Mean-field study of itinerant ferromagnetism in trapped ultracold Fermi gases: Beyond the local-density approximation

We theoretically investigate the itinerant ferromagnetic transition of a spherically trapped ultracold Fermi gas with spin imbalance under strongly repulsive interatomic interactions. Our study is based on a self-consistent solution of the Hartree-Fock mean-field equations beyond the widely used local-density approximation. We demonstrate that, while the local-density approximation holds in the paramagnetic phase, after the ferromagnetic transition it leads to a quantitative discrepancy in various thermodynamic quantities even with large atom numbers. We determine the position of the phase transition by monitoring the shape change of the free-energy curve with increasing the polarization at various interaction strengths.

Figure 1

Spin-up, spin-down and total density profiles of an imbalanced Fermi gas at $k_F^0{a}_s=2.0$ (left column) and $k_F^0{a}_s=4.0$ (right column). The solid and dashed lines refer to the Hartree-Fock MFA and LDA results, respectively. The inset in the upper, right-column plot highlights the spin-up density profile at the trap edge. The spin polarization is $p=0.10$ and the number of atoms is $N={10}^5$.

Figure 2

Magnetization profile or local spin polarization at different interaction strengths: $k_F^0{a}_s=1.8$ (dash-dotted line), $2.7$ (dashed line), and $4.0$ (solid line). These curves are calculated using the Hartree-Fock MFA theory for a total number of atoms $N={10}^5$ and a polarization $p=0.10$ at $T=0.12T_F$.

Figure 3

Atom loss rate (a), interaction energy (b), potential energy (c), kinetic energy (d), and chemical potentials (e and f) as functions of interaction strength. The Hartree-Fock MFA results (solid lines) are compared with the LDA predictions (dashed lines). The vertical dashed line marks the ferromagnetic phase transition. Here, $N={10}^5$, $p=0.10$, and $T=0.12T_F$.

Figure 4

Atom loss rate (a), interaction energy (b), potential energy (c), and kinetic energy (d) at different temperatures $T=0.22T_F$ (solid lines) and $T=0.55T_F$ (dashed lines), with a total number of atoms $N={10}^4$ and an imbalance ratio of $p=0.10$.

Figure 5

(a) Free energy as a function of spin polarization $p$ for different interaction strengths: $k_F^0{a}_s=1.94,$ $2.00$, $2.10$, $2.20$, $2.40$, and $3.0$. (b) Free energy vs interaction strength $k_F^0{a}_s$ for different spin imbalances: $p=0.05$, $0.10$, and $0.20$. Here, $N={10}^4$ and $T=0.12T_F$.