Berry Phase of Phonons and Thermal Hall Effect in Nonmagnetic Insulators

A mechanism for the phonon Hall effect (PHE) in nonmagnetic 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 discuss that the PHE also occurs in nonmagnetic band insulators subject to the magnetic field. We find that a correction to the Born-Oppenheimer approximation gives rise to a Raman-type interaction between the magnetic field and the phonons; this interaction gives rise to the Berry curvature of a phonon band. This Berry curvature results in the finite thermal Hall conductivity kappa(H) in nonmagnetic band insulators. The value of kappa(H) is calculated for square and honeycomb lattices. The order of the magnitude estimation for kappa(H) is given for Si at room temperature.