M. Gattobigio

Spectra of helium clusters with up to six atoms using soft-core potentials

In this paper, we investigate small clusters of helium atoms using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6 atoms with an interparticle potential which does not present a strong repulsion at short distances. We use an attractive Gaussian potential that reproduces the values of the dimer binding energy, the atom-atom scattering length, and the effective range obtained with one of the widely used He-He interactions, the Aziz and Slaman potential, called LM2M2. In systems with more than two atoms, we consider a repulsive three-body force that, by construction, reproduces the trimer binding energy of the LM2M2 potential. With this model, consisting of the sum of a two- and three-body potential, we have calculated the spectrum of clusters formed by four, five, and six helium atoms. We have found that these systems present two bound states, one deep and one shallow, close to the threshold fixed by the energy of the (A-1)-atom system. Universal relations between the energies of the excited state of the A-atom system and the ground-state energy of the (A-1)-atom system are extracted, as well as the ratio between the ground state of the A-atom system and the ground-state energy of the trimer.

N-boson spectrum from a discrete scale invariance

We present an analysis of the N-boson spectrum computed using a soft two-body potential, the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length a close to the unitary limit. The spectrum shows a tree structure of two states, one shallow and one deep, attached to the ground state of the system with one less particle. It is governed by a unique universal function Δ(ξ), already known in the case of three bosons. In the three-particle system the angle ξ, determined by the ratio of the two- and three-body binding energies E3/E2=tan2ξ, characterizes the discrete scale invariance of the system. Extending the definition of the angle to the N-body system as EN/E2=tan2ξ, we study the N-boson spectrum in terms of this variable. The analysis of the results, obtained for up to N=16 bosons, allows us to extract a general formula for the energy levels of the system close to the unitary limit. Interestingly, a linear dependence of the universal function as a function of N is observed at fixed values of a. We show that the finite-range nature of the calculations results in the range corrections that generate a shift of the linear relation between the scattering length a and a particular form of the universal function. We also comment on the limits of applicability of the universal relations.

Harmonic hyperspherical basis for identical particles without permutational symmetry

The hyperspherical harmonic basis is used to describe bound states in an

Implications of Efimov physics for the description of three and four nucleons in chiral effective field theory

A

Efimov Physics with {1/2} Spin-Isospin Fermions

--body system. The approach presented here is based on the representation of the potential energy in terms of hyperspherical harmonic functions. Using this representation, the matrix elements between the basis elements are simple, and the potential energy is presented in a compact form, well suited for numerical implementation. The basis is neither symmetrized nor antisymmetrized, as required in the case of identical particles; however, after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect the symmetries present in it, and the identification of the physical states is possible, as it will be shown in specific cases. We have in mind applications to atomic, molecular, and nuclear few-body systems in which symmetry breaking terms are present in the Hamiltonian; their inclusion is straightforward in the present method. As an example we solve the case of three and four particles interacting through a short-range central interaction and Coulomb potential.

Matching universal behavior with potential models

In chiral effective field theory the leading order (LO) nucleon-nucleon potential includes two contact terms, in the two spin channels

Few-nucleon bound states using the unsymmetrized HH expansion

S=0,1

Universal Behavior of Few-Boson Systems Using Potential Models

, and the one-pion-exchange potential. When the pion degrees of freedom are integrated out, as in the pionless effective field theory, the LO potential includes two contact terms only. In the three-nucleon system, the pionless theory includes a three-nucleon contact term interaction at LO whereas the chiral effective theory does not. Accordingly arbitrary differences could be observed in the LO description of three- and four-nucleon binding energies. We analyze the two theories at LO and conclude that a three-nucleon contact term is necessary at this order in both theories. In turn this implies that subleading three-nucleon contact terms should be promoted to lower orders. Furthermore this analysis shows that one single low energy constant might be sufficient to explain the large values of the singlet and triplet scattering lengths.

Energy spectra of small bosonic clusters having a large two-body scattering length

The structure of few-fermion systems having