Persistent currents in n–fold twisted Moebius strips

– We investigate the influence of the topology on generic features of the persistent current in n–fold twisted Moebius strips formed of quasi one–dimensional mesoscopic rings, both for free electrons and in the weakly disordered regime. We find that there is no generic difference between the persistent current for untwisted rings and for Moebius strips with an arbitrary number of twists. Introduction. – Persistent currents in mesoscopic rings threaded by a magnetic flux offer one of the prime examples of quantum coherence in mesoscopic physics [1]. The recent experimental realization of rings which have the shape of singly and doubly twisted Moebius strips and a diameter of about 50μm [2] raises the intriguing question whether generic features of the persistent current depend upon the topology of the ring. An affirmative answer would imply a close connection between topology and quantum coherence. This is the problem we address in the present paper. We model the ring – both in its untwisted and its twisted form – in the standard way [3]: The electrons move either freely or diffusively through a two–dimensional conducting strip (see fig. 1). We work in the regime of zero or weak disorder and assume free motion in the transverse direction. Our model applies whenever the elastic mean free path is of the order of or larger than the transverse dimension of the ring. Several theoretical papers have dealt with related problems. Predictions vary according to the dynamical regime under consideration [4,6,7,5]. In all these papers, however, the question whether there is a generic difference between the untwisted and the twisted case, seems not to have been addressed. We briefly describe the difference between our model and that of other recent work at the end of the paper. We consider a metallic or semiconducting mesoscopic quasi one–dimensional ring at low temperature. A constant homogeneous magnetic field B threads the ring in a direction perpendicular to the plane of the ring. We take account only of the Aharanov–Bohm (AB) phase φ = 2πΦ/Φ0. Here Φ = AB is the magnetic flux through the ring as given by B and by the area A of the ring, and Φ0 is the elementary flux quantum. We disregard the effect of the magnetic field on the orbital motion of the electrons in the ring. We also disregard the spin of the electrons. The ring is twisted n = 0, 1, . . . times. The case n = 0 corresponds to the case of an ordinary plane ring, the case n = 1 to an ordinary Moebius strip, and higher values