Persistent current for interacting electrons: a simple Hartree-Fock picture

The average persistent current 〈I〉 of diffusive electrons in the Hartree-Fock approximation is derived in a simple non-diagrammatic picture. The Fourier expansion directly reflects the winding number decomposition of the diffusive motion around the ring. One recovers the results of Ambegaokar and Eckern, and Schmid. Moreover one finds an expression for 〈I〉 which is valid beyond the diffusive regime. Typeset using REVTEX 1 The physics of persistent currents in mesoscopic isolated rings pierced by an AharonovBohm flux φ , has attracted a lot of interest on the description of the thermodynamic properties of mesoscopic metals, both in the non-interacting picture or in the presence of electronelectron interactions. The description of the interactions is a complicated task: although several attempts have been made to describe the role of the interactions in one-dimensional or few-channel rings, with the help of analytical arguments or numerical calculations [1], the diffusive nature of the electronic motion, which is probably essential in the experiments on metallic rings with finite thickness [2–4], has been treated only in two series of papers by Ambegaokar and Eckern and Schmid [5,6]. They calculated the average persistent current in the Hartree-Fock apprximation where the interaction is treated perturbatively in a diagrammatic picture. Although this calculation provides an average current smaller that experimentally observed [2], one can believe that it contains the essential physical ingredient namely weakly interacting diffusive electrons. Thus it may be interesting to simplify as much as possible the calculation in order to possibly generalize it, or to compare it with numerical calculations. In the above papers, the average persistent current is calculated with a diagrammatic technique where the diffusive motion is described by a Cooperon pole with a wave vector quantized by periodic boundary conditions. Then, by Poisson summation, this sum over diffusive modes is transformed into a sum over harmonics with periodicity φ0/2 where φ0 is the flux quantum. In this short note, we propose a simple derivation where the average current is directly related to the return probability for a diffusive particle. Then this return probability is expanded according to the winding number of the diffusive motion around the ring, which gives directly access to the harmonics expansion of the current. Moreover, we obtain a general expression which can be used beyond the diffusive regime. Consider a quasi-one dimensional ring of perimeter L and of transverse section S, in which the motion is supposed to be diffusive along the perimeter and uniform along the transverse section. The first step is to write the total energy ET in the Hartree-Fock approximation: 2