Grassmann integral and Balian-Br\'ezin decomposition in Hartree-Fock-Bogoliubov matrix elements

TL;DR

This paper introduces a new, compact formula for calculating matrix elements of unitary operators in Hartree-Fock-Bogoliubov states, utilizing Grassmann integrals and the Balian-Brezin decomposition, expressed via Pfaffians.

Contribution

It develops a novel formula for matrix elements in HFB states using Balian-Brezin decomposition and Grassmann integrals, extending previous methods to include multiple quasi-particle excitations.

Findings

Formula expressed in terms of Pfaffian.

Demonstrates the suitability of Balian-Brezin decomposition for Grassmann integral applications.

Shows bipartite structure similar to previous bare-particle basis results.

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-Br\'ezin 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)).