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 $d=2-ε$, we demonstrate that the number of isotropic droplet states scales as $N^{3/2}/ε^{1/2}$, and for $ε=0$, or $d=2$, scales as ${N^2}$. The ground state energies scale as $N^{2 / ε+ 1}$ in $d=2-ε$, 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 $N^{ε/2}E^{1-ε/2}$.

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