It is shown that the absolute value of the persistent current in a system with toroidal geometry is rigorously less than or equal to eħNα/4π${\mathit{mr}}_0^2$, where N is the number of electrons, ${\mathit{r}}_0^{\mathrm{-}2}$=〈${\mathit{r}}_{\mathit{i}}^{\mathrm{-}2}$〉 is the equilibrium average of the inverse of the square of the distance of an electron from an axis threading the torus, and α≤1 is a positive constant, related to the azimuthal dependence of the density. This result is valid in three and two dimensions for arbitrary interactions, impurity potentials, and magnetic fields.

• Received 7 June 1994

DOI:https://doi.org/10.1103/PhysRevB.51.2612