The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy $S_{\mathrm{topo}}$ for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy $S_{\mathrm{nonlocal}}$ (a generalization of $S_{\mathrm{topo}}$). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type-I and type-II fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that $S_{\mathrm{nonlocal}}$ could obtain geometry-dependent values, and $S_{\mathrm{nonlocal}}$ is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed.

• Received 4 February 2018
• Revised 24 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.144106