Adiabatic dynamics of coupled spins and phonons in magnetic insulators

In conventional \textit{ab initio} methodologies, phonons are calculated by solving equations of motion involving static interatomic force constants and atomic masses. The Born-Oppenheimer approximation, where all electronic degrees of freedom are assumed to adiabatically follow the nuclear dynamics, is also adopted. 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 accounts for time-reversal symmetry breaking, and can result in chiral phonon modes with non-zero angular momentum even at the zone center. However, since the energy ranges of phonons and magnons typically overlap, the spins cannot be treated as adiabatically following the lattice degrees of freedom. Instead, phonon and spins must be treated on a similar footing. Focusing on zone-center modes, we propose a method involving Hessian matrices and Berry curvature tensors in terms of both phonon and spin degrees of freedom, and describe a first-principles methodology for calculating these. We then solve Lagrange's equations of motion to determine the energies and characters of the mixed excitations, allowing us to quantify, for example, the energy splittings between chiral pairs of phonons in some cases, and the degree of magnetically induced mixing between infrared and Raman modes in others. The approach is general, and can be applied to determine the adiabatic dynamics of any mixed set of slow variables.