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Universality in Few-Body Systems with Large Scattering Length Was Originally Developed in Nuclear Physics

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Universality in Few-Body Systems with Large Scattering Length Was Originally Developed in Nuclear Physics Universality in Few-body Systems with Large Scattering Length Eric Braaten a aDepartment of Physics, The Ohio State University, Columbus, OH 43210, USA H.-W. Hammer b,c bInstitute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA cHelmholtz-Institut f¨ur Strahlen- und Kernphysik, Universit¨at Bonn, 53115 Bonn, Germany 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 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. arXiv:cond-mat/0410417v3 [cond-mat.other] 18 Aug 2006 Email addresses: [email protected] (Eric Braaten), [email protected] (H.-W. Hammer). Preprint submitted to Physics Reports March 7, 2006 Contents 1 Introduction 5 2 Scattering Concepts 10 2.1 Scatteringlength............................ 10 2.2 Natural low-energy length scale . 14 2.3 Atoms with large scattering length . 21 2.4 Particles and nuclei with large scattering length . ..... 26 3 Renormalization Group Concepts 28 3.1 Efimoveffect.............................. 28 3.2 Theresonantandscalinglimits. 30 3.3 Universality in critical phenomena . 34 3.4 Renormalization group limit cycles . 38 3.5 Universality for large scattering length . 44 4 Universality for Two Identical Bosons 46 4.1 Atom-atom scattering . 46 4.2 Theshallowdimer........................... 48 4.3 Continuousscalingsymmetry. 49 4.4 Scalingviolations ........................... 51 4.5 Theoreticalapproaches . .. .. .. .. .. .. .. 53 5 Hyperspherical Formalism 54 5.1 Hypersphericalcoordinates . 54 5.2 Low-energy Faddeev equation . 59 5.3 Hypersphericalpotentials . 64 2 5.4 Boundary condition at short distances . 69 5.5 Efimovstatesintheresonantlimit. 72 5.6 Efimov states near the atom-dimer threshold . 76 6 Universality for Three Identical Bosons 78 6.1 Discretescalingsymmetry . 78 6.2 Efimov’sradiallaw .......................... 82 6.3 Binding energies of Efimov states . 87 6.4 Atom-dimer elastic scattering . 92 6.5 Three-bodyrecombination . 99 6.6 Three-atom elastic scattering . 104 6.7 Heliumatoms ............................. 108 6.8 Universalscalingcurves . 112 7 Effects of Deep Two-Body Bound States 115 7.1 ExtensionofEfimov’sradiallaw . 115 7.2 WidthsofEfimovstates. .. .. .. .. .. .. .. 119 7.3 Atom-dimer elastic scattering . 120 7.4 Three-body recombination into the shallow dimer . 122 7.5 Three-body recombination into deep molecules . 124 7.6 Dimer relaxation into deep molecules . 126 8 Effective Field Theory 128 8.1 Effective field theories . 128 8.2 Effective theories in quantum mechanics . 130 8.3 Effective field theories for atoms . 135 8.4 Two-bodyproblem........................... 138 3 8.5 Three-bodyproblem.......................... 146 8.6 Thediatomfieldtrick ......................... 151 8.7 STMintegralequation. 155 8.8 Three-bodyobservables . 158 8.9 Renormalization group limit cycle . 161 8.10 Effects of deep 2-body bound states . 165 9 Universality in Other Three-body Systems 168 9.1 UnequalScatteringLengths. 168 9.2 Unequalmasses ............................ 172 9.3 Twoidenticalfermions . 179 9.4 Particleswithaspinsymmetry. 182 9.5 Dimensionsotherthan3 . .. .. .. .. .. .. .. 183 9.6 Few-nucleonproblem . .. .. .. .. .. .. .. 186 9.7 Halonuclei ............................... 190 10 Frontiers of Universality 191 10.1 The N-body problem for N 4 ................... 192 ≥ 10.2 Effective-range corrections . 197 10.3 Large P-wave scattering length . 200 10.4 Scatteringmodels ........................... 202 References 205 4 1 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 de- noted 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 suffi- ciently 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: 2πh¯2 /n an, (1) E ≈ m where and n are the energy density and number density, respectively. In the E literature on Bose-Einstein condensates, properties of the many-body system that are determined by the scattering length are called universal [1]. The ex- pression 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 gen- erally 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 r0 of the interaction: a r . Universality in the generic case is essentially a per- | | ∼ 0 turbative 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 aκ, where κ 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: κ = (16πan)1/2. The weak-coupling expansion parameter aκ is therefore proportional to the diluteness parameter (na3)1/2. The order (na3)1/2 and na3 ln(na3) corrections to Eq. (1) are both universal [2]. Nonuniversal effects, in the form of sensitivity to 3-body physics, appear first in the order na3 correction [3]. 5 In exceptional cases, the scattering length can be much larger in magnitude than the range of the interaction: a r . Such a scattering length necessarily | |≫ 0 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 → . 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 r0, but they can be comparable to a . 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 h¯2 E = . (2) D ma2 The corrections to this formula are parametrically small: they are suppressed by powers of r0/a. Note the nonperturbative dependence of the binding en- ergy on the interaction parameter a. This reflects the fact that universality in the case of a large scattering length is a nonperturbative strong-coupling phe- nomenon. 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 subsections 2.3 and 2.4. The first strong evidence for universality in the 3-body system was the discov- ery 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 , 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 (n) binding energies ET 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(n+1)/E(n) 1/515.03 , as n + with a = . (3) T T −→ → ∞ ±∞ 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. 1 Some early indications of universality were already observed in Refs. [5,6]. 6 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.
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