General Theory of the Zitterbewegung

TL;DR

This paper provides a general formula for the time evolution of the position operator in multi-band Hamiltonians, revealing that Zitterbewegung is a universal multi-frequency oscillation linked to the system's momentum and Berry connection.

Contribution

It introduces a unified, simple expression for Zitterbewegung in multi-band systems, applicable to arbitrary matrix elements depending on momentum.

Findings

Zitterbewegung always appears in the position operator of such systems.

The oscillatory motion is generally multi-frequency.

The amplitude relates to the Berry connection matrix.

Abstract

We derive a general and simple expression for the time-dependence of the position operator of a multi-band Hamiltonian with arbitrary matrix elements depending only on the momentum of the quasi-particle. Our result shows that in such systems the Zitterbewegung like term related to a trembling motion of the quasi-particle, always appears in the position operator. Moreover, the Zitterbewegung is, in general, a multi-frequency oscillatory motion of the quasi-particle. We derive a few different expressions for the amplitude of the oscillatory motion including that related to the Berry connection matrix. We present several examples to demonstrate how general and versatile our result is.