Anomalous sound attenuation in Weyl semimetals in magnetic and pseudomagnetic fields
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Anomalous sound attenuation in Weyl semimetals in magnetic and pseudomagneticfields
P. O. Sukhachov ∗ and L. I. Glazman Department of Physics, Yale University, New Haven, CT 06520, USA (Dated: February 8, 2021)We evaluate the sound attenuation in a Weyl semimetal subject to a magnetic field or a pseu-domagnetic field associated with a strain. Due to the interplay of intra- and inter-node scatteringprocesses as well as screening, the fields generically reduce the sound absorption. A nontrivial de-pendence on the relative direction of the magnetic field and the sound wave vector, i.e., the magneticsound dichroism, can occur in materials with non-symmetric Weyl nodes (e.g., different Fermi ve-locities and/or relaxation times). It is found that the sound dichroism in Weyl materials can bealso activated by an external strain-induced pseudomagnetic field. In view of the dependence on thefield direction, the dichroism may lead to a weak enhancement of the sound attenuation comparedto its value at vanishing fields.
I. INTRODUCTION
Weyl and Dirac semimetals are novel materials with unusual electronic properties related to their relativistic-likeenergy spectrum and nontrivial topology [1–7]. Conduction and valence bands in Dirac semimetals touch at Diracpoints. Each Dirac point can be viewed as a composition of two Weyl nodes of opposite topological charges orchiralities. The Weyl nodes in Weyl semimetals are separated in momentum and/or energy spaces. Since the totaltopological charge in the system vanishes, the Weyl nodes always appear in pairs of opposite chiralities [8–10]. One ofthe most remarkable properties of Dirac and Weyl semimetals is their ability to reproduce various effects previouslyaccessible only in high energy physics including the celebrated chiral anomaly [11, 12].In essence, the chiral anomaly is the violation of the classical conservation law of chiral or imbalance charge inquantum theory when fermions interact with electromagnetic fields. This process is often referred to as the chiralitypumping between the Weyl nodes of opposite chiralities. In the high-energy physics context, the anomaly is crucialfor the description of the neutral pion decay into two photons. The chiral anomaly plays an important role in thecondensed matter physics too. For example, it leads to the “negative” longitudinal magnetoresistance observed inDirac (Na Bi, Cd As , and ZrTe ) and Weyl (transition metal monopnictides TaAs, NbAs, TaP, and NbP) semimetals(see Refs. [13–17] for reviews on anomalous transport properties). In these materials, the resistivity decreases with amagnetic field if an electric current is driven along the field.In addition to the anomalous transport in conventional electromagnetic fields, Dirac and Weyl semimetals allowone to realize unusual chirality-dependent pseudo-electromagnetic or axial gauge fields [18–21] (see also Ref. [22] fora review). Unlike electromagnetic fields, the pseudo-electromagnetic ones act on the fermions of opposite chiralitieswith different sign. Pseudo-electromagnetic fields allow for several interesting effects such as anomalous transport [20,21, 23–29], quantum oscillations [30], collective excitations [31–35] to name a few. Of particular interest are effectsrelated to dynamical deformations, which, for example, could be induced by sound. Such effects include the soundattenuation [21, 36–38], the acoustogalvanic effect [39], the torsion-induced chiral magnetic effect current [40], andthe axial magnetoelectric effect [41]. It is worth noting that not only deformations affect electron quasiparticles, butphonons could also receive a feedback from electrons under certain conditions. In particular, the chiral anomaly forfermions is manifested in phonon dynamics [42–45].While the sound attenuation in Weyl semimetals was already investigated before, the corresponding analysis isincomplete. For example, Ref. [36] captures only the effect of the anomaly in a narrow parametric regime. Inparticular, the background contribution to the attenuation coefficient in the absence of magnetic fields, which issimilar to the case of conventional multi-valley semiconductors [46–48], was not taken into account. Furthermore, theeffects of the intra-node scattering and the electrostatic screening associated with the inter-node electron dynamicswere also neglected. Indeed, while it is common to screen the regular deformation potential [49–51], it was assumedthat the axial or chiral one, which has opposite signs at the Weyl nodes of opposite chiralities, is insensitive to thescreening since it does not lead to electric charge deviations. We show that, while this is true in the absence ofmagnetic fields, the chiral anomaly eventually allows for the oscillating electric charge even for the chiral deformation ∗ [email protected] potential. Therefore, the electric screening plays the crucial role and should be taken into account. This provided oneof the main motivations for the present work.By using the chiral kinetic theory (CKT) [52–54], we calculate the attenuation coefficient in an effective low-energymodel of Weyl semimetals subject to a scalar chirality-dependent deformation potential as well as external magneticand strain-induced pseudomagnetic fields. According to Refs. [42–44], such deformation potential could originatefrom the coupling to a pseudoscalar phonon. Indeed, the conventional momentum-independent deformation potentialis strongly screened allowing only for the valley-sensitive part to survive. In the absence of magnetic fields, ourfinal expression for the sound attenuation coefficient agrees with that for usual multi-valley semiconductors [46–48].Therefore, to activate nontrivial features of Weyl semimetals, one needs to apply external fields. Due to the chiralanomaly, the electrostatic screening, and the intra-node scattering, the sound attenuation becomes suppressed ina magnetic field. This suppression is monotonic for symmetric Weyl nodes, which are characterized by the sameparameters, and could reach several percents for sufficiently strong fields. On the other hand, in the case wherethe Weyl nodes have different Fermi velocities or are characterized by different relaxation times, the dependence onthe magnetic field becomes nonmonotonic and magnetic sound dichroism can be realized. Due to this effect, theattenuation is different for the sound propagating along or opposite to the magnetic field. A similar nonmonotonicand directional dependence occurs also if an external pseudomagnetic field is applied to the semimetal. We noticethat pseudomagnetic sound dichroism appears even if the Weyl nodes are symmetric. For realistic model parameters,sound dichroism is weak. It value for optimal (pseudo)magnetic fields is about a few percent of the attenuationcoefficient at zero fields.While we were finalizing the manuscript, a recent preprint addressing the sound attenuation in Weyl semimetals ina magnetic field [38] appeared. The authors of Ref. [38] took somewhat different route in deriving their results, butthere is a partial overlap with our findings. Specifically, the decrease of the attenuation coefficient with the magneticfield for a scalar deformation potential was predicted in Ref. [38]. In addition, the magnetic sound dichroism wasdiscussed there, albeit of a different origin, unrelated to the non-symmetric Weyl nodes. Thus, our results extend andprovide an alternative derivation for some of the conclusions reached in Ref. [38].The paper is organized as follows. We introduce the model and the sound attenuation coefficient in Sec. II.Section III is devoted to the chiral kinetic theory and the effect of propagating sound waves on the electron quasiparticledistribution function. The sound attenuation is analyzed and the numerical estimates are provided in Sec. IV. Theresults are summarized and discussed in Sec. V. Technical details concerning the derivation of the energy dissipationrate and the collision integral are given in Appendices A and B, respectively. Throughout this study, we use k B = 1. II. MODEL AND SOUND ATTENUATION COEFFICIENTA. Model of Weyl semimetal
