$N$-boson spectrum from a Discrete Scale Invariance

We present the 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 an unique universal function, $\Delta(\xi)$, already known in the case of three bosons. In the three-particle system the angle $\xi$, determined by the ratio of the two- and three-body binding energies $E_3/E_2=\tan^2\xi$, characterizes the Discrete Scale Invariance of the system. Extending the definition of the angle to the $N$-body system as $E_N/E_2=\tan^2\xi$, 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.