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The extreme-value statistics of the entanglement spectrum in disordered spinchains possessing a many-body localization transition is examined. It isexpected that eigenstates in the metallic or ergodic phase, behave as randomstates and hence the eigenvalues of the reduced density matrix, commonlyreferred to as the entanglement spectrum, are expected to follow the eigenvaluestatistics of a trace normalized Wishart ensemble. In particular, the densityof eigenvalues is supposed to follow the universal Marchenko-Pasturdistribution. We find deviations in the tails both for the disordered XXZ withtotal $S_z$ conserved in the half-filled sector as well as in a model thatbreaks this conservation. A sensitive measure of deviations is provided by thelargest eigenvalue, which in the case of the Wishart ensemble after appropriateshift and scaling follows the universal Tracy-Widom distribution. We show thatfor the models considered, in the metallic phase, the largest eigenvalue of thereduced density matrix of eigenvector, instead follows the generalizedextreme-value statistics bordering on the Fisher-Tipett-Gumbel distributionindicating that the correlations between eigenvalues are much weaker comparedto the Wishart ensemble. We show by means of distributions conditional on thehigh entropy and normalized participation ratio of eigenstates that theconditional entanglement spectrum still follows generalized extreme valuedistribution. In the deeply localized phase we find heavy tailed distributionsand L\'evy stable laws in an appropriately scaled function of the largest andsecond largest eigenvalues. The scaling is motivated by a recently developedperturbation theory of weakly coupled chaotic systems.

Probing the randomness of ergodic states: extreme-value statistics in the ergodic and many-body-localized phases