Semiclassical Dynamics of Electrons in Magnetic Bloch Bands: a Hamiltonian Approach

TL;DR

This paper rigorously derives the semiclassical equations of motion for electrons in magnetic Bloch bands, confirming their Hamiltonian structure with Berry-phase corrections and validating the Liouville theorem.

Contribution

It provides a formal diagonalization approach that clarifies the Hamiltonian nature of semiclassical electron dynamics with Berry-phase effects.

Findings

Confirmed the Hamiltonian structure of semiclassical equations with Berry-phase corrections

Established the validity of the Liouville theorem in this context

Derived equations of motion consistent with wave-packet dynamics

Abstract

y formally diagonalizing with accuracy $\hbar$ the Hamiltonian of electrons in a crystal subject to electromagnetic perturbations, we resolve the debate on the Hamiltonian nature of semiclassical equations of motion with Berry-phase corrections, and therefore confirm the validity of the Liouville theorem. We show that both the position and momentum operators acquire a Berry-phase dependence, leading to a non-canonical Hamiltonian dynamics. The equations of motion turn out to be identical to the ones previously derived in the context of electron wave-packets dynamics.