Confinement in non-Abelian lattice gauge theory via persistent homology

TL;DR

This paper applies persistent homology to analyze the topological structures in SU(2) lattice gauge theory, revealing insights into confinement, deconfinement, and the behavior of topological objects like instanton-dyons.

Contribution

It introduces a novel use of persistent homology to study gauge theory phases, providing a comprehensive, gauge-invariant topological analysis of lattice configurations.

Findings

Topological densities form spatial lumps indicating instanton-dyons.

Holonomy Lie algebra fields encode signatures of well-separated dyons.

Debye screening effects are observable in persistent homology analysis.

Abstract

We investigate the structure of confining and deconfining phases in SU(2) lattice gauge theory via persistent homology, which gives us access to the topology of a hierarchy of combinatorial objects constructed from given data. Specifically, we use filtrations by traced Polyakov loops, topological densities, holonomy Lie algebra fields, as well as electric and magnetic fields. This allows for a comprehensive picture of confinement. In particular, topological densities form spatial lumps which show signatures of the classical probability distribution of instanton-dyons. Signatures of well-separated dyons located at random positions are encoded in holonomy Lie algebra fields, following the semi-classical temperature dependence of the instanton appearance probability. Debye screening discriminating between electric and magnetic fields is visible in persistent homology and pronounced at large gauge coupling. All employed constructions are gauge-invariant without a priori assumptions on the configurations under study. This work showcases the versatility of persistent homology for statistical and quantum physics studies, barely explored to date.