Edge states, transmission matrices, and the Hall resistance

Abstract

We consider the Landauer formula, relating conductances to transmission matrices, for a two-dimensional system in a magnetic field. We argue that the magnetoresistance, R, and the Hall resistance, ${\mathrm{R}}_{\mathrm{H}}$, satisfy the sum rule (R+${\mathit{R}}_{\mathrm{H}}$${)}^{\mathrm{\ensuremath{-}}1}$=(${\mathit{e}}^{2}$/h)Tr(${\mathit{t}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$t) where t is the transmission matrix. For zero field our expressions reduce to the usual multichannel Landauer formulas. In the absence of dissipation, R approaches zero, t approaches a unit matrix, and quantized values are obtained for the Hall resistance.