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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 overviewover the used methods and an deeper insight into the implications arising fromthe treated limit. Starting from the discussion of the phase diagram obtainedby numerical means, we develop an analytic theory and derive an effectivebosonic Hamiltonian. Arising issues in the Hamiltonian are overcome byinclusion of a back-action, renormalizing the solution of our system. Based ona detailed analysis of the effective Hamiltonian, the phase diagram in thethermodynamic limit is constructed by analytic means and comparison tonumerical results obtained by density matrix renormalization group (DMRG)techniques for the full BFHM shows a very reasonable agreement. The mostprominent feature of the phase diagram, the existence of a phase separationbetween Mott insulator (MI) and charge density wave (CDW) is discussed in depthwith inclusion of important effects due to the boundary condition.

The one-dimensional Bose-Fermi-Hubbard model in the ultrafast-fermion limit: Charge density wave phase and MI - CDW phase separation