Why does the sign problem occur in evaluating the overlap of HFB wave functions?

TL;DR

This paper investigates the fundamental reason behind the sign problem in the Onishi formula for HFB wave function overlaps, contrasting it with the sign-free Pfaffian approach, and clarifies the mathematical origins of their differences.

Contribution

The paper provides a detailed explanation of why the Onishi formula has a sign problem by analyzing its derivation within a consistent framework, contrasting it with the Pfaffian formula.

Findings

The sign problem in the Onishi formula stems from its complex square root structure.

The Pfaffian formula does not suffer from the sign problem due to its different mathematical structure.

Different series expansion methods can lead to different analytical properties of the overlap formulas.

Abstract

For the overlap matrix element between Hartree-Fock-Bogoliubov states, there are two analytically different formulae: one with the square root of the determinant (the Onishi formula) and the other with the Pfaffian (Robledo's Pfaffian formula). The former formula is two-valued as a complex function, hence it leaves the sign of the norm overlap undetermined (i.e., the so-called sign problem of the Onishi formula). On the other hand, the latter formula does not suffer from the sign problem. The derivations for these two formulae are so different that the reasons are obscured why the resultant formulae possess different analytical properties. In this paper, we discuss the reason why the difference occurs by means of the consistent framework, which is based on the linked cluster theorem and the product-sum identity for the Pfaffian. Through this discussion, we elucidate the source of the sign problem in the Onishi formula. We also point out that different summation methods of series expansions may result in analytically different formulae.