Comparison between the Lipkin-Nogami and Richardson solutions with complex single particle energy in the Lipkin's model

TL;DR

This paper compares the Lipkin-Nogami and Richardson solutions for the pairing Hamiltonian, especially considering complex energies, and finds that LN closely matches the exact Richardson solution, with BCS extending into the complex plane.

Contribution

It introduces a comparison of approximate and exact pairing solutions in complex energy representations within the Lipkin model, highlighting the effectiveness of LN and BCS extensions.

Findings

LN solution agrees well with the exact Richardson solution

BCS extension to complex energies yields solutions below the critical strength

Complex energy states incorporate continuum correlations in many-body systems

Abstract

The pairing interaction is one of the most important contribution of the residual interaction and then, it is of major importance for the study of many-body systems. One can get solutions of the pairing Hamiltonian throught the Bardeen-Cooper-Schieffer (BCS) or the Lipkin-Nogami (LN) approximations but, the pairing Hamiltonian admit exact solution worked out by Richardson. Nuclei far away from the stability line have important correlations with the continuum part of the energy spectrum, due that the Fermi level is very close to the contiuum thershold. The correlations with the continuum can be included in the many-body description through the complex energy states, called Gamow states. In this work we compare the approximates and exact solutions of the pairing Hamiltonian in real and complex-energy representations. In the application of this formulation to the symmetric Lipkin model, we found that the LN solution is in a good agreement with the exact one; besides, the extension of the BCS solution to the complex energy plane gives solution even for strength below the critical one, which is purely imaginary.