Implicit renormalization approach to the problem of Cooper instability*

In the vast majority of cases, superconducting transition takes place at exponentially low temperature Tc out of the Fermi liquid regime. We discuss the problem of determining Tc from known system properties at temperatures T >> Tc, and stress that this cannot be done reliably by following the standard protocol of solving for the largest eigenvalue of the original gap-function equation. However, within the implicit renormalization approach, the gap-function equation can be used to formulate an alternative eigenvalue problem, solving which leads to an accurate prediction for both Tc and the gap function immediately below Tc. With the diagrammatic Monte Carlo techniques, this eigenvalue problem can be solved without invoking the matrix inversion or even explicitly calculating the four-point vertex function. [Phys. Rev. B 100, 064513 (2019)]