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For the overlap matrix element between Hartree-Fock-Bogoliubov states, thereare two analytically different formulae: one with the square root of thedeterminant (the Onishi formula) and the other with the Pfaffian (Robledo'sPfaffian 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-calledsign problem of the Onishi formula). On the other hand, the latter formula doesnot suffer from the sign problem. The derivations for these two formulae are sodifferent that the reasons are obscured why the resultant formulae possessdifferent analytical properties. In this paper, we discuss the reason why thedifference occurs by means of the consistent framework, which is based on thelinked cluster theorem and the product-sum identity for the Pfaffian. Throughthis discussion, we elucidate the source of the sign problem in the Onishiformula. We also point out that different summation methods of seriesexpansions may result in analytically different formulae.

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