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Concavity of the collective excitation branch of a Fermi gas in the BEC-BCS crossover

We study the concavity of the dispersion relation $q\mapsto ω\_{\mathbf{q}}$ of the bosonic excitations of a three-dimensional spin-$1/2$ Fermi gas in the Random Phase Approximation (RPA). In the limit of small wave numbers $q$ we obtain analytically the spectrum up to order $5$ in $q$. In the neighborhood of $q=0$, a change in concavity between the convex BEC limit and the concave BCS limit takes place at $Δ/μ\simeq0.869$ [$1/(k\_F a)\simeq-0.144$], where $a$ is the scattering length between opposite spin fermions, $k\_F$ is the Fermi wave number and $Δ$ the gap according to BCS theory, and $μ$ is the chemical potential. At that point the branch is concave due to a negative fifth-order term. Our results are supplemented by a numerical study which shows the evolution of the border between the zone of the $(q,Δ)$ plane where $q\mapsto ω\_{\mathbf{q}}$ is concave and the zone where it is convex.