Berry phase, hyperorbits, and the Hofstadter spectrum: semiclassical dynamics in magnetic Bloch bands

TL;DR

This paper develops a semiclassical framework incorporating Berry phase effects to analyze electron dynamics in magnetic Bloch bands, revealing connections to Hofstadter spectrum features and transport properties.

Contribution

It introduces new semiclassical equations for magnetic Bloch electrons that include Berry phase and magnetization effects, unifying various phenomena in a single formalism.

Findings

Modified electron velocity and energy due to Berry phase and magnetization.

Connection between cyclotron orbits and Hofstadter subbands.

Unified understanding of band splitting and Hall conductivities.

Abstract

We have derived a new set of semiclassical equations for electrons in magnetic Bloch bands. The velocity and energy of magnetic Bloch electrons are found to be modified by the Berry phase and magnetization. This semiclassical approach is used to study general electron transport in a DC or AC electric field. We also find a close connection between the cyclotron orbits in magnetic Bloch bands and the energy subbands in the Hofstadter spectrum. Based on this formalism, the pattern of band splitting, the distribution of Hall conduct- ivities, and the positions of energy subbands in the Hofstadter spectrum can be understood in a simple and unified picture.