if $L$ is small enough, the quantum correction in the extra dimension dominates the effective potential, and we would obtain nonzero values of $\theta_i$. Then, the next leading order term, the third term, would enforce to result in $v=0$. This simplified discussion implies that the vacuum structure depends on the size of the extra dimension. One, of course, needs to study the effective potential carefully in order to determine the vacuum structure of the model.

Let us now study the vacuum structure of the model. We follow the standard procedure to find the vacuum configuration. We first solve equations of the first derivative of the effective potential (\[effpotd\]) with respect to the order parameters, $$\begin{aligned}
{{\del \Vb}\over{\del\vbar}}&=&\vbar \left(-\mbar^2+{\lambda\over 2}\vbar^2
+\hat{\theta}_1^2\right)=0,
\label{deru}\\
{{\del \Vb}\over{\del\theta_1}}&=&
\hat{\theta}_1 \vbar^2
+{A\over \pi^2}\sum_{n=1}^{\infty}
{{-1}\over n^3}\left(\sin(n\theta_1)+\sin(n\sum_{i=1}^{N-1}\theta_i)
\right)=0,\label{derd}\\
{{\del \Vb}\over{\del\theta_k}}&=&
{A\over \pi^2}\sum_{n=1}^{\infty}{{-1}\over n^3}
\left(\sin(n\theta_k)+\sin(n\sum_{i=1}^{N-1}\theta_i)\right)=0,
\quad k=2, \cdots, N-1,
\label{dert}\end{aligned}$$ where we have used $\theta_N = - \sum_{i=1}^{N-1}\theta_i$. Solutions to the equations are candidates of the vacuum configuration. Then, we analyze the stability of the solutions against small fluctuations, and this constrains allowed regions of the solutions as the local minimum of the effective potential. Among various, if any, candidates of those configurations, the global minimum of the effective 