Hall conductivity from Kubo: Bulk or edge?

Question

Using the Kubo formula, Thouless, Kohmoto, Nightingale, and den Nijs (TKNN, PRL 49 405-408 (1982)), proved that upon summing all the contributions of the filled states of an insulator, the Hall conductivity must be an integer (the Chern number) times e^2/h.

Their theorem uses periodic boundary conditions along x and y and therefore avoids the discussion on edges. An intuitive picture of the quantum hall effect (QHE) is, however, based on currents on the edge (people always talk about skipping orbits going around the edge).

My question now is: Why do some people say that the edges are the whole story of the QHE whereas the formulation of TKNN make it look like this is not important? Alternatively, how can I convince myself that the conclusions from the TKNN derivation hold also for open boundary conditions?

Also, from the TKNN derivation, the obtained Chern number seems to me a property of the bulk. Is there a simple proof that translates this Chern number to the number of edge channels in an open boundary condition setting?

Many thanks.

Answer 1

The TKNN (bulk) and Büttiker (edge) explanations for the quantized Hall conductance correspond to different geometries.

In the TKNN theory, the "sample" consists of a torus closed on itself and therefore has no edges at all. In this case the electric potential is uniform, and the electric field is due to the time derivative of the vector potential (it lasts only as long as one varies the magnetic flux inside the torus). In this case, the Hall current is truly a bulk current.

Büttiker, on the other hand, considers a Hall bar with different electrochemical potentials on each side. If one (as does Büttiker) assumes that the electrostatic potential is uniform within the central region of the bar, and rises on the sides, then one finds that the current flows along the edges (in opposite directions), with more current flowing along one edge than along the other because of the different chemical potentials.

In a more realistic description, the electrostatic potential is not uniform in the bulk of the bar, so that the current flow takes place both along the edges and within the bulk of the bar. In any case, the net current is completely independent of the actual profile of the electrostatic potential accross the hall bar, and depends only upon the chemical potential difference. That is why why Büttuker and TKNN obtain the same answer for the (quantized) total current.

A nice discussion of this question is given by Yoshioka.