A convenient implementation of the overlap between arbitrary Hartree-Fock-Bogoliubov vacua for projection

TL;DR

This paper introduces a new, reliable, and accurate Pfaffian-based method for calculating overlaps between arbitrary Hartree-Fock-Bogoliubov vacua, effectively handling singularities caused by zero occupation numbers.

Contribution

The authors propose a novel Pfaffian formula that circumvents singularities in HFB overlaps by small occupation number adjustments, improving computational stability and applicability.

Findings

Method is reliable and accurate for arbitrary HFB overlaps.

Small occupation adjustments prevent singularities in U,V matrices.

Suitable for symmetry restoration in nuclear structure calculations.

Abstract

Overlap between Hartree-Fock-Bogoliubov(HFB) vacua is very important in the beyond mean-field calculations. However, in the HFB transformation, the $U,V$ matrices are sometimes singular due to the exact emptiness ($v_i=0$) or full occupation ($u_i=0$) of some single-particle orbits. This singularity may cause some problem in evaluating the overlap between HFB vacua through Pfaffian. We found that this problem can be well avoided by setting those zero occupation numbers to some tiny values (e.g., $u_i,v_i=10^{-8}$). This treatment does not change the HFB vacuum state because $u_i^2,v_i^2=10^{-16}$ are numerically zero relative to 1. Therefore, for arbitrary HFB transformation, we say that the $U,V$ matrices can always be nonsingular. From this standpoint, we present a new convenient Pfaffian formula for the overlap between arbitrary HFB vacua, which is especially suitable for symmetry restoration. Testing calculations have been performed for this new formula. It turns out that our method is reliable and accurate in evaluating the overlap between arbitrary HFB vacua.