On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates

TL;DR

This paper analyzes the distribution of total energy in non-interacting fermions with energies modeled by random matrix theory, revealing Gaussian behavior with variance scaling as n^2 log n, and explores semiclassical and Riemann zeta function connections.

Contribution

It provides a detailed analysis of the energy distribution for fermions in random matrix spectra, including variance growth, corrections, and semiclassical estimates, with novel insights into non-universal behavior.

Findings

Energy distribution is Gaussian for large n.

Variance of total energy scales as n^2 log n.

Semiclassical formula captures non-universal variance behavior.

Abstract

We consider a single particle spectrum as given by the eigenvalues of the Wigner-Dyson ensembles of random matrices, and fill consecutive single particle levels with n fermions. Assuming that the fermions are non-interacting, we show that the distribution of the total energy is Gaussian and its variance grows as n^2 log n in the large-n limit. Next to leading order corrections are computed. Some related quantities are discussed, in particular the nearest neighbor spacing autocorrelation function. Canonical and gran canonical approaches are considered and compared in detail. A semiclassical formula describing, as a function of n, a non-universal behavior of the variance of the total energy starting at a critical number of particles is also obtained. It is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the Riemann zeta function on the critical line.