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 (ξ ), already known in the case of three bosons. In the three-particle system the angle ξ , determined by the ratio of the twoand three-body binding energies E3/E2 = tan2 ξ , characterizes the discrete scale invariance of the system. Extending the definition of the angle to the N -body system as EN/E2 = tan2 ξ , 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.