First–principles spin–lattice coupling from downfolded electron–phonon Interaction

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 We present a method to calculate spin-phonon coupling parameters from first-principles perturbation theory by downfolding the electron-phonon coupling (EPC). We exploit the localized nature of magnetic moments and atomic displacements by working in the Wannier representation of the electronic Hamiltonian and the EPC matrix. The spin system is mapped to a classical Heisenberg Hamiltonian, whose parameters are obtained by treating local spin rotations as a perturbation within a Green's-function formalism. The spin and phonon perturbations are connected through the EPC parameters, which enter as lattice-induced perturbations to the tight-binding Hamiltonian. By combining these lattice perturbations with local spin rotations, we obtain real-space derivatives of magnetic exchange parameters without performing displaced magnetic supercell calculations. We illustrate the method on SrMnO3 and show that it can be integrated directly into standard workflows.