Understanding spin Hall effects from the motion in SU(2)XU(1) fields

TL;DR

This paper develops a classical framework for understanding spin Hall effects by analyzing particle motion in SU(2)XU(1) fields, revealing conditions for infinite spin relaxation and illustrating spin procession dynamics.

Contribution

It introduces a classical analogy to quantum spin equations, clarifies spin current non-conservation, and links spin relaxation to field conditions, providing new insights into spin dynamics in semiconductors.

Findings

Derived classical equations for spin motion in gauge fields.

Identified conditions for infinite spin relaxation time.

Analyzed spin procession in ReD field with frequency dependence.

Abstract

We derive the classical counterpart of a previously obtained quantum mechanical covariant ``continuitylike'' equation for the spin density, and present an intuitive picture for elucidating the non-conservation of the spin current. This reveals the equations of motion for a particle with spin under the Yang-Mills field (in certain semiconductors) as well as the Maxwell field, from which the condition for an infinite spin relaxation time is drawn out directly. As a concrete example, we discuss the procession of the spin orientation in spin Hall effect with the so called ReD field, which undergoes a circle with the frequency dependent on both the strength of the spin-orbit coupling and the initial velocity. The anti-commutation of the Pauli matrices is found to be crucial in simplifying the equations of motion in the view of quantum mechanism of the same topics.