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Experiments in spintronics and magnonics operate with macroscopically largenumber of localized spins within ferromagnetic (F) or antiferromagnetic (AF)materials, so that their nonequilibrium dynamics is standardly described by theLandau-Lifshitz (LL) equation treating localized spins as classical vectors offixed length. However, spin is a genuine quantum degree of freedom, and eventhough quantum effects become progressively less important for spin value$S>1$, they exist for all $S < \infty$. While this has motivated exploration oflimitations/breakdown of the LL equation, by using examples of F insulators,analogous comparison of fully quantum many-body vs. quantum (forelectrons)-classical (for localized spins) dynamics in systems wherenonequilibrium conduction electrons are present is lacking. Here we employquantum Heisenberg F or AF chains of $N=4$ sites, whose localized spinsinteract with conduction electrons via $sd$ exchange interaction, to performsuch comparison by starting from unentangled pure (at zero temperature) ormixed (at finite temperature) quantum state of localized spins as the initialcondition. This reveals that quantum-classical dynamics can faithfullyreproduce fully quantum dynamics in the F metallic case, but only when spin$S$, Heisenberg exchange between localized spins and $sd$ exchange aresufficiently small. Increasing any of these three parameters can lead tosubstantial deviations, which are explained by the dynamical buildup ofentanglement between localized spins and/or between them and electrons. In theAF metallic case, substantial deviations appear even at early times, despitestarting from unentangled N\'{e}el state, which therefore poses a challenge onhow to rigorously justify wide usage of the LL equation in phenomenologicalmodeling of antiferromagnetic spintronics experiments.

physical review. b./physical review. b

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