Hall Conductivity in the presence of repulsive magnetic impurities

The Hall conductivity of disordered magnetic systems consisting of hard-core point vortices randomly dropped on the plane with a Poissonian distribution, has a behavior analogous to the one observed experimentally by R. J. Haug, R. R. Gerhardts, K. v. Klitzling and K. Ploog, with repulsive scatterers [1]. We also argue that models of homogeneous magnetic field with disordered potential, have necessarily vanishing Hall conductivities when their Hilbert space is restricted to a given Landau level subspace. It is commonly believed [2] that in quantum Hall devices disorder plays a crucial role in the understanding of plateaus for the Hall conductivity as a function of 1/B or N(EF ), the number of electrons, at integer (or fractional) values in units of e 2/h. For an homogeneous B field, the linear response of the system to a small electric field gives no hint of such a remarkable behavior since all the states are delocalized and have the same transverse conductivity which varies linearly with 1/B or N(EF ) (classical straight line). Disorder is needed to explain why some states (in fact most of them) are localized in broadened Landau levels, thus the plateaus in the Hall conductivity, around the classical line. However, R. J. Haug et al [1] reported the experimental observation of a shifted quantized Hall conductivity with respect to the classical line, when attractive or repulsive scaterrers are considered in the Hall sample. The data Hall conductivity versus the filling factorare shifted to the left when the scaterrers are repulsive, and to the right when they are attractive. The authors in [1] were able to reproduce qualitatively these shifts within a self-consistent T-matrix approximation computation. They also argued that two-dimensional disordered δ repulsive models projected in the lowest Landau level (LLL) of an external magnetic field [3], which exhibit an asymmetrically disorder induced broadened LLL, indeed favour this phenomenon. The root of the understanding lies in the fact that conducting states are not located anymore at the center of a symmetrically broadened Landau level, as it is the case for neutral scatterers, but in one side or the other of the now asymmetric density of states (DOS). On the theoretical side on the other hand, the question was asked in [3] about the way to compute exactly the average LLL Hall conductivity. In the following, we study a model where the disorder is contained in the definition of the magnetic field itself. We consider [4] a gas of electrons coupled to hard-core point vortices, hereafter called magnetic impurities, carrying a flux φ = αφo (φo = h/e is the quantum flux) and randomly dropped on the plane according to a Poisson distribution,