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

## TL;DR

This paper investigates the confinement-deconfinement crossover in the lattice $ ext{CP}^{N-1}$ model at finite temperature, revealing a sharp transition in the Polyakov loop behavior and confirming the unbroken symmetry and degrees of freedom in different regimes.

## Contribution

The study provides the first lattice Monte Carlo analysis of the $ ext{CP}^{N-1}$ model's crossover behavior, including symmetry preservation and entropy calculations, bridging finite-$N$ and large-$N$ predictions.

## Key findings

- Polyakov loop shows a deconfinement crossover as temperature increases.
- Unbroken PSU($N$) symmetry in both small and large $L_{ au}$ regimes.
- Small $L_{ au}$ entropy matches $N-1$ free scalar fields.

## 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 $L_{\tau}$.