Review of the k-Body Embedded Ensembles of Gaussian Random Matrices

TL;DR

This paper reviews various analytical and numerical methods used to study the spectral and ergodic properties of k-Body Embedded Ensembles of Gaussian Random Matrices, which model many-body quantum systems with realistic interactions.

Contribution

It provides a comprehensive overview of multiple approaches for analyzing the spectral density, fluctuations, and ergodicity of these physically motivated random matrix ensembles.

Findings

Spectral density and fluctuation properties are characterized by various methods.

Numerical simulations support theoretical predictions.

Embedded ensembles better model realistic many-body systems than standard random matrices.

Abstract

The embedded ensembles were introduced by Mon and French as physically more plausible stochastic models of many--body systems governed by one--and two--body interactions than provided by standard random--matrix theory. We review several approaches aimed at determining the spectral density, the spectral fluctuation properties, and the ergodic properties of these ensembles: moments methods, numerical simulations, the replica trick, the eigenvector decomposition of the matrix of second moments and supersymmetry, the binary correlation approximation, and the study of correlations between matrix elements.