Average-fluctuation separation in energy levels in many-particle quantum systems with $k$-body interactions using $q$-Hermite polynomials

TL;DR

This paper demonstrates the separation of average and fluctuation parts in the state density of many-particle quantum systems with $k$-body interactions, using $q$-Hermite polynomials and spectral analysis, applicable to both fermions and bosons.

Contribution

It introduces a method to separate average and fluctuation parts in the state density using $q$-Hermite polynomials and normal mode decomposition in many-particle quantum systems.

Findings

Fluctuations are of GOE type for all $k$ values.

Smoothed state density is represented by the $q$-normal distribution.

As $k$ increases, fluctuations appear with smaller corrections.

Abstract

Separation between average and fluctuation parts in the state density in many-particle quantum systems with $k$-body interactions, modeled by the $k$-body embedded Gaussian orthogonal random matrices (EGOE($k$)), is demonstrated using the method of normal mode decomposition of the spectra and also verified through power spectrum analysis, for both fermions and bosons. The smoothed state density is represented by the $q$-normal distribution ($f_{qN}$) (with corrections) which is the weight function for $q$-Hermite polynomials. As the rank of interaction $k$ increases, the fluctuations set in with smaller order of corrections in the smooth state density. They are found to be of GOE type, for all $k$ values, for both fermion and boson systems.