Cold Fermionic Atoms in Two-Dimensional Traps: Pairing versus Hund’s Rule

The microscopic properties of few interacting cold fermionic atoms confined in a two-dimensional (2D) harmonic trap are studied by numerical diagonalization. For repulsive interactions, a strong shell structure dominates, with Hund’s rule acting at its extreme for the midshell configurations. In the attractive case, odd-even oscillations due to pairing occur simultaneously with deformations in the internal structure of the ground states, as seen from pair correlation functions.

Figure 1

${\Delta }_2(N)$ vs $N$. (a) The green, red, blue, and black curves are calculated by using $g=0.3$, 1, 3, and 5. (b) The green, red, blue, and black curves are calculated by using $g=-0.3$, $-1$, $-3$, and $-5$. (c) Results from the seniority model.

Figure 2

$E_{\mathrm{int}}/N^{3/2}$ vs $N$ for $g=\pm 0.3$.

Figure 3

Excitation energies vs $M$ for $N=8$. The top, middle, and bottom panels are obtained for $|g|=0.3$, 3, and 5, respectively. The red (blue) circles label the case of attractive (repulsive) interaction. Only the six lowest excited states are shown for each value of $M$. The closed circles mark the ground states, whose energies are taken as reference.

Figure 4

Alternative estimates of the pairing energy gap $\Delta$ vs $g$ for $N=4$ (3), 8 (7).

Figure 5

Plot of $P_{\uparrow \downarrow }(\mathit{r},{\mathit{r}}_0)$ in the $xy$ plane. The black dot at $(x,y)=(x_0,0)$ locates the spin-$\downarrow$ reference atom ($x_0$ is the average value of $r$). Top row: $g=3$. Bottom row: $g=-3$. From left to right column, $N=2$, 4, 6, 8, respectively. The squares’ size is $5\times 5$, and 15 equally spaced contour levels go from blue (minimum) to red (maximum).