Gumbel statistics for entanglement spectra of many-body localized eigenstates
Wouter Buijsman, Vladimir Gritsev, Vadim Cheianov
TL;DR
This work shows that the extreme-value statistics of entanglement spectra in many-body localized eigenstates follow Fisher–Tippett–Gumbel distributions when the largest transformed entanglement entries are properly normalized, despite entanglement spectra in ergodic states showing correlated Wishart–Laguerre behavior. By numerically analyzing disordered spin chains across system sizes up to $L=14$, the authors demonstrate that the standardized variable $\tilde{e}_{\min}=\frac{-e_{\min}-\mu}{\sigma}$ matches the Gumbel density $g(x)$ with mean $0$ and unit variance, particularly for disorder strengths $W=4$–$5$. The result implies that the physical information in the entanglement spectrum is largely contained in the few smallest $e_i$, enabling a parameter-free characterization of MBL eigenstates, though finite-size effects and the thermal side of the MBL transition remain unresolved with the studied sizes. The study opens questions about the mechanism behind Gumbel statistics and invites further exploration via entanglement-Hamiltonian models and larger-scale simulations.
Abstract
An entanglement spectrum encodes statistics beyond the entanglement entropy, of which several have been studied in the context of many-body localization. We numerically study the extreme value statistics of entanglement spectra of many-body localized eigenstates. The physical information encoded in these spectra is almost fully carried by the few smallest elements, suggesting the extreme value statistics to have physical significance. We report the surprising observation of Gumbel statistics. Our result provides an analytical, parameter-free characterization of many-body localized eigenstates.