Electron-hole symmetry and solutions of Richardson pairing model

TL;DR

This paper explores the electron-hole symmetry in the Richardson pairing model, deriving a simple ground state energy expression valid across different regimes, and confirms its accuracy through numerical solutions.

Contribution

It reveals the hidden electron-hole symmetry in Richardson equations and proposes a new, simple expression for the ground state energy applicable throughout the crossover.

Findings

Derived a simple ground state energy expression

Confirmed accuracy via numerical Richardson solutions

Unveiled hidden electron-hole symmetry

Abstract

Richardson approach provides an exact solution of the pairing Hamiltonian. This Hamiltonian is characterized by the electron-hole pairing symmetry, which is however hidden in Richardson equations. By analyzing this symmetry and using an additional conjecture, fulfilled in solvable limits, we suggest a simple expression of the ground state energy for an equally-spaced energy-level model, which is applicable along the whole crossover from the superconducting state to the pairing fluctuation regime. Solving Richardson equations numerically, we demonstrate a good accuracy of our expression.