Bloch electron in a magnetic field: Mixed dimensionality and the magnetic-field-induced generalized quantum Hall effect

The energy spectrum of a Bloch electron in a magnetic field is one-dimensional. This leads to the Peierls instability and the magnetic-field-induced transition to the quantized Hall effect. The wave function is two-dimensional. This decreases the Peierls gap and makes it exponentially vanishing with magnetic field. Disorder lifts the degeneracy and one-dimensionality of the spectrum. High disorder yields a metallic behavior. Intermediate disorder leads to the generalized quantized Hall effect. The latter has a finite magnetoresistance as a semimetal, and Hall plateaus similar to the quantized ones, but they may have any value of the effective charge.