A visualization of the SU(2) vacuum on the lattice

Abstract

Configurations of pure SU(2) gauge field theory on the lattice are transformed to Landau gauge. After Fourier transformation, large momentum amplitudes are suppressed by a variable amount, and the configurations are transformed back to x-space. Spectacular peaks in electric and magnetic field strengths are found, which share many properties with either almost pointlike instantons in regular gauge or with extended anti-instantons in singular gauge. Environments around those peaks are visualized with respect to the action, to the topological charge density, to the gauge fields and to electric and magnetic field strengths. The density of the peaks is of order 1fm −4 , and it scales according to the string tension under a variation of the coupling constant. The evolution of the peaks under the amount of Fourier filtering is visually compared to the evolution under cooling, and the gauge dependence of the peaks is discussed. Advantages and shortcomings of this method are discussed, with emphasis on possible strong distortions of the action and the topological charge density, which become gauge dependent. Finally, I compare the character of the SU(2)-configurations to those of non-compact abelian theory.

Laplacian modes probing gauge fields

“…The structure of the action density of the filtered configurations includes narrow peaks, for which we do not have a final interpretation yet. They might be close to the gauge singularities found in the Fourier-filtered Landau gauge fields [19]. Despite the appearance of the peaks we will point out a similarity to smearing/cooling in an early stage.…”

“…The Polyakov loop as the deconfinement order parameter has average 0 and therefore we choose ζ = 1/4 in the construction of the filter, Eq. (19). Indeed, the filtered Polyakov loop trP/2 averaged over the lattice has an expectation value compatible with 0 (with 50 configurations giving standard devitations from 0.015 for N = 2 to 0.083 for N = 100 whereas the original standard deviation is 0.025).…”

“…We decided not to include the highest modes and to stick to Eq. (19). Nevertheless, we have checked the consequences of such a modification for the results presented below.…”

“…(19). Obviously this determinant is given in terms of eigenmodes (and eigenvalues) of the Laplacian, too, thus it also contains information about the gauge background.…”

“…First we want to study a technical detail, namely how big the determinant is that appears in (19) to scale up the sum to an SU (2)-valued link. In other words, how much of the link is captured by the superposition of a finite number of modes (in a naive sense without inspecting any structure in the links).…”