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Hear the Sound of Weyl Fermions PHYSICAL REVIEW X 9, 021053 (2019) Featured in Physics Hear the Sound of Weyl Fermions 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 quasiparticles 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 metals due to the fact that long-range Coulomb interaction among electrons will generate a plasmon 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 thermal conductivity measurements under magnetic field. DOI: 10.1103/PhysRevX.9.021053 Subject Areas: Condensed Matter Physics I. INTRODUCTION on the surface and the negative magnetoresistance [19–23] caused by the chiral anomaly [1,24–27]. Topological semimetals are unique metallic systems with So far, the Weyl semimetal is considered a new topo- a vanishing density of states at the Fermi level [1–11]. logical state in condensed matter physics only because of Among different topological semimetals, the Weyl semi- its unique quasiparticle dynamics, which manifests itself in metal [2–5] is the most robust one because it requires no various transport experiments [19–23]. On the other hand, particular crystalline symmetry 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 state of matter 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 Weyl equation 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 temperature, 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 semiconductors 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. solids, 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 – chiral magnetic effect (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) chirality 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 plasmons, 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.
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