Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms.
We study one-dimensional strongly interacting quantum gas mixtures, including both the Bose-Fermi and spin-
1/2 Fermi-Fermi mixtures, with weak
p-wave interaction between intracomponent fermions, and demonstrate that the weak
p-wave interaction cannot be omitted in the strong interaction region where the strength of
p-wave interaction is comparable with the inverse of the strength of strongly repulsive
s-wave interaction. While the total density distribution is not sensitive to the weak
p-wave interaction, we find that the
p-wave interaction plays an important role in determining the species-dependent (or spin-dependent) density distributions and produces significant physical effects on the low-energy spin dynamics. We also derive effective spin-exchange models for strongly interacting quantum gas mixtures with weak
p-wave interaction, which indicate that a quantum phase transition from an antiferromagnetic state to a ferromagnetic state can be induced by tuning the relative strengths of intracomponent and intercomponent interactions.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a hard-wall trap or a harmonic oscillator trap with a tunable $\ensuremath{\delta}$-function barrier at the trap center. The mixture consists of a 1D Bose gas with repulsive interactions and of a 1D noninteracting spin-aligned Fermi gas, with both species interacting through hard-core interactions. Using a generalized Bose-Fermi mapping, we calculate the reduced single-particle density matrix and the momentum distribution of the gas as a function of barrier strength and the parity of particle number. The secondary peaks in the momentum distribution show remarkable correlation between particles on the two sides of the split.