Fermionic superfluidity in finite one-dimensional spin-imbalanced systems: A configuration-space Hartree-Fock-Bogoliubov approach

Ground state Hartree-Fock-Bogoliubov (HFB) theory is applied to spin imbalanced one dimensional Fermi systems with short-range interactions that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superfluid phase. The HFB energy is expressed in the single-particle basis of the non-interacting Hamiltonian. The ground-state-energy is then found via numerical minimization. In addition to allowing for arbitrary Cooper pairing, the HFB approach also fully accounts for the effects of both the inhomogeneous trapping potential and the mean-field Hartree potential. We show that the local superfluid order parameter displays an oscillating FFLO-like characteristic; although we do find an imprint of the FFLO state in the local densities, it is weak in the harmonic trap case. In contrast, in the hard-wall geometry, shows a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local densities; here the excess spins are strongly localized near regions where there is a node in the pairing amplitude, thus creating an unmistakeable crystalline modulation of the density.