Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.
The quantum theory of the third virial coefficient
C 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 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 and
He3, by means of a comparison of the corresponding expansion for the second virial coefficient
B 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
B. Then the leading term of
C is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for
C is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
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 at the Efimov point. As a by-product we are able to directly observe the divergence in
b3 due to the continuum part of the three-body resolvent together with the counterterms that render it convergent.
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.
We have shown earlier how the virial coefficients can be expressed in terms of traces taken over products of multiparticle (on-energy-shell) scattering matrices. The traces are to be taken in the angular-momentum representation instead of the momentum representation to avoid the infinite forward N-particle scattering amplitude for N > 2 caused by certain singular diagrams. Here, we analyze and sum the contribution of the singular diagrams to the third virial coefficient. It is also shown that one can get the same result if off-shell amplitudes in the momentum representation are utilized.