Zone-center chiral phonons from broken time reversal symmetry

In conventional ab initio methodologies, phonons are calculated by solving equations of motion involving static interatomic force constants and atomic masses. However, this approach does not fully account for the effects of broken time-reversal symmetry in systems with magnetic order. Recent attempts to rectify this involve the inclusion of the velocity dependence of the interatomic forces in the equations of motion, which is given by the nuclear Berry curvature. This can result in "chiral" phonon modes with non-zero angular momentum, even at the zone center. We show, via a novel density-functional theory methodology based on finite displacements, that the main contribution to this nuclear Berry curvature is from the spin rotations caused by the phonons. This requires spin and phonon degrees of freedom to be treated on the same footing. We propose a model involving Hessian matrices and Berry curvature tensors in terms of both spin and phonon degrees of freedom, and develop a first-principles methodology to calculate them. We present results on both ferromagnets and antiferromagnets, and the implication for splitting of chiral phonon modes is discussed.