This article reviews theoretical and experimental advances in Efimov physics, an array of quantum few-body and many-body phenomena arising for particles interacting via short-range resonant interactions, that is based on the appearance of a scale-invariant three-body attraction theoretically discovered by Vitaly Efimov in 1970. This three-body effect was originally proposed to explain the binding of nuclei such as the triton and the Hoyle state of carbon-12, and later considered as a simple explanation for the existence of some halo nuclei. It was subsequently evidenced in trapped ultra-cold atomic clouds and in diffracted molecular beams of gaseous helium. These experiments revealed that the previously undetermined three-body parameter introduced in the Efimov theory to stabilise the three-body attraction typically scales with the range of atomic interactions. The few- and many-body consequences of the Efimov attraction have been since investigated theoretically, and are expected to be observed in a broader spectrum of physical systems.

The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter. We show that the three-body parameter is essentially determined by the zero-energy two-body correlation. As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range. One class involves the universality of the three-body parameter recently found in ultracold atom systems. The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics.The Efimov effect is a universal low-energy quantum phenomenon, which was originally predicted in nuclear physics [1] and has rekindled considerable interest since its experimental confirmation with ultracold atoms . It is also expected to occur in solid-state physics [23,24]. This universality stems from the effective three-body attraction that occurs between particles interacting with nearly resonant short-range interactions. As a result of this attraction, three particles may bind even when the interaction is not strong enough to bind two particles. Furthermore, an infinite series of such three-body bound states exists near the unitary point where the interaction is resonant, i.e. where a two-body bound state appears and the s-wave scattering a length diverges. The typical three-body energy spectrum for such systems is represented in Fig. 1 in units of inverse length. Near zero energy and large scattering lengths, the three-body spectrum is invariant under a discrete scaling transformation by a universal factor e π/s0 ≈ 22.7 for identical bosons, where s 0 ≈ 1.00624 characterises the strength of the three-body attraction.A notable consequence of the Efimov effect is the existence of another physical scale beyond the two-body scattering length to fix the low-energy properties of the system. This scale is known as the three-body parameter. In zero-range models, it manifests itself as the necessity to introduce a momentum cutoff or a three-body boundary condition. It can be characterised, for instance, by the scattering length a − at which a trimer appears or by its binding wave number κ at unitarity, as indicated in Fig. 1. Because of the discrete scaling invariance, it is defined up to a power of e π/s0 . In this Letter, we will focus on the ground Efimov state, which slightly deviates from the discrete-scaling-invariant structure, but is more easily observed and computed, and still reveals the essence of the physics behind the three-body parameter.Three important questions can be raised concerning the three-body parameter. Is there a simple mechanism that determines the three-body parameter from the microscopic interactions? What is the microscopic length scale which determines the three-body parameter? Finally, if there is such a length scale, what are the conditions for the three-body parameter to be related to that length scale through a universal dimensionless constant,

Universality is a powerful concept in physics, allowing one to construct physical descriptions of systems that are independent of the precise microscopic details or energy scales. A prime example is the Fermi gas with unitarity limited interactions, whose universal properties are relevant to systems ranging from atomic gases at microkelvin temperatures to the inner crust of neutron stars.Here we address the question of whether unitary Bose systems can possess a similar universality. We consider the simplest strongly interacting Bose system, where we have an impurity particle ("polaron") resonantly interacting with a Bose-Einstein condensate (BEC). Focusing on the ground state of the equal-mass system, we use a variational wave function for the polaron that includes up to three Bogoliubov excitations of the BEC, thus allowing us to capture both Efimov trimers and associated tetramers. Unlike the Fermi case, we find that the length scale associated with Efimov trimers (i.e., the three-body parameter) can strongly affect the polaron's behaviour, even at boson densities where there are no well-defined Efimov states. However, by comparing our results with recent quantum Monte Carlo calculations, we argue that the polaron energy is a universal function of the Efimov three-body parameter for sufficiently low boson densities. We further support this conclusion by showing that the energies of the deepest bound Efimov trimers and tetramers at unitarity are universally related to one another, regardless of the microscopic model. On the other hand, we find that the quasiparticle residue and effective mass sensitively depend on the coherence length ξ of the BEC, with the residue tending to zero as ξ diverges, in a manner akin to the orthogonality catastrophe.

