Resolution of unity for fermionic Gaussian operators

TL;DR

This paper proves a resolution of unity for fermionic Gaussian operators with even eigenvalue distributions, impacting the mathematical treatment of strongly correlated fermion systems and random matrix theory in nonstandard symmetry classes.

Contribution

It establishes a resolution of unity for fermionic Gaussian operators in nonstandard symmetry classes, revealing new mathematical identities and implications for random matrix ensembles.

Findings

Resolution of unity exists for even eigenvalue distributions in nonstandard classes

Nontrivial results require ensembles not even in eigenvalues for nonstandard classes

Standard Wigner-Dyson classes do not have this restriction

Abstract

The fermionic Gaussian operator basis provides a representation for treating strongly correlated fermion systems, as well as playing an important role in random matrix theory. We prove that a resolution of unity exists for any even distribution of eigenvalues over hermitian fermionic Gaussian operators in the nonstandard symmetry classes. This has some important consequences. It demonstrates a useful technique for constructing fundamental mathematical identities in an exponentially complex Hilbert space. It also shows that, to obtain nontrivial results for random matrix canonical ensembles in the nonstandard symmetry classes, it is necessary to consider ensembles that are not even functions of the eigenvalues. We show that the same restriction does not apply to the standard Wigner-Dyson symmetry classes of random matrices.