BCS-BEC crossover and the disappearance of FFLO-correlations in a spin-imbalanced, one-dimensional Fermi gas

We present a numerical study of the one-dimensional BCS-BEC crossover of a spin-imbalanced Fermi gas. The crossover is described by the Bose-Fermi resonance model in a real space representation. Our main interest is in the behavior of the pair correlations, which, in the BCS limit, are of the Fulde-Ferrell-Larkin-Ovchinnikov type, while in the BEC limit, a superfluid of diatomic molecules forms that exhibits quasi-condensation at zero momentum. We use the density matrix renormalization group method to compute the phase diagram as a function of the detuning of the molecular level and the polarization. As a main result, we show that FFLO-like correlations disappear well below full polarization close to the resonance. The critical polarization depends on both the detuning and the filling.

💡 Research Summary

This paper investigates the crossover from Bardeen‑Cooper‑Schrieffer (BCS) pairing to a Bose‑Einstein condensate (BEC) in a one‑dimensional (1D) spin‑imbalanced Fermi gas using the Bose‑Fermi resonance model. The model incorporates two fermionic species in an open channel that can resonantly convert into a closed‑channel diatomic molecule, with hopping amplitudes for fermions (t) and molecules (t_mol = t/2), a detuning ν of the molecular level, and a Feshbach coupling g that controls the conversion strength. By solving the two‑body problem the authors obtain the binding energy ε_b as a function of ν and g, defining a characteristic energy ε* at resonance (ν = 0) that sets the scale for the whole crossover.

Using the density‑matrix renormalization group (DMRG) the authors compute ground‑state properties for chains up to L≈120 sites, varying the filling n = N/L, the polarization p = (N↑ − N↓)/N, and the detuning ν. Key observables are the pair‑correlation function ⟨Δ†(x)Δ(0)⟩, the momentum distribution n(k), and the molecular occupation N_mol. In the BCS regime (ν < 0) the pair correlations display oscillations with wave vector q ≈ πp, characteristic of the Fulde‑Ferrell‑Larkin‑Ovchinnikov (FFLO) state. The oscillation amplitude is large for small |ν| and low filling. As ν approaches zero and becomes positive, the amplitude diminishes rapidly; beyond a critical polarization p_c(ν,n) the oscillations vanish altogether, indicating the disappearance of FFLO correlations. The critical polarization depends sensitively on both detuning and filling: it is lower for larger positive ν (more molecular character) and higher for larger n (more fermions available to form pairs).

In the BEC side (ν ≫ 0) the system is dominated by tightly bound molecules that behave as hard‑core bosons (a Tonks‑Girardeau gas). The momentum distribution then shows a sharp peak at k = 0, signalling quasi‑condensation of molecules, while the pair‑correlation function becomes non‑oscillatory and decays algebraically at zero momentum. Between the two limits a mixed phase appears where both a finite‑momentum FFLO component and a zero‑momentum molecular condensate coexist; this regime is identified as a “Bose‑Fermi mixture superfluid”.

The authors map out the zero‑temperature phase diagram in the (ν, p) plane for several fillings. Four distinct regions emerge: (i) a balanced superfluid (p = 0) with BCS‑type pairing, (ii) a polarized FFLO‑like intermediate phase (0 < p < p_c) on the BCS side, (iii) a fully polarized normal Fermi gas (p = 1), and (iv) a mixed superfluid of molecules and partially polarized fermions for p_c < p < 1 on the BEC side. Notably, unlike single‑channel attractive Hubbard models where any finite polarization yields an FFLO state, the two‑channel resonance model predicts that FFLO correlations are suppressed near resonance because a substantial fraction of minority fermions are bound into molecules, reducing the number of pairs that can carry finite momentum.

The study provides quantitative predictions for the spin gap, the binding energy, and the critical polarization as functions of detuning and filling, offering clear experimental signatures. In ultracold‑atom setups with tight transverse confinement, one can tune ν via a magnetic Feshbach resonance and control p by preparing imbalanced spin populations. Measurements of spin‑resolved density profiles, time‑of‑flight momentum distributions, or Bragg spectroscopy could directly observe the loss of the FFLO peak and the emergence of a zero‑momentum molecular condensate, thus testing the theoretical phase diagram presented here.