Chiral phonons in time reversal broken systems from first principles

The conventional approach to the calculation of phonon modes is via eigenvalues and eigenvectors of the dynamical matrix. However, the dynamical matrix is, by construction, time reversal symmetric even for systems with known time reversal breaking in the electronic sector. Such systems have been observed to display a number of effects involving time reversal broken ion dynamics including chiral phonons, the Einstein de Haas effect, and the phonon Hall effect. In this work we develop and apply a first-principles methodology for computing the leading order corrections beyond the usual dynamical matrix which captures such time reversal broken ion dynamics. This correction comes in the form of forces proportional to ion velocities which act on ions in addition to the forces proportional to ion displacements captured by the dynamical matrix. This linear order coupling between ion velocities and forces can be expressed as a matrix of ionic Berry curvatures. An application of the formalism to CrI3 using density functional theory is presented. The addition of the velocity force results in a 6% splitting of phonons modes which would have been found to be degenerate if computed from the dynamical matrix alone.