The effective low-energy Weyl Hamiltonian in the vicinity of the Weyl node α reads as H α = χ α v F,α ( p α · σ ) , (1)where χ α = ± is the chirality or, equivalently, the topological charge of the Weyl node, v F,α is the Fermi velocity, p α is the momentum, and σ is the vector of the Pauli matrices acting in the pseudospin space. As we show in Secs. III B 3and IV C, difference between the Weyl nodes, e.g., different Fermi velocities, plays an important role in the soundattenuation in a magnetic field.To realize a Weyl semimetal, the time-reversal (TR) and/or parity-inversion (PI) symmetries should be broken.In the case of the broken TR symmetry but preserved PI symmetry, the minimal model contains two Weyl nodesseparated in momentum space by 2 b . On the other hand, the minimal number of Weyl nodes for a TR symmetric butPI symmetry-broken model is four. In the case where both symmetries are broken, the Weyl nodes in the minimalmodel could be separated in energy by 2 b and in momentum by 2 b . In material realizations of Weyl semimetals, thenumber of Weyl nodes could be significantly larger. For example, there are 24 Weyl nodes of two types in transitionmetal monopnictides (see, e.g., Ref. [4]).To describe the effects of weak dynamical strains, we introduce the deformation potential [55]. For the sake ofsimplicity, we consider only its scalar part, i.e., H strain = λ ( α ) ij ( p α ) u ij ( t, r ) . (2)Here, λ ( α ) ij ( p α ) quantifies the strength of the scalar deformation potential for the node α , u ij ( t, r ) = ( ∂ i u j + ∂ j u i ) / u ( t, r ) is the displacement vector that depends on time and coordinates. In this work, weconsider the case of plane sound waves u ( t, r ) = u e − iωt + i q · r , (3)where ω and q are the sound angular frequency and the wave vector, respectively. Henceforth, for simplicity, wesuppress the arguments of the strain tensor. In general, λ ( α ) ij ( p α ) depends on momentum. This corresponds to thehigher-order multipole modes of Fermi surface oscillations, which are the main source of the sound attenuation in usualsingle-valley metals [51]. In the present work, we focus on the momentum-independent deformation potential, i.e., weassume that λ ( α ) ij ( p α ) = λ ( α ) ij . This approximation is sufficient in order to capture the leading-order contribution tothe sound absorption in multi-valley systems.It is well-known [51] that, in the case of momentum-independent deformation potential, its valley-even component isstrongly screened in conventional metals since it causes electric charge oscillations. On the other hand, the valley-oddcomponent of the deformation potential in a multi-valley conductor does not lead to electric charge oscillations thusremaining effective in the sound attenuation [46–48]. In a conventional multi-valley conductor, sound attenuationis insensitive to a classically weak magnetic field. Situation is different in a Weyl semimetal, if the pattern of theodd-in-valley component of the deformation potential coincides with the pattern of nodes of opposite chirality. Sucha chiral component of the deformation potential induces inter-valley transitions, which, along with the magnetic field,affect the sound absorption. As was noted in Ref. [25], the chiral deformation potential can be induced in Weylsemimetals with broken PI symmetry where Weyl nodes are separated in energy. From the symmetry point of view,this part of the deformation potential requires coupling to phonon modes which are invariant under proper rotations(pseudoscalar phonons) [42–44]. Regardless of its origin, the chiral deformation potential corresponds to the antiphasemotion of the Weyl nodes of opposite chiralities in energy space.The energy spectrum in each Weyl node reads as˜ ǫ α = ǫ α + χ α b + λ ( α ) ij u ij , (4)where ǫ α = ǫ α ( p α ) is the energy dispersion in the absence of deformations at the node α and b quantifies theseparation between the Weyl nodes in energy space. For example, the Weyl nodes of opposite chiralities are locatedat different energies in SrSi [56, 57]. In the case of the effective Hamiltonian given in Eq. (1), ǫ α = v F,α p α . Finally,as in semiconductors [47], in the absence of electron transition between the valleys, each of the nodes establishes itsown effective Fermi energy µ α = µ + λ ( α ) ij u ij , (5)where µ is the equilibrium Fermi energy measured from the Weyl nodes. This corresponds to the case where the localequilibrium charge density within each valley is conserved even in the presence of dynamical deformations. It is worthnoting that such a configuration cannot exist in equilibrium for static deformations in real materials where even weakinter-node scattering processes eventually equalize the Fermi levels across all Weyl nodes.A replica of the band structure displaying two representative Weyl nodes at nonzero chiral deformation potential λ ( α ) ij u ij = χ α λ (5) ij u ij is sketched in Fig. 1. As we demonstrate below, the energy shift induced by λ (5) ij u ij leads to anontrivial scattering between Weyl nodes of opposite chiralities. B. Sound attenuation coefficient
In this Section, we discuss the sound absorption in Weyl semimetals. The sound attenuation coefficient Γ is definedas [49–51, 58] Γ =
QV I . (6)It quantifies the decay of the sound energy flux I = v s ρ m h| ∂ t u | i T v s ρ m ω u v s is the sound velocity, ρ m is the mass density, u is the magnitude of the displacementvector, h . . . i T denotes the time average, and V is the system’s volume. Further, Q is the dissipated energy per unit - - μ λ ij ( ) u ij b + λ ij (cid:0) (cid:1) u FIG. 1. Schematic band structure of a Weyl semimetal with two nodes subject to the node-dependent or chiral deformationpotential λ ( α ) ij u ij = χ α λ (5) ij u ij . In the absence of deformations, the Weyl nodes are separated by 2 b in energy. time. It can be conveniently expressed (see Appendix A) in terms of the dynamical strain tensor u ij associated withthe sound wave Eq. (2) and the perturbed electron distribution evaluated to the first order in u ij , QV = N W X α ν α ( µ )2 Re n iω (cid:16) λ ( α ) ij u ij (cid:17) ∗ n α o . (8)Here, N W is the total number of Weyl nodes and n α is the perturbed by the deformation potential electron distribution n α ( p α ) ∼ λ ( α ) ij u ij in the valley α averaged over the respective Fermi surface, n α = 1 ν α ( µ ) Z d p α (2 π ~ ) δ ( ǫ α + χ α b − µ ) n α ( p α ) . (9)We assume that temperature is small compared to the Fermi energy µ throughout this work. The density of states(DOS) in the general Eq. (8) for the specific model of Weyl nodes given in Eqs. (1) and (4) is ν α ( µ ) ≡ Z d p α (2 π ~ ) δ ( ǫ α + χ α b − µ ) = ( µ − χ α b ) π ~ v F,α . (10)The full non-equilibrium distribution function for the quasiparticles from the Weyl node α reads f α ( p α ) = f (0) α ( p α ) − ( ∂ ǫ α f (0) α ) n α ( p α ) , (11)where f (0) α ( p α ) is the Fermi-Dirac distribution