Semi-Classical Discretization and Long-Time Evolution of Variable Spin Systems

TL;DR

This paper develops a semi-classical approach to variable-spin systems, introducing a discretized Truncated Wigner Approximation and comparing it with exact and continuous methods for specific quantum models.

Contribution

It presents a novel semi-classical discretization method for variable-spin systems using the star-product asymptotics and introduces a discretized TWA for improved quantum dynamics approximation.

Findings

Discretized TWA closely matches exact quantum dynamics in tested models.

The semi-classical approach effectively captures long-time evolution of variable-spin systems.

Discretization improves computational efficiency while maintaining accuracy.

Abstract

We apply the semi-classical limit of the generalized $SO(3)$ map for representation of variable-spin systems in a four-dimensional symplectic manifold and approximate their evolution terms of effective classical dynamics on $T^{\ast }\mathcal{S}_{2}$. Using the asymptotic form of the star-product, we manage to "quantize" one of the classical dynamic variables and introduce a discretized version of the Truncated Wigner Approximation (TWA). Two emblematic examples of quantum dynamics (rotor in an external field and two coupled spins) are analyzed, and the results of exact, continuous, and discretized versions of TWA are compared.