Bloch vector, disclination and exotic quantum holonomy

TL;DR

This paper introduces a topological framework for understanding eigenspace anholonomy in quantum systems, linking it to disclinations in Bloch vectors and exploring the mathematical structure behind exotic quantum holonomy.

Contribution

It presents a novel topological formulation of eigenspace anholonomy, connecting it to disclinations and homotopy classes, and extends the approach to more complex systems.

Findings

Eigenspace anholonomy is characterized by disclinations in Bloch vectors.

The topological structure of exotic quantum holonomy is elucidated via covering maps.

Extensions to nonadiabatic cycles and N-level systems are proposed.

Abstract

A topological formulation of the eigenspace anholonomy, where eigenspaces are interchanged by adiabatic cycles, is introduced. The anholonomy in two-level systems is identified with a disclination of the director (headless vector) of a Bloch vector, which characterizes eigenprojectors. The covering map structure behind the exotic quantum holonomy and the role of the homotopy classification of adiabatic cycles are elucidated. The extensions of this formulation to nonadiabatic cycles and N-level systems are outlined.