Many-body localization transition through pairwise correlations

We investigate the phenomenon of spatial many-body localization (MBL) through pairwise correlation measures based on one- and two-point correlation functions. The system considered is the Heisenberg spin-1/2 chain with exchange interaction $J$ and random on-site disorder of strength $h$. As a representative pairwise correlation measure obtained from one-point functions, we use global entanglement. Through its finite-size-scaling analysis, we locate the MBL critical point at $h_c/J=3.8$. As for measures involving two-point functions, we analyze pairwise geometric classical, quantum, and total correlations. Similarly to what happens for continuous quantum phase transitions, it is the derivatives of these two-point correlation measures that identify the MBL critical point, which is found to be in the range $h_c/J\in [3,4]$. Our approach relies on very simple measures that do not require access to multipartite entanglement or large portions of the system.

Figure 1

Global entanglement $G_E$ as a function of the disorder strength $h/J$. Inset (a) shows the finite-size-scaling analysis, with $a=0.5$ and $h_c/J=3.8\pm 0.2$. Inset (b) shows the power-law decay of $G_E$ in the localized phase. Inset (c) shows the nonmonotonic behavior of $G_E$ for small values of $h/J$.

Figure 2

Geometric quantum discord $Q_G$ as a function of the disorder strength $h/J$ on a logarithmic scale. Inset (a) shows the first derivative $Q_G^{\prime }$ of the geometric quantum discord with respect to $h/J$. Inset (b) shows the thermodynamic limit of the maximum of $Q_G^{\prime }$, which yields $h_c/J=3.34\pm 0.03$.

Figure 3

(a) Geometric classical $C_G$ and (b) total $T_G$ correlations as functions of $h/J$ on a logarithmic scale, as well as their derivatives, (c) $C_G^{\prime }$ and (d) $T_G^{\prime }$, with respect to $h/J$. The insets in (c) and (d) exhibit the thermodynamic limit of the maxima of $C_G^{\prime }$ and $T_G^{\prime }$, respectively, yielding $h_c/J=3.43\pm 0.06$ for $C_G$ and $h_c/J=3.45\pm 0.06$ for $T_G$.