Universality in few-body systems with large scattering length
具有大散射长度的少体系统中的普遍性

Contents 目录

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
   导 言. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261
2. Scattering concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
   散射的概念. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264

3. Renormalization group concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3.1. Efimov effect ..... 275
   重正化群概念 275 3.1.275 3.1.埃菲莫夫效应 .....275
   3.2. The resonant and scaling limits ..... 276
   3.2.共振和缩放极限 .....276
   3.3. Universality in critical phenomena ..... 279
   3.3.临界现象的普遍性 .....279
   3.4. Renormalization group limit cycles ..... 281
   3.4.重正化群极限循环 .....281
   3.5. Universality for large scattering length ... ..... 285
   3.5.大散射长度的普遍性 ... .....285
   4. Universality for two identical bosons ..... 286
   4.两个相同玻色子的普遍性 .....286
   4.1. Atom-atom scattering ..... 286
   4.1.原子-原子散射 .....286
   4.2. The shallow dimer. ..... 287
   4.2.浅二聚体。.....287
   4.3. Continuous scaling symmetry ..... 287
   4.3.连续缩放对称 .....287
   4.4. Scaling violations ..... 289
   4.4.违反缩放比例 .....289
   4.5. Theoretical approaches ..... 290
   4.5.理论方法 .....290
   5. Hyperspherical formalism ..... 291
   5.超球面形式主义 .....291
   5.1. Hyperspherical coordinates ..... 291
   5.1.超球面坐标 .....291
   5.2. Low-energy Faddeev equation ... ..... 294
   5.2.低能 Faddeev 方程 ... .....294
   5.3. Hyperspherical potentials ... ..... 297
   5.3.超球面势...... .....297
   5.4. Boundary condition at short distances ..... 300
   5.4.短距离的边界条件 .....300
   5.5. Efimov states in the resonant limit ..... 302
   5.5.共振极限中的埃菲莫夫态 .....302
   5.6. Efimov states near the atom-dimer threshold ..... 305
   5.6.原子二聚体阈值附近的埃菲莫夫态 .....305
   6. Universality for three identical bosons ..... 306
   6.三个相同玻色子的普遍性 .....306
   6.1. Discrete scaling symmetry ... ..... 306
   6.1.离散缩放对称性 ... .....306
   6.2. Efimov's radial law ..... 308
   6.2.埃菲莫夫径向定律 .....308
   6.3. Binding energies of Efimov states ... ..... 311
   6.3.埃菲莫夫态的结合能 ... .....311
   6.4. Atom-dimer elastic scattering ... ..... 315
   6.4.原子-二聚体弹性散射 ... .....315
   6.5. Three-body recombination ..... 319
   6.5.三体重组 .....319
   6.6. Three-atom elastic scattering ..... 322
   6.6.三原子弹性散射 .....322
   6.7. Helium atoms ..... 325
   6.7.氦原子 .....325
   6.8. Universal scaling curves ..... 327
   6.8.通用缩放曲线 .....327
   7. Effects of deep two-body bound states ..... 329
   7.深度二体束缚态的影响 .....329
   7.1. Extension of Efimov's radial law .... ..... 329
   7.1.埃菲莫夫径向定律的扩展 .... .....329
   7.2. Widths of Efimov states ..... 332
   7.2.埃菲莫夫状态的宽度 .....332
   7.3. Atom-dimer elastic scattering ..... 332
   7.3.原子-二聚体弹性散射 .....332
   7.4. Three-body recombination into the shallow dimer ..... 333
   7.4.三体重组进入浅二聚体 .....333
   7.5. Three-body recombination into deep molecules ..... 335
   7.5.三体重组进入深分子 .....335
   7.6. Dimer relaxation into deep molecules ... ..... 336
   7.6.二聚体弛豫成深分子...... .....336
   8. Effective field theory ... ..... 338
   8.有效场论 ... .....338
   8.1. Effective field theories ..... 338
   8.1.有效场理论 .....338
   8.2. Effective theories in quantum mechanics ..... 339
   8.2.量子力学中的有效理论 .....339
   8.3. Effective field theories for atoms ... ..... 342
   8.3.原子的有效场理论 ... .....342
   8.4. Two-body problem ..... 344
   8.4.二体问题 .....344
   8.5. Three-body problem ..... 349
   8.5.三体问题 .....349
   8.6. The diatom field trick ..... 352
   8.6.硅藻领域的技巧 .....352
   8.7. STM integral equation ..... 354
   8.7.STM 积分方程 .....354
   8.8. Three-body observables ..... 357
   8.8.三体观测值 .....357
   8.9. Renormalization group limit cycle ..... 358
   8.9.重正化群极限循环 .....358
   8.10. Effects of deep 2-body bound states ..... 361
   8.10.深度二体束缚态的影响 .....361
   9. Universality in other three-body systems ..... 363
   9.其他三体系统中的普遍性 .....363
   9.1. Unequal scattering lengths ... ..... 363
   9.1.不等散射长度 ... .....363
   9.2. Unequal masses ... ..... 365
   9.2.质量不相等 ... .....365
   9.3. Two identical fermions ... ..... 370
   9.3.两个相同的费米子 ... .....370
   9.4. Particles with a spin symmetry ..... 371
   9.4.具有自旋对称性的粒子 .....371
   9.5. Dimensions other than 3 ..... 372
   9.5.3 ..... 以外的尺寸372
   9.6. Few-nucleon problem ..... 373
   9.6.少核问题 .....373
   9.7. Halo nuclei ..... 376
   9.7.晕核 .....376
   10. Frontiers of universality ..... 377
   10.普遍性的前沿 .....377
   10.1. The -body problem for ..... 377
   10.1. -body 问题的 .....377
   10.2. Effective-range corrections ..... 380
   10.2.有效范围修正 .....380
   10.3. Large P-wave scattering length ..... 382
   10.3.大 P 波散射长度 .....382
   10.4. Scattering models ..... 383
   10.4.散射模型 .....383
   Acknowledgments ..... 385
   致谢 .....385
   References ..... 385 参考文献 .....385

