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

TL;DR

This paper investigates how superconductivity emerges near a quantum-critical point in metals, emphasizing the unique role of the first Matsubara frequency in enabling a non-zero critical temperature despite fermionic incoherence.

Contribution

It demonstrates that superconductivity always occurs above a quantum-critical point and highlights the importance of the first Matsubara frequency in determining the critical temperature.

Findings

Superconductivity develops above the quantum-critical point regardless of fermionic self-energy strength.

The critical temperature $T_c$ is non-zero due to suppression of self-energy at the lowest Matsubara frequencies.

An analytic formula for $T_c$ is derived, matching numerical results for the electron-phonon model.

Abstract

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 $\omega_m = \pm \pi T$. We obtain the analytic formula for $T_c$ which reproduces well numerical results for the electron-phonon model at vanishing Debye frequency.