Theory of Energy Dispersion of Chiral Phonon

TL;DR

This paper develops a microscopic theory for phonon energy dispersion in chiral crystals, revealing that acoustic phonons do not split at linear order, while optical phonons do, and clarifies the role of specific interactions and symmetries.

Contribution

It introduces a theoretical framework explaining phonon dispersion and splitting in chiral crystals, emphasizing the role of electric toroidal quadrupoles and ruling out certain antisymmetric interactions.

Findings

Acoustic phonon splitting starts at order $k^2$ or higher.

Optical phonons exhibit $k$-linear splitting.

Antisymmetric interactions of the form $ extbf{D}_{ij} imes ( extbf{d}_i imes extbf{d}_j)$ are forbidden.

Abstract

We have developed a microscopic theory on phonon energy dispersion in chiral crystals within a harmonic approximation. One of the main issues is about the splitting of sound velocity of acoustic phonons with opposite ``crystal'' angular momentum. We have shown that the splitting must be zero even in chiral crystals and the difference starts from the order of at least $k^2$ or higher in their energy dispersion. Splitting is evident for chiral optical phonons, and we have derived a formula for their $k$-linear splitting. Another important finding is about possible interactions of atomic displacements in microscopic models. We have found that antisymmetric interactions of $\mathbf{D} _{ij} \cdot (\mathbf{d} _i \times \mathbf{d} _j) $ type are not allowed in microscopic Hamiltonians for chiral phonons in compatible with the stability against the Nambu-Goldstone mode. We have identified that the splitting in both acoustic and optical modes arises from the harmonic potentials with the electric toroidal quadrupole of $G_{u}$-type symmetry. These constraint are important for modeling real materials. Most of our microscopic calculations have been performed for (quasi-)one-dimensional systems with a trigonal crystal symmetry including Te, but these results generally hold also for other chiral phonon systems.