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arXiv:2512.03850

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

Keun-Young Kim, Kuntal Pal

quant-phcond-mat.stat-mechSubmitted Dec 3, 2025• Announced Dec 4, 2025• Revised Sep 23, 2026 (v2)

v1 · Dec 3, 2025 v2 · Sep 23, 2026

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.

Comments: Minor corrections. References added

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There is a newly emerging understanding that in the chaotic domain of isolated finite interacting many particle systems smoothed densities define the statistical description of these systems and these densities follow from embedded (two-body) random matrix ensembles and their various deformations. These ensembles predict that the smoothed form of matrix elements of a transition operator between the chaotic eigenstates weighted by the densities at the two ends (i.e. the bivariate strength density) will be a bivariate Gaussian with the bivariate correlation coefficient arising out of the non-commutability of the hamiltonian and the transition operator involved. The ensemble theory extends to systems with a mean-field and a chaos generating two-body interaction (as in nuclei, atoms and diffusive quantum dots). These developments in many-body quantum chaos are described with special reference to one-body transition operators.

Given a uniform ensemble of quantum density matrices

ρ

, it is useful to calculate the mean value over this ensemble of a product of entries of

ρ

. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions

P=(P_1, P_2, ..., P_N)

where the

P_j

are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of

P

components. The answer to the classical problem follows from an integral formula due to Dirichlet.

Random matrix theory is used to represent generic loss of coherence of a fixed central system coupled to a quantum-chaotic environment, represented by a random matrix ensemble, via random interactions. We study the average density matrix arising from the ensemble induced, in contrast to previous studies where the average values of purity, concurrence, and entropy were considered; we further discuss when one or the other approach is relevant. The two approaches agree in the limit of large environments. Analytic results for the average density matrix and its purity are presented in linear response approximation. The two-qubit system is analysed, mainly numerically, in more detail.

We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the physical eigenvalue spectrum can be compared. We explain the ideas of the symmetry classification of symmetric matrix spaces and how that yields Dyson's threefold and Altland-Zirnbauer's tenfold way. We also outline how the joint probability density function of the eigenvalues can be calculated from a given probability density function on the matrix space. Furthermore, we dive into the subtleties of the unfolding procedure. For this purpose, we explain the ideas of the local mean level spacing, the local level spacing distribution and the

k

-point correlation functions. We outline the techniques of orthogonal polynomials, determinantal and Pfaffian point processes and their related Fredholm determinants and Pfaffians as well as the supersymmetry method. Moreover, we relate the local spectral statistics to effective Lagrangians that give the relation to non-linear

σ

-models. In all these discussions, we also make brief excursions to non-Hermitian random matrix theory which are useful when studying open quantum systems, for instance.

Classical random matrix ensembles were originally introduced in physics to approximate quantum many-particle nuclear interactions. However, there exists a plethora of quantum systems whose dynamics is explained in terms of few-particle (predom-inantly two-particle) interactions. The random matrix models incorporating the few-particle nature of interactions are known as embedded random matrix ensembles. In the present paper, we provide a brief overview of these two ensembles and illustrate how the embedded ensembles can be successfully used to study decoherence of a qubit interacting with an environment, both for fermionic and bosonic embedded ensembles. Numerical calculations show the dependence of decoherence on the nature of the environment.

Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble-averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with perturbation theory and isotropic entanglement theory.