$N$-boson spectrum from a Discrete Scale Invariance

Abstract

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.

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Cited by in corpus (6)

• Generalized Efimov scenario for heavy-light mixtures
• Effective Field Theory for Few-Boson Systems
• Ground-state properties of unitary bosons: from clusters to matter
• Quantum Monte Carlo studies of a trimer scaling function with microscopic two- and three-body interactions
• Efimov universality with Coulomb interaction
• Hyperspherical cluster model for bosons: application to sub-threshold halo states in helium drops