Berry phase, hyperorbits, and the Hofstadter spectrum

TL;DR

This paper develops a semiclassical framework incorporating Berry phase effects to analyze electron dynamics in magnetic Bloch bands, explaining the Hofstadter spectrum's clustering and linking orbit quantization to Hall conductivity.

Contribution

It introduces a general, conceptually simple semiclassical theory that integrates Berry phase into electron transport and orbit quantization in high magnetic fields.

Findings

Derived an Onsager-like quantization formula for cyclotron orbits.

Connected the number of orbits to Hall conductivity.

Explained the clustering of the Hofstadter spectrum.

Abstract

We develop a semiclassical theory for the dynamics of electrons in a magnetic Bloch band, where the Berry phase plays an important role. This theory, together with the Boltzmann equation, provides a framework for studying transport problems in high magnetic fields. We also derive an Onsager-like formula for the quantization of cyclotron orbits, and we find a connection between the number of orbits and Hall conductivity. This connection is employed to explain the clustering structure of the Hofstadter spectrum. The advantage of this theory is its generality and conceptual simplicity.