Effective-interaction approach to the many-boson problem

We show that the convergence behavior of the many-body numerical diagonalization scheme for strongly interacting bosons in a trap can be significantly improved by the Lee-Suzuki method adapted from nuclear physics: One can construct an effective interaction that acts in a space much smaller than the original Hilbert space. In particular for short-ranged forces and strong correlations, the method offers a good estimate of the energy and the excitation spectrum, at a computational cost several orders of magnitude smaller than that required by the standard method.

Figure 1

(Color online) Energies for a system of $N=4$ bosons and total angular momentum $L=0\hslash$, for different cutoffs of the many-body Hilbert space (paramtrized by ${\mathcal{N}}_{\max }$). The range of the interaction is $\sigma =0.1$, and two different strengths $\left(g\right)$ are shown. The blue dashed curves are the results from standard configuration interaction calculations, while the red solid curves are energies obtained using the effective interaction approach. While the standard calculations require a large basis set to give a good energy estimate, the effective interaction provides roughly the same answer with significantly less computational effort. (Please note that for small ${\mathcal{N}}_{\max }$, there are very few states in the $P$ space, and consequently only a few eigenvalues can be obtained.)

Figure 2

(Color online) Same as Fig. 1 but for $N=9$ bosons. Note the rapid convergence of energies using the effective interaction approach. For example, we note that in this case a calculation with ${\mathcal{N}}_{\max }=6$ requires 12 basis states, while ${\mathcal{N}}_{\max }=20$ would correspond to 81 097 states.

Figure 3

(Color online) Same as Fig. 1 but for $N=20$ bosons. For $g=10$, the energy obtained using effective interaction cannot be said to be converged as a function of ${\mathcal{N}}_{\max }$.

Figure 4

(Color online) (a) Energy levels (in units of $\hslash \omega$) for a system of $N=9$ bosons and angular momentum $L=9\hslash$ (in this case there are no possible basis states for ${\mathcal{N}}_{\max }<9$). (b) Energies for a system with $N=9$ and interaction range $\sigma =1$ ($L=0\hslash$ states). See the caption of Fig. 1 for explanations.