The PEPS bulk-boundary gap and short-range conjecture

Let a PEPS have a bulk parent Hamiltonian and an associated one-dimensional boundary Hamiltonian. Bulk-boundary gap conjecture. The bulk parent Hamiltonian of the PEPS is gapped if and only if the boundary Hamiltonian is short range.

This conjecture proposes an equivalence between bulk spectral gaps and locality of the effective boundary interaction, extending the connection between bulk correlations and boundary-state properties. It was stated as an important open problem and has seen virtually no progress beyond results such as those for the AKLT model.