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Hastings's adiabatic method is a powerful tool in the analysis of gapped Hamiltonians in quantum spin systems. This method and subsequent results due to Bachmann, Michalakis, Nachtergaele and Sims give sufficient conditions for automorphic equivalence between ground state spaces along a smooth path of Hamiltonians which are uniformly locally gapped above the ground state. We develop a new adiabatic theorem for unique gapped ground states which does not require the gap for local Hamiltonians. We instead require a gap in the bulk and a smoothness of expectation values of sub-exponentially localized observables in the unique gapped ground state. The first half of this talk will be expository.

Based on joint work with Yoshiko Ogata.