Gap stability beyond controlled classes

Whether a spectral gap survives adding small local perturbations underpins the very notion of a gapped quantum phase of matter. If the gap can close under arbitrarily small perturbations, 'gapped phase' is not well defined.

What is known: The original Bravyi–Hastings–Michalakis theorem treats geometrically local commuting-projector Hamiltonians satisfying its topological-order conditions, including toric-code-type examples. Michalakis–Zwolak extend stability to frustration-free systems with local topological order and local-gap assumptions. Later work of Nachtergaele–Sims–Young develops further bulk and quasi-local stability tools. These are different sets of sufficient hypotheses, not a theorem for every frustration-free gapped model.

Residual targets include quantitatively weaker decay assumptions or a chosen frustrated family not covered by an existing criterion. They must be checked against later stability theorems: neither degeneracy nor frustration alone puts a model outside all known results. Specify, for example, a weighted interaction norm with an explicit power-law exponent, and distinguish an infinite-volume bulk gap from gaps of finite regions with physical edges. Stability of a low-energy band is not the same as keeping every finite-volume ground-state splitting zero.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.