On the off-diagonal Wick's theorem and Onishi formula

TL;DR

This paper unifies the derivation of off-diagonal Wick's theorem and Onishi formula using a single diagrammatic approach, enhancing the consistency and potential generality of kernel computations in quantum many-body methods.

Contribution

It provides a unified diagrammatic derivation for both the off-diagonal Wick theorem and Onishi formula, previously derived through separate methods.

Findings

Consistent derivation of operator and norm kernels.

Potential to extend to more general states.

Simplifies the theoretical framework for kernel calculations.

Abstract

The projected generator coordinate method based on the configuration mixing of non-orthogonal Bogoliubov product states, along with more advanced methods based on it, require the computation of off-diagonal Hamiltonian and norm kernels. While the Hamiltonian kernel is efficiently computed via the off-diagonal Wick theorem of Balian and Brezin, the norm kernel relies on the Onishi formula (or equivalently the Pfaffian formula by Robledo or the integral formula by Bally and Duguet). Traditionally, the derivation of these two categories of formulae rely on different formal schemes. In the present work, the formulae for the operator and norm kernels are computed consistently from the same diagrammatic method. The approach further offers the possibility to address kernels involving more general states in the future.