Grassmann integral and Balian–Brézin decomposition in Hartree–Fock–Bogoliubov matrix elements

Abstract

We present a new formula to calculate matrix elements of a general unitary operator with respect to Hartree-Fock-Bogoliubov states allowing multiple quasi-particle excitations. The Balian-Brézin decomposition of the unitary operator (Il Nuovo Cimento B 64, 37 (1969)) is employed in the derivation. We found that this decomposition is extremely suitable for an application of Fermion coherent state and Grassmann integrals in the quasi-particle basis. The resultant formula is compactly expressed in terms of the Pfaffian, and shows the similar bipartite structure to the formula that we have previously derived in the bare-particles basis (Phys. Lett. B 707, 305 (2012)). dξ←−The integral over η * ν (ν ∈ J) gives rise to a product of η's aswhere n = M − n. Then the integral over η ν (ν ∈ J) in the Eq.(32) can be carried out aswhere {η} J stands for the Grassmann variables η ν 1 , · · · , η ν n . After the integrals over η ν (ν ∈ J) and η * ν are performed, the integral over η ν (ν ∈ J) remains. Then, Eq.(30) is represented by the integrals over ξ , ξ * and η ν (ν ∈ J) as

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

“…It was pointed out [17] that in applying the generalized Wick's theorem, a matrix element of Eq. (7) involving n and n qps, respectively in the left-and right-side of [Ω], contains (n + n − 1)!!…”

“…In practice, the number of terms becomes so large that it is nearly impossible to write down expressions explicitly for more than 4-qp states. By using the Fermion coherent state and the Grassmann integral, a general expression for the matrix elements (7) in terms of the Pfaffian can be derived [17] …”

“…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].…”

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…”

Projected shell model for Gamow-Teller transitions in heavy, deformed nuclei

“…Therefore, state-by-state evaluations of nuclear matrix elements for GT transition are computationally feasible for heavy, deformed nuclei. Besides, GT transitions for highly-excited states are calculated directly with the resulting wave functions from diagonalization and the shell-model computation can be much facilitated by employing the modern Pfaffian algorithm [34][35][36][37][38]. Therefore, no approximations such as the Brink-Axel hypothesis are needed for highlyexcited states.…”