Universality in few-body systems with large scattering length

Abstract

Particles with short-range interactions and a large scattering length have universal low-energy properties that do not depend on the details of their structure or their interactions at short distances. In the 2-body sector, the universal properties are familiar and depend only on the scattering length a. In the 3-body sector for identical bosons, the universal properties include the existence of a sequence of shallow 3-body bound states called “Efimov states” and log-periodic dependence of scattering observables on the energy and the scattering length. The spectrum of Efimov states in the limit $a\to \pm \infty$ is characterized by an asymptotic discrete scaling symmetry that is the signature of renormalization group flow to a limit cycle. In this review, we present a thorough treatment of universality for the system of three identical bosons and we summarize the universal information that is currently available for other 3-body systems. Our basic tools are the hyperspherical formalism to provide qualitative insights, Efimov's radial laws for deriving the constraints from unitarity, and effective field theory for quantitative calculations. We also discuss topics on the frontiers of universality, including its extension to systems with four or more particles and the systematic calculation of deviations from universality.

Introduction

The scattering of two particles with short-range interactions at sufficiently low energy is determined by their S-wave scattering length, which is commonly denoted by a. 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 a 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. A many-body system has properties determined by a 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:

$$\mathcal{E}/n\approx \frac{2\pi {\hslash }^2}{m}\mathit{an}\text{,}$$

where $\mathcal{E}$ and n 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.

Generically, the scattering length a is comparable in magnitude to the range $r_0$ of the interaction: $\left|a\right|\sim r_0$. Universality in the generic case is essentially a perturbative weak-coupling phenomenon. The scattering length a plays the role of a coupling constant. Universal properties can be calculated as expansions in the dimensionless combination $a\kappa$, where $\kappa$ 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: $\kappa =(16\pi \mathit{an}{)}^{1/2}$. The weak-coupling expansion parameter $a\kappa$ is therefore proportional to the diluteness parameter $({\mathit{na}}^3{)}^{1/2}$. The order $({\mathit{na}}^3{)}^{1/2}$ and ${\mathit{na}}^3\ln ({\mathit{na}}^3)$ corrections to Eq. (1) are both universal [2]. Nonuniversal effects, in the form of sensitivity to 3-body physics, appear first in the order ${\mathit{na}}^3$ correction [3].

In exceptional cases, the scattering length can be much larger in magnitude than the range of the interaction: $\left|a\right|⪢r_0$. 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 $a\to \pm \infty$. 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 $r_0$, but they can be comparable to $\left|a\right|$. 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 $a>0$, there is a 2-body bound state near the scattering threshold with binding energy

$$E_{\mathrm{D}}=\frac{{\hslash }^2}{{\mathit{ma}}^2}\text{.}$$

The corrections to this formula are parametrically small: they are suppressed by powers of $r_0/a$. Note the nonperturbative dependence of the binding energy on the interaction parameter a. 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.

A classic example of a system with a large scattering length is ⁴He 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.

The first strong evidence for universality in the 3-body system was the discovery by Vitaly Efimov in 1970 of the Efimov effect [4],¹ a remarkable feature of the 3-body spectrum for identical bosons with a short-range interaction and a large scattering length a. In the resonant limit $a\to \pm \infty$, there is a 2-body bound state exactly at the 2-body scattering threshold $E=0$. Remarkably, there are also infinitely many, arbitrarily-shallow 3-body bound states with binding energies $E_{\mathrm{T}}^{(n)}$ that have an accumulation point at $E=0$. As the threshold is approached, the ratio of the binding energies of successive states approaches a universal constant:

$$E_{\mathrm{T}}^{(n+1)}/E_{\mathrm{T}}^{(n)}⟶1/515.03\mathrm{as}n\to +\infty \mathrm{with}a=\pm \infty \text{.}$$

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 $a=\pm \infty$. 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).

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 $a\to \pm \infty$, 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 ${\kappa }_{*}$ by expressing the asymptotic spectrum in the form

$$E_{\mathrm{T}}^{(n)}⟶{\left(\frac{1}{515.03}\right)}^{n-n_{*}}\frac{{\hslash }^2{\kappa }_{*}^2}{m}\mathrm{as}n\to +\infty \mathrm{with}a=\pm \infty$$

for some integer $n_{*}$. If the scattering length is large but finite, the spectrum of Efimov states will necessarily depend on both a and the 3-body parameter ${\kappa }_{*}$. 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 ${\kappa }_{*}$ is qualitatively different from the dependence on typical nonuniversal parameters such as the effective range. As $a\to \pm \infty$, the dependence on typical nonuniversal parameters decreases as positive powers of $r_0/a$, where $r_0$ is the range of the interaction. In contrast, the dependence on ${\kappa }_{*}$ not only does not disappear in the resonant limit, but it takes a particularly remarkable form. Few-body observables are log-periodic functions of ${\kappa }_{*}$, i.e. the dependence on ${\kappa }_{*}$ enters only through trigonometric functions of $\ln ({\kappa }_{*})$. For example, the asymptotic spectrum in Eq. (4) consists of the zeroes of a log-periodic function:

$$\sin (\frac{1}{2}{s}_0\ln [{\mathit{mE}}_{\mathrm{T}}/({\hslash }^2{\kappa }_{*}^2)])=0\text{,}$$

where $s_0\approx 1.00624$. Instead of regarding ${\kappa }_{*}$ 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 a 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 ${\mathit{na}}^3⪡1$ in a dilute system. We will refer to this extended universality simply as “universality” in the remainder of the paper.

