  The first proof of the quantum adiabatic theorem was given as early as 1928.
Today, this theorem is increasingly applied in a many-body context, e.g. in
quantum annealing and in studies of topological properties of matter. In this
setup, the rate of variation $\varepsilon$ of local terms is indeed small
compared to the gap, but the rate of variation of the total, extensive
Hamiltonian, is not. Therefore, applications to many-body systems are not
covered by the proofs and arguments in the literature. In this letter, we prove
a version of the adiabatic theorem for gapped ground states of quantum spin
systems, under assumptions that remain valid in the thermodynamic limit. As an
application, we give a mathematical proof of Kubo linear response formula for a
broad class of gapped interacting systems.
