Self-averaging in many-body quantum systems out of equilibrium: Approach to the localized phase

The self-averaging behavior of interacting many-body quantum systems has been mostly studied at equilibrium. The present paper addresses what happens out of equilibrium, as the increase of the strength of on-site disorder takes the system to the localized phase. We consider two local and two nonlocal quantities of great experimental and theoretical interest. In the delocalized phase, self-averaging depends on the observable and on the timescale, but the picture simplifies substantially when localization is reached. In the localized phase, the local observables become self-averaging at all times while the nonlocal quantities are throughout non-self-averaging. These behaviors are explained and scaling analysis is provided using the $\ell$-bit model and a toy model.

Figure 1

Left panels: Mean value of the survival probability. Right panels: Relative variance of the survival probability. The values of the disorder strength are indicated on the panels, they increase from the top to the bottom panel. The curves correspond to system sizes $L=10$ (black), 12 (blue), 14 (green), and 16 (red). The horizontal dashed line in Figs. 1, 1 and 1 indicates the saturation point, ${\sum }_{\alpha }{\left|c_{\alpha }^0\right|}^4$ [Eq. (15)], for $L=16$.

Figure 2

Left panels: Mean value of the inverse participation ratio. Right panels: Relative variance of the inverse participation ratio. The values of the disorder strength are indicated on the left panels; they increase from the top to the bottom panel. The curves correspond to system sizes $L=10$ (black), 12 (blue), 14 (green), and 16 (red).

Figure 3

Left panels: Mean value of the spin autocorrelation function. Right panels: Relative variance of the spin autocorrelation function. The values of the disorder are indicated on the left panels; they increase from the top to the bottom panel. The curves correspond to system sizes $L=10$ (black), 12 (blue), 14 (green), and 16 (red). The horizontal dashed lines in Figs. 3, 3 and 3 mark the saturation value of the spin autocorrelation function [Eq. (18)] for $L=16$.

Figure 4

Left panels: Mean value of the connected spin-spin correlation function. Right panels: Relative variance of the connected spin-spin correlation function. The values of the disorder are indicated on the left panels, they increase from the top to the bottom panel. The curves correspond to system sizes $L=10$ (black), 12 (blue), 14 (green), and 16 (red).

Figure 5

Rescaled survival probability $-L^{-1}ln\langle P_S\rangle$ in (a) and (c) and its rescaled relative variance $-L^{-1}ln(\langle P_S\rangle +1)$ in (b) and (d) for the noninteracting model $(\mathrm{\Delta }=0)$ of Eq. (4) in (a) and (b) and for the interacting case $(\mathrm{\Delta }=1)$ in (c) and (d). The system sizes are indicated.

Figure 6

Left panels: Mean value of the spin autocorrelation function. Right panels: Relative variance of the spin autocorrelation function. In both panels, the dynamics was computed using the effective $\ell$-bit Hamiltonian $H_{\text{eff}}$ in Eq. (9) with $h=6$ and $\mathrm{\Delta }=0.1$ for system sizes $L=48$, 96, 192. The dashed-line represents the noninteracting model ($\mathrm{\Delta }=0$) for $L=48$.

Figure 7

Left panels: Mean value of the connected spin-spin correlation function. Right panels: Relative variance of the connected spin autocorrelation function. In both panels, the dynamics was computed using the effective $\ell$-bit Hamiltonian $H_{\text{eff}}$ in Eq. (9) with $h=6$ and $\mathrm{\Delta }=0.1$ for system sizes $L=12$, 24, 48. The dashed-line represents the noninteracting model ($\mathrm{\Delta }=0$) for $L=12$.