Confinement-Deconfinement Crossover in the Lattice $\mathbb{C}{P}^{N-1}$ Model

Abstract

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 radii $L_s$ and $L_{\tau }$, respectively, where the ratio of the circumferences is taken to be sufficiently large ($L_s/L_{\tau }\gg 1$) to simulate 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 PSU($N$)=SU($N$)$/{\mathbb{Z}}_N$ symmetry remains unbroken at "quantum" and "classical" levels for the 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 $L_{\tau }$.

Comments: 6pages, 4 figures, (v2) references added