Thermalization of entanglement

We explore the dynamics of the entanglement entropy near equilibrium in highly entangled pure states of two quantum-chaotic spin chains undergoing unitary time evolution. We examine the relaxation to equilibrium from initial states with either less or more entanglement entropy than the equilibrium value, as well as the dynamics of the spontaneous fluctuations of the entanglement that occur in equilibrium. For the spin chain with a time-independent Hamiltonian and thus an extensive conserved energy, we find slow relaxation of the entanglement entropy near equilibration. Such slow relaxation is absent in a Floquet spin chain with a Hamiltonian that is periodic in time and thus has no local conservation law. Therefore, we argue that slow diffusive energy transport is responsible for the slow relaxation of the entanglement entropy in the Hamiltonian system.

Figure 1

Time evolution of the entanglement entropy for $L=14$ for product random (PR) initial states under Floquet dynamics (2) with $\tau =0.8$ (red line with circles) as well as Hamiltonian dynamics (1) (blue line with down triangles); “generalized Bell” initial states made from pairs of random pure (RP) states (green line with up triangles) and from “oppositely paired” (OP) states (purple line with diamonds) both under Hamiltonian dynamics. Each case is averaged over 400 initial pure states, and the error estimates are too small to be visible in this figure. See main text for more details.

Figure 2

Normalized histogram of the entanglement entropy for $L=16$ for (dashed blue line) the eigenstates of the Hamiltonian (1), the eigenstates of the Floquet operator [Eq. (2)] with $\tau =0.8$ (solid green line), and for random pure states of the full chain (very narrow red distribution), where the histogram is over 2000 randomly generated pure states.

Figure 3

Thermalization of entanglement entropy in three cases for $L=14$ (log-log scale) from random pure (RP) generalized Bell states under Hamiltonian dynamics (circular markers, green), from RP generalized Bell states under Floquet dynamics (square markers, blue), and from oppositely paired (OP) generalized Bell states under Hamiltonian dynamics (triangular markers, red). Within Hamiltonian dynamics, the larger initial energy imbalance for the OP initial states dictates a slower thermalization of the entanglement, while the absence of energy conservation for the Floquet system allows the fastest thermalization of entanglement entropy among all cases considered. See main text for description of initial states.

Figure 4

Autocorrelation of the entanglement entropy vs time for random pure states under Hamiltonian (solid lines) as well as Floquet dynamics (dashed lines) with different system sizes $L$ in log-linear scale. For each case, the autocorrelation is normalized to be one at time zero. Under Floquet dynamics, the autocorrelation decays as a simple exponential function of time, and faster than under Hamiltonian dynamics. Only weak size dependence of this normalized autocorrelation is observed under either dynamics.

Figure 5

Autocorrelation of the entanglement entropy under Hamiltonian dynamics vs square root of time for random pure states with different system sizes $L$. On this semi-log scale, all three curves roughly follow a straight line and the tail becomes more straight as system size increases.

Figure 6

Relaxation of total energy under Floquet dynamics for $L=12$ with different choices of driving period $\tau$.