  In a recent work [1] we presented results for the Bose-Fermi-Hubbard model
(BFHM) in the limit of ultrafast fermions. The present work gives an overview
over the used methods and an deeper insight into the implications arising from
the treated limit. Starting from the discussion of the phase diagram obtained
by numerical means, we develop an analytic theory and derive an effective
bosonic Hamiltonian. Arising issues in the Hamiltonian are overcome by
inclusion of a back-action, renormalizing the solution of our system. Based on
a detailed analysis of the effective Hamiltonian, the phase diagram in the
thermodynamic limit is constructed by analytic means and comparison to
numerical results obtained by density matrix renormalization group (DMRG)
techniques for the full BFHM shows a very reasonable agreement. The most
prominent feature of the phase diagram, the existence of a phase separation
between Mott insulator (MI) and charge density wave (CDW) is discussed in depth
with inclusion of important effects due to the boundary condition.
