Edge-state-transmission duality relation and its implication for 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 σ_xx and σ_xy in the diffusive limit. The edge-state scattering problem is reduced to solving the diffusion equation with two boundary conditions [∂_y-(σ_xy/σ_xx)∂_x]φ=0 and {∂_x+[(σ_xy-σ_xy^lead)/σ_xx]∂_y}φ=0. We find that the resistances in the geometry considered are not necessarily measures of the resistivity and ρ_xx=(W/L)(R/T)h/e² (R=1-T) holds only when ρ_xy 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.