The scattering of two particles with short-range interactions at sufficiently low energy is determined by their -wave scattering length, which is commonly denoted by . By low energy, we mean energy close to the scattering threshold for the two particles. The energy is sufficiently low if the de Broglie wavelengths of the particles are large compared to the range of the interaction. The scattering length is important not only for 2-body systems, but also for few-body and many-body systems. If all the constituents of a few-body system have sufficiently low energy, its scattering properties are determined primarily by . A many-body system has properties determined by if its constituents have not only sufficiently low energies but also separations that are large compared to the range of the interaction. A classic example is the interaction energy per particle in the ground state of a sufficiently dilute homogeneous Bose-Einstein condensate:
两个具有短程相互作用的粒子在足够低的能量下的散射由它们的 - 波散射长度决定，通常用 表示。所谓低能量，是指接近两个粒子散射阈值的能量。如果粒子的德布罗格利波长与相互作用的范围相比较大，则能量足够低。散射长度 不仅对二体系统很重要，对少体和多体系统也很重要。如果几体系统的所有成分都具有足够低的能量，其散射特性主要由 决定。如果多体系统的组成成分不仅具有足够低的能量，而且相较于相互作用的范围而言，它们之间的距离也很大，那么该系统的性质就由 决定。一个典型的例子是充分稀释的均相玻色-爱因斯坦凝聚态基态中每个粒子的相互作用能：

