Density and spacings for the energy levels of quadratic Fermi operators

TL;DR

This paper proves that the energy level density of quadratic Fermi operators converges to a Gaussian distribution and investigates the spacing distribution, finding Poisson-like behavior even with disorder.

Contribution

It provides a rigorous proof of Gaussian convergence for energy level densities and explores the spectral spacing distribution in disordered quadratic Fermi systems.

Findings

Energy levels follow a Gaussian distribution asymptotically.

Level spacings behave as in a Poisson process.

Disorder does not significantly alter spacing statistics.

Abstract

The work presents a proof of convergence of the density of energy levels to a Gaussian distribution for a wide class of quadratic forms of Fermi operators. This general result applies also to quadratic operators with disorder, e.g., containing random coefficients. The spacing distribution of the unfolded spectrum is investigated numerically. For generic systems the level spacings behave as the spacings in a Poisson process. Level clustering persists in presence of disorder.