Accuracy of BCS-based approximations for pairing in small Fermi systems

TL;DR

This paper evaluates the accuracy of BCS-based methods for calculating pairing correlations in small fermionic systems, finding projected BCS highly accurate in certain regimes but less so in larger, weakly paired spaces.

Contribution

It provides a systematic comparison of BCS and projected BCS approximations against exact solutions for small Fermi systems, highlighting their strengths and limitations.

Findings

Projected BCS is highly accurate for small spaces and strong pairing.

It underestimates energies in larger spaces with weak pairing.

The method performs well for realistic nuclear systems around 117Sn and 207Pb.

Abstract

We analyze the accuracy of BCS-based approximations for calculating correlation energies and odd-even energy differences in 2-component fermionic systems with a small number of pairs. The analysis is focused on comparing BCS and projected BCS treatments with the exact solution of the pairing Hamiltonian, considering parameter ranges appropriate for nuclear pairing energies. We find that the projected BCS is quite accurate over the entire range of coupling strengths in spaces of up to about 20 doubly degenerate orbitals. It is also quite accurate for two cases we considered with a more realistic Hamiltonian, representing the nuclei around 117Sn and 207Pb. However, the projected BCS significantly underestimates the energies for much larger spaces when the pairing is weak.