When construct spin liquid by projective symmetry group, we can classified spin liquids by the invariant group (IGG) of their mean field ansatze. For example, we can have Z2, U(1) and SU(2) spin liquids. In my understanding, IGG is equivalent to the low-energy gauge fluctuation. However, I am confused whether states with the same IGG actually have the same topological order? For example, do all the $Z_2$ spin liquid share the same $Z_2$ topological order? If so, is this $Z_2$ topological order corresponds to toric code or double semion?
Yes, all the $Z_2$ spin liquid share the same $Z_2$ topological order. Yes, toric code has the same $Z_2$ topological order discovered in 1991.