  The entanglement spectrum of the reduced density matrix contains information
beyond the von Neumann entropy and provides unique insights into exotic orders
or critical behavior of quantum systems. Here, we show that strongly disordered
systems in the many-body localized phase have power-law entanglement spectra,
arising from the presence of extensively many local integrals of motion. The
power-law entanglement spectrum distinguishes many-body localized systems from
ergodic systems, as well as from ground states of gapped integrable models or
free systems in the vicinity of scale-invariant critical points. We confirm our
results using large-scale exact diagonalization. In addition, we develop a
matrix-product state algorithm which allows us to access the eigenstates of
large systems close to the localization transition, and discuss general
implications of our results for variational studies of highly excited
eigenstates in many-body localized systems.
