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We study 1D fermions with photoassociation or with a narrow Fano-Feshbachresonance described by the Boson-Fermion resonance model. Using thebosonizationtechnique, we derive a low-energy Hamiltonian of the system. We show that atlow energy, the order parameters for the Bose Condensation and fermionsuperfluidity become identical, while a spin gap and a gap against theformation of phase slips are formed. As a result of these gaps, charge densitywave correlations decay exponentially in contrast with the phases where onlybosons or only fermions are present. We find a Luther-Emery point where thephase slips and the spin excitations can be described in terms ofpseudofermions. This allows us to provide closed form expressions of thedensity-density correlations and the spectral functions. The spectral functionsof the fermions are gapped, whereas the spectral functions of the bosons remaingapless. The application of a magnetic field results in a loss of coherencebetween the bosons and the fermion and the disappearance of the gap. Changingthe detuning has no effect on the gap until either the fermion or the bosondensity is reduced to zero. Finally, we discuss the formation of a Mottinsulating state in a periodic potential. The relevance of our results forexperiments with ultracold atomic gases subject to one-dimensional confinementis also discussed.Comment: 31 pages, 8 EPS figures, RevTeX 4, long version of cond-mat/050570

Phase transitions in the boson-fermion resonance model in one dimension