Effective model for pure Yang-Mills theory on ${\mathbb{T}}^2\times {\mathbb{R}}^2$ with Polyakov loops

We investigate the phase diagram and thermodynamics of $SU(N)$ pure Yang-Mills theory on a manifold ${\mathbb{T}}^2\times {\mathbb{R}}^2$ with an effective model that includes two Polyakov loops along two compactified directions. We find that a rich phase structure can appear owing to the spontaneous breaking of two center symmetries for $N=2$ and 3. Thermodynamic quantities are obtained in the model and compared with recent lattice results. It is shown that two Polyakov loops play significant roles in thermodynamics on ${\mathbb{T}}^2\times {\mathbb{R}}^2$.

Figure 1

$L_{\tau }$ dependence of $P_{\tau }$ (blue) and $P_x$ (red) at $L_x\to \infty$ for $N=2$. There exists a second-order phase transition at $L_{\tau }{T}_c^{\infty }=1$ with the critical temperature $T_c^{\infty }\approx 1/(0.874R)$. The spatial Polyakov loop $P_x$ is always zero.

Figure 2

$L_{\tau }$ dependences of $P_{\tau }$ (top panel) and $P_x$ (bottom panel) at several $L_x$ for $N=2$.

Figure 3

Polyakov loops $P(\equiv P_{\tau }=P_x)$ for the symmetric case $L=L_{\tau }=L_x$ for $N=2$ in the vicinity of the transition point. The second-order transition is found at $L{T}_c^{\infty }\approx 0.871$. The vertical line corresponds to the analytic solution for the second-order transition evaluated in Eq. (28).

Figure 4

Phase diagram on the $L_{\tau }–L_x$ plane for $N=2$. The blue (red) line separates $P_{\tau }=0$ and $P_{\tau }\ne 0$ ($P_x=0$ and $P_x\ne 0$). The dashed gray line stands for $L_{\tau }=L_x$, and the black dot is the transition point in Fig. 3. All phase transitions are of second order. The dotted line shows the asymptotic formula (c10).

Figure 5

$L_{\tau }$ dependence of $P_{\tau }$ (blue) and $P_x$ (red) at $L_x\to \infty$ for $N=3$. The temporal Polyakov loop $P_{\tau }$ shows a first-order phase transition at $T_c^{\infty }\approx 1/(0.733R)$.

Figure 6

$L_{\tau }$ dependences of $P_{\tau }$ (top panel) and $P_x$ (bottom panel) at several $L_x$ for $N=3$.

Figure 7

$L(\equiv L_{\tau }=L_x)$ dependence of the Polyakov loop $P(\equiv P_{\tau }=P_x)$ for $N=3$ in the vicinity of the phase transition point. The first-order transition occurs at $L{T}_c^{\infty }\approx 0.991$.

Figure 8

Phase diagram on the $L_{\tau }–L_x$ plane for $N=3$. The solid line shows the first-order phase transition.

Figure 9

Comparison of energy density $\epsilon$ and pressures $p_x$ and $p_z$ between our model (solid lines) and the lattice data in Ref. [26] with $N_t=16$ (circles) and 12 (squares). The dashed lines show the results with fixed values of $P_x$ and $P_t$ given in the $L_x\to \infty$ limit. The dotted lines are the results in the massless free theory and their limiting values for $L_{x}T\to \infty$ are shown by the arrows.

Figure 10

Pressure ratio $p_x/p_z$. The meanings of the symbols are the same as Fig. 9.