  We study the entanglement spectrum in the many body localizing and
thermalizing phases of one and two dimensional Hamiltonian systems, and
periodically driven `Floquet' systems. We focus on the level statistics of the
entanglement spectrum as obtained through numerical diagonalization, finding
structure beyond that revealed by more limited measures such as entanglement
entropy. In the thermalizing phase the entanglement spectrum obeys level
statistics governed by an appropriate random matrix ensemble. For Hamiltonian
systems this can be viewed as evidence in favor of a strong version of the
eigenstate thermalization hypothesis (ETH). Similar results are also obtained
for Floquet systems, where they constitute a result `beyond ETH', and show that
the corrections to ETH governing the Floquet entanglement spectrum have
statistical properties governed by a random matrix ensemble. The particular
random matrix ensemble governing the Floquet entanglement spectrum depends on
the symmetries of the Floquet drive, and therefore can depend on the choice of
origin of time. In the many body localized phase the entanglement spectrum is
also found to show level repulsion, following a semi-Poisson distribution (in
contrast to the energy spectrum, which follows a Poisson distribution). This
semi-Poisson distribution is found to come mainly from states at high
entanglement energies. The observed level repulsion only occurs for interacting
localized phases. We also demonstrate that equivalent results can be obtained
by calculating with a single typical eigenstate, or by averaging over a
microcanonical energy window - a surprising result in the localized phase. This
discovery of new structure in the pattern of entanglement of localized and
thermalizing phases may open up new lines of attack on many body localization,
thermalization, and the localization transition.
