Energy and structural properties of $N$-boson clusters attached to three-body Efimov states: Two-body zero-range interactions and the role of the three-body regulator

The low-energy spectrum of $N$-boson clusters with pairwise zero-range interactions is believed to be governed by a three-body parameter. We study the ground state of $N$-boson clusters with infinite two-body $s$-wave scattering length by performing ab initio Monte Carlo simulations. To prevent Thomas collapse, different finite-range three-body regulators are used. The energy and structural properties for the three-body Hamiltonian with two-body zero-range interactions and three-body regulator are in much better agreement with the “ideal zero-range Efimov theory” results than those for Hamiltonian with two-body finite-range interactions. For larger clusters we find that the ground-state energy and structural properties of the Hamiltonian with two-body zero-range interactions and finite-range three-body regulators are not universally determined by the three-body parameter, i.e., dependencies on the specific form of the three-body regulator are observed. For comparison, we consider Hamiltonian with two-body van der Waals interactions and no three-body regulator. For the interactions considered, the ground-state energy of the $N$-body clusters is—if scaled by the three-body ground-state energy—fairly universal, i.e., the dependence on the short-range details of the two-body van der Waals potentials is small. Our results are compared with those in the literature.

Figure 1

Schematic illustration of the energy spectrum for four identical bosons. The x marks the $(1/a_s,E)=(0,0)$ point. The dotted line shows the energy of the weakly bound dimer. The solid lines show different Efimov trimer states, which become unbound on the positive scattering length side at the atom-dimer threshold. The dashed lines show “ground state” and “excited state” tetramers that are attached to each Efimov trimer. These tetramer states hit the dimer-dimer threshold on the positive scattering length side (the energy of the two dimers is shown by the dash-dotted line). It should be noted that the excited tetramer state turns into a virtual state for a certain region of positive scattering lengths [22]; this detail is not reflected in the plot.

Figure 2

Breaking of the scale invariance for the three-boson system at unitarity with three-body hard-core regulator. The circles show the difference between the binding momentum ratio ${\kappa }_3^{\left(n\right)}/{\kappa }_3^{(n+1)}$ of the $n\mathrm{th}$ and $(n+1)\mathrm{th}$ states for the model $2\mathrm{BZR}+3\mathrm{BHC}$ and the ratio $exp(\pi /|s_0\left|\right)=22.6944$ for the model $2\mathrm{BZR}+3\mathrm{BZR}$ as a function of $n$. The solid line shows a fit to the data points. The breaking of the scale invariance becomes weaker with increasing $n$.

Figure 3

Binding momentum characteristics for the three-boson system with three-body power law regulator at unitarity. The circles show the ratio of the binding momentum of two consecutive states for the model $2\mathrm{BZR}+3\mathrm{BR}p$ as a function of $p$. Panel (a) shows the binding momentum ratio for the ground and the first excited states, while panel (b) shows the ratio for the first and the second excited states. The solid and dashed lines show the binding momentum ratio for the models $2\mathrm{BZR}+3\mathrm{BZR}$ and $2\mathrm{BZR}+3\mathrm{BHC}$, respectively.

Figure 4

Angular distributions for three identical bosons at unitarity. The circles, triangles, and squares show the angular distributions $P_{\text{tot}}\left(\theta \right),P_{\text{min}}\left(\theta \right)$, and $P_{\text{max}}\left(\theta \right)$ for the model 2BZR; these distributions are identical to those for the models $2\mathrm{BZR}+3\mathrm{BHC}$ and $2\mathrm{BZR}+3\mathrm{BR}p$. The solid, dotted, and dashed lines show the angular distributions $P_{\text{tot}}\left(\theta \right),P_{\text{min}}\left(\theta \right)$, and $P_{\text{max}}\left(\theta \right)$ for the model 2BG.

Figure 5

Radial density $\rho \left(r\right)$ for three identical bosons at unitarity $(r$ is measured relative to the center of mass of the three-body system). The dashed and solid lines show $\rho \left(r\right)$ for the models $2\mathrm{BZR}+3\mathrm{BZR}$ and $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=6$, respectively.