function, which describes electron quasiparticles in local equilibrium,and ( ∂ ǫ α f (0) α ) ≈ − δ ( ǫ α + χ α b − µ ) (12)for temperatures low compared to the Fermi energy.In the case of equal densities of states ν α ( µ ) in all Weyl nodes, ν α ( µ ) = ν ( µ ), and the deformation potential λ ( α ) ij = λ ij + χ α λ (5) ij uniform across the nodes of same chirality, Eq. (8) simplifies to QV = − ν ( µ ) ω n ( λ ij u ij ) ∗ n + (cid:16) λ (5) ij u ij (cid:17) ∗ n o at ν α ( µ ) = ν ( µ ) , λ ( α ) ij = λ ij + χ α λ (5) ij . (13)Here, the averaged distribution functions n = P N W α n α and n = P N W α χ α n α are linear combinations of the perturbedelectron distributions in valleys α averaged over the respective Fermi surfaces. The local charge density is suppressedin the limit of strong screening, n = 0, and the first term in the curly brackets in Eq. (13) vanishes.Finally, we notice that the energy dissipation rate in Eq. (8) agrees with that in Ref. [50]. We would like toemphasize that using Eq. (8) is equivalent to the conventional way of evaluation of Q via entropy production [49, 59].However, we find the form of Eq. (8) better-suited for our purposes, as it allows us to see the effect of symmetriesbetween the valleys on the energy dissipation and spares us from a separate evaluation of the Joule and inter-valleycontributions to Q . As we show in Sec. IV A, our results for the sound absorption in a magnetic field agree (in theappropriate limits) with the recent findings in Ref. [38] where the entropy production was calculated.For our qualitative estimates, we ignore the anisotropy of the deformation potential in the attenuation coefficientand estimate it as λ ( α ) ij u ij ∼ iλ ( α ) u q . Then, the final expression for the sound attenuation coefficient in Eq. (6) readsas Γ = qv s ρ m ωu N W X α ν α ( µ )Re n λ ( α ) n α o . (14)Thus, the sound attenuation is determined by the DOS ν α ( µ ), the deformation potential ∝ λ ( α ) , and the averagednon-equilibrium part of the distribution function n α . Under the simplifying assumptions regarding the DOS in Weylnodes and the chiral deformation potential, the attenuation coefficient reduces to:Γ = ν ( µ ) qv s ρ m ωu Re n λn + λ (5) n o at ν α ( µ ) = ν ( µ ) , λ ( α ) ij = λ ij + χ α λ (5) ij . (15)To describe the deviations from the equilibrium caused by dynamical deformations and determine n α , we employ thechiral kinetic theory (CKT) in the next Section. III. CHIRAL KINETIC THEORYA. General equations of chiral kinetic theory
The semiclassical kinetic equation for Weyl quasiparticles reads as [52–54] ∂ t f α + 1Θ α (cid:26)(cid:16) − e ˜ E α − ec [ v α × B α ] + e c ( ˜ E α · B α ) Ω α (cid:17) · ∂ p α f α + (cid:16) v α − e h ˜ E α × Ω α i − ec ( v α · Ω α ) B α (cid:17) · ∇ f α (cid:27) = I intra [ f α ] + I inter [ f α ] . (16)Here f α is the distribution function for electron quasiparticles at the node α defined in Eq. (11), v α = ∂ p α ˜ ǫ α is thequasiparticle velocity, − e ˜ E α is the force acting on an electron ( e > e ˜ E α = e E + ∇ ˜ ǫ α , (17)the energy spectrum ˜ ǫ α is defined in Eq. (4) and depends on spatial coordinates via the deformation potential λ ( α ) ij u ij ,vector B α combines the external magnetic and pseudomagnetic fields. The Berry curvature Ω α = Ω α ( p α ) for theHamiltonian of Eq. (1) takes the form Ω α ( p α ) = χ α ~ p α p α , (18)and Θ α = [1 − e ( B α · Ω α ) /c ] is the renormalization of the phase-space volume. The sound attenuation is determinedby the Fermi surface properties, so, in the following, we will encounter only Ω α ( p F,α ) where p F,α parametrizes pointson the Fermi spheres around the respective Weyl nodes α . In the case of a linear energy spectrum discussed inSec. II A, p F,α = ( µ − χ α b ) /v F,α . The collision integral in the right-hand side in Eq. (16) is a sum of the intra- andinter-node terms I intra [ f α ] and I inter [ f α ], respectively. Their explicit form in τ -approximation is given in Appendix B,see Eqs. (B2) and (B10).The kinetic equation (16) should be also amended with Maxwell’s equations. Since the sound and Fermi velocitiesare much smaller than the speed of light, we use a quasistatic approximation in which dynamical magnetic andsolenoidal electric fields are neglected. Then, only the Gauss law should be taken into account. It reads as ∇ · E = − πe N W X α ν α ( µ ) n α (19)and is determined by the non-equilibrium part of the distribution function (see also Eq. (11)). The combination ofthe kinetic equation (16) and the Gauss law (19) comprise the full formulation of the self-consistent problem for theresponse of Weyl semimetal to acoustic waves propagating in it.As is shown in Sec. II B, the energy dissipation rate and the sound attenuation coefficient are determined by theaveraged over the Fermi surface non-equilibrium part of the distribution function n α . To calculate this function, welinearize Eq. (16) in weak strains u ij and use the ansatz in Eq. (11). In addition, since eB α Ω α /c is small, we ignorethe renormalization of the phase-space volume Θ α ≈ n ∂ t + h v α − ec ( v α · Ω α ( p F,α )) B α i · ∇ o n ( p F,α )+ e ( v α · E ) − e c ( v α · Ω α ( p F,α )) ( E · B α ) + N W X β n ( p F,α ) − n β τ α,β = − λ ( α ) ij ( v α · ∇ u ij ) − λ ( α ) ij ec ( v α · Ω α ( p F,α )) ( B α · ∇ u ij ) + u ij N W X β λ ( α ) ij − λ ( β ) ij τ α,β . (20)Here, n ( p F,α ) is the non-equilibrium part of the distribution function at the Fermi level. In addition, we used thedispersion relation of Eq. (4) and the explicit form of the collision integrals in the τ -approximation given in Eqs. (B2)and (B11) in Appendix B. The relaxation rates 1 τ α,β = 2 π ~ | A α,β | ν β ( µ ) (21)are determined by the intra- and inter-node scattering amplitudes A α,β . In the following, we abbreviate the notationfor the intra-node relaxation time: τ α,α ≡ τ α . B. Solutions to the kinetic equation
In this Section, we consider the case of weak inter-node ( α = β ) scattering, τ α ≪ τ α,β , and τ α ω ≪
1. We checkedthat the above assumptions hold well for realistic numerical values presented in Sec. IV A. Short intra-node relaxationtime allows us to retain only the first two harmonics in the expansion of the non-equilibrium part of the distributionfunction: n α ( p α ) ≈ n (0) α + n (1) α cos θ α , (22)where θ α is the angle between v α and ∇ , and n (0) α ≈ n α . By following the standard procedure, we use Eq. (20),separate the contributions with different powers of cos θ α , and solve for n (1) α . Then, by using the obtained solution,we calculate the current density j α = j (CME) α + j (diff) α . (23)In the above equation, we separated the chiral (pseudo)magnetic effect (CME) j (CME) α [60, 61] and diffusion intra-node j (diff) α currents. They read as j (CME) α = e c Z d p α (2 π ~ ) ( v α · Ω α ( p F,α )) B α n (0) α δ ( ǫ α + χ α b − µ ) = χ α e π ~ c B α n α , (24) j (diff) α = − e Z d p α (2 π ~ ) v α n (1) α cos θ α δ ( ǫ α + χ α b − µ ) = eν α ( µ ) D α h ∇ n α + e E + λ ( α ) ij ∇ u ij i , (25)where D α = v F,α τ α h ˜ E α × Ω α ( p F,α ) i present in electron velocity, see Eq. (16), does notcontribute to the current density in the linear order in u ij .By averaging Eq. (20) over the Fermi surface and using Eq. (23), we derive the following kinetic equation, ∂ t n α − eν α ( µ ) ( ∇ · j α ) − e c (cid:18)(cid:20) E + 1 e λ ( α ) ij ∇ u ij (cid:21) · B α (cid:19) ( Ω α · v α ) = − N W X β n α − n β τ α,β − u ij N W X β λ ( α ) ij − λ ( β ) ij τ α,β , (27)which together with Eqs. (23), (24), and (25) as well as the Gauss law (19) defines the response n α to the