Universality in Efimov-associated tetramers in He 4

We calculated, using seven realistic He4-He4 potentials in the literature, the Efimov spectra of the He4 trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schrödinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the He4 trimer and tetramer [Phys. Rev. A 85, 022502 (2012)PLRAAN1050-294710.1103/PhysRevA.85.022502; Phys. Rev. A 85, 062505 (2012)PLRAAN1050-294710.1103/PhysRevA.85.062505]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length rvdW, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius R4≈3rvdW, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius R3≈2rvdW makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the He4 trimer with the realistic potentials.