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

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.

7 pages, 1 figure

References in corpus (2)

• The sign of the overlap of HFB wave functions
• On the norm overlap between many-body states. I. Uncorrelated overlap between arbitrary Bogoliubov product states

Cited by in corpus (11)

• Symmetry restoration in mean-field approaches
• Projection on particle number and angular momentum: Example of triaxial Bogoliubov quasiparticle states
• Beyond-mean-field approaches for nuclear neutrinoless double beta decay in the standard mechanism
• Variational approximations to exact solutions in shell-model valence spaces: calcium isotopes in the pf-shell
• A new and practical formulation for overlaps of Bogoliubov vacua
• Projection of Good Quantum Numbers for Reaction Fragments
• Many-body Majorana braiding without an exponential Hilbert space
• Majorana quasiparticles in atomic spin chains on superconductors
• A Pfaffian formulation for matrix elements of three-body operators in multiple quasi-particle configurations
• Symmetry-projected variational calculations with the numerical suite TAURUS II. Configuration mixing of symmetry-projected reference states
• Fock space representations of Bogolyubov transformations as spin representations