Persistent currents in an ensemble of isolated mesoscopic rings

Abstract

We calculate the persistent current induced at zero temperature by an external, constant, and homogeneous magnetic field in an ensemble of isolated mesoscopic rings. In each ring, the electrons are assumed to move independently under the influence of a Gaussian white noise random impurity potential. We account for the magnetic field only in terms of the flux threading each ring, without considering the field present in the body of the ring. We pay particular attention to the constraint of integer particle number on each ring. We evaluate the persistent current non-perturbatively, using a generating functional involving Grassmann integration. The magnetic flux threading each ring breaks the orthogonal symmetry of the formalism; forcing us to calculate explicitly the orthogonal-unitary crossover. The resulting expression for the average persistent current is paramagnetic, periodic in the flux with a period of half the elementary flux quantum φ 0, odd when the sign of the flux is reversed, and has a maximum value of about 0.3 E cd φ 0 , where d is the mean level spacing and E c the Thouless energy. This is about two orders of magnitude below experiment.