Nonadiabatic factor accompanying magnetic translation of a charged particle

TL;DR

This paper analyzes the non-adiabatic factor in the quantum evolution of a charged particle under magnetic translation symmetry, explicitly constructing all factors and discussing its impact on transition probabilities.

Contribution

It provides an explicit construction of the adiabatic, geometric, and non-adiabatic factors for a system with magnetic translation symmetry, highlighting the role of the non-adiabatic factor.

Findings

Explicit construction of all three factors in the evolution operator

Identification of the non-adiabatic factor's role in transition probabilities

Analysis of the degenerate energy eigenstates related to magnetic translation

Abstract

The quantum adiabatic theorem incorporating the Berry phase phenomenon can be characterized as a factorization of the time evolution operator into a path-dependent geometric factor, a usual dynamical factor and a non-adiabatic factor that approaches the identity in the adiabatic limit. We study a case where all these three factors can be constructed explicitly and where the instantaneous Hamiltonian has infinitely degenerate energy eigenstates associated with magnetic translation symmetry. Significance of the non-adiabatic factor in terms of transition probabilities is discussed.