Supersymmetry and non-Abelian geometric phase for a free particle on a circle with point-like interactions

TL;DR

This paper explores how supersymmetric quantum mechanics can be used to study non-Abelian geometric phases, demonstrating a model where Berry's connection resembles a Wu-Yang magnetic monopole, revealing deep geometric structures.

Contribution

The paper introduces a simple supersymmetric model for a particle on a circle with point interactions, showing the emergence of non-Abelian geometric phases linked to magnetic monopoles.

Findings

Supersymmetry guarantees doubly degenerate energy levels.

Berry's connection corresponds to a Wu-Yang-like magnetic monopole.

Model provides a clear example of non-Abelian geometric phase in quantum mechanics.

Abstract

Though not so widely appreciated in the literature, supersymmetric quantum mechanics provides an ideal playground for studying non-Abelian geometric phase, because supersymmetry always guarantees degeneracies in energy levels. In this paper we first present a simple supersymmetric model for a free particle on a circle with point-like interactions that exhibits $\mathscr{N} = 2$ supersymmetry and doubly degenerate energy levels. We then show that Berry's connection in this model is given by the Wu-Yang-like magnetic monopole in SU(2) Yang-Mills gauge theory. This article is largely based on our recent work [arXiv:1406.4857].