Special role of the first Matsubara frequency for superconductivity near a quantum critical point: Nonlinear gap equation below Tc and spectral properties in real frequencies

Near a quantum-critical point in a metal a strong fermion-fermion interaction, mediated by a soft boson, destroys fermionic coherence and also gives rise to an attraction in one or more pairing channels. The two tendencies compete with each other, and in a class of large N models, where the tendency to incoherence is parametrically stronger, one would naively expect an incoherent (non-Fermi liquid) normal state behavior to persist down to T = 0. However, this is not the case for quantum-critical systems, described by Eliashberg theory. In such systems, the part of the fermionic self-energy Sigma(omega(m)), relevant for spin-singlet pairing, is large for a generic Matsubara frequency omega(m) = pi T(2m + 1) but vanishes for fermions with omega(m )= +/-pi T, while the pairing interaction between fermions with these two frequencies remains strong. It has been shown [Y. Wang et al. Phys. Rev. Lett. 117, 157001 (2016)] that due to this peculiarity, the onset temperature for the pairing T-p is finite even at large N, when the scaling analysis predicts a non-Fermi liquid normal state. We consider the system behavior below T-p and contrast the conventional case, when omega(m )= +/-pi T are not special, and the case when the pairing is induced by fermions with omega(m) = +/-pi T. We obtain the solution of the nonlinear gap equations in Matsubara frequencies and then convert to real frequency axis and obtain the spectral function A(k, omega) and the density of states N(omega). In a conventional BCS-type superconductor, A(k, omega) and N(omega) are peaked at the gap value Delta(T), and the peak position shifts to a smaller omega as temperature increases towards T-p, i.e., the gap closes in. We show that, when the pairing is induced by fermions with omega(m) = +/-pi T, the situation is qualitatively different from the standard BCS result. Namely, the peak in N(omega) remains at a finite frequency even at T = T-p - 0, the gap just fills in near this T. The spectral function A(k, omega) either shows almost the same gap filling behavior as the density of states, or its peak position shifts to zero frequency already at a finite Delta(emergent Fermi arc behavior), depending on the position of k on the Fermi surface. As an example, we compare our results with the data for the cuprates and argue that gap filling behavior holds in the antinodal region, while the emergent Fermi arc behavior holds in the nodal region.