Confinement-deconfinement crossover in the lattice $\mathbb{C}{P}^{N-1}$ model

The $\mathbb{C}{P}^{N-1}$ sigma model at finite temperature is studied using lattice Monte Carlo simulations on $S_s^1\times S_{\tau }^1$ with circumferences $L_s$ and $L_{\tau }$, respectively, where the ratio of the circumferences is taken to be sufficiently large ($L_s/L_{\tau }\gg 1$) to approximate the model on $\mathbb{R}\times S^1$. We show that the expectation value of the Polyakov loop undergoes a deconfinement crossover as $L_{\tau }$ is decreased, where the peak of the associated susceptibility gets sharper for larger $N$. We find that the global $\mathrm{PSU}(N)=\mathrm{SU}(N)/{\mathbb{Z}}_N$ symmetry remains unbroken in different manners for small and large $L_{\tau }$, respectively: in the small $L_{\tau }$ region for finite $N$, the order parameter fluctuates extensively with its expectation value consistent with zero after taking an ensemble average, while in the large $L_{\tau }$ region the order parameter remains small with little fluctuations. We also calculate the thermal entropy and find that the degrees of freedom in the small $L_{\tau }$ regime are consistent with $N-1$ free complex scalar fields, thereby indicating a good agreement with the prediction from the large-$N$ study for small $L_{\tau }$.

Figure 1

(Left) The absolute value of expectation value of the Polyakov loop $|⟨P⟩|$ as a function of $\beta$. (Right) The volume dependences of the maximal peak height of the Polyakov loop susceptibility for each $N$ by varying $N_s$ as $N_s=40$, 80, 120, 160, 200 with $N_{\tau }=8$. We fit the four data points with the large volume, $N_s=80–200$, by a function ${\chi }_{⟨|P|⟩,\max }=a+c{V}^p$.

Figure 2

The susceptibility of the expectation value of absolute value of Polyakov loop $⟨|P|⟩$ as a function of $L_c/L_{\tau }$ for $N=3$, 5, 10, 20 with $(N_s,N_{\tau })=(200,8)$.

Figure 3

$P^{ii}$ ($i=1$, 2, 3) with $N=3$ for $\beta =0.1$ (confinement) (left) and $\beta =3.9$ (deconfinement) (right) are shown. The horizontal axis stands for the label number of configurations and we pick up one per 5000 sweeps.

Figure 4

The variance of $P^{ij}$ for $N=3$, 5, 10, and 20 as a function of $\beta$ are shown.

Figure 5

Thermal entropy density for single $\varphi$ ($s/(NT)=N_{\tau }^2⟨T_{xx}-T_{\tau \tau }⟩/N$) for $N=3$, 5, 10, 20. The dotted line denotes the large-$N$ results in the small $L_{\tau }$ regime, $2\pi /3$.