Adiabatic Perturbation Theory and Geometric Phases for Degenerate Systems

TL;DR

This paper develops an adiabatic perturbation theory for quantum systems with degenerate spectra, providing conditions for adiabatic theorem validity and calculating non-adiabatic corrections to geometric phases, with practical implications for fractional statistics.

Contribution

It introduces a formalism for degenerate quantum systems that links adiabatic conditions and non-Abelian geometric phases, including non-adiabatic corrections.

Findings

Established rigorous conditions for adiabatic theorem in degenerate systems

Derived non-adiabatic corrections to Wilczek-Zee geometric phase

Validated the formalism through exact solutions and perturbative comparison

Abstract

We introduce an adiabatic perturbation theory for quantum systems with degenerate energy spectra. This perturbative series enables one to rigorously establish conditions for the validity of the adiabatic theorem of quantum mechanics for degenerate systems. The same formalism can be used to find non-adiabatic corrections to the non-Abelian Wilczek-Zee geometric phase. These corrections are relevant to assess the validity of the practical implementation of the concept of fractional exchange statistics. We illustrate the formalism by exactly solving a time-dependent problem and comparing its solution to the perturbative one.