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Conference on "Mathematical properties of large quantum systems"
Organizers: Maria J. Esteban (CNRS), Mathieu Lewin (CNRS), Robert Seiringer (McGill, Montréal) & Jan Philip Solovej (Copenhagen)


Yvan Castin (Paris, France): "The third virial coefficient of the unitary Bose gas"
Monday 17 June 2013, 11:50 - 12:35
Amphi Hermite


By unitary Bose gas we mean a system composed of spinless bosons with s-wave interaction of infinite scattering length and almost negligible (real or effective) range. Experiments are currently trying to realize it with cold atoms. The phase transitions that may experience this system are the subject of ongoing research. In this talk, we shall focus on the non-degenerate regime for this gas, where first experimental measurements of the equation of state have the highest probability of being performed in the near future.

As we will recall, the properties of the unitary Bose gas are expected to be quite different from the ones of the better known unitary Fermi gas of spin 1/2 fermions. This is mainly due to the occurrence of the Efimov effect for three bosons, leading to a geometric spectrum of trimer states, which breaks the scaling invariance of the unitary gas and requires the introduction of a length scale characterizing the interactions, the so-called three-body parameter R_t. For spin 1/2 fermions, it is expected that such an Efimov effect is absent, not only at the three-body level but also at the N-body level, even if a mathematical proof is still lacking.

From the analytic solution of the three-boson problem in a harmonic potential, and using methods previously developed for fermions, we determine the third cumulant (or cluster integral) b_3 and the third virial coefficient a_3 of the unitary Bose gas, in the spatially homogeneous case, as a function of its temperature and the three-body parameter R_t characterizing the Efimov effect. A key point is that, converting series into integrals (by an inverse residue method), and using an unexpected small parameter (the three-boson mass angle nu=pi/6), one can push the full analytical estimate of b_3 and a_3 up to an error that is in practice negligible [1].

Considering the very few explicit analytical results on the third virial coefficient (for any interacting system) this is already a remarkable feature of the unitary Bose gas.

[1] Le troisieme coefficient du viriel du gaz de Bose unitaire/Third virial coefficient of the unitary Bose gas, Yvan Castin and Felix Werner, Rev. can. phys./Canadian Journal of Physics, 91:382 (2013), doi:10.1139/cjp-2012-0569