Dissipationless phonon Hall viscosity

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 ${\ensuremath{\omega}}_{v}$; for acoustic phonons of frequency $\ensuremath{\omega}$, it shifts the phonon spectrum by an amount of order ${(\ensuremath{\omega}/{\ensuremath{\omega}}_{v})}^{2}$ and it mixes the longitudinal and transverse acoustic phonons with a relative amplitude ratio of $\ensuremath{\omega}/{\ensuremath{\omega}}_{v}$ and with a phase shift of $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$, to lowest order in $\ensuremath{\omega}/{\ensuremath{\omega}}_{v}$. We study several examples, including the integer quantum Hall states, the quantum anomalous Hall state in Hg${}_{1\ensuremath{-}y}$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.