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Collective excitation branch in the continuum of pair-condensed Fermi gases : analytical study and scaling laws

The pair-condensed unpolarized spin-$1/2$ Fermi gases have a collective excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N. Popov, 1976). We study it at zero temperature, with the eigenenergy equation deduced from the linearized time-dependent BCS theory and extended analytically to the lower half complex plane through its branch cut, calculating both the dispersion relation and the spectral weights (quasiparticle residues) of the branch. In the case of BCS superconductors, so called because the effect of the ion lattice is replaced by a short-range electron-electron interaction, we also include the Coulomb interaction and we restrict ourselves to the weak coupling limit $Δ/μ\to 0^+$ ($Δ$ is the order parameter, $μ$ the chemical potential) and to wavenumbers $q=O(1/ξ)$ where $ξ$ is the size of a pair; when the complex energy $z_q$ is expressed in units of $Δ$ and $q$ in units of $1/ξ$, the branch follows a universal law insensitive to the Coulomb interaction. In the case of cold atoms in the BEC-BCS crossover, only a contact interaction remains, but the coupling strength $Δ/μ$ can take arbitrary values, and we study the branch at any wave number. At weak coupling, we predict three scales, that already mentioned $q\approx 1/ξ$, that $q\approx(Δ/μ)^{-1/3}/ξ$ where the real part of the dispersion relation has a minimum and that $q\approx (μ/Δ)/ξ\approx k_{\rm F}$ ($k_{\rm F}$ is the Fermi wave number) where the branch reaches the edge of its existence domain. Near the point where the chemical potential vanishes on the BCS side, $μ/Δ\to 0^+$, where $ξ\approx k_{\rm F}$, we find two scales $q\approx(μ/Δ)^{1/2}/ξ$ and $q\approx 1/ξ$. In all cases, the branch has a limit $2Δ$ and a quadratic start at $q=0$. These results were obtained for $μ>0$, where the eigenenergy equation admits at least two branch points $ε_a(q)$ and $ε_b(q)$ on the positive real axis, and for an analytic continuation through the interval $[ε_a(q),ε_b(q)] $. We find new continuum branches by performing the analytic continuation through $[ε_b(q),+\infty[$ or even, for $q$ low enough, where there is a third real positive branch point $ε_c(q)$, through $[ε_b(q),ε_c(q)]$ and $[ε_c(q),+\infty[$. On the BEC side $μ<0$ not previously studied, where there is only one real positive branch point $ ε_a(q)$, we also find new collective excitation branches under the branch cut $[ε_a (q),+\infty[$. For $μ>0$, some of these new branches have a low-wavenumber exotic hypoacoustic $z_q\approx q^{3/2}$ or hyperacoustic $z_q\approx q^{4/5}$ behavior. For $μ<0$, we find a hyperacoustic branch and a nonhypoacoustic branch, with a limit $2Δ$ and a purely real quadratic start at $q=0$ for $Δ/|μ|<0.222$.