Real-space recipes for interacting topological crystalline states

We present a unified scheme for constructing all topological crystalline states, bosonic and fermionic, free and interacting, from real-space building blocks and connectors. Building blocks are finite-size pieces of lower dimensional topological states protected by onsite symmetries alone, and connectors are ``glue'' that complete the open edges shared by two or multiple pieces of building blocks. The resulted assemblies are selected against two physical criteria we call the ``no-open-edge condition'' and the "bubble equivalence", which, respectively, ensure that each selected assembly is gapped in the bulk and cannot be deformed to a product state. The scheme is then applied to obtaining the full classification of bosonic topological crystalline states protected by several onsite symmetry groups and each of the 17 wallpaper groups in two dimensions and 230 space groups in three dimensions. We claim that our real-space recipes give the complete set of topological crystalline states for bosons and fermions, and prove the boson case analytically using a spectral sequence expansion of group cohomology.