Effective theory for trapped few-fermion systems

We apply the general principles of effective field theories to the construction of effective interactions suitable for few- and many-body calculations in a no-core shell model framework. We calculate the spectrum of systems with three and four two-component fermions in a harmonic trap. In the unitary limit, we find that three-particle results are within 10% of known semianalytical values even in small model spaces. The method is very general, and can be readily extended to other regimes, more particles, different species (e.g., protons and neutrons in nuclear physics), or more-component fermions (as well as bosons). As an illustration, we present calculations of the lowest-energy three-fermion states away from the unitary limit and find a possible inversion of parity in the ground state in the limit of trap size large compared to the scattering length. Furthermore, we investigate the lowest positive-parity states for four fermions, although we are limited by the dimensions we can currently handle in this case.

Figure 1

(Color online) Running of the scaled interaction strength $\Lambda b{\gamma }_0=\mu \Lambda C_0∕2\pi$ with the inverse of the ultraviolet cutoff (in units of the HO length) $1∕\Lambda b$, for selected values of the ratio $b∕a_2$.

Figure 2

(Color online) Dependence on the ultraviolet cutoff $N_{\text{max}}$ of selected low-lying energies (in units of the HO frequency) of three harmonically trapped fermions of spin-1/2 in the unitary regime $\left(b∕a_2=0\right)$. We present the lowest three states of $j^{\pi }={\frac{1}{2}}^{-}$ (continuous curves), $j^{\pi }={\frac{1}{2}}^{+}$ (dotted curves), and $j^{\pi }={\frac{3}{2}}^{+}$ (dashed curves).

Figure 3

(Color online) Running of the first ${\frac{1}{2}}^{-}$ excited state (circles), and three fits with the functional form in Eq. (11) taking into account all points (dotted line), only points with $N_{\text{max}}\ge 15$ (dashed line), and only points with $N_{\text{max}}\ge 21$ (dot-dashed line).

Figure 4

(Color online) Ground-state energies (in units of the HO frequency) of the three-fermion system in a harmonic trap as function of the cutoff $N_{\text{max}}$ for $b∕a_2=0$ (circles), $b∕a_2=-1$ (squares), and $b∕a_2=1$ (diamonds).

Figure 5

(Color online) Lowest (extrapolated) energy levels for $j^{\pi }={\frac{1}{2}}^{-}$ (circles) and $j^{\pi }={\frac{1}{2}}^{+}$ (squares) states of the three-fermion system in a harmonic trap as function of the ratio $b∕a_2$. Around $b∕a_2\sim 1.5$, we observe an inversion of the parity of the system’s ground state. The dashed line is a fit of the positive-parity points by a quadratic form.

Figure 6

(Color online) Same as Fig. 4 but for the $j^{\pi }=0^{+}$ states of the four-fermion system.