Entanglement Spectra of Stabilizer Codes: A Window into Fracton Order and Sub-System Symmetry Protected Topological States

We discuss the entanglement spectrum (ES) for stabilizer Hamiltonians with arbitrary weak local perturbations to contrast the entanglement features of gapped topological phases. In particular, we compare fracton order to both conventional topological order and sub-system symmetry-protected topological (SSPT) order. Our results show that non-local surface stabilizers (NLSS)—a set of symmetries of the Hamiltonian formed along the boundary of the entanglement cut—protect the universal non-local features of the ES. In conventional topological and fracton orders, some NLSS retain a form of topological invariance, whereas subsystem symmetric systems—fracton and SSPT phases—show a non-trivial geometric dependence corresponding to the sub-system symmetries. This demonstrates the interplay of geometry and topology in fracton phases as encoded in the ground states of the phase. We further show a version of the edge-entanglement correspondence in three dimensions.