Collecting all the discussions we have made in this subsection, we confirm the phase structure depicted in Fig.3. We should emphasize again that the Hosotani mechanism works as the restoration of the gauge symmetry, as in the case $N=$ odd.

Other Models
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In this section we study how phase structures change if we consider different representations of matter fields under the gauge group. Let us first introduce $N_F$ fermions in the adjoint representation under $SU(N)$ instead of those in the fundamental representation. Then, the last term in Eq.(\[effpotu\]), which stands for the fermion one-loop correction, is replaced by $${A\over{\pi^2 L^4}}\sum_{n=1}^{\infty}\sum_{i,j=1}^{N}
{1\over n^4}\cos(n(\theta_i -\theta_j)).
\label{adjfermi}$$ It has been known that the function (\[adjfermi\]) is minimized by the configuration that breaks the $SU(N)$ gauge symmetry to $U(1)^{N-1}$ [@adjointmodel], $$gL\vev{A_y}={\rm diag}\left({{N-1}\over N}\pi, {{N-3}\over N}\pi,
\cdots,
-{{N-3}\over N}\pi, -{{N-1}\over N}\pi \right).
\label{configadj}$$ Note that a zero eigenvalue located at the ${{N+1}\over 2}$th component appears for the case $N=$ odd, while all the components are nonzero for $N=$ even. This implies, again, that the phase structure is different, depending on whether $N=$ odd or even.

If $N=$ odd, it is possible for the Higgs VEV to take nonzero values, keeping the cross term vanishing 