We give exact integral expressions of the third cluster or virial coefficients of binary mixtures of ideal Bose or Fermi gases, with interspecies interactions of zero range and infinite s-wave scattering length. In general the result depends on three-body parameters Rt appearing in three-body contact conditions, because an Efimov effect is present or because the mixture is in a preefimovian regime with a mass ratio close to an Efimov effect threshold. We give a new, exact integral expression of Rt for the microscopic narrow Feshbach resonance model. A divergence of Rt in the preefimovian regime at a scaling exponent s = 1/2 is predicted and physically discussed. The analytical results are applied to typical species used in cold atom experiments.
We calculate the fourth cluster coefficients of the homogeneous unitary spin 1/2 Fermi gas as functions of the internal-state mass ratio, over intervals constrained by the 3- or 4-body Efimov effect. For this we use our 2016 conjecture (validated for equal masses by Hou and Drut in 2020) in a numerically efficient formulation making the sum over angular momentum converge faster, which is crucial at large mass ratio. The mean cluster coefficient, relevant for equal chemical potentials, is not of constant sign and increases rapidly close to the Efimovian thresholds. We also get the fourth virial coefficients, which we find to be very poor indicators of interaction-induced 4-body correlations. We obtain analytically for all
n the cluster coefficients of order
n + 1 for an infinity-mass impurity fermion, matching the conjecture for
n=3. Finally, in a harmonic potential, we predict a non-monotonic behavior of the 3 + 1 cluster coefficient with trapping frequency, near mass ratios where this coefficient vanishes in the homogeneous case.
We compute the virial coefficients, the contact parameters, and the momentum distribution of a strongly interacting three-dimensional Bose gas by means of a virial expansion up to third order in the fugacity, which takes into account three-body correlations exactly. Our results characterize the nondegenerate regime of the interacting Bose gas, where the thermal wavelength is smaller than the interparticle spacing but the scattering length may be arbitrarily large. We observe a rapid variation of the third virial coefficient as the scattering length is tuned across the three-atom and the atom-dimer thresholds. The momentum distribution at unitarity displays a universal high-momentum tail with a log-periodic momentum dependence, which is a direct signature of Efimov physics. We provide a quantitative description of the momentum distribution at high momentum as measured by P. Makotyn et al. [Nat. Phys. 10, 116 (2014)], and our calculations indicate that the lowest trimer state might not be occupied in the experiment. Our results allow for a spectroscopy of Efimov states in the unitary limit.
Abstract In this article, we revisit the heteronuclear Efimov effect in a Bose-Fermi mixture with large mass difference in the Born-Oppenheimer picture. As a specific example, we consider the combination of bosonic 133Cs and fermionic 6Li. In a system consisting of two heavy bosons and one light fermion, the fermion-mediated potential between the two heavy bosons gives rise to an infinite series of three-body bound states. The intraspecies scattering length determines the three-body parameter and the scaling factor between consecutive Efimov states. In a second scenario, we replace the single fermion by an entire Fermi Sea at zero temperature. The emerging interaction potential for the two bosons exhibits long-range oscillations leading to a weakening of the binding and a breakup of the infinite series of Efimov states. In this scenario, the binding energies follow a modified Efimov scaling law incorporating the Fermi momentum. The scaling factor between deeply bound states is governed by the intraspecies interaction, analogous to the Efimov states in vacuum.
In the free three-dimensional space, we consider a pair of identical $\ensuremath{\uparrow}$ fermions of some species or in some internal state and a pair of identical $\ensuremath{\downarrow}$ 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 $\ensuremath{\alpha}$ between $\ensuremath{\uparrow}$ and $\ensuremath{\downarrow}$ fermions and its inverse are both smaller than 13.6069\dots{}. For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schr\"odinger equation, that is, positively homogeneous functions of the coordinates of degree $s\ensuremath{-}7/2$, 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 $\ensuremath{\ell}$, 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 and numerical and limited to
s=0 and $\ensuremath{\ell}\ensuremath{\le}12$ for the discrete spectrum. We conclude that no eigenvalue of
M(0) crosses zero over the mass ratio interval $\ensuremath{\alpha}\ensuremath{\in}[1;13.6069\ensuremath{\cdots}]$, even if, in the parity sector ${(\ensuremath{-}1)}^{\ensuremath{\ell}}$, 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.