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We study the entanglement spectrum of highly excited eigenstates of two knownmodels that exhibit a many-body localization transition, namely theone-dimensional random-field Heisenberg model and the quantum random energymodel. Our results indicate that the entanglement spectrum shows a"two-component" structure: a universal part that is associated with randommatrix theory, and a nonuniversal part that is model dependent. Thenon-universal part manifests the deviation of the highly excited eigenstatefrom a true random state even in the thermalized phase where the eigenstatethermalization hypothesis holds. The fraction of the spectrum containing theuniversal part decreases as one approaches the critical point and vanishes inthe localized phase in the thermodynamic limit. We use the universal partfraction to construct an order parameter for measuring the degree of randomnessof a generic highly excited state, which is also a promising candidate forstudying the many-body localization transition. Two toy models based onRokhsar-Kivelson type wave functions are constructed and their entanglementspectra are shown to exhibit the same structure.

Two-component Structure in the Entanglement Spectrum of Highly Excited States