  We propose a simple model with the Z_N symmetry in order to answer whether
the symmetry is a good concept in QCD with light quark mass. The model is
constructed by imposing the flavor-dependent twisted boundary condition (TBC)
on the three-flavor Polyakov-loop extended Nambu-Jona-Lasinio model. In the
model, the Z_N symmetry is preserved below some temperature T_c, but
spontaneously broken above T_c. Dynamics of the simple model is similar to that
of the original PNJL model without the TBC, indicating that the Z_N symmetry is
a good concept. We also investigate the interplay between the Z_N symmetry and
the emergence of the quarkyonic phase.
