In this work 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 inter-particle 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 LM2M2 potential. 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 in 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. PACS numbers: 31.15.xj, 31.15.xt, 36.90.+f, 34.10.+x Systems composed by few helium atoms have been object of intense investigation from a theoretical and experimental point of view. The existence of the He-He molecule was experimentally established using diffraction experiments [1-4]. Its binding energy E 2b has been estimated to be around 1 mK and its scattering length a 0 around 190 a.u. This makes the He-He molecule one of the biggest diatomic molecules. On the theoretical side, several He-He potentials have been proposed; in spite of different details and derivations, all of them share the common feature of a sharp repulsion below an inter-particle distance of approximately 5 a.u..Another important characteristic of the He-He interactions is their effective range r 0 ≈ 13 a.u.. Accordingly, the ratio a 0 /r 0 is large enough (> 10) to place small helium clusters into the frame of Efimov physics [5,6]. As shown by Efimov, when at least two of the two-body subsystems present an infinitely large scattering length (or zero binding energy) an infinite sequence of bound states (called Efimov states) appear in the three-body system; their binding energies scale in a geometrical way and they accumulate at zero energy. The scaling factor, e −2π/s 0 ≈ 1/515.03, is universal and depends only on the ratio between particle masses (for three identical bosons s 0 ≈ 1.00624), not on the details of the two-body interaction (see Ref.[7] for a review). For a finite a 0 /r 0 ratio, the number of the Efimov states has been estimated to be N = (s 0 /π)ln|a 0 /r 0 | [6]; for example, the (bosonic) three 4 He system presents an excited state just below the atom-dimer threshold that has been identified as an Efimov state.Triggered by this interesting fact, several investigations of the helium trimer have been produced, establishing that its ex...

We present the exact solution for the many-body wavefunction of a one-dimensional mixture of bosons and spin-polarized fermions with equal masses and infinitely strong repulsive interactions under external confinement. Such a model displays a large degeneracy of the ground state. Using a generalized Bose-Fermi mapping we find the solution for the whole set of ground-state wavefunctions of the degenerate manifold and we characterize them according to group-symmetry considerations. We find that the density profile and the momentum distribution depends on the symmetry of the solution. By combining the wavefunctions of the degenerate manifold with suitable symmetry and guided by the strong-coupling form of the Bethe-Ansatz solution for the homogeneous system we propose an analytic expression for the many-body wavefunction of the inhomogeneous system which well describes the ground state at finite, large and equal interactions strengths, as validated by numerical simulations.PACS numbers: 67.85.Pq

In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with A = 2,3,4,5,6 particles interacting through a soft interparticle potential. In order to make contact with a real system, we use an attractive Gaussian potential that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with one of the most widely used 4 He-4 He interactions, the LM2M2 potential of Aziz and Slaman. The intensity of the potential is varied in order to explore the clusters' spectra in different regions with large positive and large negative values of the two-body scattering length. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two-and three-body potential, we have calculated the spectrum of the four-, five-, and six-particle systems. In all the regions explored, we have found that these systems present two states, one deep and one shallow close to the A − 1 threshold. Some universal relations between the energy levels are extracted; in particular, we have estimated the universal ratios between thresholds of the three-, four-, and five-particle continua using the two-body Gaussian potential. They agree with recent measurements and theoretical predictions.

Universal behaviour has been found inside the window of Efimov physics for systems with N = 4, 5, 6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced in [A. Kievsky and M. Gattobigio, Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground-and excited-energy of N ≤ 6 systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.Universality is one of the concepts that have attracted physicists along the years. Different systems, having even different energy scales, share common behaviours. The most celebrated example of universality comes from the investigation of critical phenomena [1,2]: at the critical point, materials that are governed by different microscopic interactions share the same macroscopic laws, for instance the same critical exponents. The theoretical framework to understand universality has been provided by the renormalization group (RG); the critical point is mapped onto a fixed point of a dynamical system, the RG flow, whose phase space is represented by Hamiltonians. At the critical point the systems have scale-invariant (SI) symmetry, forcing all of the observables to be exponential functions of the control parameter. A consequence of SI symmetry is the scaling of the observables: for different materials, in the same class of universality, a selected observable can be represented as a function of the control parameter and, provided that both the observable and the control parameter are scaled by some materialdependent factor, all representations collapse onto a single universal curve [3].