Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap

We present a solution of the three-fermion problem in a harmonic potential across a Feshbach resonance. We compare the spectrum with that of the two-body problem and show that it is energetically unfavorable for the three fermions to occupy one lattice site rather than two. We also demonstrate the existence of an energy level crossing in the ground state with a symmetry change of its wave function, suggesting the possibility of a phase transition for the corresponding many-body case.

Figure 1

Energy vs inverse scattering length.

Figure 2

Difference between three-fermion energy and two-fermion energy plus one-fermion energy vs inverse scattering length.

Figure 3

(Color online) Relative coordinates for the three-fermion problem. In terms of variables used in Eq. (6), $r=\mid \mathbf{r}\mid$, $R=\sqrt{3}\mid {\mathbf{r}}_{\perp }\mid ∕2$.

Figure 4

Contour plots of $r^2{\mid \Psi (r,R,\theta )\mid }_{r\to 0}^2$ as a function of $R$ and $\theta$ for $r∕d\simeq 0$ and $d∕a\simeq 0$.

Figure 5

Contour plots of $r^2{R}^2{\mid \sin \theta \Psi (r,R,\theta )\mid }^2$ as a function of $R$ and $\theta$ for $r∕d=2$ and $d∕a\simeq 0$.

Figure 6

(Color online) Normalized probability density distribution functions of variables defined in Fig. 3 for various scattering lengths. The solid line is for the $n=0$, $l=0$, $m=0$ state; the dashed line is for the $n=0$, $l=1$, $m=\pm 1$ state; and the dotted line is for the $n=0$, $l=1$, $m=0$ state.