When can localized spins interacting with conduction electrons in ferro- or antiferromagnets be described classically via the Landau-Lifshitz equation: Transition from quantum many-body entangled to quantum-classical nonequilibrium states*

The nonequilibrium dynamics of localized spins within magnetic materials are standardly described by the Landau-Lifshitz (LL) equation. However, spin is a quantum degree of freedom, and its effects exist for all spin S less than infinity. To understand the limits of LL equation, we compare LL trajectories to quantum expectation values of localized spin operators in the presence of electrons. We start from the unentangled ground state of localized spins as the initial condition and apply a magnetic field to initiate dynamics. This reveals that quantum-classical dynamics can faithfully reproduce fully quantum dynamics in the FM case, but only when spin S, Heisenberg exchange between localized spins and sd exchange are sufficiently small. Increasing any of these parameters can lead to deviations, which are explained by the growth of entanglement between localized spins and between them and electrons. Including thermal fluctuations only delays the time at which entanglement grows. In the AFM case, substantial deviations appear even at early times, despite starting from unentangled Neel state.