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.PACS numbers: 21.60Ev, 21.60.Jz

Projection of good quantum numbers for reaction fragments

“…(45) and (47) I took into account that the transformation from field to quasiparticle operators is unitary. Using the technology described by Balian and Brezin [4], Ring and Schuck [5] in Appendix E, and particularly the method introduced by Mizusaki et al [7] one can show that…”

“…Recently Mizusaki et al [7] clarified the reasons why the Onishi and Yoshida formula does not have a sign ambiguity, particularly in the case when the size of the Fock space is finite, as is the case in the overwhelming majority of numerical implementations. They have proven that Onishi and Yoshida [8] and Robledo [10] formulas for the norm overlaps are identical in this case.…”

Recent progress of the sign problem in the Onishi formula

“…where Y = Z (0) * Z (1) and the subscript c denotes the connected term. Note that the summation up to the infinite number of terms appears in Eq.…”

“…On the other hand, the overlap is obtained by expanding the exponential function as φ 0 |φ 1 = −| exp(Ẑ (0) † ) exp(Ẑ (1)…”

“…In Eq. 3, the overlap is provided by an N/2-th order polynomial of the matrix elements of Z (1) and Z (0) * . Thus, this formula apparently does not have the sign problem and is simple without numerical instability.…”