Quantum Hall Effect — Questions and Answers

Abstract

1. Dr. Laughlin, your gauge theory of the quantum Hall effect (QHE) is based on the equilibrium current formula

$$ J_Y = - \frac{1} {{Ar}}\left( {\frac{{\partial {\text{F}}}} {{\partial {\text{AY}}}}} \right)_{T,\zeta } = - \frac{\partial } {{Ar}} < V_Y > _{eq} $$

(1)

where F is the free energy and Ar is the area of the system, and v_y is the single-electron velocity. How do we know that the isothermal current (1) is the one which is measured in OHE experiments? This question arises because (1) is in general not equal to the current of an isolated system as given by the Kubo formula. In the case of an electron-impurity system which is infinite in the direction along the electric field (no edges), the two currents coincide if the Fermi energy is situated in any gap (zero density of states). Is it possible to demonstrate that the two types of current are equal as long as the Fermi energy lies in a mobility gap (σ_xx = O)? The situation seems to be even more difficult to survey if the system is finite in the direction of the electric field and, therefore, in (1), edge current contributions have to be taken into account. Or is this a trivial matter?