Electron-phonon-coupled Langevin dynamics for strongly-correlated insulators

Abstract

The Landau-Lifshitz-Gilbert (LLG) equations are widely used to study spin dynamics in Mott insulators. However, because energy damping is typically introduced phenomenologically, their validity for describing nonequilibrium processes and their connection to the microscopic origin of dissipation in real materials remains unclear. In this paper, we derive generalized stochastic LLG equations from first principles for spin-orbital coupled Mott insulators, explicitly incorporating the coupling between electronic degrees of freedom and lattice vibrations. Our approach is based on a path-integral formalism formulated along the Keldysh contour, which naturally accounts for dissipation and thermal fluctuations through interactions with a phonon bath and emergent stochastic noise. We benchmark our theoretical framework by numerically integrating the equations of motion for a two-orbital spin chain coupled to Einstein phonons. The resulting energy relaxation mimics realistic cooling dynamics, exhibits nontrivial transient behavior during thermalization, and accurately reproduces thermodynamic properties upon equilibration. We further demonstrate how electron-phonon coupling induces hybridization between electronic and phononic modes in the excitation spectrum and show that the conventional LLG equations are recovered as a limiting case of our microscopic theory. These results establish a robust and reliable framework for capturing dissipative spin dynamics in strongly correlated systems, both in and out of equilibrium.