# Multiband Effects in Equations of Motion of Observables Beyond the Semiclassical Approach

## TL;DR

This paper develops a Hamiltonian-based framework for multiband wave packets in Bloch electrons, incorporating interband Berry phase effects to improve understanding of their equations of motion beyond semiclassical models.

## Contribution

It introduces a projection operator method derived from adiabatic perturbation theory to derive multiband equations of motion including non-Abelian Berry phase effects.

## Key findings

- Derived generalized equations of motion for multiband wave packets.
- Applied the approach to single, degenerate, and crossing bands.
- Potential implications for transport and Hall effects.

## Abstract

The equations of motion for the position and gauge invariant crystal momentum are considered for multiband wave packets of Bloch electrons. For a localized packet in a subset of bands well-separated from the rest of the band structure of the crystal, one can construct an effective electromagnetic Hamiltonian with respect to the center of the packet. We show that the equations of motion can be obtained via a projection operator procedure, which is derived from the adiabatic approximation within perturbation theory. These relations explicitly contain information from each band captured in the expansion coefficients and energy band structure of the Bloch states as well as non-Abelian features originating from interband Berry phase properties. This general and transparent Hamiltonian-based approach is applied to a wave packet spread over a single band, a set of degenerate bands, and two linear crossing bands. The generalized equations of motion hold promise for novel effects in transport currents and Hall effect phenomena.