where and are the energy density and number density, respectively. In the literature on Bose-Einstein condensates, properties of the many-body system that are determined by the scattering length are called universal [1]. The expression for the energy per particle in Eq. (1) is an example of a universal quantity. Corrections to such a quantity from the effective range and other details of the interaction are called nonuniversal. Universality in physics generally refers to situations in which systems that are very different at short distances have identical long-distance behavior. In the case of a dilute Bose-Einstein condensate, the constituents may have completely different internal structure and completely different interactions, but the many-body systems will have the same macroscopic behavior if their scattering lengths are the same.
其中 和 分别是能量密度和数量密度。在有关玻色-爱因斯坦凝聚态的文献中，由散射长度决定的多体系统特性被称为普适性[1]。式（1）中每个粒子的能量表达式就是通用量的一个例子。由有效范围和相互作用的其他细节对这种量的修正称为非普遍性。物理学中的普遍性通常是指在短距离上有很大差异的系统在长距离上具有相同的行为。在稀释玻色-爱因斯坦凝聚态的情况下，各成分可能具有完全不同的内部结构和完全不同的相互作用，但如果它们的散射长度相同，多体系统将具有相同的宏观行为。

Generically, the scattering length is comparable in magnitude to the range of the interaction: . Universality in the generic case is essentially a perturbative weak-coupling phenomenon. The scattering length plays the role of a coupling constant. Universal properties can be calculated as expansions in the dimensionless combination , where is an appropriate wave number variable. For the energy per particle in the dilute Bose-Einstein condensate, the wave number variable is the inverse of the coherence length: . The weak-coupling expansion parameter is therefore proportional to the diluteness parameter . The order and corrections to Eq. (1) are both universal [2]. Nonuniversal effects, in the form of sensitivity to 3-body physics, appear first in the order correction [3].
一般来说，散射长度 的大小与相互作用的范围 相当： 。一般情况下的普遍性本质上是一种微扰弱耦合现象。散射长度 起着耦合常数的作用。普遍性可以通过无量纲组合 的展开来计算，其中 是一个合适的波数变量。对于稀释玻色-爱因斯坦凝聚态中每个粒子的能量，波数变量是相干长度的倒数： 。因此，弱耦合膨胀参数 与稀释参数 成正比。 和 对式（1）的修正都是普遍的[2]。非普遍效应，即对三体物理的敏感性，首先出现在 修正[3]。

In exceptional cases, the scattering length can be much larger in magnitude than the range of the interaction: . Such a scattering length necessarily requires a fine-tuning. There is some parameter characterizing the interactions that if tuned to a critical value would give a divergent scattering length . Universality continues to be applicable in the case of a large scattering length, but it is a much richer phenomenon. We continue to define low energy by the condition that the de Broglie wavelengths of the constituents be large compared to , but they can be comparable to . Physical observables are called universal if they are insensitive to the range and other details of the short-range interaction. In the 2-body sector, the consequences of universality are simple but nontrivial. For example, in the case of identical bosons with , there is a 2-body bound state near the scattering threshold with binding energy
在特殊情况下，散射长度可以比相互作用的范围大得多： 。这样的散射长度必然需要微调。有一些表征相互作用的参数，如果调整到临界值，就会产生发散的散射长度： 。在散射长度较大的情况下，普遍性仍然适用，但它是一种更为丰富的现象。我们继续用这样一个条件来定义低能：与 相比，成分的德布罗格利波长要大，但它们可以与 相媲美。如果物理观测值对短程相互作用的范围和其他细节不敏感，那么它们就被称为普适观测值。在二体部门，普遍性的后果很简单，但并不复杂。例如，在相同玻色子的情况下， ，在散射阈值附近存在一个结合能为

The corrections to this formula are parametrically small: they are suppressed by powers of . Note the nonperturbative dependence of the binding energy on the interaction parameter . This reflects the fact that universality in the case of a large scattering length is a nonperturbative strong-coupling phenomenon. It should therefore not be a complete surprise that counterintuitive effects can arise when there is a large scattering length.
该公式的修正参数很小：它们被 的幂所抑制。请注意结合能对相互作用参数的非全扰动依赖性 。这反映了一个事实，即大散射长度情况下的普遍性是一种非微扰的强耦合现象。因此，当散射长度较大时，出现反直觉效应也就不足为奇了。

