Localization of topological charge density near $T_c$ in quenched QCD with Wilson flow

We smear quenched lattice QCD ensembles with lattice volume $32^3\times 8$ by using Wilson flow. Six ensembles at temperature near the critical temperature $T_c$ corresponding to the critical inverse coupling ${\beta }_c=6.06173(49)$ are used to investigate the localization of topological charge density. If the effective smearing radius of Wilson flow is large enough, the density, size and peak of Harrington-Shepard (HS) caloron-like topological lumps of ensembles are stable when $\beta \le 6.050$, but start to change significantly when $\beta \ge 6.055$. The inverse participation ratio (IPR) of topological charge density shows similar results, it begins to increase when $\beta \ge 6.055$ and is stable when $\beta \le 6.050$. The pseudoscalar glueball mass is extracted from the topological charge density correlator (TCDC) of ensembles at $T=1.19T_c$, and $1.36T_c$, the masses are 1.915(98) and 1.829(123) GeV respectively, they are consistent with results from conventional methods.

Figure 1

The topological charges $Q$ of ten configurations versus Wilson flow, $\lambda$ runs from 0.3 to 0.9 fm. $Q$ of original configurations have also been presented. The topological charges approach to integers and do not drop down to the value zero during the Wilson flow.

Figure 2

The parameters setting: ${\epsilon }_R=0.5$, ${\epsilon }_S=0.4$, left panels: $\epsilon =1.0$, right panels: $\epsilon =0.7$. From top to bottom: the average density $⟨N⟩$, the average size $⟨\rho ⟩$, and the average absolute value $⟨q_c(x)⟩$.

Figure 3

$⟨\mathrm{IPR}⟩$ of topological charge density with Wilson flow versus the inverse coupling $\beta$, the effective smearing radius $\lambda$ of Wilson flow runs from 0.3 to 0.9 fm.

Figure 4

The pseudoscalar glueball mass M with Wilson flow and fixed ending point, the horizontal axis $r/a$ is the starting point of the preliminary fitting range. Left: $T=1.19T_c$, right: $T=1.36T_c$.