Pairing in quantum critical systems: Transition temperature, pairing gap, and their ratio

We compute the ratio of the pairing gap Δ at T=0 and Tc for a set of quantum-critical models in which the pairing interaction is mediated by a gapless boson with local susceptibility χ(Ω)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/Tc has been recently computed numerically for 0<γ<2 within Eliashberg theory and was found to increase with increasing γ [T.-H. Lee, Phys. Rev. Lett. 121, 187003 (2018)10.1103/PhysRevLett.121.187003]. We argue that the origin of the increase is the divergence of 2Δ/Tc at γ=3. We obtain an approximate analytical formula for 2Δ/Tc for γ≤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 Tc and Δ, including numerical prefactors. We show that these prefactors depend on fermionic self-energy in a rather nontrivial way. The ratio 2Δ/Tc approaches the BCS value 3.53 at γ→0.