# Resurgence for superconductors

## TL;DR

This paper explores the connection between the energy gap in superconductors and the mathematical theory of resurgence, proposing that the perturbative series diverges factorially with a singularity linked to the energy gap.

## Contribution

It conjectures a deep link between the superconducting energy gap and the divergence of perturbative series, supported by exact calculations in the Gaudin-Yang model.

## Key findings

- Perturbative series for the ground state energy diverges factorially.
- Leading Borel singularity is governed by the superconducting energy gap.
- Identified a class of diagrams responsible for factorial growth.

## Abstract

An important non-perturbative effect in quantum physics is the energy gap of superconductors, which is exponentially small in the coupling constant. A natural question is whether this effect can be incorporated in the theory of resurgence. In this paper we take some steps in this direction. We conjecture that the perturbative series for the ground state energy of a superconductor is factorially divergent, and that its leading Borel singularity is governed by the superconducting energy gap. We test this conjecture in detail in the attractive Gaudin-Yang model, an exactly solvable model in one dimension with a BCS-like ground state. In order to do this, we develop techniques to calculate the exact perturbative series of its ground state energy up to high order. We also argue that the Borel singularity is of the renormalon type, and we identify a class of diagrams leading to factorial growth. We give additional evidence for the conjecture in other models.