Degenerate Fermions and Wilson Loop in $1+1$ Dimensions

We investigate the effect of finite fermion density on symmetry breaking by Wilson loops in $(1+1)$ dimensions. We find the breaking and restoration of symmetry at finite density in the models with $SU(2)$ and $SU(3)$ gauge symmetries, in the presence of the adjoint fermions. The transition can occur at a finite density of fermions, regardless of the periodic or antiperiodic boundary condition of the fermion field; this is in contrast to the finite-temperature ease examined by Ho and Hosotani (Nucl. Phys. B345 (1990) 445) where the boundary condition of fractional twist is essential to the occurrence of the phase transition.