量子N玻色子态与孤子液滴的量子化运动：低维普适标度性质
Quantum N-Boson States and Quantized Motion of Solitonic Droplets: Universal Scaling Properties in Low Dimensions
摘要：
在这篇文章中，我们说明了一族的N吸引玻色子原子在大$N$的限制下的解的标度性质。这些解代表了原子液滴中孤子自由度的量子化动力学。在低于2的维度或$d=2-ε$中，我们证明了各向同性液滴状态的数量缩放为$N^{3/2}/ε^{1/2}$，并且对于$ε=0$或$d=2$，缩放为${N^2}$。在$d=2-ε$中，基态能量按$N^{2 / ε+ 1}$标度，而在$d=2$中，按N的指数函数标度。我们获得了普适能谱和广义Tjon关系;它们的标度性质是由短距离量子玻色子场的渐进自由唯一确定的，这是低维中的一个独特特征。我们还研究了量子回路校正的效果，从各种虚拟的过程中产生的，并表明所得到的寿命范围广泛的激发态尺度为$N^{ε/2}E^{1-ε/2}$。
英文摘要：
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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