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 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 $L_{tau}$.