Field theory in a strong magnetic field and the quantum Hall effect: Integer Hall effect

Abstract

A field theory of two-dimensional continuum electrons in a strong magnetic field is formulated based on a magnetic lattice representation that preserves translational invariance. The gauge invariance, which is described by the Ward-Takahashi identity, leads to the topologically invariant expression of the Hall conductance and to the exact low-energy theorem. Electrons are localized around a short-range impurity potential and a plateau of the Hall conductance with an integer multiple of ${\mathit{e}}^{2}$/h is realized in the localized state regions. In the presence of extended states at the Fermi energy, ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{y}}$ differs from the quantized value by an amount proportional to the transverse conductance ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}\mathit{x}}$.