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

TL;DR

The paper surveys how free probability theory, via the DOS of sums of non-commuting operators, can approximate spectral densities in quantum many-body and random-matrix models. It introduces the Cauchy and R-transforms, free additive convolution, and subordination formulas as central tools, then applies them to local Haar-Ising Hamiltonians and disordered systems such as the Anderson model. A perturbative scheme based on subordination is developed to compute analytical DOS corrections for models like Rosenzweig-Porter and Anderson with diagonal disorder, including cases where the perturbation DOS is semicircular or arcsine. The work also discusses partial freeness, fourth-moment matching, and the role of free compression/decompression, highlighting both successes and limitations of the approach and outlining avenues for further analytic and numerical development.

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.

Figures (4)

• Figure 1: DOS of the Anderson model with high off-diagonal interaction strength and its comparison with the free probability approximation. Here we have set $J=10$ (left plot) or $J=5$ (right plot), and the size of the Hamiltonian is $N=2000$. The numerical results show an average over 1000 independent realisations of the Hamiltonian. The red curve indicates $\rho_{H}(\lambda)$ obtained from the second-order Cauchy transform in \ref{['GH_2_diso']}, while the black dashed curve represents the arcsine distribution. For comparison, with the brown curves in both the plots, we have also shown a free compressed version of the arcsine distribution (with the associated Cauchy transform in \ref{['Cauchy_KM_com']}, $\eta=2$) with free compression parameter, for the left plot $0.895$, and $0.82$ for the right plot.
• Figure 2: DOS of the Anderson model with low off-diagonal interaction strength and its comparison with the approximate DOS obtained using the subordination formula. Here we have set $J=0.2$ (left plot) or $J=0.3$ (right plot), and the size of the Hamiltonian is $N=2000$. The numerical results show an average over 1000 independent realisations of the Hamiltonian. The red curve indicates $\rho_{H}(\lambda)$ obtained from the second-order Cauchy transform in \ref{['GH_2_diso']}, while the black curve represents the Wigner semicircle distribution. The approximate formula provides a good description of the exact DOS in the bulk of the spectrum; however, it fails near the edges (we have not shown the sharp edges in the red curves for clarity).
• Figure 3: DOS of the Anderson model with Gaussian on-site disorder ($\sigma=1$) and low off-diagonal interaction strength, and its comparison with the approximate DOS obtained using the subordination formula. Here we have set $J=0.2$ (left plot) or $J=0.3$ (right plot), and the size of the Hamiltonian is $N=2000$. The numerical results show an average over 1000 independent realisations of the Hamiltonian. The brown curve indicates $\rho_{H}(\lambda)$ obtained from the Cauchy transform in \ref{['GC_approx']}, while the black dashed curve represents the Gaussian distribution. In the right panel, the red curve represents the DOS obtained from \ref{['GC_kappa_3']} by retaining up to $n=2$ terms.
• Figure 4: Plots of the compressed arcsine and the Kesten-McKay distribution (with generic $\eta$), and their comparison with the DOS obtained from the approximate Cauchy transform in \ref{['Cauc_com_KM_app']}. Plot in the left panel shows the free compression of the arcsine distribution with $\alpha=0.8$, while the plot in the right panel shows the free compression of the Kesten-McKay distribution with $\eta=3$, and $\alpha=0.89$. The red curves show the approximate DOS, while the black dashed curves represent the exact analytical compressed distributions \ref{['comr_KM']}. The brown curve in the left panel indicates the uncompressed arcsine distribution, while the brown curve in the right panel shows the uncompressed Kesten-McKay distribution with $\eta=3$.