# Hall conductivity from Kubo: Bulk or edge?

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.

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This seems to be what you want: journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.3697 –  Jia Yiyang Mar 16 at 2:15
Also this: arxiv.org/abs/1207.5989 and this: de.arxiv.org/abs/cond-mat/9912186v1 –  PPR Oct 26 at 23:10