Automorphic equivalence within gapped phases in the bulk

Understand

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 $\varphi_s(A)$.
• This requirement is weaker than the requirement of the gap of the local Hamiltonians, since a uniform spectral gap for finite dimensional ground states implies a gap in the bulk for unique gapped ground states, as well as the smoothness.