Effective 3-Body Interaction for Mean-Field and Density-Functional Theory

Density functionals for nuclei usually include an effective 3-body interaction that depends on a fractional power of the density. Using insights from the many-body theory of the low-density two-component Fermi gas, we consider a new, nonlocal, form for the energy functional that is consistent with the Fock-space representation of interaction operators. In particular, there is a unique spatially nonlocal generalization of the contact form of the interaction that preserves the ${\rho }^{7/3}$ density dependence required by the many-body theory. We calculate the ground-state energies for particles in a harmonic trap by using the nonlocal induced 3-body interaction and compare them to numerically accurate Green’s function Monte Carlo calculations. Using no free parameters, we find that a nonlocality in the space domain provides a better description of the weak-coupling regime than the local-density approximation.

Figure 1

(a) A contribution to the effective two-particle interaction in second-order perturbation. (b) The same perturbation contribution, in a many-body context with orbitals $k_1$, $k_2$, and $k_3$ occupied.

Figure 2

The ground-state energy (in units of $\hslash \omega$) of a system of 8 particles plotted as a function of the scattering length (in units of $\sqrt{\hslash /m\omega }$).