Preprint number ADP-01-46/T478

Low-lying eigenmodes of the Wilson-Dirac operator and correlations with topological objects

Daniel-Jens Kusterer, John Hedditch, Waseem Kamleh, Derek B. Leinweber and Anthony G. Williams
Special Research Center for the Subatomic Structure of Matter (CSSM) and Department of Physics and Mathematical Physics, University of Adelaide 5005, Australia

Keywords: Lattice, Eigenmodes, Wilson-Dirac operator, Topology, Instantons

Abstract:
The probability density of low-lying eigenvectors of the hermitian Wilson-Dirac operator $H(\kappa )=\gamma_5 D_{\mathrm{W}}(\kappa )$ are examined. Comparisons in position and size between eigenvectors, topological charge and action density are made. We do this for standard Monte-Carlo generated SU(3) background fields and for single instanton background fields. Both hot and 12-sweep-cooled SU(3) background fields are considered. An instanton model is fitted to eigenmodes and topological charge density and the sizes and positions of these are compared.

Fig. 1:

Fig. 1 caption:
(a) Action density of a single instanton configuration on a $16^4$ lattice.
(b) First eigenvector for the single instanton configuration for $\kappa = 0.19$. Note the very strong correlation with the instanton on the action density.
(c) First eigenvector for the single instanton configuration for $\kappa = 0.25$. Strong correlation between the eigenvector and the action density.
(d) Second eigenvector for the single instanton configuration for $\kappa = 0.25$. The localization has a prolate shape compared to the spherical instanton.
(e) Second eigenvector for the single instanton configuration for $\kappa = 0.19$. The localization has a wall like shape with a prolate correlation to the instanton.
(f) Third eigenvector for the single instanton configuration for $\kappa = 0.19$. The eigenmode has a wall like shape and is not correlated with the instanton.

Fig. 4:

Fig. 4 caption:
(a) Topological charge density on of a 12-sweep cooled $16^3\times 32$ configuration.
(b) Action density of the same configuration. There is clearly less structure than on the topological charge density to be seen.
(c) Eigenvector density of the same configuration with correlation to the object seen on the action density and the topological charge density at $\kappa = 0.14$.
(d) Eigenvector density of the same configuration at $\kappa = 0.16$ showing correlation to the same topological object.
(e) Eigenvector density of the same configuration at $\kappa = 0.18$ showing correlation to the same topological object. The localization is minimal for this value of $\kappa$.
(f) Eigenvector density of the same configuration at $\kappa = 0.19$ showing correlation to the same topological object. The localization is about to become bigger again.

More images are available here.