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The large-$N$ nonlinear $\mathrm{O}(N)$, ${\mathrm{CP}}^{N\ensuremath{-}1}\ensuremath{\sigma}$ models are studied on ${R}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$. The $N$-component scalar fields of the models are supposed to acquire a phase ${e}^{i2\ensuremath{\pi}\ensuremath{\delta}}(0\ensuremath{\le}\ensuremath{\delta}<1)$ along the circulation of the circle ${S}^{1}$. We evaluate the effective potentials to the leading order of the $\frac{1}{N}$ expansion. It is shown that on ${R}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ …