Berry phase of phonons and thermal Hall effect in nonmagnetic insulators

A mechanism for phonon Hall effect (PHE) in non-magnetic insulators under an external magnetic field is theoretically studied. PHE is known in (para)magnetic compounds, where the magnetic moments and spin-orbit interaction play an essential role. In sharp contrast, we here show that a non-zero Berry curvature of acoustic phonons is induced by an external magnetic field due to the correction to the adiabatic Born-Oppenheimer approximation. This results in the finite thermal Hall conductivity ${\kappa }_H$ in nonmagnetic band insulators. Our estimate of ${\kappa }_H$ for a simple model gives $\kappa_H \sim 1.0\times 10^{-5} $[W\mathrm{/}Km]at$ B=10 $[T]and$ T=150 $[K].