Semiclassical Diagonalization of Quantum Hamiltonian and Equations of Motion with Berry Phase Corrections

TL;DR

This paper introduces a general semiclassical diagonalization method at order  for quantum Hamiltonians, naturally incorporating Berry phase corrections into equations of motion for various physical systems.

Contribution

It provides a novel, systematic diagonalization procedure that accounts for Berry phase effects in a broad class of quantum Hamiltonians, leading to new semiclassical equations of motion.

Findings

Derived new equations of motion with Berry phase corrections

Applied method to Dirac particles in electromagnetic and gravitational fields

Analyzed Bloch electron propagation in external electromagnetic fields

Abstract

It has been recently found that the equations of motion of several semiclassical systems must take into account terms arising from Berry phases contributions. Those terms are responsible for the spin Hall effect in semiconductor as well as the Magnus effect of light propagating in inhomogeneous media. Intensive ongoing research on this subject seems to indicate that a broad class of quantum systems may be affected by Berry phase terms. It is therefore important to find a general procedure allowing for the determination of semiclassical Hamiltonian with Berry Phase corrections. This article presents a general diagonalization method at order $\hbar $ for a large class of quantum Hamiltonians directly inducing Berry phase corrections. As a consequence, Berry phase terms on both coordinates and momentum operators naturally arise during the diagonalization procedure. This leads to new equations of motion for a wide class of semiclassical system. As physical applications we consider here a Dirac particle in an electromagnetic or static gravitational field, and the propagation of a Bloch electrons in an external electromagnetic field.