We address the microscopic origin of the universal three-body parameter that fixes the spectrum of few-atom systems in the Efimov regime. We identify it with a nonadiabatic deformation of the three-atom system which occurs when three atoms come within the distance of the van der Waals length. This deformation explains the universal ratio of the scattering length at the triatomic resonance to the van der Waals length observed in several experiments and confirmed by numerical calculations.

In the free three-dimensional space, we consider a pair of identical ↑ fermions of some species or in some internal state, and a pair of identical ↓ fermions of another species or in another state. There is a resonant s-wave interaction (that is of zero range and infinite scattering length) between fermions in different pairs, and no interaction within the same pair. We study whether this 2 + 2 fermionic system can exhibit (as the 3+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is the mass ratio α between ↑ and ↓ fermions and its inverse are both smaller than 13.6069. . . . For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schrödinger equation, that is positively homogeneous functions of the coordinates of degree s − 7/2, where s is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of s are such that M (s) has a zero eigenvalue; here the operator M (s), that depends on the total angular momentum ℓ, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of M (s), analytical and for an arbitrary imaginary s for the continuous spectrum, numerical and limited to s = 0 and ℓ ≤ 12 for the discrete spectrum. We conclude that no eigenvalue of M (0) crosses zero over the mass ratio interval α ∈ [1, 13.6069 . . .], even if, in the parity sector (−1) ℓ , the continuous spectrum of M (s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the 2+2 fermions.We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-1/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary s-axis of s times the logarithmic derivative of the determinant of M (s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results. PACS numbers: 67.85.-d, 21.45.-v, 34.50.-s R dxdx ′ 1 −1 dudu ′ mz,m ′ z (−1) ℓ+1 x, u, ℓ, m z |[M (ℓ) (iS)] −1 |x ′ , u ′ , ℓ, m ′ z × x ′ , u ′ , ℓ, m ′ z | d dS M (ℓ) (iS)|x, u, ℓ, m z (B21)

The zero-energy universal properties of scattering between a particle and a dimer that involves an identical particle are investigated for arbitrary scattering angular momenta. For this purpose, we derive an integral equation that generalises the Skorniakov -Ter-Martirosian equation to the case of non-zero angular momentum. As the mass ratio between the particles is varied, we find various scattering resonances that can be attributed to the appearance of universal trimers and Efimov trimers at the collisional threshold.

For a system of two identical fermions and one distinguishable particle interacting via a short-range potential with a large s-wave scattering length, the Efimov trimers and Kartavtsev-Malykh trimers exist in different regimes of the mass ratio. The Efimov trimers are known to exhibit a discrete scaling invariance, while the Kartavtsev-Malykh trimers feature a continuous one. We point out that a third type of trimers, "crossover trimers", exist universally regardless of short-range details of the potential. These crossover trimers have neither discrete nor continuous scaling invariance. We show that the crossover trimers continuously connect the discrete and continuous scaling regimes as the mass ratio and the scattering length are varied. We identify the regions for the KartavtsevMalykh trimers, Efimov trimers, crossover trimers, and non-universal trimers in terms of the mass ratio and the s-wave scattering length by investigating the scaling property and universality (i.e. independence on short-range details) of the trimers.

Abstract. We consider a two-component ideal Fermi gas in an isotropic harmonic potential.Some eigenstates have a wavefunction that vanishes when two distinguishable fermions are at the same location, and would be unaffected by s-wave contact interactions between the two components. We determine the other, interactionsensitive eigenstates, using a Faddeev ansatz. This problem is nontrivial, due to degeneracies and to the existence of unphysical Faddeev solutions. As an application we present a new conjecture for the fourth-order cluster or virial coefficient of the unitary Fermi gas, in good agreement with the numerical results of Blume and coworkers.