Coupling of acoustic phonon to a spin-orbit entangled pseudospin

We consider coupling of acoustic phonon to pseudospins consisting of electronic spins locked to orbital angular momentum states. We show that a Berry phase term arises from projection onto the time-dependent lowest energy manifold. We examine consequences on the phonon modes, in particular mode splitting, induced chirality and Berry curvatures under an external magnetic field which Zeeman couples to the pseudospin.