Approaching Thouless Energy and Griffiths Regime in Random Spin Systems By Singular Value Decomposition

TL;DR

This paper uses singular value decomposition to analyze the eigenvalue spectra of random spin systems, revealing universal behaviors related to Thouless energy and Griffiths regimes, and providing insights into many-body localization transitions.

Contribution

It introduces a novel SVD-based method to study eigenvalue spectra, identifying universal power-law behaviors and their relation to ergodic and Griffiths regimes in random spin systems.

Findings

Eigenvalue spectra exhibit two power-law branches with different exponents.

The part of the spectrum beyond Thouless energy follows random matrix theory.

The exponent relates to the exponential part of level spacing distribution, not symmetry.

Abstract

We employ singular value decomposition (SVD) to study the eigenvalue spectra of random spin systems. By SVD, eigenvalue spectrum is decomposed into orthonormal modes $W_k$ with weight $\lambda_k$. We show that the scree plot ($\lambda_k$ with respect to $k$) in the ergodic phase contains two branches that both follow power-law $\lambda_k\sim k^{-\alpha}$ but with different exponents $\alpha$. By evaluating $W_k$, it's verified the part of $\lambda_k $ with $k>k_{\text{Th}}$ is universal that follows random matrix theory, where $k_{\text{Th}}$ is related to the Thouless energy. We further demonstrate that $\alpha$ corresponds only to the exponential part of the level spacing distribution while being insensitive to the level repulsion, or equivalently the system's symmetry. Consequently, $\alpha$ gives an underestimation for the many-body localization transition point, which suggests a non-ergodic behavior that may be attributed to the Griffiths regime.