Extended states in systems with odd-rank transfer matrices

The localization properties are investigated within a transfer matrix formulation. We find that there may exist one extended eigenstate when the rank of the transfer matrix is odd. An edge state in a quantum Hall system is such an example. It is a chiral state that can always carry a current. PACS numbers: 03.65.−w, 72.10.Bg, 73.20.Jc A non-interacting Schrödinger equation can be formulated into a transfer matrix form. This method has been used to numerically study Anderson localization and to investigate electron transport properties of mesoscopic systems [1–4]. Anderson et al [5] used this method to study the one-dimensional localization problem. Shapiro generalized the formulation into a high dimensional system [6]. In the Anderson localization model, an electron in a channel can move in two opposite directions due to the time reversal symmetry. The rank of an m-channel transfer matrix is 2m, an even number, because there are two modes on each channel. The transfer matrix method is also used to investigate transport properties of a two-dimensional electron gas under a strong magnetic field, a quantum Hall (QH) system in which the time reversal symmetry is broken. According to a semiclassical model, an electronic state in a strong magnetic field and a smooth potential can be decomposed into a rapid cyclotron motion and a slow drifting motion of its guiding centre. The direction of the drifting motion is unidirectional, i.e., electrons are in chiral states [7]. One can use a so-called Chalker–Coddington (CC) network model to investigate a QH system [8]. In this model, electrons on each link can move in only one direction. Thus there is only one mode on each channel. Consequently, the rank of the transfer matrix in the CC model, which is the number of independent modes, may be odd because there is no restriction on the channel number. This is different from the transfer matrix in the Anderson localization model, whose 3 Current address: Institut für Theoretische Physik, Universitat Regensburg, 93040 Regensburg, Germany. 0305-4470/04/041337+07$30.00 © 2004 IOP Publishing Ltd Printed in the UK 1337