Categorizing Fracton Topological Order

We employ quantities generalizing the two-dimensional S-matrix invariant to categorize 3D stabilizers models into type-I fracton models, type-II fracton models, TQFTs and stacks of 2D Toric codes. These quantities count the numbers of anticommuting string-string, string-membrane and membrane-membrane operators. The scaling behavior of these quantities with the size of the strings and membranes gives a signature of the class of topological order. We also investigate two-dimensional compactifications of three-dimensional fracton models. We find the two-dimensional topological phases produced as a function of compactification radius and uncover translation symmetry-enrichment that leads to twisted boundary conditions. This allows us to interpret the complicated ground space degeneracy of type-II 3D fracton models in terms of the symmetry enriched anyons in the compactified model.