dynamicstrain u ij .The first two terms on the left-hand side of Eq. (27) in the above equation correspond to the conventional continuityequation. The chiral anomaly is described by the third term, where( Ω α · v α ) = | Ω α ( p F,α ) | v F,α = χ α π ~ ν α ( µ ) , (28)cf. Eq. (18). The collision integral in the right-hand side contains the term describing usual inter-node scattering (thefirst term). The second term in the collision integral originates from the different effective Fermi energies of the Weylnodes of opposite chiralities (see also Fig. 1 for the schematic band structure). It is worth noting that this term andthe diffusion current were not accounted for in Ref. [36].In the case of sound-induced dynamical strains, the deformation potential and the non-equilibrium part of thedistribution function have the plane-wave dependence on time and coordinates (see Eq. (3)). Therefore, by using theexplicit form of the currents given in Eqs. (24) and (25), we rewrite Eq. (27) as (cid:2) q D α − iχ α ( v Ω ,α · q ) − iω (cid:3) n α − ieD α ( q · E ) − χ α ( v Ω ,α · E ) + N W X β n α − n β τ α,β = − u ij λ ( α ) ij q D α − iχ α λ ( α ) ij ( v Ω ,α · q ) + N W X β λ ( α ) ij − λ ( β ) ij τ α,β . (29)Here, to simplify the notations, we omitted the overbar in the averaged distribution function, i.e., n α → n α , andintroduced the anomalous velocity v Ω ,α = | Ω α ( p F,α ) | v F,α c e B α = e π c ~ ν α ( µ ) B α . (30)As one can see from Eq. (29), diffusion (see the second term in Eq. (29)) and the chiral anomaly (see the third termin Eq. (29)) introduce the dependence on the electric field E . The Gauss law (19) allows one to express E as alinear combination of n α , thus completing the formulation of the linear response problem. We will provide its explicitsolution under simplifying assumptions.Let us consider Weyl semimetals where there is a symmetry between the pairs ( α, − α ) of Weyl nodes. Specifically,we assume that the deformation potential can be parametrized as λ ( α ) ij = λ ij + χ α λ (5) ij , (31)the two relaxation times 1 τ , ± α = N W X β | χ ± α − χ β | τ ± α,β (32)are the same for all pairs, and B α = B + χ α B . (33)Here the pseudomagnetic field B can be generated by static strains such as bending or torsion [18–21] (see also Ref. [22]for a review). As an example of the corresponding system with such a symmetry, we mention Weyl semimetals wherethe pairs of Weyl nodes are well separated in momentum space. Then, only the scattering rates between the nodes α and − α of opposite chiralities inside each pair α contribute to τ ,α , i.e., τ ,α ≈ τ α, − α . In the presence of saidsymmetry, the matrix of the linear system in Eq. (29) becomes block-diagonal and one can consider kinetic equationsonly for a single pair of Weyl nodes with chiralities χ − α = − χ α .To shorten the notations, we introduce the square of the inverse length [62] q = 4 πe N W X α ν α ( µ ) (34)of the Thomas-Fermi screening in Eq. (29). By expressing the electric field in terms of n α with the help of the Gausslaw (19) and using Eq. (34), we rewrite the kinetic equations for a single pair α = ± (cid:20) τ ,α + q D α − iχ α ( v Ω ,α · q ) − iω (cid:21) n α + (cid:2) q D α − iχ α ( v Ω ,α · q ) (cid:3) q q ν α ( µ ) n α + ν − α ( µ ) n − α ν α ( µ ) + ν − α ( µ ) − n α + n − α τ ,α = − λ ij u ij (cid:2) q D α − iχ α ( v Ω ,α · q ) (cid:3) − χ α λ (5) ij u ij (cid:20) τ ,α + q D α − iχ α ( v Ω ,α · q ) (cid:21) . (35)In what follows, we solve Eq. (35) for symmetric Weyl nodes ( b = 0, v F,α = v F , τ α = τ , and τ ,α = τ ) in twocases: (i) B = 0 and B = 0 as well as (ii) B = 0 and B = 0. In addition, the solutions for non-symmetric nodes( b = 0, v F,α = v F, − α , τ α = τ − α , and τ ,α = τ , − α ) at B = 0 and B = 0 are also considered.
1. Solution of the kinetic equations at a finite magnetic field
In the case of symmetric Weyl nodes and B = 0, one can simplify Eq. (35) to (cid:20) τ q − iχ α ( v Ω · q ) − iω (cid:21) n α − (cid:20) τ − q D + iχ α ( v Ω · q ) q q (cid:21) ( n α + n − α )2= − λ ij u ij (cid:2) q D − iχ α ( v Ω · q ) (cid:3) − χ α λ (5) ij u ij (cid:20) τ q − iχ α ( v Ω · q ) (cid:21) , (36)where v Ω = v Ω ,α and the effective scattering rate 1 τ q = 1 τ + q D (37)includes both intra- and inter-node scattering processes. Notice that v Ω ∝ B is a TR-odd axial vector. The fullsolution to Eq. (36) reads n α = − λ ij u ij M (cid:20) q D (cid:18) τ q − iω (cid:19) − χ α ω ( v Ω · q ) + ( v Ω · q ) (cid:21) − χ α λ (5) ij u ij M " (cid:0) q + q (cid:1) D − iωτ q − χ α ω ( v Ω · q ) + ( q + q ) ( v Ω · ˆ q ) , (38)where ˆ q = q /q and M = (cid:2)(cid:0) q + q (cid:1) D − iω (cid:3) (1 /τ q − iω ) + ( q + q ) ( v Ω · ˆ q ) (39)is proportional to the determinant of the system in Eq. (36).In material realizations of Weyl semimetals, the Fermi energy µ is small compared to that in metals. For example, µ = 10 −
30 meV in a typical Weyl semimetal TaAs [63, 64]. Still the free-carrier density is high enough to result ina fairly short Thomas-Fermi screening length, so that the conditions q TF ≫ q, σ ≫ ω, σ ≫ τ ω , (40)are easily satisfied. Here σ = q D/ (4 π ) is the electric conductivity at B = B = 0. The first of the conditions (40)allows one to disregard the effect of the valley-even part of the deformation potential λ ij in Eq. (38): in agreementwith expectations [46] (see also Refs. [47, 50]), its contribution to n α scales as ( q/q TF ) . Using Eq. (40), we simplifythe general solution in Eqs. (38), (39) and find n α as: n α = − χ α λ (5) ij u ij /τ q + ( v Ω · ˆ q ) /D /τ q + ( v Ω · ˆ q ) /D − iω , n = N W X α n α = 0 , n = N W X α χ α n α = N W χ α n α , (41)As we showed in Sec. II B, since the attenuation coefficient is determined by the real part of n , see Eq. (15), it isalready evident that the combination of the electrostatic screening and the intra-node momentum relaxation has aprofound effect on how the magnetic field affects the sound dissipation. We discuss this in detail in Sec. IV A.
2. Solution of the kinetic equation at a finite pseudomagnetic field
Let us now consider the case in which the system is subject only to a pseudomagnetic field B . The kineticequation (35) at B = 0 and symmetric Weyl nodes reads as (cid:20) τ q − i ( v Ω , · q ) − iω (cid:21) n α − (cid:20) τ − q D + i ( v Ω , · q ) q q (cid:21) ( n α + n − α )2= − λ ij u ij (cid:2) q D − i ( v Ω , · q ) (cid:3) − χ α λ (5) ij u ij (cid:20) τ q − i ( v Ω , · q ) (cid:21) , (42)where v Ω , = ( v Ω ,α − v Ω , − α ) / v Ω , ∝ B , where B breaks both TR and PIsymmetries. Under conditions of Eq. (40), its solution is: n α = − χ α λ (5) ij u ij /τ q − i ( v Ω , · q )1 /τ q − i ( v Ω , · q ) − iω , n = N W X α n α = 0 , n = N W X α χ α n α . (43)Notice that the direction of the pseudomagnetic field depends on the pattern of Weyl node pairs. Therefore, thesummation over all Weyl nodes should be performed with care. We discuss a few corresponding examples in Sec. IV B.By comparing Eqs. (41) and (43), it is evident that the magnetic and pseudomagnetic fields affect the distributionfunctions differently. In particular, the distribution function n α in Eq. (43) depends on the direction of the pseu-domagnetic field B . This observation is not surprising because B , which can be induced by static strains suchas torsion or bending, breaks both the symmetry between the Weyl nodes and crystal symmetries. As we show inSec. IV B, the pseudomagnetic field has a profound effect on the sound attenuation in Weyl semimetals leading todistinct attenuation coefficients for the sound propagating along and opposite to the pseudomagnetic field.
