Signatures of Nonergodicity in Sparse Random Matrices

Abstract

The prevalence of sparsity in interacting many-body systems motivates an investigation into the spectral statistics of sparse random matrices with on-site disorder. We numerically demonstrate that the Anderson transition can be identified through the statistical properties of the ground state. By analytically deriving the energy moments and calculating the shifted kurtosis, we estimate the critical sparsity threshold for this localization-delocalization transition. The short-range energy correlation in the bulk indicates that the Anderson transition at infinite temperature coincides with the quantum phase transition. Furthermore, long-range energy correlations in the bulk spectrum reveal a Thouless energy scale, suggesting a broad nonergodic regime within the delocalized phase.

Figures (7)

• Figure 1: Matrix representations of various sparse Hamiltonians: (a) 1D Heisenberg model with nearest neighbor interactions and periodic boundary condition in the zero magnetization sector for 8 spin-$\frac{1}{2}$ particles in the $\hat{\sigma}^z$ basis Das2025. (b) Anderson model on $4\times 4\times 4$ cubic lattice with nearest neighbor hopping and periodic boundary condition in the site basis Lydzba2021. (c) Transverse field Ising model with long-range interaction in the odd-parity sector for 7 spin-$\frac{1}{2}$ particles in the $\hat{\sigma}^z$ basis Das2025. (d) Adjacency matrix of an Erdős-Rényi-Gilbert graph on 64 nodes where the probability of finding a non-zero edge is 0.1 Gilbert1959.
• Figure 2: Ground state energies of sGOE. (a) density of the ground state energies obtained using the wall-Chebyshev projector Zhao2024 for $N = 16384$ and different values of $\gamma$. The energy levels are shifted and scaled to ensure zero mean and unit variance. Dashed red (blue) curve denotes the Gumbel (Tracy-Widom) distribution. (b) Rényi divergence of order $q = \frac{1}{2}$ between the numerical histogram of the ground state and the Gumbel (Tracy-Widom) distribution as a function of $\gamma$ for different system sizes shown via solid (dashed) lines.
• Figure 3: Ground state eigenfunctions of sGOE. (a) Fractal dimensions of the ground state for different values of $\gamma$. Horizontal dashed lines denote the smallest fractal dimension, $D_\infty$. Inset shows $D_\infty$ as a function of $\gamma$. (b) Bipartite von Neumann entanglement entropy as a function of system size for various $\gamma$. Gray dashed line denotes the Page value Page1993. Inset shows the system size scaling $\mathcal{S}_{vN} \propto \alpha_{vN} \ln N$. Error-bars denote 95% confidence interval.
• Figure 4: Energy moments and DOS of sGOE. (a) shifted kurtosis (Eq. \ref{['eq:Kurt_def']}) as a function of $\gamma$ for different system sizes (values given in the legend) where sparsity $p \equiv N^{-\frac{\gamma}{2}}$. Markers denote numerical data obtained using Girard-Hutchinson estimator (Eq. \ref{['eq:GH_estimator_def']}) and the solid lines denote the analytical expressions from Eq. \ref{['eq:kurt']}. Horizontal dashed line correspond to $\left\langle K \right\rangle = \frac{25}{4}$. The inset shows the collapsed data assuming 2nd order transition ansatz (Eq. \ref{['eq:2nd_ansatz']}) where $\gamma' \equiv (\gamma - 2)\ln N ^{\frac{1}{\nu}}$, $\nu \approx 1.0464$. (b) ensemble averaged DOS obtained using the Lanczos method Lin2016 for $N = 1024$ and different values of $\gamma$. We fix the energy variance $\sigma_E ^2 = \frac{1}{4}$ for all values of $\gamma$. Dashed red (blue) curve denotes the Gaussian (semicircle) distribution. Inset shows the average DOS at $\gamma = 2$ for different system sizes.
• Figure 5: Short-range energy correlation of sGOE. (a) density of the ratio of level spacings for middle 40% spectrum, $N = 1024$ and different values of $\gamma$. Dashed blue (red) curve denotes the analytical expressions of GOE (Poisson ensemble). (b) average ratio of level spacings as a function of $\gamma$ for different system sizes. Inset shows the collapsed data following Eq. \ref{['eq:2nd_ansatz']} for a critical parameter $\gamma_c = 2$ and exponent $\nu \approx 0.9316$.