Density of states of quantum systems from free probability theory: a brief overview

TL;DR

This paper reviews how free probability theory can be used to approximate the density of states in quantum systems and random matrices, highlighting successes, limitations, and perturbative improvements.

Contribution

It introduces a perturbation scheme based on subordination formulas to improve density of states calculations in complex models.

Findings

Free probability provides accurate density of states approximations in many models.

Limitations occur when the free approximation fails to match exact results.

Perturbation methods can refine density of states estimates in specific models.

Abstract

We provide a brief overview of approaches for calculating the density of states of quantum systems and random matrix Hamiltonians using the tools of free probability theory. For a given Hamiltonian of a quantum system or a generic random matrix Hamiltonian, which can be written as a sum of two non-commutating operators, one can obtain an expression for the density of states of the Hamiltonian from the known density of states of the two component operators by assuming that these operators are mutually free and by using the free additive convolution. In many examples of interacting quantum systems and random matrix models, this procedure is known to provide a reasonably accurate approximation to the exact numerical density of states. We review some of the examples that are known in the literature where this procedure works very well, and also discuss some of the limitations of this method in situations where the free probability approximation fails to provide a sufficiently accurate description of the exact density of states. Subsequently, we describe a perturbation scheme that can be developed from the subordination formulas for the Cauchy transform of the density of states and use it to obtain approximate analytical expressions for the density of states in various models, such as the Rosenzweig-Porter random matrix ensemble and the Anderson model with on-site disorder.