Error Analysis of Free Probability Approximations to the Density of States of Disordered Systems

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.

Figure 1

Calculation of the DOS, $\rho (\xi )$, of the Hamiltonian $H$ of (8) with $M=5000$ samples of $2000\times 2000$ matrices for (a) low, (b) moderate and (c) high noise ($\sigma /J=0.1$, 1 and 10, respectively, with $\sigma =1$). For each figure we show the results of free convolution defined in scheme I (${\rho }^{(A_1⊞B_1)}$; black solid line), scheme II (${\rho }^{(A_2⊞B_2)}$; green dashed line) and exact diagonalization (${\rho }^{(H)}$; red dotted line).

Figure 2

Diagrammatic expansion of the term $⟨A_1{B}_1{A}_1{B}_1{A}_1{B}_1{A}_1{B}_1⟩$ in terms of allowed paths dictated by the matrix elements of $A_1$ and $B_1$ of scheme I in (9a).

Figure 3

DOS, $\rho (\xi )$, of the Hamiltonian (8) with $M=5000$ samples of $2000\times 2000$ matrices with (a) low, (b) moderate and (c) high semicircular on-site noise ($\sigma /J=0.1$, 1 and 10, respectively, with $\sigma =1$), as calculated with exact diagonalization (red dotted line), free convolution (black solid line), and perturbation theory with $A_1$ as reference (blue dashed line) and $B_1$ as reference (gray dash-dotted line). The partitioning scheme is scheme I of (9a).