Nonreciprocal Acoustic and Optical Phonon Dispersion Mediated by Berry Curvature in Chiral Weyl Semimetals

TL;DR

This work addresses how Berry curvature and orbital magnetic moment in chiral Weyl semimetals imprint onto phonon dispersions under a magnetic field, producing a phonon magnetochiral effect (PMCE). It develops a semiclassical framework combining the Boltzmann kinetic equation for Weyl electrons with elasticity theory, incorporating node-dependent Weyl properties and magnetic-field–driven chiral imbalance to compute both acoustic and non-polar optical phonon dispersions. The results show zero-field renormalizations of acoustic and optical branches and magnetic-field–induced nonreciprocity, with acoustic effects dominated by attenuation and a small real shift, while optical modes exhibit linear-in-wavevector PMCE. The findings advance phonons as probes of topological band geometry and dynamical anomaly physics and point to polar-mode extensions and strain-engineering possibilities for enhanced nonreciprocal phononics.

Abstract

We investigate the phonon magnetochiral effect (PMCE) in chiral Weyl semimetals by deriving the nonreciprocal dispersion relations of both acoustic and non-polar optical phonons in the presence of a magnetic field. Using a semiclassical Boltzmann kinetic framework that incorporates Berry curvature, orbital magnetic moment, and node-dependent electronic structure, we obtain analytic expressions for the magnetic-field-induced corrections to the phonon dynamical matrix. Inequivalent Weyl nodes with distinct Fermi velocities, Fermi energies, and relaxation times generate a dynamical chiral imbalance that alters the phonon dispersion. For acoustic phonons, the formalism yields the magnetic-field-dependent corrections to the longitudinal mode, while for optical phonons we identify an optical analogue of the PMCE that produces a corresponding shift in the optical branch. Together, these results provide a unified theoretical description of how band-geometric properties of Weyl fermions influence both acoustic and optical phonon dispersions in chiral Weyl semimetals.

Figures (3)

• Figure 1: (a) Geometry of the phonon magnetochiral effect (PMCE): the relative orientation of the magnetic field $\mathbf{B}$ and the phonon wave vector $\mathbf{q}$ determines the nonreciprocal correction to the phonon dispersion. Reversing the direction of $\mathbf{B}$ (or equivalently $\mathbf{q}$) leads to unequal phonon frequencies, $\omega \neq \omega'$, reflecting the odd-in-$B$ and odd-in-$q$ nature of the effect. (b) Schematic depiction of a chiral Weyl semimetal with two inequivalent Weyl nodes characterized by different Fermi velocities, Fermi energies, and strong Berry curvature and orbital magnetic moments. The quantities $\Gamma_{A}$ and $\Gamma_{E}$ denote the intra-node and inter-node relaxation rates, respectively, which govern local equilibration and chiral charge transfer between Weyl nodes.
• Figure 2: Acoustic phonon dispersion in a chiral Weyl semimetal for zero magnetic field ($\omega^{(0)}$) and for finite fields $B>0$ and $B<0$, the latter corresponding to phonon propagation parallel and antiparallel to $\mathbf{B}$. The top-left panel displays the real part of the dispersion, where the zero-field curve $\omega^{(0)}$ represents the electron–phonon renormalized longitudinal mode. A finite magnetic field introduces a small magnetochiral splitting between the $\omega_{\pm}$ (for $B>0$) and $\omega_{\pm}$ (for $B<0$) branches. The magnitude of this shift, shown in the top-right panel, is small but experimentally resolvable senguptapmce. In contrast, the imaginary part of the dispersion (bottom-left) exhibits a substantially larger nonreciprocal correction, as quantified in the bottom-right panel.
• Figure 3: Optical phonon dispersion for zero magnetic field and for finite fields $B>0$ and $B<0$ in a chiral Weyl semimetal. The zero-field dispersion ($\omega^{(0)}$) represents the electron–phonon renormalized non-polar optical mode. Applying a magnetic field produces a linear magnetochiral splitting between the $\omega(B>0)$ and $\omega(B<0)$ branches: the real part (top-left) shifts by a small but finite amount, while the imaginary part (bottom-left) exhibits a significant directional asymmetry in attenuation. The right-hand panels quantify the magnetochiral effects, showing that the optical-phonon PMCE increases linearly with $q$.