  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/Km] at $ B=10 $[T] and $ T=150 $[K].
