Quantum Hamiltonian diagonalization and Equations of Motion with Berry Phase Corrections

TL;DR

This paper reviews a new diagonalization method for matrix Hamiltonians that incorporates Berry phase effects, showing how these corrections significantly alter the dynamics of various quantum systems in the semiclassical regime.

Contribution

It introduces a formal expansion approach in powers of  for diagonalizing matrix Hamiltonians, capturing Berry phase effects in the equations of motion.

Findings

Berry phase terms modify semiclassical dynamics of quantum systems

Diagonalization method systematically includes Berry corrections

Applicable to relativistic particles and electrons in solids

Abstract

It has been recently found that the equations of motion of several semiclassical systems must take into account anomalous velocity terms arising from Berry phase contributions. Those terms are for instance responsible for the spin Hall effect in semiconductors or the gravitational birefringence of photons propagating in a static gravitational field. Intensive ongoing research on this subject seems to indicate that actually a broad class of quantum systems might have their dynamics affected by Berry phase terms. In this article we review the implication of a new diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of $\hbar$. In this approach both the diagonal energy operator and dynamical operators which depend on Berry phase terms and thus form a noncommutative algebra, can be expanded in power series in \hbar $. Focusing on the semiclassical approximation, we will see that a large class of quantum systems, ranging from relativistic Dirac particles in strong external fields to Bloch electrons in solids have their dynamics radically modified by Berry terms.