Entanglement complexity in quantum many-body dynamics, thermalization, and localization

Entanglement is usually quantified by von Neumann entropy, but its properties are much more complex than what can be expressed with a single number. We show that the three distinct dynamical phases known as thermalization, Anderson localization, and many-body localization are marked by different patterns of the spectrum of the reduced density matrix for a state evolved after a quantum quench. While the entanglement spectrum displays Poisson statistics for the case of Anderson localization, it displays universal Wigner-Dyson statistics for both the cases of many-body localization and thermalization, albeit the universal distribution is asymptotically reached within very different time scales in these two cases. We further show that the complexity of entanglement, revealed by the possibility of disentangling the state through a Metropolis-like algorithm, is signaled by whether the entanglement spectrum level spacing is Poisson or Wigner-Dyson distributed.

Figure 1

Comparison between ESS and energy level spacing statistics after a quantum quench at $t_0=1000$ starting from a random product state in systems that are Anderson localized [(a) and (b)], nonintegrable and featuring ETH [(c) and (d)], and featuring MBL [(e) and (f)]. ESS follows three different distributions, namely, Poisson (a), WD (c), and a nonuniversal one (e), thus perfectly classifying the three different dynamical phases. On the other hand, the distribution of the energy level spacings is always Poisson in all three cases. It becomes WD in the nonintegrable, ETH case shown in the inset of (d) only if the conservation of the total magnetization $S_z$ is broken by a field in the $x$ direction. In the MBL case, the ESS approaches WD upon discarding the largest eigenvalues values of the spectrum [inset of (e)]. All simulations are done with 2000 realizations of disorder and $L=12$ unless otherwise specified.

Figure 2

(a) The KL divergence $D_{\mathrm{KL}}$ as a function of the fraction of truncation of the full spectrum for different total evolution times ($L=14$ and $z_i\in [-8,8]$). The data are averaged over 100 realizations of disorder and 2000 realizations of the initial product state, evolved for times $t=100,500,1000$, and ${10}^6$. (b) Scaling of $D_{\mathrm{KL}}$ with $1/ln\left(t\right)$ for the full spectrum and for the truncated spectrum at fraction 0.1875, consistent with the KL divergence vanishing at long times and the ESS asymtoptically reaching the WD distribution.

Figure 3

Attempt of disentangling using the entanglement cooling algorithm starting from the states at $t_0=1000$. $\overline{S}$ is the von Neumann entropy averaged over all possible bipartitions of the system with $L=12$.