# Multiple solutions for the equilibrium populations in BCS superconductors

## TL;DR

This paper explores multiple solutions for the energy gap and quasiparticle distributions in BCS superconductors, revealing new solutions with different gaps and phase transition characteristics depending on the attraction band symmetry.

## Contribution

It introduces the existence of multiple solutions for BCS energy gaps and quasiparticle populations, including solutions with non-zero populations at zero temperature.

## Key findings

- Two sets of solutions for energy gaps below the phase transition temperature.
- A second solution with a smaller energy gap and non-zero quasiparticle populations at zero temperature.
- In symmetric attraction bands, the second solution's gap is one third of the standard BCS gap.

## Abstract

It was recently shown that the BCS formalism leads to several solutions for the energy gap and the equilibrium quasiparticle distribution, with a phase transition temperature which depends on the position of the chemical potential within the attraction band (the attraction band AB is defined as the single-particle energy interval in which the pairing interaction is manifested). Moreover, in some cases, the phase transition may be of the first, not of the second order. Here I will find two sets of solutions for any temperature below the phase transition temperature. I will also show that, when the AB is symmetric with respect to the chemical potential (the textbook BCS problem) there are still two solutions, with different energy gaps: one solution is the typical (textbook) BCS solution, whereas the other one has a smaller energy gap and non-zero quasiparticle populations down to zero temperature. At zero temperature, the energy gap corresponding to the second solution is one third of the typical BCS solution.