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The entanglement spectrum (ES) provides a barometer of quantum entanglementand encodes physical information beyond that contained in the entanglemententropy. In this paper, we explore the ES of stabilizer codes, which furnishexactly solvable models for a plethora of gapped quantum phases of matter.Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weaklocal perturbations thus allows us to develop a general framework within whichthe entanglement features of gapped topological phases can be computed andcontrasted. In particular, we study models harboring fracton order, both type-Iand type-II, and compare the resulting ES with that of both conventionaltopological order and of (strong) subsystem symmetry protected topological(SSPT) states. We find that non-local surface stabilizers (NLSS), a set ofsymmetries of the Hamiltonian which form on the boundary of the entanglementcut, act as purveyors of universal non-local features appearing in theentanglement spectrum. While in conventional topological orders and fractonorders, the NLSS retain a form of topological invariance with respect to theentanglement cut, subsystem symmetric systems---fracton and SSPTphases---additionally show a non-trivial geometric dependence on theentanglement cut, corresponding to the subsystem symmetry. This sheds furtherlight on the interplay between geometric and topological effects in fractonphases of matter and demonstrates that strong SSPT phases harbour a measure ofquasi-local entanglement beyond that encountered in conventional SPT phases. Wefurther show that a version of the edge-entanglement correspondence,established earlier for gapped two-dimensional topological phases, also holdsfor gapped three-dimensional fracton models.

Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter