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

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.

Comments: 7 pages, 5 figures, proceedings of the 37th International Symposium on Lattice Field Theory (LATTICE2019), 16-22 June 2019, Wuhan, China