Hence, there is no coexisting phase for $\bar{m}<\pi$, so that the Hosotani phase must be the vacuum configuration for $\bar{m}<\pi$. Thus, we obtain that $$\mbox {vacuum configuration}=\left\{\begin{array}{lll}
\mbox{Hosotani phase} &\mbox{for}& \mbar < \pi,\\
\mbox{coexisting phase} &\mbox{for}& \mbar > \pi.
\end{array}\right.
\label{vaceven1}$$ Since the order parameters are connected continuously at the phase boundary $\bar{m}=\pi$, the phase transition is the second order.

(ii)  $t < 48\pi^2 {{N-1}\over N}$

In this parameter region of $t$, we observe in Fig.5 that there is one solution denoted by $\theta_c$ for $\bar{m}>\pi$. The solution satisfies the condition (\[restricteven\]) and is stable, so that the coexisting phase is the vacuum configuration for $\bar{m}>\pi$, as in the case (i).

Unlike the case (i), $\theta_c$ does not approach $\pi$ as $\bar{m}\rightarrow \pi$. When the scale $\bar{m}$ is equal to $\pi$, there appears a new solution denoted by $\theta_{c}^{\prime} = \pi$, while the $\theta_c$ still lies between $\pi$ and $2\pi$, as shown in Fig.5. If we go to smaller scales than $\pi$, there are two solutions $\theta_{c}^{\prime}$ and $\theta_c$ with $\pi<\theta_{c}^{\prime}\le \theta_c <2\pi$ for $\bar{m}_1^{\prime} \le \bar{m} < \pi$, where the two solutions coincide at $\bar{m}=\bar{m}_1^{\prime}$ (see Fig.5). Since there are no solutions in the required region of $\theta_1$ below 