Rigorous upper bound for the persistent current in systems with toroidal geometry
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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/4πmr02, where N is the number of electrons, r0-2= 〈ri-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.