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

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.