Inequivalence of Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics for a single quantum spin

TL;DR

This paper investigates the fundamental differences between quantum Landau-Lifshitz and Landau-Lifshitz-Gilbert equations for a single spin-1 system, revealing inequivalence in their dynamics despite similar classical counterparts.

Contribution

It demonstrates that quantum Landau-Lifshitz and LLG equations are inequivalent for spin-1 particles, highlighting fundamental differences in quantum dissipation mechanisms.

Findings

Quantum $q$-LL and $q$-LLG generate inequivalent evolution for spin-1.

Equivalence holds only for pure states or spin-1/2 states.

Trajectories are qualitatively similar despite inequivalence.

Abstract

We examine the relation between the quantum Landau-Lifshitz equation ($q$-LL) [Phys. Rev. Lett. 110, 147201 (2013)] and quantum Landau-Lifshitz-Gilbert equation ($q$-LLG) [Phys. Rev. Lett. 133, 266704 (2024)]; two non-linear purity preserving master equations that extend classical atomistic spin dynamics into the quantum regime. While the classical LL and LLG counterparts for any number of spins are known to be equivalent, i.e., give identical spin trajectories up to a rescaling of the time parameter, the quantum formulations are equivalent only in certain cases, such as for pure states or for arbitrary single spin-$\frac{1}{2}$ states. Here, we demonstrate that this equivalence breaks down even at the level of a single spin, provided $s \geq 1$. Focusing on a spin-1 particle in an anisotropic crystal field, we show that the $q$-LL and $q$-LLG equations generate inequivalent time evolution. We introduce temporal rescaling misfits that quantify the inequivalence of the two types of dynamics. Although our results highlight fundamental differences in dissipation mechanisms encoded in these equations, the resulting trajectories remain qualitatively similar for this system.