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

TL;DR

This paper derives a formula for calculating the norm overlap between Hartree-Fock-Bogoliubov states with odd parity, using Grassmann numbers and Fermion coherent states, enhancing computational methods in nuclear physics.

Contribution

The paper introduces a novel formula for norm overlaps involving odd-parity HFB states, extending existing methods for even-even systems.

Findings

Derived a Pfaffian-based formula for odd-parity HFB overlaps

Expressed the formula using Bogoliubov transformation matrices

Facilitated calculations for one quasi-particle excited states

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.