Early Breakdown of Area-Law Entanglement at the Many-Body Delocalization Transition

We introduce the numerical linked cluster expansion as a controlled numerical tool for the study of the many-body localization transition in a disordered system with continuous nonperturbative disorder. Our approach works directly in the thermodynamic limit, in any spatial dimension, and does not rely on any finite size scaling procedure. We study the onset of many-body delocalization through the breakdown of area-law entanglement in a generic many-body eigenstate. By looking for initial signs of an instability of the localized phase, we obtain a value for the critical disorder, which we believe should be a lower bound for the true value, that is higher than current best estimates from finite size studies. This implies that most current methods tend to overestimate the extent of the localized phase due to finite size effects making the localized phase appear stable at small length scales. We also study the mobility edge in these systems as a function of energy density, and we find that our conclusion is the same at all examined energies.

Figure 1

The $n$th order area-law contribution $a_n$ [Eq. (5)] at $T=\infty$. Near $h_c$, data has been averaged over more than $3\times 10^5$ disorder realizations of the chain, and error bars show the standard error of the mean. The dashed lines are a demonstration of the linear extrapolation to $a_{\infty }$ by fitting the last four terms in the series.

Figure 2

Plot of the ratios $r_n=a_n/a_{n-1}$ versus $1/n$. Also shown is the line $\exp (-1/n)$, above which we argue that the expansion cannot converge exponentially.

Figure 3

The phase diagram with (scaled) energy and disorder on the axes, identified by interpolation of the intercept method (Fig. 1) up to order 10. Error bars represent confidence in our interpolation. Also shown is the $h_c$ obtained by finite size scaling of the Lanczos results from Ref. [33] (error bars are not shown).