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We study the superconducting properties of population-imbalanced ultracoldFermi mixtures in one-dimensional (1D) optical lattices that can be effectivelydescribed by the spin-imbalanced attractive Hubbard model (AHM) in the presenceof a Zeeman magnetic field. We use the mean-field theory approach to obtain theground state phase diagrams including some unconventional superconductingphases such as the Fulde--Ferrell--Larkin--Ovchinnikov (FFLO) phase, and the$\eta$ phase (an extremal case of the FFLO phase), both for the case of a fixedchemical potential and for a fixed number of particles. It allows to determineoptimal regimes for the FFLO phase as well as $\eta$-pairing stability. We alsoinvestigate the evolution from the weak coupling (BCS-like limit) to the strongcoupling limit of tightly bound local pairs (BEC) with increasing attraction,at $T=0$. Finally, the obtained results show that despite of the occurrence ofthe Lifshitz transition induced by an external magnetic field, thesuperconducting state can still exist in the system, at higher magnetic fieldvalues.

Critical behaviour in one dimension: unconventional pairing, phase separation, BEC-BCS crossover and magnetic Lifshitz transition