New Effective Interaction for the Trapped Fermi Gas

We apply the configuration-interaction method to calculate the spectra of two-component Fermi systems in a harmonic trap, studying the convergence of energies at the unitary interaction limit. We find that for a fixed regularization of the two-body interaction the convergence is exponential or better in the truncation parameter of the many-body space. However, the conventional regularization is found to have poor convergence in the regularization parameter, with an error that scales as a low negative power of this parameter. We propose a new regularization of the two-body interaction that produces exponential convergence for systems of three and four particles. We estimate the ground-state energy of the four-particle system to be $(5.045\pm 0.003)\hslash \omega$.

Figure 1

Convergence in $N_{\max }$ for the $A=3$ ground-state energy. (a) $E_{N_{\max }}^{(q)}$ versus $N_{\max }$ for $q=3$. Open circles correspond to the renormalized contact interaction and solid circles to the interaction defined by (7, 9). (b) $\Delta E_{N_{\max }}^{(3)}$ versus $N_{\max }$ in a logarithmic scale. All energies are in units of $\hslash \omega$.

Figure 2

Convergence of the $q$-regulated energies for the $A=3$ ground state. (a) $E^{(q)}$ versus $q$ for both interactions (symbols and units as in Fig. 1). The dotted line is the exact ground-state energy. (b) The error $|\delta E^{(q)}|$ in a logarithmic scale.

Figure 3

(a) $E^{(q)}$ versus $q$ for the lowest $L=0$ excited state of the $A=3$ system. The inset shows $\delta E^{(q)}$ versus $q$ in a logarithmic scale. Symbols and units as in Fig. 2. (b) $E^{(q)}$ versus $q$ for the $L=0$ ground state of the $A=4$ system. The inset shows $\Delta E^{(q)}$ versus $q$ for the new interaction in a logarithmic scale. The solid line is a linear fit to $q=1,2,3$.