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Superconductivity near a quantum-critical point: The special role of the first Matsubara frequency

Near a quantum-critical point in a metal strong fermion-fermion interaction mediated by a soft collective boson gives rise to incoherent, non-Fermi liquid behavior. It also often gives rise to superconductivity which masks the non-Fermi liquid behavior. We analyze the interplay between superconductivity and fermionic incoherence for a set of quantum-critical models with effective dynamical interaction between low-energy fermions. We argue that superconductivity always develops above a quantum-critical point, no matter how strong the fermionic self-energy is. We argue that superconductivity should not be viewed as a pairing of incoherent fermions, as previously thought. Rather, $T_c$ is non-zero due to the fact that the self-energy is suppressed at the two lowest fermionic Matsubara frequencies $ω_m = \pm πT$. We obtain the analytic formula for $T_c$ which reproduces well numerical results for the electron-phonon model at vanishing Debye frequency.