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Small clusters of He atoms in finite-cutoff EFT

Betzalel Bazak

TL;DR

This work investigates universal few-body physics in weakly bound $^4$He clusters using a finite-cutoff EFT. By calibrating LO two- and three-body interactions to LM2M2 observables and solving the $A$-body Schrödinger equation up to $A=8$ with the stochastic variational method, the authors obtain binding energies and atom-cluster scattering parameters that agree with LM2M2 results to a few percent. Scattering data are extracted from trapped spectra via the Busch relation and effective-range expansion, enabling insight into atom–dimer, atom–trimer, and larger-cluster interactions within the EFT framework. The results demonstrate that finite-cutoff EFT provides an efficient, predictive connection between realistic helium potentials and universal few-body physics, including the Efimov-related cluster structure, and establish a foundation for extending the approach to higher orders and larger systems.

Abstract

Small clusters of $^4$He atoms provide a paradigmatic setting for exploring universal phenomena in few-body quantum systems with large scattering length. Their weakly bound states serve as ideal test cases for studying Efimov physics and the emergence of universality beyond the three-body sector. In this work, we investigate few-$^4$He systems within a finite-cutoff effective field theory (EFT) framework. The EFT interactions are calibrated to reproduce low-energy observables obtained from the realistic LM2M2 potential, enabling a direct and systematic comparison between the two approaches. We demonstrate that, for suitably chosen finite cutoffs, the empirical effective range is accurately reproduced already at leading order, achieving next-to-leading-order precision without explicit higher-order corrections. Using these interactions, we solve the Schrödinger equation for systems of a few atoms, obtaining binding energies and scattering observables in excellent agreement with results derived from realistic interatomic potentials. In particular, we compute atom--tetramer scattering parameters and binding energies of clusters up to eight atoms, thereby extending the EFT description to larger helium systems. Our findings establish a quantitative bridge between realistic helium potentials and finite-cutoff EFT, showing that the latter provides an efficient and predictive framework for describing few-body universality in weakly bound quantum systems.

Small clusters of He atoms in finite-cutoff EFT

TL;DR

This work investigates universal few-body physics in weakly bound He clusters using a finite-cutoff EFT. By calibrating LO two- and three-body interactions to LM2M2 observables and solving the -body Schrödinger equation up to with the stochastic variational method, the authors obtain binding energies and atom-cluster scattering parameters that agree with LM2M2 results to a few percent. Scattering data are extracted from trapped spectra via the Busch relation and effective-range expansion, enabling insight into atom–dimer, atom–trimer, and larger-cluster interactions within the EFT framework. The results demonstrate that finite-cutoff EFT provides an efficient, predictive connection between realistic helium potentials and universal few-body physics, including the Efimov-related cluster structure, and establish a foundation for extending the approach to higher orders and larger systems.

Abstract

Small clusters of He atoms provide a paradigmatic setting for exploring universal phenomena in few-body quantum systems with large scattering length. Their weakly bound states serve as ideal test cases for studying Efimov physics and the emergence of universality beyond the three-body sector. In this work, we investigate few-He systems within a finite-cutoff effective field theory (EFT) framework. The EFT interactions are calibrated to reproduce low-energy observables obtained from the realistic LM2M2 potential, enabling a direct and systematic comparison between the two approaches. We demonstrate that, for suitably chosen finite cutoffs, the empirical effective range is accurately reproduced already at leading order, achieving next-to-leading-order precision without explicit higher-order corrections. Using these interactions, we solve the Schrödinger equation for systems of a few atoms, obtaining binding energies and scattering observables in excellent agreement with results derived from realistic interatomic potentials. In particular, we compute atom--tetramer scattering parameters and binding energies of clusters up to eight atoms, thereby extending the EFT description to larger helium systems. Our findings establish a quantitative bridge between realistic helium potentials and finite-cutoff EFT, showing that the latter provides an efficient and predictive framework for describing few-body universality in weakly bound quantum systems.

Paper Structure

This paper contains 12 sections, 20 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Model and Methods
  3. Two-body interaction
  4. Three-body interaction
  5. Few-body calculations
  6. Results
  7. Two-atom system
  8. Three-atom system
  9. Four-atom system
  10. Five-atom system
  11. Larger clusters
  12. Conclusion

Figures (2)

  • Figure 1: Effective-range expansion for $S$-wave atom--dimer scattering. Results obtained from calculations in a weak harmonic trap using Eq. \ref{['Busch']} are shown, together with second-order polynomial fits (solid lines).
  • Figure 2: Energy spectrum of three $^4$He atoms in a harmonic trap as a function of the trap frequency. The nearly constant ground-state energy corresponds to the compact trimer, whereas the excited trimer exhibits a stronger dependence on the trap strength due to its extended structure. Higher-lying states correspond to atom--dimer excitations and to the three-free-atom continuum. The free-space binding energies are indicated by dashed lines, denoted as $t$ for the deeply bound trimer, $t^*$ for the shallow trimer, $d{+}a$ for the atom--dimer threshold, and $a{+}a{+}a$ for the three-atom breakup threshold.