Localized eigenmodes of covariant Laplacians in the Yang-Mills vacuum

TL;DR

This study investigates the localization properties of eigenmodes of the covariant Laplacian in the Yang-Mills vacuum, revealing representation-dependent localization behaviors and their relation to confinement mechanisms.

Contribution

It provides the first detailed numerical analysis of eigenmode localization in different gauge representations and their connection to vortex structures and confinement.

Findings

Localized modes are finite in volume and insensitive to lattice size.

Localization volume depends on gauge representation and scales differently in the continuum limit.

Vortex removal affects localization in the fundamental representation but not in the adjoint.

Abstract

As a probe of the Yang-Mills vacuum, we study numerically the eigenmode spectrum of the covariant lattice Laplacian operator. We find that the eigenmodes at the low and high ends of the spectrum are localized in finite regions whose volume is insensitive to the lattice volume. We also find that the vacuum is seen very differently by localized modes of the covariant Laplacian in different representations of the gauge group. In the fundamental representation, the data suggests that the localization volume is finite in physical units set by the string tension, and localization disappears when center vortices are removed. In the adjoint and j=3/2 representations the low and high-lying modes are far more localized, and the localization volume appears to scale to zero, in physical units, in the continuum limit. The adjoint Laplacian is insensitive to vortex removal, but we find that exponential localization is absent for adjoint eigenmodes in the Higgs phase of a gauge-Higgs theory. Localization is also absent in the spectrum of the Coulomb gauge Faddeev-Popov operator, as required in Coulomb gauge confinement scenarios.