A Unified Theory of Quantum Holonomies

TL;DR

This paper develops a unified theoretical framework for understanding both phase and eigenspace quantum holonomies, including exotic cases with spectral degeneracy, using a novel gauge connection approach.

Contribution

It introduces a gauge connection formalism that treats phase and eigenspace holonomies equally, extending the understanding of quantum holonomies beyond traditional methods.

Findings

Unified description of phase and eigenspace holonomies

Development of a gauge invariant formalism for eigenspace holonomy

Examples demonstrating adiabatic quantum holonomy, including exotic cases with spectral degeneracy

Abstract

A periodic change of slow environmental parameters of a quantum system induces quantum holonomy. The phase holonomy is a well-known example. Another is a more exotic kind that exhibits eigenvalue and eigenspace holonomies. We introduce a theoretical formulation that describes the phase and eigenspace holonomies on an equal footing. The building block of the theory is a gauge connection for an ordered basis, which is conceptually distinct from Mead-Truhlar-Berry's connection and its Wilczek-Zee extension. A gauge invariant treatment of eigenspace holonomy based on Fujikawa's formalism is developed. Example of adiabatic quantum holonomy, including the exotic kind with spectral degeneracy, are shown.