Exotic Z_N generalization of 3D fractons

In this work, we introduce a set of Z_N generalizations of a few 3D Z₂ stabilizer models including fracton models, known as the toric code, X-cube model, and Haah code. We carefully investigate their elementary excitations and ground state degeneracies on a three-torus T³ and find that the Z_N generalizations exhibit unusual excitations and ground state degeneracies, which are strongly dependent on the system sizes. For example, the immobile fracton excitation in the original Z₂ X-cube model becomes mobile in some Z_N generalizations.