Three and four harmonically trapped particles in an effective-field-theory framework

We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body interactions, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the no-core shell model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by other methods (in the four-body case). We also present results for finite scattering lengths and nonzero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.

Figure 1

Energy in units of the HO frequency, $E/\omega$, of the ground state $L^{\pi }=1^{-}$ of the $A=3$ system at unitarity, as a function of the three-body model-space size, $N_3^{\max }$: (a) $N_2^{\max }=10$, (b) $N_2^{\max }=18$. Circles correspond to LO, (red) squares to NLO, and (green) diamonds to NNLO. The dashed line marks the exact value [18].

Figure 2

Energy in units of the HO frequency, $E/\omega$, of the ground state $L^{\pi }=1^{-}$ of the $A=3$ system at unitarity, as a function of the two-body cutoff, $N_2^{\max }$. Notation as in Fig. 1.

Figure 3

Same as Fig. 1, but for the first excited state with $L^{\pi }=1^{-}$.

Figure 4

Same as Fig. 2, but for the first excited state with $L^{\pi }=1^{-}$.

Figure 5

Same as Fig. 2, but for (left) the ground state and (right) the first excited state with $L^{\pi }=0^{+}$.

Figure 6

Energy in units of the HO frequency, $E/\omega$, of the lowest states with $L^{\pi }=1^{-}$ (circles) and $L^{\pi }=0^{+}$ (squares) at NNLO, as a function of the ratio $b/a_2$ between the HO length and the two-body scattering length, when $r_2/b=0$. At $b/a_2\simeq 1.34$ there is an inversion of the ground state.

Figure 7

Atom-dimer scattering length in units of the two-body scattering length, $a_{\mathrm{ad}}/a_2$, as a function of the two-body cutoff, $N_2^{\max }$. NNLO results for $b/a_2=3$ (black circles) and $b/a_2=4$ (red squares) are compared with the value (blue line) of Ref. [20].

Figure 8

Same as Fig. 6, but for $r_2/b=0.1$. In this case, the crossing point between the $L=1^{-}$ and $L=0^{+}$ lowest states moves to $b/a_2\simeq 1.75$, a larger value than in the absence of effective range.

Figure 9

Energy in units of the HO frequency, $E/\omega$, of the ground state $L^{\pi }=0^{+}$ of the $A=4$ system at unitarity, as a function of the four-body model space size, $N_4^{\max }$: (a) $N_2^{\max }=4$ and (b) $N_2^{\max }=8$. Notation as in Fig. 1.

Figure 10

Same as Fig. 2, but for (left) the ground state and (right) the first excited state with $L^{\pi }=0^{+}$ for the $A=4$ system at unitarity. Notation as in Fig. 1.