Langevin dynamics of generalized spins as SU($N$) coherent states

TL;DR

This paper develops a Langevin dynamics framework for generalized spin states represented as SU(N) coherent states, enabling efficient simulation of complex spin systems with higher-order multipole moments and topological defects.

Contribution

It reformulates spin dynamics as Langevin equations for SU(N) coherent states, extending classical models to include quantum multipole components and topological solitons.

Findings

Efficient numerical sampling of thermal equilibrium spin configurations.

Simulation of relaxation processes creating CP² Skyrmions with dipole and quadrupole features.

Demonstration of the approach on non-equilibrium topological defect formation.

Abstract

Classical models of spin systems traditionally retain only the dipole moments, but a quantum spin state will frequently have additional structure. Spins of magnitude $S$ have $N=2S+1$ levels. Alternatively, the spin state is fully characterized by a set of $N^{2}-1$ local physical observables, which we interpret as generalized spin components. For example, a spin with $S=1$ has three dipole components and five quadrupole components. These components evolve under a generalization of the classical Landau-Lifshitz dynamics, which can be extended with noise and damping terms. In this paper, we reformulate the dynamical equations of motion as a Langevin dynamics of SU($N$) coherent states in the Schr\"odinger picture. This viewpoint is especially useful as the basis for an efficient numerical method to sample spin configurations in thermal equilibrium and to simulate the relaxation and driven motion of topological solitons. To illustrate the approach, we simulate a non-equilibrium relaxation process that creates CP$^{2}$ Skyrmions, which are topological defects with both dipole and quadrupole character.