Calorons and dyons at the thermal phase transition analyzed by overlap fermions

In a pilot study, we use the topological charge density defined by the eigenmodes of the overlap Dirac operator (with ultraviolet filtering by mode truncation) to search for lumps of topological charge in $SU(2)$ pure gauge theory. Augmenting this search with periodic and antiperiodic temporal boundary conditions for the overlap fermions, we demonstrate that the lumps can be classified either as calorons or as separate caloron constituents (dyons). Inside the topological charge clusters, the (smeared) Polyakov loop is found to show the typical profile characteristic for calorons and dyons. This investigation, motivated by recent caloron/dyon model studies, is performed at the deconfinement phase transition for $SU(2)$ gluodynamics on ${20}^3\times 6$ lattices described by the tadpole-improved Lüscher-Weisz (LW) action. The transition point has been carefully located. As a necessary condition for the caloron/dyon detection capability, we check that the LW action, in contrast to the Wilson action, generates lattice ensembles, for which the overlap Dirac eigenvalue spectrum smoothly behaves under smearing and under the change of the boundary conditions.

Figure 1

The eigenvalue spectra of the Dirac overlap operator with (a) periodic and (b) antiperiodic temporal boundary conditions for equilibrium ($N_{\mathrm{APE}}=0$) and smeared ($N_{\mathrm{APE}}=5$ and 10) pure $SU(2)$ gauge configurations on a ${20}^3\times 6$ lattice, generated with the standard Wilson action at $\beta =2.40$.

Figure 2

The eigenvalue spectra of the Dirac overlap operator with (a) periodic and (b) antiperiodic temporal boundary conditions for equilibrium ($N_{\mathrm{APE}}=0$) and smeared ($N_{\mathrm{APE}}=5$ and 10) pure $SU(2)$ gauge configurations on a ${20}^3\times 6$ lattice, generated with the tadpole-improved Lüscher-Weisz action at ${\beta }_{\mathrm{imp}}=3.20$.

Figure 3

(a) The ensemble average of the modulus $|L|$ of the average Polyakov loop $L=\frac{1}{V}\sum _{\overrightarrow{x}}\mathrm{Tr}\mathcal{P}(\overrightarrow{x})$ as a function of $\beta$ for the tadpole-improved Lüscher-Weisz action on a ${20}^3\times 6$ lattice. (b) The susceptibility of $|L|$ as function of $\beta$.

Figure 4

(a) The gluonic topological charge density $q_{\mathrm{gluon}}(x)$ (left) and the Polyakov loop $p(\overrightarrow{x})$ (right) for a classical $Q=2$ configuration generated at maximally nontrivial holonomy [asymptotically $p(\overrightarrow{x})=0$] on a ${16}^3\times 4$ lattice [with four dyons maximally separated in the $(x,z)$-plane]. (b) The fermionic topological charge density $q^{(p/a)}(x)$ reconstructed out of the 20 lowest eigenmodes of the overlap Dirac operator with periodic (right) and antiperiodic (left) temporal boundary conditions.

Figure 5

The maxima of clusters of the fermionic $|q(x)|$ seen under periodic boundary condition (filled circles) and under antiperiodic boundary condition (filled triangles) for two configurations (a) and (b) in the sample, shown in the $(q_{\max \mathrm{cluster}},P_{\mathrm{extr}\mathrm{cluster}})$ plane (the precise meaning is explained in the text). Peaks at opposite sign of $P_{\mathrm{extr}\mathrm{cluster}}$, that are connected by dashed lines, have appeared under different boundary conditions at the same space-time position (“not jumping”) and are interpreted as calorons. Isolated peaks have appeared only once under the respective boundary condition at the given position (jumping) and are interpreted as dyons. The marked objects D and CAL in (b) are portrayed in detail in Fig. 6.

Figure 6

The fermionic topological charge density $q^{(p/a)}(x)$ (left) and the Polyakov loop $p(\overrightarrow{x})$ (right): (a) for a typical caloron cluster [when $q^{(p)}(x)\approx q^{(a)}(x)$] and (b) for a typical dyon cluster [which was visible only in $q^{(p)}(x)$] from Fig. 5b. The topological density and the Polyakov loop are represented as a function over part of the $(x,y)$-plane. Please notice the different scales for the topological charge density and for the Polyakov loop. The Polyakov loop is measured after $N_{\mathrm{APE}}=10$ smearing steps.

Figure 7

The histogram of dyon-dyon distances in lattice units. The first two bins correspond to calorons which are unambiguously paired within distances $d<2a$. The rest of the histogram with $d\ge 2a$ refers to the remaining lumps grouped in suitable dyon-dyon pairs according to the closest distance.

Figure 8

The topological clusters of the whole sample shown analogously to Fig. 5. The unpaired dyons are placed at their original $(q_{\max \mathrm{cluster}},P_{\mathrm{extr}\mathrm{cluster}})$ positions. The dyon pairs identified as calorons are finally relocated according to the average ${\overline{P}}_{\mathrm{cluster}}$ over their original, opposite sign values $P_{\max }$ and $P_{\min }$.