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

Wen-Jia Rao

doi:10.1103/PhysRevB.105.054207

paper

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

arXiv:2110.05120 · doi:10.1103/PhysRevB.105.054207

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 $λ_{k}$ . We show that the scree plot ( $λ_{k}$ with respect to $k$ ) in the ergodic phase contains two branches that both follow power-law $λ_{k} \sim k^{- α}$ but with different exponents $α$ . By evaluating $W_{k}$ , it's verified the part of $λ_{k}$ with $k > k_{Th}$ is universal that follows random matrix theory, where $k_{Th}$ is related to the Thouless energy. We further demonstrate that $α$ 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, $α$ gives an underestimation for the many-body localization transition point, which suggests a non-ergodic behavior that may be attributed to the Griffiths regime.

arXiv comment7 pages, 5 figures

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