Closed-form formulas in number-conserved pairing theory

TL;DR

This paper derives closed-form formulas for norms and density matrices in number-conserved pairing theory, enabling more efficient calculations in many-body quantum systems with exact particle numbers.

Contribution

It introduces novel formulas expressed as minors and Pfaffians for systems with fixed particle numbers, applicable to both even and odd systems, enhancing computational methods in pairing theory.

Findings

Formulas expressed as sums of minors and Pfaffians.

Applicable to both even and odd particle-number systems.

Facilitates applications in symmetry restoration and generator coordinate methods.

Abstract

In this work, I present closed-form formulas for the norm and many-body density matrices between general wave functions with exact particle numbers in pairing theory, using properties of the generalized Kronecker delta. These formulas, expressed as sums of minors and Pfaffians, apply to both even and odd particle-number systems and accommodate pair condensate as well as broken-pair configurations. This formalism directly facilitates applications in the generator coordinate method and symmetry restoration techniques, including angular momentum projection.