Universal entanglement spectra in random quantum circuits*

We will discuss the time evolution of the entanglement spectra in a spin-1/2 chain
under random unitary dynamics. We find that the reduced density matrix develops
level repulsion on an O(1) time scale while the bandwidth of the entanglement
spectrum is rapidly changing, well before the subsystem has fully thermalized. In
this locally thermal regime, random matrix theory describes the correlations
between nearby entanglement energy levels but fails to capture global level
correlations in the entanglement spectral form factor or the entanglement density of
states. We find this behavior is universal and holds for a multitude of other quantum
circuits. We will provide a heuristic explanation of our results.