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

From real materials to model Hamiltonians with density matrix downfolding