Hear the Sound of Weyl Fermions

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.

Popular Summary

Sound in hot air travels much faster than in cold air because at high temperature the atoms collide faster and more often. One may then wonder if sound can travel at zero temperature, where atoms seldom collide. Because of short-range forces among atoms, neutral fermionic liquids such as helium-3 can carry a so-called “zero sound” at absolute zero. But in charged fermionic liquids, such as normal metals, Coulomb interactions between electrons make it hard to detect such exotic behavior. Here, we predict in a Weyl semimetal—materials where valence and conduction bands cross in single points—a special type of zero sound can exist even in the presence of Coulomb interactions.

This chiral zero sound (CZS) is a new type of acoustic collective mode carried by the electrons in the semimetal. We predict that this mode will appear if the Weyl semimetal is exposed to a magnetic field in what is known as the “chiral limit.” In this limit, electrons within valleys (electronic band minimums) scatter more often than electrons outside of those valleys. In this CZS, the number of electrons in different valleys oscillates in such a way that the total number of electrons does not evolve over time. Thus, the CZS carries no charge, which means that it can contribute to the thermal conductivity but not the electric conductivity of a Weyl semimetal in a magnetic field.

The CZS could lead to several interesting phenomena, among which the dramatic oscillations in thermal conductivity with a changing magnetic field is the most striking and can be viewed as smoking-gun evidence of its presence.

Figure 1

Chiral zero sound and chiral plasmon modes in the minimal model with four Weyl points. The symmetry group of the model is $C_{2v}$ consisting of a twofold rotation axis in the $z$ direction and two mirror planes in the $x\text{−}z$ plane and $y\text{−}z$ 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 $(s,a)$, with $s=\pm$ the chirality and $a=1$, 2 the subvalley index. (a) The volume of Fermi surfaces as functions of space and time in the chiral plasmon mode, where $(s,1)$ and $(s,2)$ are always in phase, making the mode even under $C_2$ 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 $(-,1)$ [$(-,2)$] and $(+,1)$ [$(+,2)$] 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 $(s,1)$ and $(s,2)$ are always out of phase, making the mode odd under $C_2$ 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 $(-,1)$, $(+,1)$, $(-,2)$, and $(+,2)$ Fermi surfaces are represented by the red solid, blue solid, red dashed, and blue dashed lines, respectively. In the chiral zero sound mode, both the net current and the net density vanish.

Figure 2

(a) The specific heat (per unit volume) in the four-WPs model is plotted as a function of temperature. The specific heat is plotted in the unit of $k_B{\mathrm{\Lambda }}^3$, where $k_B$ is Boltzmann’s constant, and $\mathrm{\Lambda }$ is the cutoff in the momentum integral. The temperature $T$ is plotted in units of Debye temperature for the CZS mode, ${\mathrm{\Theta }}_{\mathrm{CZS}}=c(B)\mathrm{\Lambda }/k_B$, where $c(B)$ is the speed of the CZS. (b) The thermal conductivity in the four-WPs model is plotted as a function of magnetic field. Here, $S_{\mathrm{ex}}(\mu )$ is the area enclosed by the extreme circle on the Fermi surface that is perpendicular to the magnetic field. The parameters are set as ${\omega }_B=0.2\mu$, $k_{B}T=1/(2{\tau }_0)=0.001\mu$, and $T/{\mathrm{\Theta }}_{\mathrm{CZS}}=10$ and 0.1 for the blue line and red line, respectively.