the scale $\bar{m}_1^{\prime}$, the coexisting phase disappears and the Hosotani phase must be the vacuum configuration for $\bar{m}<\bar{m}_1^{\prime}$.

One has to take care about what is happening in the region of $\mbar_1^{\prime} \leq \mbar \leq \pi$. For this region, there are two solutions, $\theta_c$ and $\theta_{c}^{\prime}$, to Eq.(\[eqevendt\]) or the equation $F(\theta_1)=G(\theta_1)$. It turns out that the solution $\theta_c$ is stable but the other one is unstable, as shown in the appendix. Thus, there are two candidates for the vacuum configuration, [*i.e.*]{} the Hosotani phase and the coexisting phase given by $\theta_c$. Since $\theta_{c}^{\prime}=\theta_c$ ($\theta_{c}^{\prime}=\pi$) at $\bar{m}=\bar{m}_1^{\prime}$ ($\bar{m}=\pi$), we find that the unstable solution $\theta_{c}^{\prime}$ becomes identical to the coexisting phase given by $\theta_c$ (the Hosotani phase) at $\bar{m}=\bar{m}_1^{\prime}$ ($\bar{m}=\pi)$. This shows that the coexisting phase (the Hosotani phase) is not the vacuum configuration at the boundary $\bar{m}=\bar{m}_1^{\prime}$ $(\bar{m}=\pi)$. This observation implies that there exists a critical scale $\bar{m}_4^{\prime}$ such that[^6] $$\Vb(\theta_i, \vbar,\mbar_{4}^{\prime})\bigg|_{\rm type~I}
=\Vb(\theta_i, \vbar,\mbar_{4}^{\prime})\bigg|_{\rm type~III(\theta_c)},$$ at which the first-order phase transition must occur. Since the above equation is satisfied only once for $\bar{m}_1^{\prime}\le\bar{m}\le\pi$, we obtain the phase structure for $t<48\pi^2\frac{N-1}{N}$ as $$\mbox {vacuum configuration}=\left\{\begin{array}{lll}
\mbox{Hosotani phase} &\mbox{for}& \mbar < \bar{m}_{4}^{\prime},\\
\mbox{coexisting phase} &\mbox{for}& \mbar > \bar{m}_{4}^{\prime}.
\end{array}\right.
\label{vaceven2}$$

