Roman Höllwieser

Center vortices and the Dirac spectrum

Institute of Physics, Slovak Academy of Sciences, SK–845 11 Bratislava, Slovakia(Dated: June 20, 2008)We study correlations between center vortices and the low-lying eigenmodes of the Dirac operator, in boththe 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-lyingDirac eigenmodes required for chiral symmetrybreaking do not appear to be present in center-projected configurations. We show that the low-lying modes are infact 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 e xtend the results of Kovalenko et al. [Phys.Lett. B 648, 383 (2007)], showing that there is a correlation between center vortex locations, and the scalardensity of low-lying Dirac eigenmodes derived from unprojected configurations. This correlation is strongestat points which are associated, in the vortex picture, with non-vanishing topological charge density, such asvortex intersection and “writhing” points. We present supp orting evidence that the lowest Dirac eigenmodes,in both asqtad and overlap formulations, have their largest concentrations in point-like regions, rather than onsubmanifolds of higher dimensionality.I. INTRODUCTION

Lattice Index Theorem and Fractional Topological Charge

American Physical Society, One Research Road, Ridge, NY 11961, USA(Dated: October 12, 2010)We study topological properties of classical spherical center vortices with the low-lying eigenmodesof the Dirac operator in the fundamental and adjoint representations using both the overlap andasqtad staggered fermion formulations. In particular we address the puzzle raised in a previouswork of our group [Phys. Rev. D 77, 14515 (2008)], where we found a violation of the lattice indextheorem with the overlap Dirac operator in the fundamental representation even for “admissible”gauge fields. Here we confirm the discrepancy between the topological charge and the index of theDirac operator also for asqtad staggered fermions and the adjoint representation. Numerically, thediscrepancy equals the sum of the winding numbers of the spheres when they are regarded as mapsWe report on our studies of topological properties of classi cal spherical center vortices with the low-lying eigenmodes of the Dirac operator in the fundament al and adjoint representations using both the overlap and asqtad fermion formulations. We confirm the discrepancy between the topological charge and the index of the Dirac operator, which was already found in a previous work of our group [Phys. Rev. D 77, 14515 (2008)] for overlap fermi ons, also for staggered fermions and adjoint representations. Furthermore, the index theor em of the adjoint fermions gives some evidence for fractional topological charge. During coolin g the spherical center vortex on a 40 3×2 lattice we find an object with topological charge Q= 1/2 which we identify as a Dirac monopole without an antimonopole. For more details see [arXiv: hep-lat:1005.1015 ].

Intersections of thick center vortices, Dirac eigenmodes and fractional topological charge in SU(2) lattice gauge theory

Intersections of thick, plane SU(2) center vortices are characterized by the topological charge |Q| = 1/2. We compare such intersections with the distribution of zeromodes of the Dirac operator in the fundamental and adjoint representation using both the overlap and asqtad staggered fermion formulations in SU(2) lattice gauge theory. We analyze configurations with four intersections and find that the probability density distribution of fundamental zeromodes in the intersection plane differs significantly from the one obtained analytically in [1]. The Dirac eigenmodes are clearly sensitive to the traces of the Polyakov (Wilson) lines and do not exactly locate topological charge contributions. Although, the adjoint Dirac operator is able to produce zeromodes for configurations with topological charge |Q| = 1/2, they do not locate single vortex intersections, as we prove by forming arbitrary linear combinations of these zeromodes — their scalar density peaks at least at two intersection points. With pairs of thin and thick vortices we realize a situation similar to configurations with topological charge |Q| = 1/2. For such configurations the zeromodes do not localize in the regions of fractional topological charge contribution but spread over the whole lattice, avoiding regions of negative traces of Polyakov lines. This sensitivity to Polyakov lines we also confirm for single vortex-pairs, i.e., configurations with nontrivial Polyakov loops but without topological charge.

Tests of the lattice index theorem

We investigate the lattice index theorem and the localization of the zero modes for thick classical center vortices. For nonorientable spherical vortices, the index of the overlap Dirac operator differs from the topological charge although the traces of the plaquettes deviate only by a maximum of 1.5% from trivial plaquettes. This may be related to the fact that even in Landau gauge some links of these configuration are close to the nontrivial center elements.

Center vortices and chiral symmetry breaking in S U ( 2 ) lattice gauge theory

We investigate the chiral properties of near-zero modes for thick classical center vortices in

Colorful SU(2) center vortices in the continuum and on the lattice

SU(2)

Critical analysis of topological charge determination in the background of center vortices in SU(2) lattice gauge theory

lattice gauge theory as examples of the phenomena which may arise in a vortex vacuum. In particular we analyze the creation of near-zero modes from would-be zero modes of various topological charge contributions from center vortices. We show that classical colorful spherical vortex and instanton ensembles have almost identical Dirac spectra and the low-lying eigenmodes from spherical vortices show all characteristic properties for chiral symmetry breaking. We further show that also vortex intersections are able to give rise to a finite density of near-zero modes, leading to chiral symmetry breaking via the Banks-Casher formula. We discuss the mechanism by which center vortex fluxes contribute to chiral symmetry breaking.

Random center vortex lines in continuous 3D space-time

The spherical vortex as introduced in [G. Jordan et al., Phys. Rev. D 77, 014515 (2008)] is generalized. A continuum form of the spherical vortex is derived and investigated in detail. The discrepancy between the gluonic lattice topological charge and the index of the lattice Dirac operator described in previous papers is identified as a discretization effect. The importance of the investigations for Monte Carlo configurations is discussed.

Model of random center vortex lines in continuous

We analyze topological charge contributions from classical SU(2) center vortices with shapes of planes and spheres using different topological charge definitions, namely the center vortex picture of topological charge, a discrete version of F\~{F} in the plaquette and hypercube definitions and the lattice index theorem. For the latter the zeromodes of the Dirac operator in the fundamental and adjoint representations using both the overlap and asqtad staggered fermion formulations are investigated. We find several problems for the individual definitions and discuss the discrepancies between the different topological charge definitions. Our results show that the interpretation of topological charge in the background of center vortices is rather subtle.

2+1

We present a model of center vortices, represented by closed random lines in continuous 2+1-dimensional space-time. These random lines are modeled as being piece-wise linear and an ensemble is generated by Monte Carlo methods. The physical space in which the vortex lines are defined is a cuboid with periodic boundary conditions. Besides moving, growing and shrinking of the vortex configuration, also reconnections are allowed. Our ensemble therefore contains not a fixed, but a variable number of closed vortex lines. This is expected to be important for realizing the deconfining phase transition. Using the model, we study both vortex percolation and the potential V(R) between quark and anti-quark as a function of distance R at different vortex densities, vortex segment lengths, reconnection conditions and at different temperatures. We have found three deconfinement phase transitions, as a function of density, as a function of vortex segment length, and as a function of temperature. The model reproduces the ...