Phase space methods for the spin dynamics in condensed matter systems

TL;DR

This paper develops a phase-space quantum framework for spin-1/2 fermions, deriving coupled Wigner, Vlasov, and hydrodynamic equations that incorporate Zeeman and spin-orbit effects for condensed matter systems.

Contribution

It introduces a self-consistent mean-field model using phase-space methods for spin dynamics, including new semiclassical and hydrodynamic equations with spin effects.

Findings

Derived a four-component Wigner equation with spin effects.

Formulated coupled Maxwell-Wigner equations for self-consistent modeling.

Proposed a closure relation for hydrodynamic equations.

Abstract

Using the phase-space formulation of quantum mechanics, we derive a four-component Wigner equation for a system composed of spin-1/2 fermions (typically, electrons) including the Zeeman effect and the spin-orbit coupling. This Wigner equation is coupled to the appropriate Maxwell equations to form a self-consistent mean-field model. A set of semiclassical Vlasov equations with spin effects is obtained by expanding the full quantum model to first order in the Planck constant. The corresponding hydrodynamic equations are derived by taking velocity moments of the phase-space distribution function. A simple closure relation is proposed to obtain a closed set of hydrodynamic equations.