Crystals that lack mirror and inversion symmetries are known as chiral crystals; that is, their atomic structures cannot be superimposed onto their mirror images. The collective vibrations of the atoms in a crystal are quantized as bosonic quasiparticles called phonons. Each phonon carries an energy ℏω, where ℏ is the reduced Planck constant, and ωis the angular eigenfrequency of vibration. The distinct patterns of the lattice vibrations are classified into phonon modes or branches, which are determined by the symmetry and structure of the crystal.

In chiral crystals, phonons in certain modes possess an internal angular momentum and are referred to as chiral phonons. This angular momentum originates from the atomic circular motion, in which atoms rotate clockwise or counterclockwise around the phonon propagation direction. Moreover, when chiral phonons propagate along a symmetry axis of the crystal, their angular momentum aligns with the propagation direction and takes quantized values in units of ℏ; this quantity is known as the crystal angular momentum (CAM). While this angular momentum resembles that of circularly-polarized-light photons, unlike photons in vacuum, chiral phonons exhibit energy splitting between opposite CAM modes. This energy splitting, which does not occur in nonchiral crystals, enables the selective excitation and control of phonons with a chosen CAM. 

In this study, we address two fundamental problems. First, we investigate how angular momentum quantization changes when phonons propagate along the directions away from the symmetry axes. Second, we aim to identify the essential physical factors that govern the energy splitting between opposite CAM modes. A simple symmetry consideration implies that this factor must change sign under spatial inversion. However, its microscopic origin, in terms of interatomic interactions, remains unclear. 

For these analyses, we construct simple a cubic lattice model with multiple chiral axes and examine its phonon energy dispersion while continuously tuning the degree of lattice chirality. The results show that chiral phonons always carry quantized angular momentum (±1) aligned with the propagation direction, even away from the symmetry axes, provided that the phonon’s wavelength is sufficiently long. Furthermore, we demonstrate that energy splitting is governed by a few pseudoscalars that characterize the chirality of the system. In addition, we derive an explicit expression in terms of the spatial distribution of the interatomic force constants. These results provide distinct guidelines for first-principles material design and for the experimental identification of chiral crystals exhibiting large CAM energy splitting.  

(Written by Hirokazu Tsunetsugu on behalf of all authors)

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