A new formulation to calculate general HFB matrix elements through Pfaffian

TL;DR

This paper introduces a Pfaffian-based formula for calculating matrix elements between multi-quasiparticle HFB states, simplifying the process and unifying key tensors, thus advancing computational methods in many-body quantum systems.

Contribution

The paper presents a novel Pfaffian formula that extends generalized Wick's theorem, simplifying calculations of HFB matrix elements and unifying transition tensors.

Findings

The new formula is more compact than traditional methods.

It unifies transition density and pairing tensors in HFB.

The Pfaffian version of Lewis Carroll's formula relates to Balian and Brezin's conjecture.

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.