Quark-gluon thermodynamics with the $\mathbb{Z}_{3}$ symmetry

We propose a simple model with the 3 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 3 symmetry is preserved below some temperature Tc, but spontaneously broken above Tc. Dynamics of the simple model is similar to that of the original PNJL model without the TBC, indicating that the 3 symmetry is a good concept. We also investigate the interplay between the 3 symmetry and the emergence of the quarkyonic phase.