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An entanglement spectrum encodes statistics beyond the entanglement entropy,of which several have been studied in the context of many-body localization. Wenumerically study the extreme value statistics of entanglement spectra ofmany-body localized eigenstates. The physical information encoded in thesespectra is almost fully carried by the few smallest elements, suggesting theextreme value statistics to have physical significance. We report thesurprising observation of Gumbel statistics. Our result provides an analytical,parameter-free characterization of many-body localized eigenstates.

Gumbel statistics for entanglement spectra of many-body localized eigenstates