3. Solution of the kinetic equations at a finite magnetic field: non-symmetric Weyl nodes
Finally, we analyze the case of non-symmetric Weyl nodes, i.e., b = 0, v F,α = v F, − α , τ α = τ − α , and τ ,α = τ , − α .The solution to Eq. (35) reads n α = − χ α λ (5) ij u ij ν − α ( µ ) ν α ( µ ) + ν − α ( µ ) 1˜ M ( τ ,α (cid:2) q D − α + i ( v Ω , − α · q ) (cid:3) + τ , − α (cid:2) q D α − i ( v Ω ,α · q ) (cid:3) + 2 τ ,α τ , − α (cid:2) q D α − i ( v Ω ,α · q ) (cid:3) (cid:2) q D − α + i ( v Ω , − α · q ) (cid:3) ) , (44)with ˜ M = τ ,α (cid:2) q D − α + i ( v Ω , − α · q ) (cid:3) + τ , − α (cid:2) q D α − i ( v Ω ,α · q ) (cid:3) + 2 τ ,α τ , − α ν α ( µ ) + ν − α ( µ ) ( ν α ( µ ) (cid:2) q D α − i ( v Ω ,α · q ) (cid:3) (cid:2) q D − α + i ( v Ω , − α · q ) − iω (cid:3) + ν − α ( µ ) (cid:2) q D α − i ( v Ω ,α · q ) − iω (cid:3) (cid:2) q D − α + i ( v Ω , − α · q ) (cid:3) ) . (45)In deriving Eqs. (44) and (45), we used Eq. (40) and retained only the terms surviving in the limit 1 /q TF →
0. Thesolution remains cumbersome even in that limit. Therefore, in the following, we consider small deviations from thesymmetry: D α = D + χ α δD = D (cid:20) − χ α δν ( µ ) ν ( µ ) (cid:21) + χ α δ ˜ D, ν α ( µ ) = ν ( µ ) + χ α δν ( µ ) . (46)Here, D = ( D α + D − α ) /
2, while the difference δD = χ α ( D α − D − α ) / δD ≪ D . Notice alsothat while D is a true scalar, δD is a pseudoscalar. Similar definitions and assumptions are made for other variables.In particular, δτ and δ v Ω depend only on the DOS deviations δν ( µ ) similarly to the second term in the squarebrackets in Eq. (46)). [We assume also the case of well-separated Weyl node pairs, as discussed after Eq. (33).] It isimportant that δD contains an additional term δ ˜ D , which is determined by different Fermi velocities and/or intra-node0scattering amplitudes in the nodes α and − α . As we show below, this term is responsible for the odd-in-magneticfield term in n α .Expanding the solution (44) up to the first order in δν ( µ ) and δ ˜ D , n α = n (0) α + n (1) α + . . . , we obtain: n (1) α = − χ α n (0) α δν ( µ ) ν ( µ ) − χ α λ (5) ij u ij ( v Ω · q ) ω h /τ q + ( v Ω · ˆ q ) /D − iω i δ ˜ DD , (47) n (1)5 = N W X α χ α n (1) α = − N W λ (5) ij u ij ( v Ω · q ) ω h /τ q + ( v Ω · ˆ q ) /D − iω i δ ˜ DD , (48)where the zeroth-order solution n (0) α is given by Eq. (41) found in the limit of symmetric Weyl nodes. While thezeroth-order solution is even in B · ˆ q , odd powers of B · ˆ q appear in the first-order correction given in Eq. (47) at δ ˜ D = 0. The corresponding component survives the summation over the nodes, see Eq. (48). It is easy to check that,as expected for the conditions (40), deviations of electric charge vanish, i.e., P N W α ν α ( µ ) n (0) α + P N W α ν ( µ ) n (1) α = 0.In the next Section, by using the obtained solutions, we discuss the sound attenuation in Weyl semimetals. IV. SOUND ATTENUATION
In parallel with Sec. III B, we evaluate the sound attenuation in three limits. We start with the results for symmetricWeyl nodes and nonzero magnetic field (while B = 0), then consider the effect of the pseudomagnetic field (at B = 0),and conclude this Section with the results specified for non-symmetric Weyl nodes at B = 0. A. Sound attenuation in a magnetic field
By using Eqs. (15) and (41), we find the following attenuation coefficient for the nonzero magnetic field B andsymmetric Weyl nodes: Γ( B , ω, ˆ q ) = N W ν ( µ ) | λ (5) | v s ρ m q h /τ q + ( v Ω · ˆ q ) /D ih /τ q + ( v Ω · ˆ q ) /D i + ω , (49)where q = ω/v s . In the absence of magnetic fields, B = 0 (i.e., v Ω = 0), the sound attenuation coefficient reads asΓ( ω ) = (cid:2) N W ν ( µ ) | λ (5) | / ( v s ρ m ) (cid:3) n τ q q / h τ q ω ) io . It agrees with the results obtained in Refs. [46–48] for multi-valley semiconductors and in Ref. [38] for Weyl semimetals. At τ q ω ≪ ω . The same scaling is valid alsofor single-valley conductors with momentum-dependent deformation potential [49, 51].As one can see from Eq. (49), the sound attenuation coefficient monotonically decreases with the increase of magneticfield B . By using the typical parameter values, see Eq. (51) below, it is easy to check that the condition τ q ω ≪ B , ω, ˆ q ) = N W ν ( µ ) | λ (5) | v s ρ m q /τ q + ( v Ω · ˆ q ) /D , (50)where the dependence of v Ω on B is given by Eq. (30). Further expansion of Eq. (50) in B would reproduce the resultof Ref. [38].We used semiclassical kinetic equations to derive Eqs. (49) and (50). Therefore, we confine our consideration tonon-quantizing magnetic fields. Furthermore, we did not account for the effect of magnetic field on the intra-valleyelectron dynamics. At an arbitrary angle between B and ˆ q , this limits our consideration to classically weak magneticfields, cf. Ref. [38]. The latter constraint is eased for the field B aligned with ˆ q , as we are considering spherical Fermisurfaces. In that respect, there is a similarity between the evaluation of Γ( B ) and the evaluation of the conductivitytensor component along the direction of a non-quantizing magnetic field [65]: in a metal with a spherical Fermi surface,cyclotron motion of electrons does not affect these two quantities.We present the dependence of the normalized magnetic-field dependent part of the sound attenuation coefficient inFig. 2. In plotting it, we used Eqs. (49) and (50) as well as parameters v F ≈ × cm/s , µ ≈
20 meV , τ ≈ × − s , τ ≈ . × − s , v s ≈ . × cm/s , (51)1quoted [64, 66, 67] for the Weyl semimetal TaAs. As one can see from Fig. 2, the suppression of the sound attenuationcoefficient by magnetic field is significant for experimentally feasible frequencies of ultrasound and for B approachingthe regime of classically strong magnetic fields. (cid:1) =
100 MHz (cid:1) = - - (cid:2) [ T ] | (cid:3) ( B ) - (cid:3) ( ) | / (cid:3) ( ) (cid:4) =
20 meV (cid:5) =
60 ps (cid:5) = FIG. 2. The magnetic-field dependent part of the sound attenuation coefficient for the field B k applied along the soundpropagation direction. For the parameters given in Eq. (51), one may use Eq. (50) at frequencies ν ≡ ω/ (2 π ) .
