  Due to advances in computer hardware and new algorithms, it is now possible
to perform highly accurate many-body simulations of realistic materials with
all their intrinsic complications. The success of these simulations leaves us
with a conundrum: how do we extract useful physical models and insight from
these simulations? In this article, we present a formal theory of
downfolding--extracting an effective Hamiltonian from first-principles
calculations. The theory maps the downfolding problem into fitting information
derived from wave functions sampled from a low-energy subspace of the full
Hilbert space. Since this fitting process most commonly uses reduced density
matrices, we term it density matrix downfolding (DMD).
