Three attractively interacting fermions in a harmonic trap: Exact solution, ferromagnetism, and high-temperature thermodynamics

Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism. Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance. We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive. When the $s$-wave scattering length $a$ is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment. We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion. The second and third virial coefficients are calculated. The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.

Figure 1

Energy spectrum of the relative motion of a trapped two-fermion system near a Feshbach resonance (i.e, $d/a=0$, where $d$ is the characteristic harmonic oscillator length). For a positive scattering length $a>0$ in the right part of the figure, the ground state is a molecule with size $a$, whose energy diverges as $E_{\mathrm{rel}}\simeq -\hslash {}^2/({\mathit{ma}}^2)$. The excited states or the upper branch of the resonance may be viewed as the Hilbert space of a repulsive Fermi gas with the same scattering length $a$. In this two-body picture, the level from the point 2 to 3 is the ground-state energy level of the repulsive two-fermion subspace, whose energy initially increases linearly with increasing $a$ from $1.5\hslash \omega$ at the point 2 and finally saturates toward $2.5\hslash \omega$ at the resonance point 3. For comparison, we illustrate as well the ground-state energy level in the case of a negative scattering length and show how the energy increases with increased scattering length from point 1 to 2.

Figure 2

Configuration of three interacting fermions, two spin-up and one spin-down.

Figure 3

Relative energy spectrum of three interacting fermions at different subspaces or relative angular momenta $l$. On the positive scattering length (BEC) side of the resonance, there are two types of energy levels: one (is vertical and) diverges with decreasing the scattering length $a$ and the other (is horizontal) converges to the noninteracting spectrum. The latter may be viewed as the energy spectrum of three repulsively interacting fermions. In analogy with the two-fermion case, we show in the ground-state subspace ($l=1$) the ground-state energy level of the repulsive three-fermion system (i.e, from point 2 to point 3), as well as the ground-state energy level of the attractive three-fermion system for $a<0$ (i.e., from the point 1 to point 2). In the unitarity limit, we show by the circles the energy levels that should be excluded when we identify the energy spectrum for infinitely large repulsive interactions.

Figure 4

Energy per particle $E/({\mathit{NE}}_F)$ as a function of reduced temperature $T/T_F$ for a homogeneous Fermi gas with infinitely attractive and repulsive interactions. The predictions of quantum virial expansion up to the second- and third-order are shown by solid lines and dashed lines, respectively. For comparison, we plot the ideal gas result with the dot-dashed line.

Figure 5

Entropy per particle $S/({\mathit{NE}}_F)$ as a function of reduced temperature $T/T_F$ for a homogeneous Fermi gas with infinitely attractive and repulsive interactions. The other details are the same as in Fig. 4.

Figure 6

Energy per particle $E/({\mathit{NE}}_F)$ as a function of reduced temperature $T/T_F$ for a trapped Fermi gas with infinitely attractive and repulsive interactions. The predictions of quantum virial expansion up to the second and third order are shown by solid lines and dashed lines, respectively. For comparison, we plot the ideal gas result with the dot-dashed line. We show also the experimental data measured at ENS by empty squares for an attractive Fermi gas at unitarity [4, 7], which agree extremely well with the prediction from quantum virial expansion.

Figure 7

Entropy per particle $S/({\mathit{NE}}_F)$ as a function of reduced temperature $T/T_F$ for a trapped Fermi gas with infinitely attractive and repulsive interactions. The other details are the same as in Fig. 6.