The driven Harper model

TL;DR

This paper studies the driven Harper model, extending the classic model to include electric fields, and finds that eigenstates are either extended or localized based on Bloch frequency relationships, with implications for wave-function localization.

Contribution

It introduces the driven Harper model with two Bloch frequencies and analyzes the conditions for eigenstate localization or extension, including incommensurate frequency cases.

Findings

Eigenstates are either extended or localized depending on Bloch frequency commensurability.

The model applies to electrons in a magnetic field with an in-plane electric field.

Localization length estimates are provided for incommensurate Bloch frequencies.

Abstract

We analyze the driven Harper model, which appears in the problem of tight-binding electrons in the Hall configuration (normal to the lattice plane magnetic field plus in-plane electric field). The presence of an electric field extends the celebrated Harper model, which is parametrized by the Peierls phase, into the driven Harper model, which is additionally parametrized by two Bloch frequencies associated with the two components of the electric field. We show that the eigenstates of the driven Harper model are either extended or localized, depending on the commensurability of Bloch frequencies. This results holds for both rational and irrational values of the Peierls phase. In the case of incommensurate Bloch frequencies we provide an estimate for the wave-function localization length.