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arXiv:2608.15324

Three-body contact for fermions. II. Non-degenerate limit

X. Leyronas, Félix Werner

Published Aug 15, 2026• 0 citations
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Abstract

A fundamental quantity characterizing Fermi gases with zero-range interactions is the three-body contact $C_3$, which determines several observables including the number of nearby triplets of fermions and the three-body loss rate in cold atom experiments, as shown in a companion article. Here, we compute $C_3$ to leading order in the non-degenerate limit for the homogeneous gas with negative or infinite scattering length $a$. At $a=\infty$, using a wavefunction approach, we obtain the analytical expression of $C_3$, which has a remarkably slow $1/T^{0.22728}$ dependence on the temperature $T$. In the Feynman diagram technique, which we use for $a<0$, the correct three-body short-distance correlations emerge only after a non-trivial cancellation between leading order two-body and three-body correlations, and the resulting power-law scaling comes from the large-wavevector tail of the 3-body T-matrix, which we derive inspired by the analytical solution of the three-body problem at the unitary limit.

DOI
10.48550/arxiv.2608.15324

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We consider the resonant Fermi gas, that is, two-component fermions in three dimensions interacting by a short-range potential of large scattering length. We introduce a quantity, the three-body contact, that determines several observables. Within the zero-range model, the number of nearby fermion triplets, the large-momentum tail of the center-of-mass momentum distribution of nearby fermion pairs, as well as the large-momentum tail of the two-particle momentum distribution, are expressed in terms of the three-body contact. For a small finite interaction range, the formation rate of deeply bound dimers by three-body recombination, as well as the three-body contribution to the finite-range correction to the energy, are expressed in terms of the three-body contact and of a three-body parameter. This three-body parameter, which vanishes in the zero-range limit, is defined through the asymptotic behavior of the zero-energy scattering state at distances intermediate between the range and the two-body scattering length. In general, the three-body contact has different contributions labeled by spin and angular momentum indices, and the three-body parameter can depend on those indices. We also include the generalization to unequal masses for ↑ and ↓ particles. With respect to the relation between three-body loss rate and number of nearby triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93 , 090404 (2004)], the present work adds a derivation, expresses the proportionality factor in terms of the three-body parameter, and includes the general case where there are several contributions to the three-body contact and several three-body parameters.
In a recent experiment with ultracold trapped Rb85 atoms, Makotyn et al. studied a quantum-degenerate Bose gas in the unitary limit where its scattering length is infinitely large. We show that the observed momentum distributions are compatible with a universal relation that expresses the high-momentum tail in terms of the two-body contact C2 and the three-body contact C3. We determine the contact densities for the unitary Bose gas with number density n to be C2 ≈ 20 n(4/3) and C3 ≈ 2n(5/3). We also show that the observed atom loss rate is compatible with that from 3-atom inelastic collisions, which gives a contribution proportional to C3, but the loss rate is not compatible with that from 2-atom inelastic collisions, which gives a contribution proportional to C2. We point out that the contacts C2 and C3 could be measured independently by using the virial theorem near and at unitarity, respectively.
The aim of this thesis was to measure the three-body contact in the unitary Fermi gas for the first time. The three-body contact is a physical observable that quantifies the three-body short-range correlations. Our probe, sensitive to three fermions with spins ↑↑↓ or ↓↓↑ at short distances, is three-body loss from the optical trap, caused by recombination into a deeply bound molecule and a free atom. In our system, the two spins signify two different hyperfine states of 6Li. We calibrate that recombination process by measuring three-body loss in the motional ground state of three fermions in an optical tweezer at unitarity. The numerically determined three-body contact C_3 (3) 3 of the three fermion system and the experimentally obtained loss rate allow us to extract Im[a_3], which quantifies the coupling to the deeply bound molecular state. Depending on the initial spin configuration ↑↑↓ or ↓↓↑, we find two different values, differing by almost a factor of two. This difference can only be explained by microscopic details of the interatomic coupling. In the many-body unitary Fermi gas, we use the coupling parameter Im[a_3] to extract the zero-temperature dimensionless three-body contact density ζ3\zeta_3ζ3​ from three-body loss measurements. We obtain a value of ζ3\zeta_3ζ3​ = 0.22(4) and observe that the loss rate scales with a characteristic three-body constant s = 1.77, which stems from an underlying SO(2,1) symmetry. We heat the gas and determine ζ3\zeta_3ζ3​ in the thermal regime, observing agreement with the theoretical prediction based on the virial expansion. The extracted value of ζ3\zeta_3ζ3​ > 0, which eludes theory, indicates three-body correlations. At the same time, the experimentally confirmed constant s proves the existence of irreducible three-body correlations that cannot be explained by two-body correlations alone.
We derive exact general relations between various observables for NNN bosons with zero-range interactions, in two or three dimensions, in an arbitrary external potential. Some of our results are analogous to relations derived previously for two-component fermions and involve derivatives of the energy with respect to the two-body sss-wave scattering length aaa. Moreover, in the three-dimensional case, where the Efimov effect takes place, the interactions are characterized not only by aaa, but also by a three-body parameter Rt{R}_{t}Rt​. We then find additional relations which involve the derivative of the energy with respect to Rt{R}_{t}Rt​. In short, this derivative gives the probability of finding three particles close to each other. Although it is evaluated for a totally lossless model, it also gives the three-body loss rate always present in experiments (due to three-body recombination to deeply bound diatomic molecules), at least in the limit where the so-called inelasticity parameter $\ensuremath{\eta}$ is small enough. As an application, we obtain, within the zero-range model and to first order in $\ensuremath{\eta}$, an analytic expression for the three-body loss rate constant for a nondegenerate Bose gas at thermal equilibrium with infinite scattering length. We also discuss the generalization to arbitrary mixtures of bosons and/or fermions.
The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems known to date, as it saturates the unitarity bound on the quantum mechanical scattering cross section. The UFG corresponds to a two-component Fermi gas in the limit of short interaction range and large scattering length, and is currently realized in ultracold-atom experiments via Feshbach resonances. While easy to define, the UFG poses a challenging quantum many-body problem, as it lacks any characteristic scale other than the density. As a consequence, accurate quantitative predictions of the thermodynamic properties of the UFG require Monte Carlo calculations. However, significant progress has also been made with purely analytical methods. Notably, in 2005 Tan derived a set of exact thermodynamic relations in which a universal quantity known as the "contact" C plays a crucial role. Recently, C has also been found to determine the prefactor of the high- frequency power-law decay of correlators as well as the right-hand-sides of shear- and bulk viscosity sum rules. The contact is therefore a central piece of information on the UFG in equilibrium as well as away from equilibrium. In this talk we describe some of the known aspects of Fermi gases at and around unitarity, show our latest Monte Carlo results for the contact at finite temperature, and summarize the open questions in the field, some of which we are starting to answer using large-scale Monte Carlo calculations by adapting methods from Lattice QCD.