The $\mathbb{Z}_2$ toric-code and the double-semion topological order of hardcore Bose-Hubbard-type models in the strong-interaction limit

We present a generic framework for the emergence of the $\mathbb{Z}_2$ toric-code and the double-semion topological order in a wide class of hardcore Bose-Hubbard-type models governed by density-density interaction and in the strong-interaction regime. We fix fractional filling factor and study under which conditions the density-density interaction gives rise to topological degeneracy. We further specify which dynamics determines the toric-code and the double-semion topological order. Our results indicate that the specifics of the density-density interaction determine the long-range entanglement of the model which possesses "restricted patterns" of the long-range entanglement realized in corresponding string-net models with the same topological order.