Gauge Covariance and Spin Current Conservation in the Gauge Field Formulation of Systems with Spin-Orbit Coupling

Abstract

The question of gauge-covariance in the non-Abelian gauge-field formulation of two space-dimensional systems with spin-orbit coupling relevant to spintronics is investigated. Although, these are generally gauge-fixed models, it is found that for the class of gauge fields that are space-time independent and satisfy a U(1) algebra, thus having a vanishing field strength, there is a residual gauge freedom in the Hamiltonian. The gauge transformations assume the form of a space-dependent rotation of the transformed wave functions with rotation angles and axes determined by the specific form of the gauge-field, i.e., the spin-orbit coupling. The fields can be gauged away, reducing the Hamiltonian to one which is isospectral to the free-particle Hamiltonian, and giving rise to the phenomenon of persistent spin helix reported first by B.~A.~Bernevig \emph{et al.} [Phys.~Rev.~Lett. \textbf{97}, 236601 (2006)]. The investigation of the global gauge transformations leads to the derivation of a continuity equation where the component of the spin-density along given directions, again fixed by the specific form of the gauge field, is conserved.