An Angular Dependent Supersymmetric Quantum Mechanics with a ${\bf Z}_2$-invariant Potential

TL;DR

This paper introduces a novel supersymmetric quantum mechanics model with an angular-dependent potential that breaks continuous symmetries to a discrete ${f Z}_2$ symmetry, revealing new self-dual equations and spontaneous symmetry breaking phenomena.

Contribution

It generalizes conformally invariant topological quantum mechanics by incorporating an angular-dependent potential, leading to new self-dual equations and symmetry breaking insights.

Findings

Presence of an order parameter and spontaneous symmetry breaking

Discovery of saddle points instead of maxima in the potential

Construction of the model in both path integral and operator frameworks

Abstract

We generalize the conformally invariant topological quantum mechanics of a particle propagating on a punctured plane by introducing a potential that breaks both the rotational and the conformal invariance down to a ${\bf Z}_2$ angular-dependent discrete symmetry. We derive a topological quantum mechanics whose localization gauge functions give interesting self-dual equations. The model contains an order parameter and exhibits a spontaneous symmetry breaking with two ground states above a critical scale. Unlike the ordinary $O(2)$-invariant Higgs potential, an angular-dependence is found and saddle points, instead of local maxima, appear, posing subtle questions about the existence of instantons. The supersymmetric quantum mechanical model is constructed in both the path integral and the operatorial frameworks.