Spin-orbit induced equilibrium spin currents in materials

The existence of pure spin currents in absence of any driving external field is commonly considered an exotic phenomenon appearing only in quantum materials, such as topological insulators. We demonstrate instead that equilibrium spin currents are a rather general property of materials with non-negligible spin-orbit coupling (SOC). Equilibrium spin currents can be present at the surfaces of a slab. Yet, we also propose the existence of global equilibrium spin currents, which are net bulk spin currents along specific crystallographic directions of solid-state materials. Equilibrium spin currents are allowed by symmetry in a very broad class of systems having gyrotropic point groups. The physics behind equilibrium spin currents is uncovered by making an analogy between electronic systems with SOC and non-Abelian gauge theories. The electron spin can be seen as analogous to the color degree of freedom in SU(2) gauge theories and equilibrium spin currents can then be identified with diamagnetic color currents appearing as the response to a effective non-Abelian magnetic field generated by the SOC. Equilibrium spin currents are not associated with spin transport and accumulation, but they should nonetheless be carefully taken into account when computing transport spin currents. We provide quantitative estimates of equilibrium spin currents for a number of different systems, specifically the Au(111) and Ag(111) metallic surfaces presenting Rashba-type surface states, nitride semiconducting nanostructures, and bulk materials, such as the prototypical gyrotropic medium tellurium. In doing so, we also point out the limitations of model approaches showing that first-principles calculations are needed to obtain reliable predictions. We therefore use density functional theory computing the so-called bond currents, which represent a powerful tool to deeply understand the relation between equilibrium currents, electronic structure, and crystal point group.