A classic example of a system with a large scattering length is atoms, whose scattering length is more than a factor of 10 larger than the range of the interaction. More examples ranging from atomic physics to nuclear and particle physics are discussed in detail in Sections 2.3 and 2.4.
散射长度大的系统的一个典型例子是 原子，其散射长度比相互作用的范围大 10 倍以上。第 2.3 节和第 2.4 节将详细讨论从原子物理到核物理和粒子物理的更多例子。

The first strong evidence for universality in the 3-body system was the discovery by Vitaly Efimov in 1970 of the Efimov effect a remarkable feature of the 3-body spectrum for identical bosons with a short-range interaction and a large scattering length . In the resonant limit , there is a 2-body bound state exactly at the 2-body scattering threshold . Remarkably, there are also infinitely many, arbitrarily-shallow 3-body bound states with binding energies that have an accumulation point at . As the threshold is approached, the ratio of the binding energies of successive states approaches a universal constant:
三体系统普遍性的第一个有力证据是维塔利-埃菲莫夫 1970 年发现的埃菲莫夫效应 ，这是具有短程相互作用和大散射长度的相同玻色子的三体频谱的一个显著特点 。在共振极限 ，恰好在二体散射阈值处存在一个二体束缚态 。值得注意的是，还有无限多的任意浅的 3 体结合态，其结合能为 ，其累积点为 。随着阈值的接近，连续状态的结合能之比会接近一个普遍常数：

The universal ratio in Eq. (3) is independent of the mass or structure of the identical particles and independent of the form of their short-range interaction. The Efimov effect is not unique to the system of three identical bosons. It can also occur in other 3-body systems if at least two of the three pairs have a large S-wave scattering length. If the Efimov effect occurs, there are infinitely many, arbitrarily-shallow 3-body bound states in the resonant limit . Their spectrum is characterized by an asymptotic discrete scaling symmetry, although the numerical value of the discrete scaling factor may differ from the value in Eq. (3).
式（3）中的普遍比率与相同粒子的质量或结构无关，也与它们的短程相互作用形式无关。埃菲莫夫效应并非三个相同玻色子系统所独有。如果三对粒子中至少有两对具有较大的 S 波散射长度，它也会出现在其他三体系统中。如果发生了埃菲莫夫效应，那么在共振极限 中就会存在无限多的任意浅的三体束缚态。它们的频谱具有渐近离散缩放对称性的特征，尽管离散缩放因子的数值可能与公式 (3) 中的数值不同。

For systems in which the Efimov effect occurs, it is convenient to relax the traditional definition of universal which allows dependence on the scattering length only. In the resonant limit , the scattering length no longer provides a scale. However, the discrete Efimov spectrum in Eq. (3) implies the existence of a scale. For example, one can define a wave number by expressing the asymptotic spectrum in the form
对于发生埃菲莫夫效应的系统，放宽传统的普遍性定义是很方便的，因为传统定义只允许依赖于散射长度。在共振极限 中，散射长度不再提供尺度。然而，公式 (3) 中的离散埃菲莫夫谱意味着尺度的存在。例如，我们可以通过表达渐近谱的形式来定义波数

for some integer . If the scattering length is large but finite, the spectrum of Efimov states will necessarily depend on both and the 3-body parameter . Thus although the existence of the Efimov states is universal in the traditional sense, their binding energies are not. The dependence of few-body observables on is qualitatively different from the dependence on typical nonuniversal parameters such as the effective range. As , the dependence on typical nonuniversal parameters decreases as positive powers of , where is the range of the interaction. In contrast, the dependence on not only does not disappear in the resonant limit, but it takes a particularly remarkable form. Few-body observables are log-periodic functions of , i.e. the dependence on enters only through trigonometric functions of . For example, the asymptotic spectrum in Eq. (4) consists of the zeroes of a log-periodic function:
对于某个整数 。如果散射长度很大但有限，那么埃菲莫夫态的谱必然取决于 和三体参数 。因此，尽管埃菲莫夫态的存在在传统意义上是普遍的，但它们的结合能却不是。少子体观测值对 的依赖与对典型非普遍参数（如有效范围）的依赖有着质的不同。当 时，对典型非通用参数的依赖随着 的正幂而减小，其中 是相互作用的范围。相反，对 的依赖不仅不会在共振极限消失，而且会以一种特别显著的形式出现。少体观测值是 的对数周期函数，即对 的依赖只通过 的三角函数进入。例如，式 (4) 中的渐近谱由对数周期函数的零点组成：

