Quantum Bose and Fermi gases with large negative scattering length in the two-bodyS-matrix approximation

Abstract

We study both Bose and Fermi gases at finite temperature and density in an approximation that sums an infinite number of many-body processes that are reducible to two-body scatterings. This is done for arbitrary negative scattering length, which interpolates between the ideal and unitary gas limits. In the unitary limit, we compute the first four virial coefficients within our approximation. The second virial coefficient is exact, and we extend the previously known result for fermions to bosons and also for both bosons and fermions for the upper branch on the other side of unitarity (infinitely large positive scattering length). Assuming bosons can exist in a metastable state before undergoing mechanical collapse, we map out the critical temperatures for strongly coupled Bose-Einstein condensation as a function of scattering length.

Virial coefficients for trapped Bose and Fermi gases beyond the unitary limit: AnS-matrix approach

“…The method is a "foam diagram" approximation which is valid for high temperatures and low densities and considers only contributions from two-body processes to the free energy. It is explained in [16] and has been already used to study the thermodynamical and critical properties of quantum gases in two and three dimensions in the unitary limit [17,18] and beyond the unitary limit in three dimensions [19]. In [20] the method was used to calculate the ratio of the viscosity to entropy density and the results were in well agreement with experimental data [21].…”

“…In [19] it was shown how this method may be used to obtain the coefficients of the virial expansion for quantum gases and the first four virial coefficients were calculated in three dimensions in the unitary limit. The second coefficient provided by this method is exact and agrees with the result in [22].…”

Metastability of Bose and Fermi gases on the upper branch

“…Both of the above expressions also hold for bosons [19]. Before moving on to a discussion of our results on the upper branch we will put the integral equation and q in more convenient forms.…”

“…Before moving on to a discussion of our results on the upper branch we will put the integral equation and q in more convenient forms. Rotational invariance demands y be a function of |k| 2 , thus after rescaling k → √ 2mT k, the angular integrals in the integral equation (6) can be performed analytically (see appendix A in [19]). The result is the following:…”

“…For both bosons and fermions, the limit a s → ±∞ is the so-called unitary limit, where the theory is scale invariant. The unitary limit has been explored extensively within this formulation of statistical mechanics in [18][19][20], and will therefore not be discussed in this work. Although we will restrict our analysis of the upper branch to a s > 0, it is nevertheless important to mention that for bosons the upper branch phase is believed to extend smoothly across unitarity.…”