Lowest Eigenvalues of Random Hamiltonians

TL;DR

This paper develops empirical formulas to accurately estimate the lowest eigenvalues of random Hamiltonians across various quantum systems, improving previous methods by incorporating higher moments and extending applicability to large matrices.

Contribution

Introduces new empirical formulas for estimating lowest eigenvalues of random Hamiltonians, including higher moments and large-dimension matrices, applicable to fermion and boson systems.

Findings

Empirical formulas effectively estimate lowest eigenvalues in diverse systems.

Adding higher moments improves estimation accuracy.

Formulas are applicable to excited energy evaluations.

Abstract

In this paper we present results of the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues are applicable to many different systems (except for $d$ boson systems). We improve the accuracy of the formula by adding moments higher than two. We suggest another new formula to evaluate the lowest eigenvalues for random matrices with large dimensions (20-5000). These empirical formulas are shown to be applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.