Weak localization and integer quantum Hall effect in a periodic potential

We consider magnetotransport in a disordered two-dimensional electron gas in the presence of a periodic modulation in one direction. Existing quasiclassical and quantum approaches to this problem account for Weiss oscillations in the resistivity tensor at moderate magnetic fields, as well as a strong modulation-induced modification of the Shubnikov–de Haas oscillations at higher magnetic fields. They do not account, however, for the operation at even higher magnetic fields of the integer quantum Hall effect, for which quantum interference processes are responsible. We introduce a field-theory approach, based on a nonlinear $\sigma$ model, which encompasses naturally both the quasiclassical and quantum-mechanical approaches, as well as providing a consistent means of extending them to include quantum interference corrections. A perturbative renormalization-group analysis of the field-theory shows how weak localization corrections to the conductivity tensor may be described by a modification of the usual one-parameter scaling, such as to accommodate the anisotropy of the bare conductivity tensor. We also show how the two-parameter scaling, conjectured as a model for the quantum Hall effect in unmodulated systems, may be generalized similarly for the modulated system. Within this model we illustrate the operation of the quantum Hall effect in modulated systems for parameters that are realistic for current experiments.