Fock space representations of Bogolyubov transformations as spin representations

TL;DR

This paper explores the Fock space representation of Bogolyubov transformations as spin representations, deriving formulas for overlap amplitudes of quasi-fermion vacua and clarifying the scope of the Onishi formula.

Contribution

It provides new derivations and clarifications of formulas related to Bogolyubov transformations, including a missing proof and a combinatoric proof of recent results.

Findings

Derived more complete formulas for overlap amplitudes

Clarified differences between two Onishi formulas

Provided new proofs for existing formulas

Abstract

The representation on a Fock space of the group of Bogolyubov transformations is recognized as the spin representation of an orthogonal group. Derivations based on this observation of some known formulas for the overlap amplitude of two Bogolyubov quasi-fermion vacuum states that are in some cases more complete than those in the literature are shown. It is pointed out that the name of an "Onishi formula" is assigned in the literature to two different expressions which are related but have different scopes. One of them has what has been described as a sign problem, the other one, due to Onishi and Yoshida, has a more limited scope and no sign problem. I give a short proof of the latter, whose derivation is missing in the paper by Onishi and Yoshida, and a new, combinatoric, proof of an equivalent formula recently derived by Robledo.