Probing center vortices and deconfinement in $\mathrm{SU}(2)$ lattice gauge theory with persistent homology

TL;DR

This paper introduces a topological data analysis method using persistent homology to detect center vortices in SU(2) lattice gauge theory, providing a gauge-invariant phase indicator for deconfinement transition with accurate critical parameter estimates.

Contribution

It develops a novel, gauge-invariant approach using persistent homology to identify vortices and define phase indicators in lattice gauge theory, improving understanding of deconfinement.

Findings

Persistent homology detects explicit vortices in the deconfined phase.

New phase indicator accurately estimates critical parameters.

Finite-size scaling confirms the method's effectiveness.

Abstract

We investigate the use of persistent homology, a tool from topological data analysis, as a means to detect and quantitatively describe center vortices in $\mathrm{SU}(2)$ lattice gauge theory in a gauge-invariant manner. We provide evidence for the sensitivity of our method to vortices by detecting a vortex explicitly inserted using twisted boundary conditions in the deconfined phase. This inspires the definition of a new phase indicator for the deconfinement phase transition. We also construct a phase indicator without reference to twisted boundary conditions using a simple $k$-nearest neighbours classifier. Finite-size scaling analyses of both persistence-based indicators yield accurate estimates of the critical $\beta$ and critical exponent of correlation length $\nu$ of the deconfinement phase transition.