Dynamics of many-body localization

Following the field theoretic approach of Basko et al. [Ann. Phys. 321, 1126 (2006)], we study in detail the real-time dynamics of a system expected to exhibit many-body localization. In particular, for time scales inaccessible to exact methods, we demonstrate that within the second-Born approximation the temporal decay of the density-density correlation function is consistent with a finite value for $t\to \infty$, as expected in a nonergodic state. This behavior persists over a wide range of disorder and interaction strengths and suggests that the nonergodic phase is surprisingly robust compared to the prediction of Basko et al. We argue that this discrepancy may be resolved by the presence of a nonergodic, but metallic, phase.

Figure 1

Dynamical phase boundaries of the system as a function of $W/t$ and $V/t$. The red star represents the transition point obtained in Ref. [13] and the gray solid line an extension of their calculation to a critical line ($L=12$; see their Fig. 6). The blue crosses denote the parameters used in this work. Vertical cut: $W=3.5$ and $V=0.1–0.4$. Horizontal cut: $W=1,3,5$ and $V=0.25$. The solid and dotted black lines are the solution of Eq. (3), demarcating the boundary between metal from insulator and ergodic and nonergodic metals, correspondingly. The dashed lines are extrapolations to regions below which such equations are not valid.

Figure 2

(a) Density-density correlation function as a function of time averaged over 128 disorder realizations $(V=0.25)$. The black lines designate the exact solution calculated using ED $(L=12)$ while other lines designate the solution using QME. (b) Finite size behavior for the same parameters using QME. The system size increases for darker tones. The system sizes used: $W=0,$ $L=2^{14}$ (black dashed); $W=1,$ $L=12,48,96$ (red); $W=3,$ $L=12,28,36$ (green); and $W=5,$ $L=12,28$ (blue). Shaded areas designate uncertainty bounds.

Figure 3

(a) Modified density-density correlation $[\delta \overline{\rho }\left(t\right)\equiv 1-4\delta \rho \left(t\right)]$ as a function of $Vt$, averaged over 128 disorder realizations, for $W=3.5,L=36$. (b) Left panel rescaled by plotting $V^{-0.78}\delta \overline{\rho }$ as a function of $Vt$ (see text). Shaded areas designate uncertainty bounds.