Mesoscopic conductance and its fluctuations at a nonzero Hall angle

We consider the bilocal conductivity tensor and the two-probe conductance and its fluctuations for a disordered phase-coherent two-dimensional system of noninteracting electrons in the presence of a magnetic field, including correctly the edge effects. Analytical results are obtained by perturbation theory in the limit ${\sigma }_{\mathrm{xx}}\gg 1$. For mesoscopic systems the conduction process is dominated by diffusion, but we show that, due to the lack of time-reversal symmetry, the boundary condition for diffusion is altered at the reflecting edges. Instead of the usual condition that the derivative along the direction normal to the wall of the diffusing variable vanishes, the derivative at the Hall angle to the normal vanishes. We demonstrate the origin of this boundary condition in several approaches. Within the standard diagrammatic perturbation expansion, we evaluate the bilocal conductivity tensor to leading order in $1/{\sigma }_{\mathrm{xx}},$ exhibiting the edge currents and the boundary condition. We show how to calculate conductivity and conductance using the nonlinear $\sigma$ model with the topological term, to all orders in $1/{\sigma }_{\mathrm{xx}}.$ Edge effects are related to the topological term, and there are higher-order corrections to the boundary condition. We discuss the general form of the current-conservation conditions. We evaluate explicitly the mean and variance of the conductance, to leading order in $1/{\sigma }_{\mathrm{xx}}$ and to order $({\sigma }_{\mathrm{xy}}/{\sigma }_{\mathrm{xx}}{)}^2$, and find that the variance of the conductance increases with the Hall ratio. Thus the conductance fluctuations are no longer simply described by the unitary universality class of the ${\sigma }_{\mathrm{xy}}=0\mathrm{}$ case, but instead there is a one-parameter family of probability distributions. Our results differ from previous calculations, which neglected ${\sigma }_{\mathrm{xy}}$-dependent effects other than the leading-order boundary condition. In the quasi-one-dimensional limit, the usual universal result for the conductance fluctuations of the unitary ensemble is recovered, in contrast to results of previous authors.