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In conventional \textit{ab initio} methodologies, phonons are calculated bysolving equations of motion involving static interatomic force constants andatomic masses. The Born-Oppenheimer approximation, where all electronic degreesof freedom are assumed to adiabatically follow the nuclear dynamics, is alsoadopted. This approach does not fully account for the effects of brokentime-reversal symmetry in systems with magnetic order. Recent attempts torectify this involve the inclusion of the velocity dependence of theinteratomic forces in the equations of motion, which accounts for time-reversalsymmetry breaking, and can result in chiral phonon modes with non-zero angularmomentum even at the zone center. However, since the energy ranges of phononsand magnons typically overlap, the spins cannot be treated as adiabaticallyfollowing the lattice degrees of freedom. Instead, phonon and spins must betreated on a similar footing. Focusing on zone-center modes, we propose amethod involving Hessian matrices and Berry curvature tensors in terms of bothphonon and spin degrees of freedom, and describe a first-principles methodologyfor calculating these. We then solve Lagrange's equations of motion todetermine the energies and characters of the mixed excitations, allowing us toquantify, for example, the energy splittings between chiral pairs of phonons insome cases, and the degree of magnetically induced mixing between infrared andRaman modes in others. The approach is general, and can be applied to determinethe adiabatic dynamics of any mixed set of slow variables.

Adiabatic dynamics of coupled spins and phonons in magnetic insulators