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

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 e²/h is realized in the localized state regions. In the presence of extended states at the Fermi energy, σxy differs from the quantized value by an amount proportional to the transverse conductance σₓₓ.