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We present a detailed study of the population imbalanced three-componentHubbard chain with attractive interactions. Such a system can be realizedexperimentally with three different hyperfine states of ultra cold $^6$Li atomsin an optical lattice. We find that there are different phases that competewith each other in this system: A molecular superfluid phase in which the threefermion species pair up to form molecules (trions), a usual pairing phaseinvolving two species with exactly opposite momenta, and a more exoticgeneralized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase consisting of threecompeting pairing tendencies with different non-zero center-of-mass momenta. Atlarge attractive interactions the system exhibits strong tendencies towardscollapse and phase separation. Employing thedensity-matrix-renormalization-group-method (DMRG) to determine the decayexponents of the various correlators we establish the phase diagram of thismodel for different fillings and interactions. We also discuss theexperimentally relevant situation in a trap and report the existence of anadditional region where two species are dynamically balanced.

Imbalanced thee-component Fermi gas with attractive interactions:  Multiple FFLO-pairing, Bose-Fermi and Fermi-Fermi mixtures versus collapse  and phase separation