Inhomogeneous Fermi mixtures at Unitarity: Bogoliubov-de Gennes vs. Landau-Ginzburg

We present an inhomogeneous theory for the low-temperature properties of a resonantly interacting Fermi mixture in a trap that goes beyond the local-density approximation. We compare the Bogoliubov-de Gennes and a Landau-Ginzburg approach and conclude that the latter is more appropriate when dealing with a first-order phase transition. Our approach incorporates the state-of-the-art knowledge on the homogeneous mixture with a population imbalance exactly and gives good agreement with the experimental density profiles of Shin {\it et al}. [Nature {\bf 451}, 689 (2008)]. We calculate the universal surface tension due to the observed interface between the equal-density superfluid and the partially polarized normal state of the mixture. We find that the exotic and gapless superfluid Sarma phase can be stabilized at this interface, even when this phase is unstable in the bulk of the gas.

💡 Research Summary

The paper tackles the challenging problem of describing a resonantly interacting, population‑imbalanced Fermi mixture confined in a harmonic trap at unitarity, where the usual local‑density approximation (LDA) fails to capture sharp spatial variations associated with a first‑order phase transition. The authors develop an inhomogeneous theory that explicitly goes beyond LDA by employing two complementary theoretical frameworks: the Bogoliubov‑de Gennes (BdG) mean‑field equations and a phenomenological Landau‑Ginzburg (LG) functional.

In the BdG approach, the authors solve the full set of coupled differential equations for the quasiparticle amplitudes (u_n(\mathbf{r})) and (v_n(\mathbf{r})) together with the spatially varying order parameter (\Delta(\mathbf{r})). To make the calculation realistic, they incorporate the exact equation of state of the homogeneous, imbalanced unitary gas obtained from recent quantum‑Monte‑Carlo simulations and experiments. This ensures that the bulk chemical potentials and pressures are correctly reproduced. However, the BdG method shows intrinsic limitations when a first‑order transition is present: the self‑consistent iteration can become trapped in a metastable solution, and the method does not naturally generate the double‑well structure of the free energy that characterizes a first‑order transition. Consequently, the BdG results reproduce the overall shape of the density profiles but underestimate the steepness of the interface between the superfluid core and the normal shell.

The LG approach circumvents these problems by constructing a free‑energy density functional (f