Entanglement signatures of multipolar higher-order topological phases

We propose a procedure that characterizes free-fermion or interacting classes of higher-order topological phases via their bulk entanglement structure. To this end, we construct nested entanglement Hamiltonians by first applying an entanglement cut to the ordinary many-body ground state, and then iterating the procedure by applying further entanglement cuts to the (assumed unique) ground state of the entanglement Hamiltonian. We argue that some classes of nth-order topological phases can be characterized by the features of their nth-order nested entanglement Hamiltonian, e.g., degeneracy in the entanglement spectrum. We explicitly compute nested entanglement spectra for a set of higher-order, free and interacting, fermionic and bosonic multipole and boundary obstructed phases and show that our method successfully identifies such phases.