Zero-pairing limit of Hartree-Fock-Bogoliubov reference states

TL;DR

This paper proves that the zero-pairing limit of Hartree-Fock-Bogoliubov states is mathematically well-defined regardless of shell structure, revealing new features about the relationship between HFB and Hartree-Fock theories.

Contribution

It demonstrates the well-defined nature of the zero-pairing limit in HFB states and explores how the limit depends on shell structure and naive filling, challenging common assumptions.

Findings

Zero-pairing limit is mathematically well-defined for both closed- and open-shell systems.

The nature of the limit state depends on shell structure and naive filling.

HFB does not reduce to Hartree-Fock even when pairing is driven to zero.

Abstract

The variational Hartree-Fock-Bogoliubov (HFB) mean-field theory is the starting point of various (ab initio) many-body methods dedicated to superfluid systems. While taking the zero-pairing limit of HFB equations constitutes a text-book problem when the system is of closed-(sub)shell character, it is typically, although wrongly, thought to be ill-defined whenever the naive filling of single-particle levels corresponds to an open-shell system. The present work demonstrates that the zero-pairing limit of an HFB state is mathematically well-defined, independently of the closed- or open-shell character of the system in the limit. Still, the nature of the limit state strongly depends on the underlying shell structure and on the associated naive filling reached in the zero-pairing limit for the particle number A of interest. All the analytical findings are confirmed and illustrated numerically. While HFB theory has been intensively scrutinized formally and numerically over the last decades, it still uncovers unknown and somewhat unexpected features. From this general perspective, the present analysis demonstrates that HFB theory does not reduce to Hartree-Fock theory even when the pairing field is driven to zero in the HFB Hamiltonian matrix.