Relation between current and density profiles of interacting electronic systems in a magnetic field

An exact formal relationship between the equilibrium orbital-current and density profiles of a nonuniform electronic system in the presence of a magnetic field is derived from current-density-functional theory. In the local-density approximation we obtain an explicit formula, including exchange and correlation effects, for the equilibrium current in terms of the gradient of density. This formula is shown to reduce to exact asymptotic results in the limit of very large magnetic fields or in the linear-response regime. A surprising consequence of our formula is that in three dimensions, the direction of circulation of the current, relative to the gradient of the density, can have either sign, depending on the local value of the density. A nonlocal linear relationship between current and density for rapidly varying density profiles and/or weak magnetic fields is also derived.