Interplay between sign problem and $Z_3$ symmetry in three-dimensional Potts models

We construct four kinds of Z(3)-symmetric three-dimensional (3D) Potts models, each with a different number of states at each site on a 3D lattice, by extending the 3D 3-state Potts model. Comparing the ordinary Potts model with the four Z(3)-symmetric Potts models, we investigate how Z(3) symmetry affects the sign problem and see how the deconfinement transition line changes in the mu-k plane as the number of states increases, where mu(k) plays a role of chemical potential (temperature) in the models. We find that the sign problem is almost cured by imposing Z(3) symmetry. This mechanism may happen in Z(3)-symmetric QCD-like theory. We also show that the deconfinement transition line has stronger mu dependence with respect to increasing the number of states.