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

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.