Topology across the finite temperature transition studied by overimproved cooling in gluodynamics and QCD

Gluodynamics and two-flavor QCD at nonzero temperature are studied with the so-called overimproved cooling technique under which caloron solutions may remain stable. We consider topological configurations either at the first occuring stable plateau of topological charge or at the first (anti-) self-dual plateau and find the corresponding topological susceptibility at various temperatures on both sides of the thermal transition or crossover. In pure gluodynamics the topological susceptibility drops sharply at the deconfinement temperature, while in full QCD it decreases smoothly at temperatures above the pseudocritical one. The results are close to those calculated by other methods. We interpret our findings in terms of the (in)stability of calorons with nontrivial holonomy and their dyon constituents against overimproved cooling.

Figure 1

Cooling histories of $|Q|$ corresponding to $\epsilon =1$ (Wilson action, red dotted line), $\epsilon =0$ (slightly over-improved action, blue dashed line) and $\epsilon =-1$ (over-improved action as used in the rest of this paper, black solid line). Left: different cooling histories for one configuration from the confined phase. Right: different cooling histories for a (conditionally stable) caloron in a configuration taken from the transition region.

Figure 2

Typical cooling histories of configurations from the confined phase (left). The cooling history of a (finally unstable) caloron in a configuration taken from the transition region (right). The red dashed lines represent $|Q|$, the blue solid lines represent $S$.

Figure 3

The distributions of topological charges (left) and the Monte Carlo time histories along the Monte Carlo chain (right) of pure gluodynamics are shown, for the confined phase in the upper part and for the deconfined phase in the lower part. Confinement is represented by a sample at $T=0.883T_c$, deconfinement is illustrated by a sample at $T=1.117T_c$, in both cases with an actual statistics of 1000 $L_t=4$, $L_s=16$ lattice configurations (only partly shown in the history).

Figure 4

Left: topological susceptibility for gluodynamics on the lattices ${16}^3*4$ (red circles), ${24}^3*4$ (blue up triangles) and ${24}^3*6$ (green down triangles). Right: susceptibility for full QCD on the lattices ${16}^3*8$ (red up triangles) and ${24}^3*8$ (blue down triangles).

Figure 5

Left: comparison of the topological susceptibility for gluodynamics obtained in Ref. 46 (blue circles) with the topological susceptibility calculated in the present paper (red up triangles). Right: comparison of topological susceptibility for full QCD from Ref. 47 (blue circles) with the topological susceptibility calculated in the present paper (red triangles).

Figure 6

Comparison of the topological susceptibility for gluodynamics from Ref. 45 (blue up triangles) and from Ref. 56 (black stars) with the topological susceptibility calculated in the present paper on the lattices ${24}^3*6$ (green down triangles) and ${16}^3*4$ (red circles).

Figure 7

For gluodynamics, scatter plots of holonomy (spatially averaged Polyakov loop) for thermalized configurations and after cooling. Left panels: for the confined phase ($T/T_c=0.97$) and at the corresponding stable, nontrivial (anti-)self-duality plateaus; right panels: for the deconfined phase ($T/T_c=1.09$) and correspondingly after 1500 cooling sweeps at (almost) vanishing action values.

Figure 8

For full QCD, scatter plots of holonomy (spatially averaged Polyakov loop) for thermalized configurations and after cooling at stable, nontrivial (anti-)self-duality plateaus. Left panels: for the confined phase ($T/T_c=0.96$); right panels: for the deconfined phase ($T/T_c=1.05$).

Figure 9

The regular triangle with inner point $O$ and three perpendiculars to triangle sides. The sum of them $O{A}_1+O{A}_2+O{A}_3\equiv m_1+m_2+m_3=1$ is constant for all inner points.