Figure 6

Energy per particle of $N$-boson clusters at unitarity. (a) Summary of literature results. The dashed and dotted lines show the analytical prediction by Gattobigio and Kievsky [21] and Nicholson [43], respectively. The triangles show the diffusion Monte Carlo (DMC) energies for a Hamiltonian with two-body square-well interaction and repulsive three-body hard-core regulator [23]. The diamonds show the energy for the model 2BG [28]. (b) Summary of our PIMC calculations. The circles and pluses are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=4$ and 8, respectively; the error bars (not shown) are of the order of the symbol sizes. The squares, diamonds, and triangles are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=5,6$, and 7, respectively; the error bars (not shown) are smaller than the symbol sizes. (c) Summary of our calculations for two-body van der Waals models. The circles, crosses, and squares show our DMC results for the models 2BLJ, 2B10-6, and 2B8-6, respectively.

Figure 7

Comparison of our PIMC energies (left) and literature results (right) for selected $N$. Panels (a)–(c) show our PIMC energy per particle for $N$-boson clusters interacting through the model $2\mathrm{BZR}+3\mathrm{BR}p$ as a function of $p$ for $N=6,10,$ and 13, respectively. For comparison, panels (d)–(f) show the energy per particle from the literature for the same $N$. The triangles, diamonds, and squares are from von Stecher [23], Nicholson [43], and Gattobigio et al. [21], respectively. Since the work by Nicholson is restricted to even $N$, comparison for $N=13$ cannot be made.

Figure 8

Expectation value $\overline{r}$ of the pair distance as a function of $N$ for $N$-boson systems interacting through various models. (a) The squares are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=6$. (b) The triangles are for the model 2BG. (c) The circles are for the model 2BLJ. The error bars show the variance of the pair distance. The pair distances are plotted using two different units: (i) the inverse three-body binding momentum (left axis) and (ii) the characteristic length scale of the model Hamiltonian (right axis).

Figure 9

(a) Expectation value $\overline{R}$ of the sub-three-body hyperradius (triple size) as a function of $N$ for $N$-boson systems interacting through the model $2\mathrm{BZR}+3\mathrm{BR}6$. The error bars show the variance of the triple size. (b) Triple distribution function $P_{\text{triple}}\left(R\right)$ for the $N=13$ cluster scaled using the three-body binding momentum ${\kappa }_3$. The solid lines from top to bottom at ${\kappa }_{3}R=0.6$ are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=4,5,6,7,$ and 8. The inset replots the triple distribution functions using the binding momentum ${\kappa }_{13}$ of the $N=13$ droplet. In these units, the triple distribution functions for different $p$ collapse.

Figure 10

(a) Maximum density ${\rho }_{\text{max}}$ as a function of $N$ for $N$-boson systems interacting through various models. The circles, squares, and diamonds are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=5$ (lowest data set), $6,$ and 7 (highest data set), respectively. For comparison, the line is for the model 2BG. (b) Same data as in (a) but replotted as the minimum average interparticle distance ${\left({\rho }_{\text{max}}\right)}^{-1/3}$. The right axis shows the data for the model $2\mathrm{BZR}+3\mathrm{BR}6$ in units of $L_6$.

Figure 11

Scaled pair distribution function $4\pi r^2{P}_{\text{pair}}\left(r\right)$ for $N=13$ bosons interacting through various models. (a) The solid lines from top to bottom at ${\kappa }_{3}r=0.8$ are for the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=4$–8, scaled using the three-body binding momentum ${\kappa }_3$. The inset replots the pair distribution functions scaled using the binding momentum ${\kappa }_{13}$ of the $N=13$ droplet. In these units, the pair distribution functions for different $p$ collapse. (b) The dashed and dotted lines show the scaled distribution functions for the models 2BLJ and 2BG, respectively, using the three-body binding momentum ${\kappa }_3$.

Figure 12

Angular distribution $P_{\text{tot}}\left(\theta \right)$ for $N$-boson clusters interacting through the model $2\mathrm{BZR}+3\mathrm{BR}p$ with $p=6$. The lines from top to bottom at $\theta =0$ are for $N=5,6,7,9,$ and 13.

Figure 13

Assessing the applicability of Eq. (4) for $N$-boson systems with two-body finite-range interactions at unitarity. Circles and triangles show the normalized difference $({\kappa }_N^{\text{appr}}-{\kappa }_N)/{\kappa }_N$ for the models 2BG and 2BLJ, respectively, as a function of the number of particles $N$.