Entanglement entropy of topological phases with multipole symmetry

We study entanglement entropy of unusual ${\mathbb{Z}}_N$ topological stabilizer codes which admit fractional excitations with restricted mobility constraint in a manner akin to fracton topological phases. It is widely known that the subleading term of the entanglement entropy of a disk geometry in conventional topologically ordered phases is related to the total number of the quantum dimension of the fractional excitations. We show that, in our model, such a relation does not hold, i.e., the total number of the quantum dimension varies depending on the system size, whereas the subleading term of the entanglement entropy takes a constant number irrespective to the system size. We give a physical interpretation of this result in the simplest case of the model. More thorough analysis on the entanglement entropy of the model on generic lattices is also presented.