100 MHz (redsolid line). For illustration, we also plot the attenuation coefficient given in Eq. (49) for a much higher frequency, ν = 1 GHz(blue dashed line), at which q Dτ >
1. The gray shaded area corresponds to a classically strong magnetic field. The crossoverto a quantizing magnetic field region is indicated by the black dotted line.
Our Eqs. (49) and (50) differ substantially from the sound attenuation predictions of Ref. [36]. There are two majordifferences distinguishing the current work from the simplifications accepted in Ref. [36], as we account for: (i) a smallbut finite intra-node relaxation time and (ii) electrostatic screening of the electric fields accompanying sound wavesin a Weyl semimetal subject to a magnetic field. It is the combination of these effects that leads to the discrepancy.To demonstrate this explicitly, let us consider the model case of a very fast intra-node relaxation, D →
0. Then, theattenuation coefficient due to the valley-dependent part of the deformation potential readslim D → Γ( B , ω, ˆ q ) = N W ν ( µ ) | λ (5) | v s ρ m τ q ( τ ω ) h − ( q + q ) ( v Ω · ˆ q ) /ω i + 1 , (52)where we used Eqs. (38) and (39) without any assumption about the ratio q/q TF . The magnetic field dependence firstappears here in the second order in magnetic field:lim D → Γ( B , ω, ˆ q ) = lim D → Γ( ω ) + δ Γ( B , ω, ˆ q ) , δ Γ( B , ω, ˆ q ) = 2 N W ν ( µ ) | λ (5) | v s ρ m ( τ ω ) v s τ (cid:0) q + q (cid:1) ( v Ω · ˆ q ) . (53)To obtain the final form of δ Γ( B , ω, ˆ q ) in the above equation, we considered the limit of low frequency, τ ω ≪
1. Atno screening, q TF = 0, we find a qualitative [68] agreement with the conclusions of Ref. [36]: magnetic field wouldenhance sound attenuation. Furthermore, as follows from Eq. (53), screening ( q TF = 0) strengthens the chiral anomalyeffect. Thus, the qualitative difference between the results in Eqs. (49) and (52) stems from the intra-node diffusionat finite D , along with the presence of electric fields accompanying non-uniform chiral currents. Regretfully, even ina strongly-disordered material with the mean free path of the order of Fermi wavelength the diffusion coefficient ishigh enough to invalidate Eq. (52). By the same token, the associated with the chiral anomaly propagating collectivemode [36] turns into an overdamped one due to diffusion. One can see this by examining the roots of the polynomial M ( ω ) in Eq. (39). B. Sound attenuation in a pseudomagnetic field
In this Section, we consider the sound attenuation in the pseudomagnetic field B that was not discussed inthe literature before. As in Sec. III B 2, we assume that the Weyl nodes are symmetric. Furthermore, since the2pseudomagnetic field depends on the configuration of the Weyl nodes, we focus on a system with a single pair ofnodes. The extension to the case of multiple Weyl nodes is discussed at the end of this Section. By using Eqs. (15)and (43), we derive Γ( B , ω, ˆ q ) = 2 ν ( µ ) | λ (5) | v s ρ m q /τ q /τ q + [ ω + ( v Ω , · q )] . (54)Here, ( v Ω , · q ) is a true scalar because v Ω , ∝ B is a polar vector. Among the most notable features of Γ( B , ω, ˆ q ) isits dependence on the direction of the pseudomagnetic field with respect to the sound wave vector, see also Eq. (30)for the dependence of v Ω , on B . Indeed, depending on the sign of ( v Ω , · q ), the pseudomagnetic field can eitherreduce or enhance the sound absorption. Furthermore, Γ( B , ω, ˆ q ) is a nonmonotonic function of B which has amaximum, Γ max ( ω, ˆ q ) = 2 ν ( µ ) | λ (5) | τ q ω / ( v s ρ m ), at ω = − ( v Ω , · q ).One can easily extract the odd-in-pseudomagnetic field B part of the attenuation coefficient. The result at τ q | ω + ( v Ω , · q ) | ≪ B , ω, ˆ q ) − Γ( B , ω, − ˆ q ) ≈ − ν ( µ ) (cid:12)(cid:12) λ (5) (cid:12)(cid:12) v s ρ m ( τ q ω ) ( v Ω , · q ) (55)The above equation clearly shows that the sound attenuation is different when the sound propagates along or oppositeto the pseudomagnetic field. Therefore, we dub this effect the pseudomagnetic sound dichroism .To estimate the effects of the pseudomagnetic field on the sound absorption, we show the odd- and even-in-pseudomagnetic field parts of the attenuation coefficient in Figs. 3(a) and 3(b), respectively. Our numerical estimatessuggest that the dichroism is weak and reaches ∼ B ∼ B , cf. Figs. 2 and 3(b). Indeed, the pseudomagnetic fieldgenerally reduces the sound attenuation coefficient. The reduction is strong for large fields comparable to classicallystrong B . It is worth noting that attainable values of strain-induced pseudomagnetic fields are estimated to be about0 . (cid:1) =
100 MHz (cid:1) = - - (cid:2) [ T ] | (cid:3) ( B ) - (cid:3) ( - B ) | / [ (cid:3) ( ) ] (cid:4) =
20 meV (cid:5) =
60 ps (cid:5) = (a) (cid:1) =
100 MHz (cid:1) = - - (cid:2) [ T ] | (cid:3) ( B ) + (cid:3) ( - B ) | / [ (cid:3) ( ) ] - (cid:4) =
20 meV (cid:5) =
60 ps (cid:5) = (b) FIG. 3. Dependence of the (a) odd- and (b) even-in-pseudomagnetic field parts of the attenuation coefficient on the pseu-domagnetic field applied along the sound propagation direction B , k . We used the following sound frequencies ν ≡ ω/ (2 π ): ν = 100 MHz (red solid line) and ν = 1 GHz (blue dashed line), which correspond to the regimes q Dτ < q Dτ > Finally, let us comment on the sound absorption in a system with multiple Weyl nodes. Unlike the case witha nonzero magnetic field considered in Sec. IV A, the pseudomagnetic field itself depends on the configuration ofWeyl nodes. Therefore, in general, one needs to use Eqs. (14) and (29) to obtain the sound attenuation coefficient.However, as we discussed after Eq. (30), certain symmetries can be used to identify the pairs of Weyl nodes and thecorresponding pseudomagnetic fields. For example, in the simplest model of TR symmetric Weyl semimetals, thereare two pairs of Weyl nodes such that the pseudomagnetic fields in each pair have opposite directions. Therefore, the3sound attenuation coefficient is given by the sum of the attenuation coefficients obtained in Eq. (49) with the oppositesigns of the field B . No pseudomagnetic sound dichroism appears in this case. The dichroism could be observed ifthe TR symmetry is broken, e.g., in an intrinsically magnetic material such as EuCd As [69, 70]. C. Sound attenuation dichroism in a magnetic field
In this Section, we calculate the sound attenuation in Weyl semimetals with non-symmetric Weyl nodes subject toa magnetic field. As in Section III B 2, we focus on the stemming from the difference between the Weyl nodes first-order correction Γ (1) ( B , ω, ˆ q ) to the attenuation coefficient. In particular, we assume that the differences between thedensities of states [ ν α ( µ ) − ν − α ( µ )] /ν ( µ ) ≪
1, the Fermi velocities ( v F,α − v F, − α ) /v F ≪
1, and diffusion constants( D α − D − α ) /D ≪ (1) ( B , ω, ˆ q ) = 2 N W | λ (5) | ν ( µ ) ωv s ρ m q h /τ q + ( v Ω · ˆ q ) /D i(cid:26)h /τ q + ( v Ω · ˆ q ) /D i + ω (cid:27) δ ˜ DD ( v Ω · q ) , (56)where δ ˜ D is determined by the difference of the Fermi velocities and/or the intra-node scattering amplitudes aroundthe Weyl nodes α and − α . Because δ ˜ D and ( v Ω · q ) are pseudoscalars, the resulting attenuation coefficient is a scalar.As one can see from Eq. (56), there is a dependence of the sound absorption on the relative orientation of the magneticfield and the sound wave vector. Indeed, the sign of the coefficient Γ (1) ( B , ω, ˆ q ) is flipped together with the directionof the magnetic field with respect to the sound wave vector, resulting in the magnetic sound dichroism . A similareffect was predicted in Refs. [37, 38], whose origin, however, relies on the difference of the deformation potentialsbetween different nodes. In the limit τ q ω ≪ ω and τ agrees withthat of Ref. [38].It is interesting to compare the odd-in-magnetic field attenuation coefficient with its pseudomagnetic counterpart.Upon expanding Eq. (56) in τ q ω ≪ q · v Ω ) ≪ D/τ q , its comparison with Eq. (55) yieldsΓ (1) ( B , ω, ˆ q ) − Γ (1) ( B , ω, − ˆ q )Γ( B , ω, ˆ q ) − Γ( B , ω, − ˆ q ) = − ( v Ω · q )( v Ω , · q ) δ ˜ DD . (57)Therefore, the magnetic sound dichroism is expected to be weaker than its pseudomagnetic counterpart at equalrespective fields.
