  We prove an adiabatic theorem for general densities of observables that are
sums of local terms in finite systems of interacting fermions, without
periodicity assumptions on the Hamiltonian and with error estimates that are
uniform in the size of the system. Our result provides an adiabatic expansion
to all orders, in particular, also for initial data that lie in eigenspaces of
degenerate eigenvalues. Our proof is based on ideas from a recent work of
Bachmann et al. who proved an adiabatic theorem for interacting spin systems.
  As one important application of this adiabatic theorem, we provide the first
rigorous derivation of the so-called linear response formula for the current
density induced by an adiabatic change of the Hamiltonian of a system of
interacting fermions in a ground state, with error estimates uniform in the
system size. We also discuss the application to quantum Hall systems.
