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

Abstract

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