Dissipationless Phonon Hall Viscosity

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 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 sound waves with a relative amplitude ratio of $\omega/\omega_v$ and with a phase shift of $\pm \pi/2$, 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 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 gapped sector of a system.