离散标度不变性的$N$-玻色子谱
$N$-boson spectrum from a Discrete Scale Invariance
摘要：
我们提出了分析的$N$-玻色子光谱计算使用软二体势的强度已被改变，以覆盖扩展范围的正，负值的二体散射长度$a$接近酉极限。光谱显示了一个树结构的两个状态，一个浅，一个深，连接到基态的系统少一个粒子。它由一个独特的普适函数$Δ(ξ)$控制，这个函数在三玻色子的情况下已经为人所知。在三粒子系统中，由二体和三体结合能之比$E_3/E_2=\tan^2ξ$确定的角度$ξ$表征了系统的离散标度不变性。将角的定义推广到$N$-体系统为$E_N/E_2=\tan^2ξ$，我们用这个变量研究了$N$-玻色子谱.分析的结果，获得高达$N=16$玻色子，使我们能够提取一个通用的公式接近酉极限的系统的能级。有趣的是，在$a$的固定值处观察到作为$N$的函数的通用函数的线性依赖性。我们表明，有限范围的计算结果的范围内的修正，产生的散射长度$a$和一个特定形式的通用功能之间的线性关系的移位。我们还评论的普遍关系的适用性的限制。
英文摘要：
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, $Δ(ξ)$, already known in the case of three bosons. In the three-particle system the angle $ξ$, determined by the ratio of the two- and three-body binding energies $E_3/E_2=\tan^2ξ$, characterizes the Discrete Scale Invariance of the system. Extending the definition of the angle to the $N$-body system as $E_N/E_2=\tan^2ξ$, 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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