Norm-overlap formula for Hartree–Fock–Bogoliubov states with odd number parity

Abstract

A formula to calculate a norm overlap between Hartree-Fock-Bogoliubov (HFB) states with the odd number parity (one quasi-particle excited states) is derived with help of the Grassmann numbers and the Fermion coherent states. The final form of the formula is expressed in terms of a product of the Pfaffian for a neighboring even-even system (the zero quasi-particle state), and an extra factor consisting of the Bogoliubov transformation matrix and the anti-symmetric matrix in Thouless' HFB ansatz for the even-even system. I. INTRODUCTIONThe Hartree-Fock-Bogoliubov (HFB) method has been a powerful method in descriptions of nuclear states [1]. The reason is that the method can deal with the two most important correlations in interacting many-body nuclear systems, that is, deformation and pairing.These correlations are effectively taken into account by breaking relevant symmetries (the rotational and gauge symmetries). As a consequence, however, the conservation laws of angular momentum and the particle numbers are violated.Restorations of these broken symmetries are achieved through quantum number projections, but there have been difficulties to overcome in carrying out practical calculations of the projection. The major problem lies in the calculation of the so-called norm overlap kernels, which is necessary in the projections. The evaluation of the norm overlap is particularly difficult in angular momentum projection because the rotational symmetry is associated with the non-Abelian SO(3) group. * Electronic address: m.oi@isc.senshu-u.ac.jp

Systematic study of multi-quasiparticleK-isomeric bands in tungsten isotopes by the extended projected shell model

“…The proposal was largely inspired by the initial introduction of the Pfaffian idea by Robledo [30] to treat the sign problem in calculations of the overlap of Hartree-Fock-Bogoliubov (HFB) wave-functions. It was realized later [31][32][33][34][35] that the method can be applied to evaluation of general HFB matrix elements. The new Pfaffian method has turned out to be a feasible and very efficient algorithm for calculations with projected multi-qp states.…”

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…”

“…(9) can be easily evaluated as a special case of the formula derived in Ref. [7], where one-quasiparticle excited states are expressed as…”

“…(12), can be evaluated easily with the help of the formula derived in Ref. [7] (for the first three terms of one-body parts in Eq. (12)) and Robledo's original Pfaffian formula [4] (for the last two terms in Eq.(12).…”

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].…”