Generalization of Balian-Brezin decomposition for exponentials with linear fermionic part

TL;DR

This paper extends the Balian-Brezin decomposition to include exponentials with linear fermionic components, providing new formulas for overlaps and generalized Wick's theorem, thus broadening the analytical tools for fermionic Gaussian states.

Contribution

The work introduces a comprehensive extension of the Balian-Brezin decomposition to linear fermionic exponentials, including overlap formulas and a generalized Wick's theorem.

Findings

Extended Balian-Brezin decomposition to linear terms

Derived overlap formulas for Gaussian states with linear parts

Generalized Wick's theorem for linear fermionic scenarios

Abstract

Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick's theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian-Brezin decomposition (BBD) to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick's theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the $\mathfrak{so}(N)$ Lie algebra.