The fact that the Hall resistance c

The fact that the Hall resistance can be expressed in terms of the transmission probabilities of edge states at the Fermi level does not imply that these few states carry a macroscopic current, nor does it imply that the current flows at the edges. A determination of the spatial current distribution i(r), rather than just the total current I,requires consideration of all the states below the Fermi level, which acquire a net drift velocity because of the Hall field. As we discussed in Section III.A.2, knowledge of i(r) is not necessary to know the resistances in the regime of linear response, because the Einstein relation allows one to obtain the resistance from the diffusion constant. Edge channels tell you where the current flows if the electrochemical potential difference δμ is entirely due to a density difference, relevant for the diffusion problem. Edge channels have nothing to say about where the current flows if δμ is mainly of electrostatic origin, relevant for the problem of electrical conduction. The ratio δμ/I is the same for both problems, but i(r) is not.