  The extreme-value statistics of the entanglement spectrum in disordered spin
chains possessing a many-body localization transition is examined. It is
expected that eigenstates in the metallic or ergodic phase, behave as random
states and hence the eigenvalues of the reduced density matrix, commonly
referred to as the entanglement spectrum, are expected to follow the eigenvalue
statistics of a trace normalized Wishart ensemble. In particular, the density
of eigenvalues is supposed to follow the universal Marchenko-Pastur
distribution. We find deviations in the tails both for the disordered XXZ with
total $S_z$ conserved in the half-filled sector as well as in a model that
breaks this conservation. A sensitive measure of deviations is provided by the
largest eigenvalue, which in the case of the Wishart ensemble after appropriate
shift and scaling follows the universal Tracy-Widom distribution. We show that
for the models considered, in the metallic phase, the largest eigenvalue of the
reduced density matrix of eigenvector, instead follows the generalized
extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution
indicating that the correlations between eigenvalues are much weaker compared
to the Wishart ensemble. We show by means of distributions conditional on the
high entropy and normalized participation ratio of eigenstates that the
conditional entanglement spectrum still follows generalized extreme value
distribution. In the deeply localized phase we find heavy tailed distributions
and L\'evy stable laws in an appropriately scaled function of the largest and
second largest eigenvalues. The scaling is motivated by a recently developed
perturbation theory of weakly coupled chaotic systems.

Disordered Systems and Neural Networks;Statistical Mechanics;Quantum Physics