where . Instead of regarding as a nonuniversal parameter, it is more appropriate to regard it as a parameter that labels a continuous family of universality classes. Thus for systems in which the Efimov effect occurs, it is convenient to relax the definition of universal to allow dependence not only on the scattering length but also on the 3-body parameter associated with the Efimov spectrum. This definition reduces to the standard one in the 2-body system, because the 3-body parameter cannot affect 2-body observables. It also reduces to the standard definition for dilute systems such as the weakly-interacting Bose gas, because 3-body effects are suppressed by at least in a dilute system. We will refer to this extended universality simply as "universality" in the remainder of the paper.
其中 。与其把 看作是一个非普遍性参数，不如把它看作是一个标示连续普遍性类族的参数更为合适。因此，对于发生埃菲莫夫效应的系统，放宽普遍性的定义是很方便的，不仅允许依赖于散射长度 ，还允许依赖于与埃菲莫夫谱相关的三体参数。这一定义简化为二体系统中的标准定义，因为三体参数不会影响二体观测值。对于稀释系统（如弱相互作用的玻色气体），它也可以简化为标准定义，因为在稀释系统中，三体效应至少会被 所抑制。在本文的其余部分，我们将把这种扩展的普遍性简称为 "普遍性"。

If the problem of identical bosons with large scattering length is formulated in a renormalization group framework, the remarkable behavior of the system of three identical bosons in the resonant limit is associated with a renormalization group limit cycle. The 3-body parameter associated with the Efimov spectrum can be regarded as parameterizing the position along the limit cycle. The asymptotic behavior of the spectrum in Eq. (3) reflects a discrete scaling symmetry that is characteristic of a renormalization group limit cycle. In contrast to renormalization group fixed points, which are ubiquitous in condensed matter physics and in high energy and nuclear physics, few physical applications of renormalization group limit cycles have been found. Consequently, the renormalization group theory associated with limit cycles is largely undeveloped. The development of such a theory could prove to be very helpful for extending universality into a systematically improvable calculational framework.
如果把具有大散射长度的相同玻色子问题放在重正化群框架中进行表述，那么三个相同玻色子系统在共振极限中的显著行为就与重正化群极限周期有关。与埃菲莫夫谱相关的三体参数可被视为沿极限周期的位置参数。式（3）中频谱的渐近行为反映了重正化群极限周期所特有的离散缩放对称性。与凝聚态物理、高能物理和核物理中无处不在的重正化群定点相比，重正化群极限周期的物理应用还很少被发现。因此，与极限循环相关的重正化群理论在很大程度上还没有发展起来。事实证明，这种理论的发展将非常有助于把普遍性扩展到一个可系统改进的计算框架中。

Since universality has such remarkable consequences in the 2-body and 3-body sectors, we expect it to also have important implications in the -body sector with . This is still mostly unexplored territory. Universality may also have important applications to the many-body problem. These applications are particularly topical, because of the rapid pace of experimental developments in the study of ultracold atoms. By cooling an atom with a large scattering length to sufficiently low temperature, one can reach a regime where universality is applicable. Fascinating many-body phenomena can occur within this regime, including Bose-Einstein condensation in the case of bosonic atoms and
既然普遍性在二体和三体部门有如此显著的影响，我们期待它在 - 体部门也有重要影响， 。这仍然是一个尚未探索的领域。普遍性还可能在多体问题上有重要应用。由于超冷原子研究实验的快速发展，这些应用尤其具有现实意义。通过将具有较大散射长度的原子冷却到足够低的温度，就可以达到普遍性适用的状态。在这一体系中会出现令人着迷的多体现象，包括玻色-爱因斯坦凝聚（玻色-爱因斯坦凝聚是玻色原子的情况）和超冷原子的多体现象。