Equality of the bulk and edge Hall conductances in 2D

Von Klitzing [15] observed that a two dimensional electron gas at very low temperatures and strong magnetic field displays a quantization of the Hall conductance, that is the conductance measured in the direction transversal to the applied current. Specifically, the conductance plotted as a function of the magnetic field shows extremely flat plateaux at integer multiples of e²/h (e is a charge of electron and h is Planck's constant). Two pictures were introduced for a description of the Quantum Hall Effect: "Edge currents picture" and "Bulk currents picture". The edge current picture suggests that the Hall current flows in the narrow regions along the sample boundaries (we will denote the corresponding conductance by σ_E), so that the Hall voltage drops entirely in these regions. On the other hand, the description in terms of bulk currents suggests that the Hall voltage drops gradually across the sample (and let sB denote the Hall conductance associated with this regime). It was proposed by Halperin [13] that in reality one should expect an intermix of these two pictures, and that σ_E = σ_B. In [14], σ_E,B were linked for a Harper's model with rational flux. In more general setup, the equality of the edge and the bulk conductances was recently rigorously established [10,16,18], provided that there is a spectral gap Δ at Fermi energy of the single-particle (bulk) Hamiltonian H_B.