# Dirac Eigenmodes and Center Vortices

The aim of the project is to analyse relations between certain properties of center vortices and Dirac eigenmodes. In the preceding project, Mechanism of chiral symmetry breaking, we found good correlations between low-lying eigenmodes and center vortices, now we want to study their interplay in more detail. Therefore we systematically analyse the following aspects:

**Topology and staggered fermions**Staggered fermions don't have exact zeromodes, but a separation between ``would be'' zeromodes and non-chiral modes is observed for improved staggered quark actions [Wong et al.]. The chirality (pseudo-scalar density) for staggered fermions is defined by <ψϒ5ψ>, where ϒ5 corresponds to a displacement along the diagonal of a hypercube. To ensure gauge invariance the product has to include gauge field multiplications along all shortest paths connecting opposite corners of the hypercube. The lattice index theorem shall be tested with asqtad staggered fermions for plane and spherical vortices.

**Index theorem and adjoint fermions**

The correlation of the number of zero modes and topological charge on the lattice was actually discussed at recent lattice conferences with respect to adjoint fermions. Adjoint modes can be used to trace the topological structures present in the vacuum, with less sensitivity to the ultraviolet fluctuations of the field [Gonzales et al.]. The index of the massless Dirac operator in the adjoint representation of the SU(N) gauge group in a background field of topological charge Q is equal to 2NQ. The lattice index theorem shall be tested for overlap and asqtad staggered fermions in the adjoint color representation for our artificial vortex configurations. Since the fermion is in the real representation, the spectrum of the adjoint Dirac operator is doubly degenerate. Therefore, the index can only be even valued. It will be interesting whether we find all possible even values or only multiples of four, resulting in a gauge field background made up of classical instantons.

**Adjoint zero modes and fractional topological charge**

Classical instantons carry an integer topological charge, thus, in case of

SU(2) and of a quark in a fundamental representation there is exactly one

zero mode for an instanton. Now, if the actual consituents of the QCD

vacuum have topological charge Q=1/2 of that of instantons, no

zero mode is produced. However, if one changes to adjoint fermions, then

the topological charge 1/2 is able to create a zero mode. Edwards et al. presented some evidence for fractional topological charge on the lattice, whereas Garcìa Pérez et al. associate this to lattice artefacts, i.e. instantons of size of the order of the lattice spacing. Preliminary measurements with adjoint fermions on our spherical vortices also showed fractional topological charge. These methods should also be applied to Monte Carlo configurations and possible evidences of fractional topological charge have to be analysed due to its vortex structure.

**Cooling of vortex configurations**

The artificial configurations have to be analysed in more detail, in order to find the responsible constituents of the QCD vacuum. During cooling we already found some evidence for fractional topological charge for a spherical vortex on a 40^{3}x2 lattice. This issue has to be analysed with respect to adjoint zero modes, which seem to be able to measure fractional topological charge. Nevertheless there we face the problem of identification of the vortex structure after cooling. For very smooth (cooled) gauge fields, the gauge fixing procedure breaks down and the center projection fails to identify correct vortex positions.

Another ansatz would be to use different coupling constants in time and space direction during cooling, in order to decide whether the problem is a finite size or finite temperature effect.

**Overlap vs. staggered Dirac operator**

There is a big discussion about staggered fermions going on in the lattice community (see e.g. [Creutz]). The rooting procedure and flavour mixing are in fact not rigorously proved to be valid yet and this debate will not be settled within this project. Nevertheless, we want to compare different fermion representations with respect to topological aspects.

**Eigenmode peaks and vortex structure**

In the preceding project, Mechanism of chiral symmetry breaking, we presented strong correlations between vortex surface points and Dirac eigenmode densities, which peak in point-like regions. By looking inside the vortex structures at eigenmode density peaks, we can get an idea of the essential components of the gauge field for Dirac eigenmodes to peak at these positions. In particular we want to identify these structures which of course produce topological charge on the one hand, but also seem to be relevant for chiral symmetry breaking since the low-lying eigenmodes are present at these regions. In particular we think about vortex sheets that seem to disturb or reflect Dirac fermions in a way that chiral symmetry is broken. We are trying to probe these ideas by looking at "chiral currents". In fact, fermions bouncing back of some barrier without changing their spin are able to explain the mechanism of chiral symmetry breaking. This analysis may also reveal some relevant differences between the fermion representations.

