Relation between persistent currents and the scattering matrix.

Predictions of Akkermans et al. are essentially changed when the Krein spectral displacement operator is regularized by means of zeta function. Instead of piecewise constant persistent current of free electrons on the plane one has a current which varies linearly with the flux and is antisymmetric with regard to all time preserving values of α including 1/2. Different self-adjoint extensions of the problem and role of the resonance are discussed. PACS : 72.10.Bg, 72.20.My e-mail address : [email protected] [email protected] Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS In [1] a persistent current around the origin, pierced by a flux tube, for free electrons in the plane when all states below the Fermi energy EF are occupied, was calculated. The persistent current was defined with respect to a point. It was given by the total current through a line that extends from that point to infinity, in the absence of currents through the external leads. Remarkably enough, a very compact formula for the scattering-state contribution to the persistent current was derived, dI(E, α) = (2πi)∂α[ln det S(E, α)]dE, (1) where dI(E, α) is the differential contribution to the persistent current at energy E, S(E, α) is the on-shell scattering matrix, and α is the total flux Φ in the units of the flux quantum Φo = hc/e [1]. We shall write α as α = n+η where n = [α] is the nearest integer smaller than α, and η is the nonintegral part of α, 0 ≤ η < 1. It is well known that the scattering in the presence of the Aharonov-Bohm potential is singular since the phase shifts δl(E) do not decay as the orbital quantum number l → ∞ [2]. Therefore in [1] limL→∞ ∑ |l|<L δl was used to define ln det S. However, recently it has been shown [3] that when ln det S is regularized by means of zeta function regularization the Krein-Friedel formula [4] (2πi) ln det S(E, α) = △N(E, α), (2) where △N(E, α) is the change of the integrated density of scattering states caused by the presence of a scatterer, remains also valid in the singular scattering problem. The results for △N(E, α) for different self-adjoint extensions of the problem can be read off from [3]. The most general expression for the contribution of scattering states to Nα for E ≥ 0 which includes the situation when bound states are present and confirms calculations of [5] is △N(E, α) = − 1 2 η(1− η) + 1 π arctan ( sin(ηπ) cos(ηπ)− (|E−n|/E) )