Ground state degeneracy on torus in a family of $\mathbb{Z}_N$ toric code

Abstract: Topologically ordered phases in $2+1$ dimensions are generally characterized by three mutually-related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such degeneracy is known as topological degeneracy and usually can be seen under the periodic boundary condition regardless of the choice of the system size $L_1$ and $L_2$ in each direction. In this work we introduce a family of extensions of the Kitaev toric code to $N$ level spins ($N\geq2$). The model realizes topologically ordered phases or symmetry-protected topological phases depending on parameters in the model. The most remarkable feature of the topologically ordered phases is that the ground state may be unique, depending on $L_1$ and $L_2$, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.