**Dirac eigenmode correlations**

Another way is to directly correlate the different fermion representations, by means of their localisation properties. Low-lying eigenmodes show one main peak containing more than eighty percent of the eigenmode density, for overlap fermions we get one peak per chirality (except for zero modes). It is natural to identify eigenmodes which peak at the same positions with each other. It is interesting to compare the eigenvalues of these modes, whether they show up at approximately equal energies. For comparison, also Wilson fermions shall be included in this analysis.

**Fermions in the adjoint representation**

Finally, also adjoint overlap, Wilson and staggered fermions shall be considered with respect to center vortices. Adjoint fermions do not feel the center, but it would be interesting if there were some dramatic change in the behavior of the low-lying Dirac modes for the full configurations. We plan to measure the spectra, localisation and correlation properties of eigenmodes in the adjoint color representation.

### Contributors

### Collaborators

- Jeff Greensite, Physics and Astronomy Dept., San Francisco State University
- Urs M. Heller, American Physical Society
- Ŝtefan Olejník, Institute of Physics, Slovac Academy of Sciences

### Publications

Center vortices and topological charge

Roman Höllwieser, Thomas Schweigler, Manfried Faber (Vienna, Tech. U., Atominst. & Vienna, Tech. U.), Urs M. Heller (APS, New York). 2012. 8 pp.

Published in **PoS ConfinementX (2012) 078**

Conference: C12-10-08.1 Proceedings

Center Vortices and Chiral Symmetry Breaking in SU(2) Lattice Gauge Theory

Roman Höllwieser, Thomas Schweigler, Manfried Faber, Urs M. Heller. Apr 2, 2013. 19 pp.

e-Print: **arXiv:1304.1277 [hep-lat] | PDF**

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

Thomas Schweigler, Roman Hollwieser, Manfried Faber (TU Vienna), Urs M. Heller (APS, New York). Dec 2012. 8 pp.

DOI: 10.1103/PhysRevD.87.054504

e-Print: **arXiv:1212.3737 [hep-lat] | PDF**

**Critical analysis of topological charge determination in the background of center vortices in SU(2) lattice gauge theory.** Roman Hollwieser, Manfried Faber, Urs M. Heller. Feb 2012. 7 pp. e-Print: arXiv:1202.0929 [hep-lat] PDF

**Vortex Intersections, Dirac Eigenmodes and Fractional Topological Charge in SU(2) Lattice Gauge Theory.**

Roman Hollwieser, Manfried Faber, Urs M. Heller. 2011. 7pp.

Prepared for 29th International Symposium on Lattice Field Theory (Lattice 2011), Squaw Valley, California, 10-16 July 2011.

Published in **PoS LATTICE2011:269,2011**. Proceedings of Science Server

**Intersections of thick Center Vortices, Dirac Eigenmodes and Fractional Topological Charge in SU(2) Lattice Gauge Theory.**

Roman Hollwieser, Manfried Faber, Urs M. Heller, . Mar 2011. 18pp.

Published in **JHEP 1106:052,2011**.

e-Print: **arXiv:1103.2669** [hep-lat] Abstract, PDF

**Spherical vortices, fractional topological charge and lattice index theorem in SU(2) LGT.**

Roman Hollwieser, Manfried Faber, Urs M. Heller. 2010. 7pp.

Prepared for 28th International Symposium on Lattice Field Theory (Lattice 2010), Villasimius, Sardinia, Italy, 14-19 Jun 2010.

Published in **PoS LATTICE2010:276,2010**. Proceedings of Science Server

**Violations of the Lattice Index Theorem for spherical center vortices.**

Roman Hollwieser, Manfried Faber, Urs M. Heller. 2011. 3pp. Published in **AIP Conf.Proc. 1343 (2011) 227-229** Prepared for Conference: C10-08-30.1, Journal Server - AIP Conf.Proc.; AIP Conference Server

Lattice Index Theorem and Fractional Topological Charge.

Roman Hollwieser, Manfried Faber, Urs M. Heller. May 2010. 9pp.

e-Print: **arXiv:1005.1015** [hep-lat] Abstract, PDF