Large-$N$ nonlinear $\sigma$ models on $R^2\times S^1$

TL;DR

This paper investigates the phase structure of large-$N$ nonlinear sigma models on a cylindrical space, revealing complex phases in the $O(N)$ model and temperature-like behavior in the $CP^{N-1}$ model, using leading order $1/N$ expansion.

Contribution

It provides a detailed analysis of the phase structure of large-$N$ sigma models on $R^2 imes S^1$, highlighting differences between $O(N)$ and $CP^{N-1}$ models.

Findings

$O(N)$ model exhibits rich phase structure on $R^2 imes S^1$.

$CP^{N-1}$ model's phase is analogous to finite temperature.

Effective potentials evaluated at leading order of $1/N$ expansion.

Abstract

The large-$N$ nonlinear $O(N)$, $CP^{N-1}$ $\sigma$ models are studied on $R^2 \times S^1$. The $N$-components scalar fields of the models are supposed to acquire a phase $e^{i2\pi\delta}$ $(0\leq \delta <1)$, along the circulation of the circle, $S^1$. We evaluate the effective potentials to the leading order of the $1/N$ expansion. It is shown that, on $R^2\times S^1$ the $O(N)$ model has rich phase structure while the phase of $CP^{N-1}$ model is just that of the model at finite temperature.