MESOSCOPIC CONDUCTANCE AND ITS FLUCTUATIONS AT A NONZERO HALL ANGLE

We consider the bilocal conductivity tensor and the two-probe conducta nce and its fluctuations for a disordered phase-coherent two-dimension al system of noninteracting electrons in the presence of a magnetic fi eld, including correctly the edge effects. Analytical results an obtai ned by perturbation theory in the limit sigma(xx)much greater than 1. For mesoscopic systems the conduction process is dominated by diffusio n, but we show that, due to the lack of time-reversal symmetry, the bo undary condition for diffusion is altered at the reflecting edges. Ins tead of the usual condition that the derivative along the direction no rmal to the wall of the diffusing variable vanishes, the derivative at the Hall angle to the normal vanishes. We demonstrate the origin of t his boundary condition in several approaches. Within the standard diag rammatic perturbation expansion, we evaluate the bilocal conductivity tensor to leading order in 1/sigma(xx), exhibiting the edge currents a nd the boundary condition. We show how to calculate conductivity and c onductance using the nonlinear sigma model with the topological term, to all orders in 1/sigma(xx). Edge effects are related to the topologi cal term, and there are higher-order corrections to the boundary condi tion. We discuss the general form of the current-conservation conditio ns. We evaluate explicitly the mean and variance of the conductance, t o leading order in 1/sigma(xx) and to order (sigma(xy)/sigma(xx))(2), and find that the variance of the conductance increases with the Hall ratio. Thus the conductance fluctuations are no longer simply describe d by the unitary universality class of the sigma(xy)=0 case, but inste ad there is a one-parameter family of probability distributions. Our r esults differ from previous calculations, which neglected sigma(xy)-de pendent effects other than the leading-order boundary condition. In th e quasi-one-dimensional limit, the usual universal result for the cond uctance fluctuations of the unitary ensemble is recovered, in contrast to results of previous authors.