Subsystem Symmetry Enriched Topological Order in Three Dimensions

We construct a model of three dimensional (3D) topological order enriched by planar subsystem symmetries by decorating the 3D toric code with two-dimensional subsystem symmetry-protected topological orders. This decoration causes the loop-like excitations of the 3D toric code to fractionalize under the planar subsystem symmetries, resulting in an extensive degeneracy of the excitation and also an increased value of the topological entanglement entropy for certain bipartitions. Our model can be obtained by gauging the global symmetry of a short-range entangled model which has symmetry-protected topological order coming from an interplay of global and subsystem symmetries. We further study this interplay by gauging various subgroups of the symmetry, resulting in a network of models, including models with fracton topological order, which showcases more of the possible types of subsystem symmetry enrichment that can occur in 3D.