Pairing in quantum-critical systems: $T_c$, $\Delta$, and their ratio

Abstract

We compute the ratio of the pairing gap $\Delta$ at $T=0$ and $T_c$ for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson with local susceptibility $\chi (\Omega) \propto 1/|\Omega|^\gamma$ (the $\gamma$ model). The limit $\gamma = 0+$ ($\chi (\Omega) =\log {|\Omega|}$) describes color superconductivity, and models with $\gamma >0$ describe superconductivity in a metal at the onset of charge or spin order. The ratio $2\Delta/T_c$ has been recently computed numerically for $0<\gamma <2$ within Eliashberg theory and was found to increase with increasing $\gamma$ [T-H Lee et al, arXiv:1805.10280]. We argue that the origin of the increase is the divergence of $2\Delta/T_c$ at $\gamma =3$. We obtain an approximate analytical formula for $2\Delta/T_c$ for $\gamma \leq 3$ and show that it agrees well with the numerics. We also consider in detail the opposite limit of small $\gamma$. Here we obtain the explicit expressions for $T_c$ and $\Delta$, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather non-trivial way. The ratio $2\Delta/T_c$ approaches the BCS value $3.53$ at $\gamma \to 0$.