Extreme value statistics and eigenstate thermalization in kicked quantum chaotic spin-$1/2$ chains

It is often expected (and assumed) for a quantum chaotic system that the presence of correlated eigenvalues implies that all the other properties as dictated by random matrix theory are satisfied. We demonstrate using the spin-$1/2$ kicked field Ising model that this is not necessarily true. We study the properties of eigenvalues of the reduced density matrix for this model, which constitutes the entanglement spectrum. It is shown that the largest eigenvalue does not follow the expected Tracy--Widom distribution even for the large system sizes considered. The distribution instead follows the extreme value distribution of Weibull type. Furthermore, we also show that such deviations do not lead to drastic change in the thermalization property of this system by showing that the models satisfy the diagonal and off-diagonal eigenstate thermalization hypothesis. Finally, we study the spin-spin autocorrelation function and numerically show that it has the characteristic behavior for chaotic systems: it decreases exponentially and saturates to a value at late time that decreases with system size.