TY - CHAP

T1 - Equality of the bulk and edge Hall conductances in 2D

AU - Elgart, A.

PY - 2006/10/2

Y1 - 2006/10/2

N2 - 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 σE), 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 σE = σB. In [14], σE,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 HB.

AB - 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 σE), 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 σE = σB. In [14], σE,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 HB.

UR - http://www.scopus.com/inward/record.url?scp=33749034476&partnerID=8YFLogxK

U2 - 10.1007/3-540-34273-7_23

DO - 10.1007/3-540-34273-7_23

M3 - Chapter

AN - SCOPUS:33749034476

SN - 3540310266

SN - 9783540310266

T3 - Lecture Notes in Physics

SP - 325

EP - 332

BT - Mathematical Physics of Quantum Mechanics

A2 - Asch, Joachim

A2 - Joye, Alain

ER -