Gauge-invariant formulation of spin-current density-functional theory

Spin-currents and non-Abelian gauge potentials in electronic systems can be treated by spin-current densityfunctional theory, whose main input is the exchange-correlation xc energy expressed as a functional of spin-currents. Constructing a functional of spin-currents that is invariant under U 1 SU 2 transformations is a long-standing challenge. We solve the problem by expressing the energy as a functional of a variable we call “invariant vorticity.” As an illustration we construct the xc energy functional for a two-dimensional electron gas with linear spin-orbit coupling and show that it is proportional to the fourth power of the spin-current.