Center Vortices and the Dirac Spectrum

TL;DR

This study investigates the relationship between center vortices and low-lying Dirac eigenmodes across different fermion formulations, revealing formulation-dependent presence of modes and their correlation with vortex structures, impacting understanding of chiral symmetry breaking.

Contribution

It clarifies the presence of low-lying Dirac modes in various formulations and their correlation with vortex structures, addressing previous discrepancies and extending existing results.

Findings

Low-lying modes are present in the asqtad formulation but not in overlap.

Strong correlation between vortex locations and Dirac eigenmode densities.

Lowest eigenmodes concentrate in point-like regions rather than extended submanifolds.

Abstract

We study correlations between center vortices and the low-lying eigenmodes of the Dirac operator, in both the overlap and asqtad formulations. In particular we address a puzzle raised some years ago by Gattnar et al. [Nucl. Phys. B 716, 105 (2005)], who noted that the low-lying Dirac eigenmodes required for chiral symmetry breaking do not appear to be present in center-projected configurations. We show that the low-lying modes are in fact present in the staggered (asqtad) formulation, but not in the overlap and ``chirally improved'' formulations, and suggest a reason for this difference. We also confirm and extend the results of Kovalenko et al. [Phys. Lett. B 648, 383 (2007)], showing that there is a correlation between center vortex locations, and the scalar density of low-lying Dirac eigenmodes derived from unprojected configurations. This correlation is strongest at points which are associated, in the vortex picture, with non-vanishing topological charge density, such as vortex intersection and ``writhing'' points. We present supporting evidence that the lowest Dirac eigenmodes, in both asqtad and overlap formulations, have their largest concentrations in point-like regions, rather than on submanifolds of higher dimensionality.