In this way, 2D Ising model is mapped to the Toric code state as an entangled state with topological order which shows a natural robustness against local perturbation [@rob1; @rob2; @zare16; @zareiprb]. In order to understand topological order of this state, let us consider the above mapping for a lattice with a periodic boundary condition, i.e., a torus. With such a topology, it is necessary to reconsider the constraints which have been considered in Eq.(\[cons0\]). In fact, the constraints associated with faces of the lattice are not independent when we consider a periodic boundary condition. As a result, the product of all face operators in Toric code on a torus is equal to Identity operator. Furthermore, there are other constraints which should be added because of the topology of the torus. In particular, consider two non-trivial loops as shown in Fig.(\[k1\]) around the torus in two different directions. Since $Z$-type loop operators, denoted by $T^1 _z$ and $T^2 _z$, corresponding to the above non-trivial loops, can not be constructed by a product of face operators, they should be considered as two independent constraints. Consequently, state $|G\ra$ includes both trivial and non-trivial loop operators. The role of non-trivial loop operators in 