Edge state transmission, duality relation and its implication to measurements

The duality in the Chalker-Coddington network model is examined. We are able to write down a duality relation for the edge state transmission coefficient, but only for a specific symmetric Hall geometry. Looking for broader implication of the duality, we calculate the transmission coefficient T in terms of the conductivity σxx and σxy in the diffusive limit. The edge state scattering problem is reduced to solving the diffusion equation with two boundary conditions (∂y − σxy σxx ∂x)φ = 0 and [∂x + σxy−σ xy σxx ∂y]φ = 0. We find that the resistances in the geometry considered are not necessarily measures of the resistivity and ρxx = L W R T h e (R = 1 − T ) holds only when ρxy is quantized. We conclude that duality alone is not sufficient to explain the experimental findings of Shahar et al and that Landauer-Buttiker argument does not render the additional condition, contrary to previous expectation.