One-dimensional three-boson problem with two- and three-body interactions

We solve the three-boson problem with contact two- and three-body interactions in one dimension and analytically calculate the ground and excited trimer-state energies. Then, by using the diffusion Monte Carlo technique, we calculate the binding energy of three dimers formed in a one-dimensional Bose-Bose or Fermi-Bose mixture with attractive interspecies and repulsive intraspecies interactions. Combining these results with our three-body analytics, we extract the three-dimer scattering length close to the dimer-dimer zero crossing. In both considered cases, the three-dimer interaction turns out to be repulsive. Our results constitute a concrete proposal for obtaining a one-dimensional gas with a pure three-body repulsion.

Figure 1

The trimer energy in units of $|E_2|=1/m{a}_2^2$ vs $ln(a_3/|a_2\left|\right)$ for positive $a_2$ (solid black). The red filling indicates the atom-dimer scattering continuum, the blue dash-dotted lines correspond to $E_3=4E_2$ valid in the absence of the three-body force, and the red dotted line shows the asymptote $E_3=-4e^{-2\gamma }/m{a}_3^2$ valid in the absence of the two-body force. The black dashed curve is the trimer energy for $a_2<0$. The repulsive two-body interaction in this case pushes the trimer into the three-atom continuum at a finite value of $ln(a_3/|a_2\left|\right)$ (see text).

Figure 2

$E_3/\left|E_2\right|$ vs $1/ln\left(\sqrt{2|E_2|}{a}_3\right)$ (same as Fig. 1 except for the inverse of the horizontal axis) for one-dimensional dimers. Here $E_2$ and $E_3$ are the tetramer and hexamer energies measured relative to the two- and three-dimer thresholds, respectively. The solid curve is the prediction of Eq. (6) and the dashed curve is a fit, which includes finite-dimer-size effects into account (see text). The dash-dotted line is the McGuire result $E_3=4E_2$ for three pointlike bosons with no three-body interaction. The red squares are the DMC data for case BB plotted using $a_3=0.01a_{\uparrow \downarrow }$ and the blue circles stand for case FB with $a_3=0.03a_{\uparrow \downarrow }$. The error bars are larger in the latter case because of the larger statistical noise induced by the nodal surface imposed by the Fermi statistics.