V. SUMMARY AND DISCUSSION
We investigated the anomalous sound attenuation in Weyl semimetals in external magnetic and pseudomagneticfields. It was found that the nontrivial topology of Weyl semimetals activated by these fields has an unusual man-ifestation in the sound absorption. In addition to presenting new results, our work extends, corrects, and providesalternative derivation for some of the conclusions reached in the literature.We develop an effective way of evaluating the sound attenuation coefficient, by directly relating it to the solution ofa linearized kinetic equation, see Eqs. (14), (15), and (29). Under rather natural assumptions regarding the symmetryof Weyl nodes, a compact form of the solutions to kinetic equations allows us to elucidate the dependence on thematerial parameters as well as the fields strength and orientation, see Eqs. (41), (43), (47), and (48).The expression for the sound attenuation coefficient for vanishing magnetic and pseudomagnetic fields agrees withthat in multi-valley semiconductors [46–48]. However, unlike the earlier studies in Ref. [36], we found that soundattenuation coefficient is generically suppressed by the magnetic field, see Eqs. (49) and (50). Indeed, the chiralanomaly activated by this field necessary leads to the deviations of electric charge density and, consequently, to theappearance of induced electric field. This electric field alone, however, cannot explain the suppression of the soundabsorption. As we explicitly showed, the other important ingredient is a nonvanishing intra-node relaxation time τ .The combination of finite τ and the electrostatic screening strongly affects the sound absorption and, contrary toRef. [36], leads to a negative sign of the anomalous part of the sound attenuation coefficient. These findings agreewith the recent results presented in Ref. [38]. Our estimates suggest that the suppression could be noticeable andreaches several percent of the attenuation coefficient at zero field for sufficiently strong magnetic fields, see Fig. 2.4The nontrivial dependence on the magnetic field direction with respect to the sound wave vector appears whenthe Weyl nodes of opposite chiralities are non-symmetric. In particular, different Fermi velocities and/or intra-node scattering amplitudes for the Weyl nodes of opposite chiralities make the dependence on the magnetic fieldnonmonotonic and lead to the magnetic sound dichroism, see Eq. (56). This effect, however, is estimated to beweak. For optimal magnetic fields and small difference between the nodes, it reaches a few percent of the attenuationcoefficient at zero field, see Fig. 3. The magnetic sound dichroism for non-symmetric Weyl nodes is one of our mainresults.The sound absorption in an external strain-induced pseudomagnetic field, which is unique for relativistic-like energyspectrum, is also nontrivial. Similar to the case of magnetic field, the sound attenuation coefficient decreases in largepseudomagnetic fields. The dependence is, however, nonmonotonic and demonstrates the pseudomagnetic sounddichroism even when the Weyl nodes in the undeformed material are symmetric, see Eqs. (54) and (55). Like magneticsound dichroism, its pseudomagnetic counterpart is estimated to be weak. The decrease of the sound attenuationcoefficient and the sound dichroism in an external pseudomagnetic field is another major result of our study.The proposed effects provide a way to probe the anomalous properties of Weyl semimetals and the effects of magneticand strain-induced pseudomagnetic fields via sound absorption experiments. It would be especially interesting to seeboth the magnetic and pseudomagnetic sound dichroism by flipping the direction of the sound propagation with respectto the field, as well as to observe the reduction of the attenuation coefficient for the sound propagating along thefield. As possible material candidates, transition metal monopnictides TaAs, TaP, NbAs, and NbP [4], the magneticcompound EuCd As [69, 70], and SrSi [56, 57] could be used. The magnetic material EuCd As is a promisingcandidate for the investigation of the pseudomagnetic sound dichroism since it breaks the time-reversal symmetry andcontains only a single pair of Weyl nodes. As for the magnetic sound dichroism, transition metal monopnictides couldbe useful since they contain two types of Weyl nodes with different Fermi velocities. The Weyl nodes of oppositechiralities are separated in energy in SrSi also leading to non-symmetric nodal parameters. ACKNOWLEDGMENTS
The Authors acknowledge useful communications with E. V. Gorbar, D. Pesin, H. Rostami, and I. A. Shovkovy. Thiswork is supported by NSF DMR-2002275 (LG). P.O.S. acknowledges the support through the Yale Prize PostdoctoralFellowship in Condensed Matter Theory.
