# Implicit renormalization approach to the problem of Cooper instability

## TL;DR

This paper introduces an implicit renormalization method to accurately predict superconducting transition temperature and gap function from high-temperature data, bypassing traditional eigenvalue approaches and complex vertex calculations.

## Contribution

It proposes an alternative eigenvalue problem within the implicit renormalization framework that improves the prediction of $T_c$ and the gap function without explicit vertex computations.

## Key findings

- Accurate $T_c$ prediction from high-temperature data.
- Eigenvalue problem formulation within implicit renormalization.
- Solution via diagrammatic Monte Carlo without matrix inversion.

## Abstract

In the vast majority of cases, superconducting transition takes place at exponentially low temperature $T_c$ out of the Fermi liquid regime. We discuss the problem of determining $T_c$ from known system properties at temperatures $T \gg T_c$, 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 $T_c$ and the gap function immediately below $T_c$. 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.