A new formulation to calculate general HFB matrix elements through the Pfaffian

Abstract

A new formula is presented for the calculation of matrix elements between multi-quasiparticle Hartree-Fock-Bogoliubov (HFB) states. The formula is expressed in terms of the Pfaffian, and is derived by using the Fermion coherent states with Grassmann numbers. It turns out that the formula corresponds to an extension of generalized Wick's theorem and simplifies the combinatorial complexity resulting from practical applications of generalized Wick's theorem by unifying the transition density and the transition pairing tensor in the HFB theory. The resultant formula is simpler and more compact than the traditional description of matrix elements of general many-body operators. In addition, through the derivation of our new formula, we found that the Pfaffian version of the Lewis Carroll formula corresponds to the relation conjectrured by Balian and Brezin for the HFB theory in 1969.

New Pfaffian formulae for an overlap of multiple quasiparticle states

“…Many theoretical works followed Robledo's pioneering work [5][6][7][8][9] in order to extend Robledo's formula to multiple quasiparticle excited states. The extension essentially corresponds to finding a formula to evaluate a quantity such as…”

“…(11) can be evaluated with an aid of the formula derived in [8] to be Pf(X)Pf(X −1 { j,i,k,l} ), which was demonstrated explicitly as an example in Eq. (46) of Ref.…”

“…(44) in Ref. [8] for the definition of X −1 , which can be regarded as the "antisymmetrised" bipartite density matrix. The expectation value of the second term in Eq.…”

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

“…His proof is mathematically very elegant and interesting [5]. Moreover, these techniques led us to a relation to the generalized Wick's theorem and its related topics [9], [10], [11], [12], [13]. The proof by Robledo is, however, rather abstract and it somewhat keeps us from an intuitive understanding of the reason for the disappearance of the sign ambiguity.…”

Recent developments of the projected shell model based on many-body techniques

“…In nuclear structure physics, the Pfaffian concept has been introduced [9] as a key mathematical tool for solving the long-standing problem in the phase determination of the Onishi formula [10]. Moreover, it has been shown that the Pfaffian algorithm is very efficient also for calculating overlap matrix elements [11][12][13][14][15][16]. In particular, by means of Fermion coherent states and the Grassmann integral, one can derive an alternative approach to calculate the rotated matrix element for general qp states [17], which serves as a theoretical framework to extend the PSM model space [18].…”