Lattice study on the twisted ${\mathbb C}P^{N-1}$ models on ${\mathbb R}\times S^{1}$

TL;DR

This study uses lattice simulations to explore the phase structure and topological features of the ${ m CP}^{N-1}$ sigma model on a space with one large and one small circle, revealing deconfinement crossover and fractional instantons.

Contribution

It provides new insights into the phase transitions and topological objects in the ${ m CP}^{N-1}$ model with twisted boundary conditions on ${ m R} imes S^1$ using lattice methods.

Findings

Deconfinement crossover observed with periodic boundary conditions.

Unbroken ${ m Z}_N$ symmetry maintained with twisted boundary conditions.

Evidence for fractional instantons and bions causing transitions between ${ m Z}_N$ vacua.

Abstract

We report the results of the lattice simulation of the ${\mathbb C} P^{N-1}$ sigma model on $S_{s}^{1}$(large) $\times$ $S_{\tau}^{1}$(small). We take a sufficiently large ratio of the circumferences to approximate the model on ${\mathbb R} \times S^1$. For periodic boundary condition imposed in the $S_{\tau}^{1}$ direction, we show that the expectation value of the Polyakov loop undergoes a deconfinement crossover as the compactified circumference is decreased, where the peak of the associated susceptibility gets sharper for larger $N$. For ${\mathbb Z}_{N}$ twisted boundary condition, we find that, even at relatively high $\beta$ (small circumference), the regular $N$-sided polygon-shaped distributions of Polyakov loop leads to small expectation values of Polyakov loop, which implies unbroken ${\mathbb Z}_{N}$ symmetry if sufficient statistics and large volumes are adopted. We also argue the existence of fractional instantons and bions by investigating the dependence of the Polyakov loop on $S_{s}^{1}$ direction, which causes transition between ${\mathbb Z}_{N}$ vacua.