α − δ 0 nνð Þ¼ nνð Þ nν; ðH7Þ ∞ X∞ Z þ 2ℜ dαFðαÞe2πilα: ðI2Þ where 1 l¼ 0 Z X We define β eB dk δ ϵ0 α − μ νðBÞ¼2π 2π T½ νðk; Þ ðH8Þ α Z∞ 1 βð0Þ 0 − 0þ α α ν ðBÞ¼Fð Þ 2 Fð Þþ d Fð ÞðI3Þ is the compressibility at finite temperature. Here we assume 0 that eB > 0. Then, the kinetic equation of collective modes can be written as as the nonoscillating component and

021053-19 ZHIDA SONG and XI DAI PHYS. REV. X 9, 021053 (2019)

Z∞ where Sν is the area enclosed by the fixed-energy circle in ðlÞ 2π α βν ðBÞ¼2ℜ dαFðαÞe il ;l≥ 1 ðI4Þ the k plane. ϕðϵ;kÞ includes the Maslov index plus the 0 Berry phase. For linear isotropic WPs, we always have ϕνðϵ; kÞ¼0, and so in the following, we omit ϕνðϵ;kÞ. ð0Þ ϵ as the oscillating components. βν ðBÞ is just the com- Expanding Sνð ;kÞ as pressibility in the weak field limit [Eq. (B4)]. Now let us 2 ðlÞ 1 ∂ Sν calculate βν ðBÞ. We use the relation ϵ ≈ ϵ − 2 Sνð ;kÞ Sex;νð Þþ2 ∂ 2 ðk kexÞ ; ðI6Þ k ex Sνðϵ;kÞ α ¼ − ϕνðϵ;kÞ; ðI5Þ 2πeB we then have

Z∞ Z βðlÞ ℜ eB α dk 2π α δ ϵ0 α − μ ν ðBÞ¼ π d 2π expði l Þ T½ νðk; Þ 0 Z Z  2  ϵ 1 ∂ Sν − 2 1 dk dS νðϵÞ Sex;νð Þþ2 2 j ðk kexÞ ¼ ℜ dϵ ex; exp i2πl ∂k ex δ ðϵ − μÞ: ðI7Þ 2π2 2π dϵ 2πeB T R ∞ 2 1 Applying the Gaussian integral formula −∞ exp½ði=2Þax dx ¼½ð2πiÞ=a2, we get Z     2 −1 1 ∂ 1 2 ϵ ϵ π ðlÞ Sν dSex;νð Þ Sex;νð Þ βν B ℜ dϵ l i2πl − i δ ϵ − μ ; ð Þ¼ 4π3 ∂ 2 2π ϵ exp 2π 4 Tð Þ ðI8Þ k ex eB d eB

2 2 where we assume ½ð∂ SνÞ=ð∂k Þ < 0. Substituting Eq. (H5) into the above equation, we get Z Z     2 −1 1 ∂ 1 2 ϵ ϵ π ðlÞ Sν dSex;νð Þ Sex;νð Þ βν B ℜ dϵ dω l i2πl − i ð Þ¼ 4π3 ∂ 2 2π ϵ exp 2π 4 k ex eB d eB

1=ð2kBTÞ 1 1=2τ0 × ω 2 2 : ðI9Þ 1 þ coshð Þ π ðϵ − μ − ωÞ þ 1=ð2τ0Þ kBT

Using the contour integral in the upper half plane of ϵ, we get Z     2 −1 i 1 ∂ 1 2 ϵ S νðμ þ ω þ Þ π ðlÞ Sν dSex;νð Þ ex; 2τ0 βν B ℜ dω l i2πl − i ð Þ¼ 4π3 ∂ 2 2π ϵ exp 2π 4 k ex eB d ϵ¼μþωþi=ð2τ0Þ eB 1=ð2k TÞ × B : ðI10Þ 1 þ cosh ω kBT μ ω 2τ μ μ μ ω 2τ We approximate Sex;νf þ þ½i=ð 0Þg as Sex;νð Þþf½dSex;νð Þ=ðd Þgf þ½i=ð 0Þg, then we have       2 −1 1 μ ∂ 1 2 μ μ π ðlÞ dSex;νð Þ Sν dSex;νð Þ l Sex;νð Þ βν B ≈ ℜ l − i2πl − i ð Þ 8π3 μ ∂ 2 2π exp μ 2 τ exp 2π 4 d k ex eB d eB 0 eB μ Z dSex;νð Þ 2ωl 1 expði μ 2 Þ × dω d eB : ðI11Þ k T 1 þ cosh ω B kBT

Now we need to calculate the integral on the second line of the above equation. We denote this integral as I. We choose the contour −∞ → ∞ → ∞ þ i2π → −∞ þ i2π → −∞, and then we have

     dS νðμÞ k Tl dS νðμÞ k Tl dS νðμÞ k Tl 1 − exp −2π ex; B I ¼ 4π ex; B exp −π ex; B ; ðI12Þ dμ eB dμ eB dμ eB

021053-20 HEAR THE SOUND OF WEYL FERMIONS … PHYS. REV. X 9, 021053 (2019) and thus, 2 I ¼ μ ; ðI13Þ dSex;νð Þ πkBTl sinchð dμ eB Þ where sinchðxÞ¼ðex − e−xÞ=ð2xÞ. Therefore, we get

  μ   2 −1 dSex;νð Þ l 1 μ ∂ 1 2 expð− Þ μ π ðlÞ dSex;νð Þ Sν dμ 2eBτ0 Sex;νð Þ βν ðBÞ ≈ l cos 2πl − : ðI14Þ 3 2 dS νðμÞ π 4π dμ ∂k μ 2πeB ex; kBTl 2πeB 4 ex; sinchð dμ eB Þ

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