Interacting ensemble of the instanton-dyons and the deconfinement phase transition in the SU(2) gauge theory

Instanton-dyons, also known as instanton-monopoles or instanton-quarks, are topological constituents of the instantons at nonzero temperature and holonomy. We perform numerical simulations of the ensemble of interacting dyons for the SU(2) pure gauge theory, using standard Metropolis Monte Carlo and integration over parameter methods. We calculate the free energy as a function of the holonomy (logarithm of the Polyakov line), the dyon densities, and the Debye mass, and find its minima as a function of those parameters. We show that the backreaction on the holonomy potential does generate confinement, provided the density is sufficiently high (or the temperature sufficiently low). We then report various properties of the self-consistent ensembles as a function of temperature.

Figure 1

Free energy density $f$ as function of holonomy $\nu$, for $\mathrm{\Lambda }=0.5$ and ${\tilde{V}}_0=0.3$, for upper and ${\tilde{V}}_0=0.6$ for the lower plot. Three curves correspond to $g=4$, 3.5, 3, bottom to top, in the upper figure and $g=4$, 3.5, 3.25 in the bottom one. It is seen how the maximum as a function of $g$ goes further and further toward the confining value of $1/2$ as g goes up, and $S$ and $T$ go down.

Figure 2

Polyakov loop $P$ as a function of action parameter $S$ for $\mathrm{\Lambda }=0.5$ and ${\tilde{V}}_0=0.3$.

Figure 3

Densities $n_i$ of $i=M$ or $i=L$ dyons as a function of the action parameter $S$, for $\mathrm{\Lambda }=0.5$ and ${\tilde{V}}_0=0.3$. Note that the two densities are different at $S>6$.

Figure 4

Free energy density $f$ as a function of density $n$ for $N_{\text{Total}}=64$ (filled blue circle) and $N_{\text{Total}}=128$ (yellow square) at $\nu =0.5$, $M_D=2$, $S=6$ and $n=n_M=n_L$. Volume effects are seen to not be important in the region of interest around $n=0.3$ and difference is expected to come from the sharper shape of $⟨S(\lambda )⟩$ in the case of $N_{\text{Total}}=128$, which require a higher amount of points to obtain the free energy F.

Figure 5

Free energy density $f$ as a function of $\nu$ at $S=6$, $M_D=2$ and $N_M=N_L=16$. The different curves corresponds to different densities. Filled circle $n=0.53$, square $n=0.37$, diagonal $n=0.27$, upward triangle $n=0.20$, downward triangle $n=0.15$, open circle $n=0.12$. Not all densities are shown.

Figure 6

Self-consistent value of the holonomy $\nu$ (upper plot) and Polyakov line (lower plot) as a function of action $S$ (lower scales), which is related to $T/T_c$ (upper scales). The error bars are estimates based on the fluctuations of the numerical data.

Figure 7

Self-consistent value of the Debye Mass $M_D$ as a function of action $S$ (lower scale) which is related to $T/T_c$ (upper scale). The error bars are estimates based on the fluctuations of the numerical data. Points represent lattice data from [21] as a function of $T/T_c$.

Figure 8

Density $n$ (of an individual kind of dyons) as a function of action $S$ (lower scale) which is related to $T/T_c$ (upper scale) for M dyons(higher line) and L dyons (lower line). The error bars are estimates based on the density of points and the fluctuations of the numerical data.

Figure 9

Self-consistent value of the Free energy density $f$ as a function of action $S$ (lower scale) which is related to $T/T_c$ (upper scale).

Figure 10

(Not-self-consistent in holonomy $\nu$) free energy density $f$, here shown as a function of the value of the holonomy (in form of the Polyakov loop $P$) at $S=6$, 7, 9. The lower the action the lower the minimum of the free energy.

Figure 11

Correlation function $C_{ij}$ for MM and ML for $S=6$ (upper) and $S=9$ (lower). In the $MM$ case the correlation function vanishes at small distances due to the core.

Figure 12

Action $S_3$ of a single dyon as a function of Debye mass over holonomy $M_D/v$ in the potential described by Eq. (26) normalized by the action $S_0$ for $M_D=0$.

Figure 13

Higgs field $A_4^3$ of a single dyon, along the z-axis, through the center of the dyon for different Debye masses. From top to bottom: $M_D/v=0$, $M_D/v=0.45$ and $M_D/v=1.41$.