Monopole content of topological clusters: Have Kraan-van Baal calorons been found?

Using smearing of equilibrium lattice fields generated at finite temperature in the confined phase of $SU(2)$ lattice gauge theory, we have investigated the emerging topological objects (clusters of topological charge). Analyzing their monopole content according to the Polyakov gauge and the maximally Abelian gauge, we characterize part of them to correspond to nonstatic calorons or static dyons in the context of Kraan-van Baal caloron solutions with nontrivial holonomy. The behavior of the Polyakov loop inside these clusters and the (model-dependent) topological charges of these objects support this interpretation.

Figure 1

For confining configurations on a ${20}^3\times 6$ lattice at $\beta =2.3$ after 4D smearing, we show in (a) the distribution of the Polyakov loop in lattice points where timelike Abelian monopole currents are present after 100 smearing steps (solid histogram for the Polyakov gauge, dashed histogram for MAG) compared with the overall distribution of the Polyakov loop (dotted histogram) which is well described by the Haar measure (solid curve). All distributions are normalized. (b) The distributions of the Polyakov loop in points where timelike Abelian monopole currents are present in Polyakov gauge comparing 100 and 50 smearing steps (solid and dashed histograms correspondingly). (c) The distribution of “asymptotic holonomy” (as defined in the text) after 100 smearing steps.

Figure 2

(a) The distribution of the Polyakov loop averaged over the monopole skeleton of topological clusters with static monopoles (dashed histogram) and for clusters with monopole-antimonopole pairs (solid histogram) (both according to the Polyakov gauge after 100 smearing steps). (b) The same for the maximally Abelian gauge and 100 smearing steps. (c) The same for the Polyakov gauge after only 50 smearing steps.

Figure 3

(a) The scatter plot shows the concentration of topological clusters after 100 smearing steps in the $(q_{\max }{)}_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$-$⟨PL(\mathrm{\text{Abelian}}\mathrm{\text{monopoles}}){⟩}_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$ plane, depending on the monopole content in the Polyakov gauge: Clusters with static monopoles (circles) and clusters with monopole-antimonopole pairs (triangles) are forming different clusters in this plot. (b) The same for the maximally Abelian gauge. (c) The same for the Polyakov gauge after only 50 smearing steps.

Figure 4

Scatter plots as in Figs. 3a, 3b, 3c, respectively, but for the $Q_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$-$⟨PL(\mathrm{\text{Abelian}}\mathrm{\text{monopoles}}){⟩}_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$ plane.

Figure 5

(a) The distribution of the topological charge per cluster, $Q_{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$, of topological clusters after 100 smearing steps with static monopoles (dashed histogram) and for clusters with monopole-antimonopole pairs (solid histogram) in the Polyakov gauge. (b) The same for the maximally Abelian gauge. (c) The same for the Polyakov gauge after only 50 smearing steps.

Figure 6

Scatter plots as in Figs. 3, 4 but for $O(100)$ of artificially and randomly generated single caloron configurations. The monopole content is defined in the maximally Abelian gauge.