Appendix A: Energy dissipation rate
In this Section, we calculate the energy dissipation rate required for the sound attenuation coefficient defined inEq. (6). The energy dissipation rate is defined as Q = 1 T Z T dt h ddt ˆ H i . (A1)Here, T is the period of sound waves and ˆ H = ˆ H ee + ˆ H ep is the full Hamiltonian that includes the electron-electronˆ H ee and the electron-phonon ˆ H ep parts. We use the standard definition of the ensemble averaging h ddt ˆ H i = Tr (cid:26) ˆ ρ ∂∂t ˆ H (cid:27) . (A2)In the case of the sound attenuation, only ˆ H ep explicitly depends on time. Therefore, h ∂ t ˆ H i = h ∂ t ˆ H ep i = N W X α Z d r Z d p α (2 π ~ ) λ ( α ) ij ( p α ) [ ∂ t u ij ( t, r )] f α ( r , p α ) , (A3)where the sum runs over all N W Weyl nodes, the valley-dependent deformation potential is λ ( α ) ij ( p α ) u ij ( t, r ), u ij ( t, r ) =( ∂ i u j + ∂ j u i ) / u ( t, r ) is the displacement vector.The distribution function f α ( r , p α ) = f (0) α ( p α ) + δf α ( r , p α ) contains the time- and coordinate-independent equilib-rium part f (0) α ( p α ) and the oscillating non-equilibrium correction determined by the deformation potential δf α ( r , p α ) ≈ δ ( ǫ α + χ α b − µ ) n α ( r , p α ) . (A4)5Here, µ is the equilibrium Fermi energy measured from the Weyl nodes and we assume that temperature is smallcompared to µ . In addition, ǫ α = ǫ α ( p α ) is the energy dispersion relation and b quantifies the separation betweenthe Weyl nodes of opposite chiralities in energy.Due to the time-averaging in Eq. (A1), only the non-equilibrium part of the distribution function δf α ( r , p α ) ∝ λ ( α ) ij ( p α ) u ij ( t, r ) contributes to the dissipation rate. The final expression for the energy dissipation rate is QV = N W X α
12 Re (cid:26)Z d p α (2 π ~ ) iω h λ ( α ) ij u ij i ∗ δf α ( p α ) (cid:27) = N W X α ν α ( µ )2 Re n iω h λ ( α ) ij u ij i ∗ n α o . (A5)Here, we used the plane-wave dependence for the displacement vector u ( t, r ) = u e − iωt + i q · r , where ω and q are thesound frequency and wave vector, respectively, u is the displacement magnitude, and V is the spatial integrationvolume. In addition, we defined the averaged over the Fermi surface solution of the kinetic equations (at given q ) as n α , n α = 1 ν α ( µ ) Z d p α (2 π ~ ) δ ( ǫ α + χ α b − µ ) n α ( p α ) . (A6)Here ν α ( µ ) = Z d p α (2 π ~ ) δ ( ǫ α + χ α b − µ ) (A7)is the density of states (DOS) per Weyl node.The dissipation rate in Eq. (A5) agrees with the result in Ref. [58] if the external electric field in that paper isignored. It is in agreement also with the general definition of the absorbed power in Ref. [50]. Appendix B: Collision integrals
In this Section, we consider the collision integral on the right-hand side of the kinetic equation (16). It is convenientto split the integral into the intra- and inter-node parts that are discussed in Secs. B 1 and B 2, respectively.
1. Intra-node scattering
By using the Fermi golden rule (see, e.g., Ref. [51]), the collision integral for the intra-node scattering processes isdefined as I intra [ f α ( p α )] = − Z d p ′ α (2 π ~ ) Θ α ( p ′ α ) 2 π ~ | A α,α | δ (˜ ǫ α ( p α ) − ˜ ǫ α ( p ′ α )) [ f α ( p α ) − f α ( p ′ α )] , (B1)where f α ( p α ) is the distribution function for the electron quasiparticles from node α (we suppressed the explicitdependence on spatial coordinates), Θ α ( p α ) = [1 − e ( B α · Ω α ) /c ] is the renormalization of the phase-space vol-ume, B α is an external (pseudo)magnetic field, Ω α = Ω α ( p α ) is the Berry curvature given in Eq. (18), and | A α,β | is the scattering amplitude between the Weyl nodes α and β . In the case of weak (non-quantizing) magneticfields, v F ~ | eB α | / (2 cµ ) ≪
1, we neglect the magnetic moment ∝ ( B α · Ω α ) in the energy dispersion ˜ ǫ α and setΘ α ( p α ) ≈ f (0) α ( p α ) ≈ f (0) α ( p α ) and is given by the standard Fermi-Dirac distribution. Then, we can useEq. (A4) for the deviations from the equilibrium state δf α ( p α ). This allows us to rewrite the intra-node collisionintegral as I intra [ f α ( p α )] ≈ − Z d p ′ α (2 π ~ ) π ~ | A α,α | δ ( ǫ α ( p α ) − ǫ α ( p ′ α )) [ δf α ( p α ) − δf α ( p ′ α )]= − n ( p α ) − n α τ α δ ( ǫ α ( p α ) + χ α b − µ ) . (B2)Here, we introduced the intra-node relaxation time1 τ α ≡ τ α,α = Z d p α (2 π ~ ) π ~ | A α,α | δ ( ǫ α ( p α ) + χ α b − µ ) = 2 π ~ | A α,α | ν α ( µ ) . (B3)6
2. Inter-node scattering
Further, we proceed to the inter-node collision integral I inter [ f α ( p α )]. Because the deformation potential dependson the chirality of the Weyl nodes (see the discussion in Sec. II A), there are two types of contributions in the collisionintegral, i.e., I inter [ f α ( p α )] = N W X β = α | χ α − χ β | I (1) α,β + N W X β = α | χ α + χ β | I (2) α,β , (B4)where P N W β = α runs over all nodes excluding β = α . Here, I (1) α,β corresponds to the scattering between the Weyl nodesof the opposite chiralities and I (2) α,β describes the transfer between the nodes of the same chirality χ α . Let us beginwith the former part, I (1) α,β = − Z d p β (2 π ~ ) Θ β ( p β ) 2 π ~ | A α,β | δ (˜ ǫ α ( p α ) − ˜ ǫ β ( p β )) [ f α ( p α ) − f β ( p β )] ≈ − Z d p β (2 π ~ ) π ~ | A α,β | δ (cid:16) ǫ α ( p α ) − ǫ β ( p β ) + 2 χ α b + λ ( α ) ij u ij − λ ( β ) ij u ij (cid:17) × h δf α ( p α ) − δf β ( p β ) + f (0) α ( p α ) − f (0) β ( p β ) i ≈ − Z d p β (2 π ~ ) π ~ | A α,β | δ ( ǫ α ( p α ) − ǫ β ( p β ) + 2 χ α b ) × h δf α ( p α ) − δf β ( p β ) − (cid:16) λ ( α ) ij − λ ( β ) ij (cid:17) u ij ( ∂ ǫ β f (0) β ) i , (B5)where we used δ (cid:16) ǫ α ( p α ) − ǫ β ( p β ) + 2 χ α b + λ ( α ) ij u ij − λ ( β ) ij u ij (cid:17) h f (0) α ( p α ) − f (0) β ( p β ) i ≈ δ ( ǫ α ( p α ) − ǫ β ( p β ) + 2 χ α b ) h f (0) α ( p α ) − f (0) β ( p α ) − (cid:16) λ ( α ) ij − λ ( β ) ij (cid:17) u ij ( ∂ ǫ β f (0) β ) i (B6)and neglected the second-order terms ∝ ( λ ( α ) ij u ij ) in the last line in Eq. (B5).By introducing the following inter-node relaxation time:1 τ α,β = Z d p β (2 π ~ ) π ~ | A α,β | δ ( ǫ β ( p β ) + χ β b − µ ) = 2 π ~ | A α,β | ν β ( µ ) , (B7)we rewrite Eq. (B5) as I (1) α,β = − n α ( p α ) − n β τ α,β δ ( ǫ α + χ α b − µ ) − (cid:16) λ ( α ) ij − λ ( β ) ij (cid:17) u ij τ α,β δ ( ǫ α + χ α b − µ ) . (B8)The result for the inter-node scattering integral between the nodes of the same chirality can be obtained in thesame way. It reads as I (2) α,β = − n α ( p α ) − n β τ α,β δ ( ǫ α + χ α b − µ ) . (B9)Then, the inter-node collision integral in τ -approximation reads as I inter [ f α ( p α )] = − N W X β = α n α ( p α ) − n β τ α,β δ ( ǫ α + χ α b − µ ) − u ij N W X β λ ( α ) ij − λ ( β ) ij τ α,β δ ( ǫ α + χ α b − µ ) . (B10)After averaging over the Fermi surface, one obtains I inter [ f α ( p α )] = − N W X β n α − n β τ α,β − u ij N W X β λ ( α ) ij − λ ( β ) ij τ α,β . (B11)This collision integral is used in Eq. 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