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 ωᵥ; for acoustic phonons of frequency ω, it shifts the phonon spectrum by an amount of order (ω/ωᵥ)² and it mixes the longitudinal and transverse acoustic phonons with a relative amplitude ratio of ω/ωᵥ and with a phase shift of ±π/2, to lowest order in ω/ωᵥ. We study several examples, including the integer quantum Hall states, the quantum anomalous Hall state in Hg_1-yMnyTe quantum wells, and a mean-field model for pₓ+ipy 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.