Phonon-mediated spin-spin interaction: A general theory and application to diamond nitrogen vacancy centers

Interaction between spins forms diverse classes of magnetic materials. The interaction can be carried out not only by electrons and photons but also by phonons. We consider a general symmetry-allowed spin-phonon Hamiltonian in ${C}_{3v}$ systems which consists of spins interacting with symmetric and antisymmetric lattice deformations. The derived model is used to calculate the phonon-mediated spin-spin interaction---the interaction is a quadrupole-quadrupole interaction. We first consider phonons in a homogeneous elastic medium, which leads to a spin-spin interaction with unphysical oscillations as a function of distance between the spins. We overcome this by extending the theory for lattice systems and confirm that the interaction strength decays as $1/{r}^{3}$. The effect of surface acoustic waves is also considered. The general theory is demonstrated for nitrogen vacancy centers in diamond as a prominent example. Furthermore, we provide an estimation of the magnitude of the spin-spin interaction and discuss the effect of the phonon-mediated spin-spin interaction on the energy level scheme of two interacting spins.