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Physical Review A

Virial coefficients in the presence of an infinite number of bound states and strongly repulsive potentials: Application to the Efimov point

Carmelo Pisani, Bruce H. J. McKellar

Published Dec 1, 1989• 4 citations
Publisher

Abstract

Extending some previous work of Boll\'e [Phys. Rev. A 36, 3259 (1987)], we construct expressions for the n-particle cluster coefficients that remedy difficulties associated with the presence of both an infinite number of bound states and strongly repulsive potentials. Applying the existence theorems for cluster coefficients implicit in the work of Ginibre [J. Math. Phys. 6, 238 (1965); 6, 252 (1965); 6, 1432 (1965)], we demonstrate the validity of these expressions. We apply the formalism to obtain a well-behaved expression for ${b}_{3}$ at the Efimov point. As a by-product we are able to directly observe the divergence in ${b}_{3}$ due to the continuum part of the three-body resolvent together with the counterterms that render it convergent.

DOI
10.1103/physreva.40.6597

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Extending some previous work of Boll'e [Phys. Rev. A 36, 3259 (1987)], we construct expressions for the n-particle cluster coefficients that remedy difficulties associated with the presence of both an infinite number of bound states and strongly repulsive potentials. Applying the existence theorems for cluster coefficients implicit in the work of Ginibre [J. Math. Phys. 6, 238 (1965); 6, 252 (1965); 6, 1432 (1965)], we demonstrate the validity of these expressions. We apply the formalism to obtain a well-behaved expression for b3{b}_{3}b3​ at the Efimov point. As a by-product we are able to directly observe the divergence in b3{b}_{3}b3​ due to the continuum part of the three-body resolvent together with the counterterms that render it convergent.
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.
Cluster coefficients for a quantum gas can be related by means of a Laplace transform to the resolvent of an interacting system. Techniques developed by Faddeev are applied in order to express resolvents in terms of quantities which satisfy coupled integral equations. The resulting theory for the cluster coefficients is free of convergence difficulties encountered in series expansions of those coefficients in terms of a binary-collision kernel or two-body scattering matrix. Present computational difficulties necessitate an approximate solution of the Faddeev equations. The assumption of a separable two-body scattering matrix makes possible such a solution and a subsequent calculation of cluster coefficients.
The Efimov effect for a three-body system was studied previously in an adiabatic representation, where the distance between two ``heavy'' particles was represented by x, and the third ``light'' particle was described by coordinate y relative to the center of mass of the heavy ones. When x is held fixed, an exact solution for the light particle can be obtained if the interaction is assumed to be of separable form. The resulting adiabatic potential between the heavy particles shows the critical $\ensuremath{-}{1/x}^{2}$ behavior that leads to an infinite number of bound states. However, subsequently the leading nonadiabatic effect was shown to generate an undesirable correction of 1/x1/x1/x type, the pseudo-Coulomb disease (PCD). To remedy the PCD, the Efimov effect is reexamined in the adiabatic state representation, but with the new Jacobi coordinates (r,R), where r describes the distance between one of the heavy particles and the light one and R is the position of the other heavy particle with respect to the center of mass of the subsystem of light-heavy particles. It is then shown that the adiabatic potential in this distorted cluster state representation behaves as $\ensuremath{-}{1/R}^{2}$ at large R if the pair described by r has a zero-energy bound state. Moreover, in this case the leading nonadiabatic correction term does not manifest the PCD, thus explicitly showing that the PCD is due to the ``wrong'' choice of coordinates (x,y). Alternatively, a particle boost factor is introduced to eliminate the PCD in the treatment with the original (x,y) coordinates. This is shown to be equivalent to the change in coordinates, described above. Such a factor is usually associated with high energy collisions, but for the first time shown here to play also an important role in near zero-energy situations.
We study the behaviour of weakly bound clusters and their relation to the well-known three-body Efimov states. We adopt a model to describe the universal behaviour of strongly interacting bosonic systems, and we test its validity by reproducing predictions of three- and four-body universal states. Then, we extend our study to larger systems and identify a series of universal cluster states that can be qualitatively interpreted as adding one particle at a time to an Efimov trimer. The properties of these cluster states and their experimental signatures are discussed.
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 nnn the cluster coefficients of order nnn + 1 for an infinity-mass impurity fermion, matching the conjecture for n=3n=3n=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.
The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.