From Elastic Spin to Phonon Spin: Symmetry and Fundamental Relations

Abstract

This note is mainly based on postgraduate lectures at Tongji University since 2020 spring. We firstly revisit the elastic spin and orbital angular momentum [PNAS 115, 9951 (2018)] but more general by applying the Noether's theorem to the Lagrangian of the elastic system, and by applying the symmetry argument in the field theory. Then, fundamental relations between elastic energy flux and elastic spin are uncovered. In particular cases, the wave spin is closely related to the vorticity of energy flux and momentum. Secondly, we move forward from the elastic spin to revisit the phonon spin [Fizika Tverdogo Tela 3, 2160 (1961] by applying the second quantization to elastic fields. We show that the uncovered phonon spin, a polarized elastic-vibration quanta, is generally not restricted to transverse phonon modes, but applying to general phonon modes, like the longitudinal phonon modes, surface phonon modes, hybridized phonon modes and so on, regarded as a consequence of mode interferences. The elastic spin and phonon spin originate from the local rotating of the field polarization in time domain, not the local circulation (vorticity) of displacement or velocity in space domain.