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

TL;DR

This paper explores the universal scaling laws of quantum N-boson droplet states in low dimensions, revealing how their energies and state counts depend on particle number and dimensionality, with implications for quantum field behavior.

Contribution

It introduces universal scaling properties and energy spectra of bosonic droplet states in low dimensions, highlighting the role of asymptotic freedom and quantum corrections.

Findings

Number of droplet states scales as N^{3/2}/ε^{1/2} in d=2-ε

Ground state energies scale as N^{2/ε + 1} in d=2-ε

Lifetime of excited states scales as N^{ε/2}E^{1-ε/2}

Abstract

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 $N^{\epsilon/2}E^{1-\epsilon/2}$.