Quantum N-Boson States and Quantized Motion of Solitonic Droplets: Universal Scaling Properties in Low Dimensions

In this article, we illustrate the scaling properties of a family of solutions for N attractive bosonic atoms in the limit of large $N$. These solutions represent the quantized dynamics of solitonic degrees of freedom in atomic droplets. In dimensions lower than two, or $d=2-\epsilon$, we demonstrate that the number of isotropic droplet states scales as $N^{3/2}/\epsilon^{1/2}$, and for $\epsilon=0$, or $d=2$, scales as ${N^2}$. The ground state energies scale as $N^{2 / \epsilon + 1}$ in $d=2-\epsilon$, and when $d=2$, scale as an exponential function of N. We obtain the universal energy spectra and the generalized Tjon relation; their scaling properties are uniquely determined by the asymptotic freedom of quantum bosonic fields at short distances, a distinct feature in low dimensions. We also investigate the effect of quantum loop corrections that arise from various virtual processes and show that the resultant lifetime for a wide range of excited states scales as $N^{\epsilon/2}E^{1-\epsilon/2}$.