  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.
