The third virial coefficient of a two-component unitary Fermi gas across an Efimov-effect threshold

We consider a mixture of two single-spin-state fermions with an interaction of negligible range and infinite s-wave scattering length. By varying the mass ratio α across αc ≃ 13.6069 one can switch on-and-off the Efimov effect. We determine analytically the third cluster coefficient of the gas. We show that it is a smooth function of α across αc since, unexpectedly, the three-body parameter characterizing the interaction is relevant even on the non-Efimovian side α < αc. Introduction. – A powerful theory tool in the statistical physics of interacting quantum systems is the socalled cluster or virial expansion, where the thermodynamic potentials are expanded in powers of the small degeneracy parameter [1]. Whereas the second cluster coefficient b2 had a known general expression since the 1930s [2], it has been a long-lasting challenge to determine the third cluster coefficient b3 explicitly. Starting from the late 1950s, analytical results for b3 have been obtained for the two-body hard-core model, the archetype of nonresonant interactions where the s-wave scattering length a is at most of the order of the interaction range, in the form of expansions in powers of a small parameter λ/a [3] or a/λ [4], where λ is the thermal de Broglie wavelength. Interest in b3 was reactivated by recent experimental breakthrough with cold atoms: long-lived spin 1/2 Fermi gases can be prepared in the resonantly interacting regime (|a|≫ interaction range) via Feshbach resonances [5]. This motivated numerical calculation of b3 in maximally interacting, unitary limit 1/a=0, with the harmonic regulator technique of [6] as done in [7], or with diagrams [8]. Due to scaling invariance in the unitary limit, b3 is just a number, and via a precise measurement of the gas equation of state [9, 10], its predicted value was confirmed [10]. Physics is richer when the Efimov effect [11] sets in: the continuous scaling invariance is broken, there appears a length scale Rt characterizing the interaction, the threebody parameter, and there exists an infinite number of trimer states with an asymptotically geometric spectrum. The third cluster coefficient b3 becomes a function of temperature. In a spinless bosonic gas with zero-range interactions, it was determined analytically [12]. Within the three-body hard-core model that fixes Rt [13], Quantum Monte Carlo simulations have confirmed this analytical prediction and have shown that the third order cluster expansion can provide a good description of the gas down to the liquid-gas transition [14], exemplifying its usefulness. The problem is even more intriguing when a system parameter allows to switch on-and-off the Efimov effect, as in the two-component Fermi gas with adjustable mass ratio. For two identical fermions and a distinguishable particle, there is an Efimov effect if the fermion-other particle mass ratio α exceeds αc = 13.6069 . . . [11, 15]. Up to now, the calculation of b3 is numerical and limited to α < αc [16]. Strikingly it predicts that b3 has an infinite derivative at α = αc. As b3 is a coefficient in the grand potential Ω, this would imply a singular derivative of Ω as a function of α, i.e. a first order phase transition, subsisting at arbitrarily low phase space density, i.e. at temperatures T arbitrarily higher than the Fermi temperature TF , contrarily to common expectations for phase transitions. The present work determines b3 analytically and solves this paradox. The cluster expansion. – We consider a mixture of two fully polarized fermionic species, with single particle masses m1 and m2, with no intraspecies interaction and a purely s-wave interspecies interaction, of negligible range and infinite scattering length (unitary limit). At thermal equilibrium in a cubic box, the total pressure P admits in the thermodynamic limit the cluster expansion