The formation of bound states of fermions in one dimension has always been one of the key topics in condensed matter physics. Motivated by recent experimental progresses in Prof. Jeremy Levy's group, we study the interplay of both species (spin and transverse band index) and mass imbalance in a mixture of two or more species of fermions with at-tractive interactions in one dimension. Previous theoretical and experimental efforts have shown the existence of a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase for the case of two species with equal mass, in addition to the fully paired and fully polarized phases. For the unequal mass case, there are signatures of trion phases as well. We use DMRG to ex-plore the rich possibilities of quantum phases and their transport signatures for the cases of two and more species of Fermions as we vary the interaction strengths and mass imbalanc-es. With this we can gain insights into ongoing experiments with sketched nanowires in LAO/STO and ultracold atoms confined to one-dimensional tubes. 

We also study the formation of bound states in a single component Fermi chain with attrac-tive interactions. The phase diagram, computed from DMRG (density matrix renormaliza-tion group), shows not only a superfluid of paired fermions (pair phase) and a liquid of fer-mion triplets (trion phase), but also a phase with two gapless modes. We show that the lat-ter phase is described by a 2-component Tomonaga-Luttinger liquid (TLL) theory, consist-ing of one charged and one emergent neutral mode. We argue based on our numerical data, that the single, pair, and trion phases are descendants of the 2-component TLL theory.  We speculate on the nature of the phase transitions amongst these phases.