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 spacetime 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 wavefunctions 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 Bernevig et al (2006 Phys. Rev. Lett. 97 236601). 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.