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.

Since universality has such remarkable consequences in the 2-body and 3-body sectors, we expect it to also have important implications in the N-body sector with $N⩾4$. 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 superfluidity in the case of fermionic atoms. Universality has particularly exciting applications to these many-body phenomena, but they are beyond the scope of this review.

Most of this review is focused on the problem of identical bosons, because this is the system for which the consequences of universality have been most thoroughly explored. Identical bosons have the advantage of simplicity while still exhibiting the nontrivial realization of universality associated with the Efimov effect. However, we also summarize the universal results that are currently known for other few-body systems, including ones that include identical fermions. We hope this review will stimulate the further development of the universality approach to such systems.

The idea of universality in systems with a large scattering length has its roots in low-energy nuclear physics and has many interesting applications in particle and nuclear physics. Most of these applications are to systems with fortuitously large scattering lengths that arise from some accidental fine-tuning. Universality also has many interesting applications in atomic and molecular physics. There are some atoms that have fortuitously large scattering lengths, but there are also atoms whose scattering lengths can be tuned experimentally to arbitrarily large values. This makes universality particularly important in the field of atomic and molecular physics. We will therefore develop the ideas of universality using the language of atomic physics: “atom” for a particle, “dimer” for a 2-body bound state, etc.

We begin by introducing some basic scattering concepts in Section 2. We introduce the natural low-energy length scale and use it to define large scattering length. In Section 3, we describe the Efimov effect and introduce some renormalization group concepts that are relevant to the few-body problem with large scattering length. We define the scaling limit in which universality becomes exact and the resonant limit in which $a\to \pm \infty$. We point out that these limits are associated with a renormalization group limit cycle that is characterized by a discrete scaling symmetry. The simple and familiar features of universality in the problem of two identical bosons are described in Section 4. We point out that there is a trivial continuous scaling symmetry in the scaling limit and we calculate the leading scaling violations which are determined by the effective range.

In Section 5, we develop the hyperspherical formalism for three identical bosons. We use this formalism to deduce some properties of Efimov states in the resonant limit and near the atom–dimer threshold. In Section 6, we describe the most important features of universality for three identical bosons in the scaling limit. Logarithmic scaling violations reduce the continuous scaling symmetry to a discrete scaling symmetry. They also imply that low-energy 3-body observables depend not only on a but also on an additional scaling-violation parameter. We then present explicit results for 3-body observables, including the binding energies of Efimov states, atom–dimer scattering, 3-body recombination, and 3-atom scattering. We illustrate these results by applying them to the case of helium atoms, which have a large scattering length. We use universal scaling curves to illustrate the nontrivial realization of universality in the 3-body sector for identical bosons. In Section 7, we describe how the predictions from universality are modified by effects from deep 2-body bound states.

In Section 8, we describe a powerful method called effective field theory for calculating the predictions of universality. Using an effective field theory that describes identical bosons in the scaling limit, we derive a generalization of the Skorniakov–Ter-Martirosian (STM) integral equation and use it to calculate the most important low-energy 3-body observables. We also show that the renormalization of the effective field theory involves an ultraviolet limit cycle.

Sections 4–8 are focused exclusively on the problem of identical bosons. In Section 9, we discuss universality for other 3-body systems. We summarize what is known about the generalizations to distinguishable particles, fermions, unequal scattering lengths, and unequal masses. In Section 10, we discuss some of the most important frontiers of universality in few-body systems. They include power-law scaling violations such as those associated with the effective range, the N-body problem with $N⩾4$, unnaturally large low-energy constants in other angular momentum channels such as P-waves, and the approach to universality in scattering models.

Some of the sections of this review could be omitted by the first-time reader. He or she should begin by reading Section 2 on Scattering Concepts and Sections 3.1 and 3.2 on Renormalization Group Concepts. The first-time reader should then read Section 4 on Universality for Two Identical Bosons, Sections 5.1, 5.5, and 5.6 on the Hyperspherical Formalism, and Section 6 on Universality for Three Identical Bosons. The reader who is primarily interested in systems for which there are no tightly-bound 2-body bound states could skip Section 7 on Effects of Deep Two-body Bound States. The first-time reader could also skip Section 8 on Effective Field Theory. A first pass through the review could be completed by reading Section 9 on Universality in Other Three-body Systems. We hope this will whet the reader's appetite for a more thorough reading of all the sections.