Dissipationless Phonon Hall Viscosity

TL;DR

This paper investigates how gapped electronic states in crystals induce a dissipationless Hall viscosity affecting acoustic phonons, with implications for probing electronic wave function curvature through phonon measurements.

Contribution

It introduces the concept of dissipationless Hall viscosity in phonon response and explores its effects in various quantum states, linking phonon behavior to electronic Berry curvature.

Findings

Hall viscosity causes phonon spectrum shifts proportional to (/_v)^2

Mixing of longitudinal and transverse phonons with phase shifts of /2

Proposes phonon measurements as probes of electronic wave function curvature

Abstract

We study the acoustic phonon response of crystals hosting a gapped time-reversal symmetry breaking electronic state. The phonon effective action can in general acquire a dissipationless "Hall" viscosity, which is determined by the adiabatic Berry curvature of the electron wave function. This Hall viscosity endows the system with a characteristic frequency, \omega_v; for acoustic phonons of frequency \omega, it shifts the phonon spectrum by an amount of order (\omega/\omega_v)^2 and it mixes the longitudinal and transverse acoustic phonons with a relative amplitude ratio of \omega/\omega_v and with a phase shift of +/- \pi/2, to lowest order in \omega/\omega_v. We study several examples, including the integer quantum Hall states, the quantum anomalous Hall state in Hg_{1-y}Mn_{y}Te quantum wells, and a mean-field model for p_x + i p_y superconductors. We discuss situations in which the acoustic phonon response is directly related to the gravitational response, for which striking predictions have been made. When the electron-phonon system is viewed as a whole, this provides an example where measurements of Goldstone modes may serve as a probe of adiabatic curvature of the wave function of the gapped sector of a system.