Calorons and monopoles from smeared $SU(2)$ lattice fields at nonzero temperature

In equilibrium, at finite temperature below and above the deconfining phase transition, we have generated lattice $SU(2)$ gauge fields and have exposed them to smearing in order to investigate the emerging clusters of topological charge. Analyzing in addition the monopole clusters according to the maximally Abelian gauge, we have been able to characterize part of the topological clusters to correspond either to nonstatic calorons or static dyons in the context of Kraan-van Baal caloron solutions with nontrivial holonomy. We show that the relative abundance of these calorons and dyons is changing with temperature and offer an interpretation as dissociation of calorons into dyons with increasing temperature. The profile of the Polyakov loop inside the topological clusters and the (model-dependent) accumulated topological cluster charges support this interpretation. Above the deconfining phase transition light dyons (according to Kraan-van Baal caloron solutions with almost trivial holonomy) become the most abundant topological objects. They are presumably responsible for the magnetic confinement in the deconfined phase.

Figure 1

Left: two static dyons accompanied by two static monopoles; right: a single caloron accompanied by a monopole loop. The classical $SU(2)$ configurations have been generated on a ${16}^3\times 4$ lattice and then Abelian-projected. In the $(z,y,t)$ lattice at $x=0$ we show the sites with action density $s>s_{\max }/5$ (left, for dyons) and $s>s_{\max }/40$ (right, for calorons) together with the emerging monopole trajectories.

Figure 2

The effect of smearing on the “string tension” $\sigma$. In confinement ($\beta =2.23$, 2.3, 2.37) $\sigma$ was obtained from the correlator of Polyakov loops. In deconfinement ($\beta =2.5$, 2.6) $\sigma$ was obtained from the behavior of spacelike Wilson loops. The lattice size was ${20}^3\times 6$.

Figure 3

The distribution of the Polyakov loop over sites where timelike Abelian monopole currents are detected (thick line). Monopoles are obtained by Abelian projection in MAG. For comparison, the distribution of Polyakov loops over all sites is shown (thin line). The upper row of figures shows the results for $\beta =2.2$, 2.3, 2.4 (from left to right, confinement). The lower row of figures shows the results for $\beta =2.5$, 2.6 (from left to right, deconfinement).

Figure 4

The distribution of the “asymptotic” holonomy $H$ as defined in the text. The upper row of figures shows the results for $\beta =2.2$, 2.3, 2.4 (from left to right, confinement). The lower row of figures shows the results for $\beta =2.5$, 2.6 (from left to right, deconfinement).

Figure 5

Scatter plots in the ($Q_{\mathrm{cluster}}$, $⟨PL(\mathrm{\text{Abelian monopoles}}){⟩}_{\mathrm{cluster}}$) plane for $\beta =2.2$, 2.3, 2.4 (upper row from left to right, confinement) and for $\beta =2.5$, 2.6 (lower row from left to right, deconfinement). The circles represent dyon clusters, the triangles undissociated calorons.

Figure 6

Scatter plots in the ($Q_{\mathrm{cluster}}$, $⟨PL(\mathrm{\text{Abelian monopoles}}){⟩}_{\mathrm{cluster}}$) plane for $\beta =2.6$. The left figure is the sum of the other two. The figure in the center shows configurations (45 from 200 configurations) with topological charge $|Q|=1$ ($0.5\le |Q|\le 1.5$), the right figure the complementary 155 configurations with zero topological charge $Q=0$ ($|Q|\le 0.5$). The meaning of the symbols is the same as in Fig. 5.

Figure 7

Left: distributions of the radii $\overline{r}$ of dyonic clusters for $\beta =2.3$ (solid lines, confinement phase) and $\beta =2.6$ (dashed/dotted lines, deconfinement phase). The dashed line corresponds to heavy dyon clusters, whereas the dotted line refers to light dyons. Right: distributions of the scale-sizes $\rho$ of isolated caloron clusters for $\beta =2.2$ (solid line), $\beta =2.3$ (dashed line), and $\beta =2.4$ (dotted line). Both the values for $\overline{r}$ and $\rho$ are presented in units of the lattice spacing $a(\beta )$.