possible for the component gauge field for the compactified direction to acquire nonvanishing vacuum expectation values (VEV). The Hosotani mechanism has been studied extensively in (supersymmetric) gauge models[@HosotaniS2; @HosotaniT2; @HP; @Hoso-ft; @Pattern-Matter; @Pattern-BC; @HIL] and, in particular, paid much attention in the context of orbifold compactifications[@orbifold; @Hoso-orbi]. On the other hand, the Higgs mechanism also breaks gauge symmetry by the nonvanishing VEV for the Higgs field even at the tree level. This suggests that if we consider the gauge-Higgs model on such the space-time, the gauge symmetry can be broken by both or either of the two mechanisms due to the existence of the two kinds of the order parameters for gauge symmetry breaking.

In a previous paper[@previous] we showed that phase structures of gauge-Higgs models on $M^3\otimes S^1$ are nontrivial, where $M^3 (S^1)$ is three-dimensional Minkowski space-time (a circle). In the paper we studied the phase structure of the simplest $SU(2)$ gauge-Higgs model and found three different phases called Hosotani, Higgs and coexisting phases. In each phase the VEVs for the two order parameters take the different forms and values. The structure depends on a parameter of the model and the size of $S^1$. The critical radius and the 