Evolution of Entanglement Spectra under Generic Quantum Dynamics

We characterize the early stages of the approach to equilibrium in isolated quantum systems through the evolution of the entanglement spectrum. We find that the entanglement spectrum of a subsystem evolves with three distinct timescales. First, on an $o(1)$ timescale, independent of system or subsystem size and the details of the dynamics, the entanglement spectrum develops nearest-neighbor level repulsion. The second timescale sets in when the light cone has traversed the subsystem. Between these two times, the density of states of the reduced density matrix takes a universal, scale-free $1/f$ form; thus, random-matrix theory captures the local statistics of the entanglement spectrum but not its global structure. The third time scale is that on which the entanglement saturates; this occurs well after the light cone traverses the subsystem. Between the second and third times, the entanglement spectrum compresses to its thermal Marchenko-Pastur form. These features hold for chaotic Hamiltonian and Floquet dynamics as well as a range of quantum circuit models.

Figure 1

Spectral properties of the reduced density matrix under generic time evolution. (a) Spectral density of the reduced density matrix for random unitary ($R$) and chaotic Hamiltonian ($H$) dynamics at early times; this follows a $1/f$ distribution (dashed line). (b) Adjacent gap ratio of the entanglement spectrum, as a function of time, comparing $R$ and $H$ dynamics. Colors denote the model (red for $R$ and blue for $H$); for each color, empty symbols are for system size $L=12$ and subsystem size $l_A=6$, whereas filled symbols are for $L=16$, $l_A=8$. The random-matrix prediction [$⟨r⟩\approx 0.599$] is marked with a dashed black line. (c),(d) Evolution of entanglement bandwidth $w$ and von Neumann entanglement entropy $S_1$, for $R$ and $H$ dynamics with $L=16$, $l_A=8$. For both $R$ and $H$ dynamics, the entanglement bandwidth grows until $t=l_A/2$, then shrinks, whereas the entanglement entropy keeps growing.

Figure 2

Pure random unitary circuits: (a) The disconnected ESFF $D(\theta )$ and (b) the connected ESFF $D_c(\theta )$ for various different times. The legend in (a) is shared across these figures.

Figure 3

Random unitary circuits with a single conservation law. (a) Evolution of the ESFF in the conserving case for an initial state with definite particle number, at $L=16$, $l_A=6$, averaged over 600 samples. Note the appearance of ramp-plateau structure despite the Poisson level statistics in (b). (b) Level statistics parameter $r$ for the conserving circuit with fixed- and variable-number initial states, which, respectively, approach Poisson and random-matrix behavior (dashed lines show the exact distributions [34] of the $r$ ratio for Poisson and GUE distributions, respectively).

Figure 4

Dependence on the local Hilbert space $q$. (a) Entanglement DOS at a fixed time, $t=2$, as a function of local Hilbert space dimension $q$. The shape of the DOS does not seem to change much with $q$, though the average entanglement energy goes down as one might expect. (b) Adjacent gap ratio $r$ vs $q$ at a fixed time $t=1$ and $t=2$, these are the only times at which there are appreciable deviations from GUE level statistics for $q>2$ (the dashed line marks the exact GUE value $r\approx 0.599$).