Quantum Mutual Information as a Probe for Many-Body Localization

We demonstrate that the quantum mutual information (QMI) is a useful probe to study many-body localization (MBL). First, we focus on the detection of a metal-insulator transition for two different models, the noninteracting Aubry-André-Harper model and the spinless fermionic disordered Hubbard chain. We find that the QMI in the localized phase decays exponentially with the distance between the regions traced out, allowing us to define a correlation length, which converges to the localization length in the case of one particle. Second, we show how the QMI can be used as a dynamical indicator to distinguish an Anderson insulator phase from a MBL phase. By studying the spread of the QMI after a global quench from a random product state, we show that the QMI does not spread in the Anderson insulator phase but grows logarithmically in time in the MBL phase.

Figure 1

Qualitative behavior of the QMI in the two different phases of the interacting disorder model $\mathcal{H}$ (1) for a fixed disorder configuration. $t=V=W=1$ (left) and $t=V=1$ $W=5$ (right). The red dots represent the sites of the chain, and the thickness of the blue bonds between sites $\{i,j\}$ is proportional to the magnitude of $\{\mathcal{I}([i],[j])\}/\{{\max }_{i,j}\mathcal{I}([i],[j])\}$ averaged over 16 eigenstates in the middle of the spectrum.

Figure 2

(AAH model) The upper left panel shows the localization length $\xi$ for different system sizes as a function of disorder strength $W$. The dashed line at $W_c=2$ represents the known transition point between the extended and localized states [49]. For values below $W_c$, $\xi$ increases with system size, while for values above $W_c$ it saturates. The right upper panel shows $\overline{\sigma }$ for different system sizes as a function of disorder strength $W$; for values of $W$ below $W_c$ $\overline{\sigma }$ grows with system size while for values above $W_c$ it saturates. The lower panels show ${\xi }_j^{-1}$ in two different phases: for $W=1.5$ in the extended phase ${\xi }_j^{-1}$ goes to zero as a function of $j$, while for $W=2.2$ in the localized phase it saturates to a positive value implying a finite correlation length $\xi$.

Figure 3

(Spinless Hubbard chain) The top left panel shows the localization length $\xi$ for different system sizes as a function of disorder strength $W$. The top right panel shows $\overline{\sigma }$ for different system sizes as a function of disorder strength $W$, for values $W<4$ it grows with system size, while for larger values it saturates. The vertical dashed line at $W_c=3.5$ is the value for the expected transition [11, 17]. The bottom lower panels show ${\xi }_j^{-1}$ in the two different phases. For $W=1$ in the extended phase, ${\xi }_j^{-1}$ goes to zero as a function of $j$, for $W>4$ in the localized phase it starts to saturate to a positive value implying a finite correlation length.

Figure 4

$\overline{⟨⟨X^2⟩⟩}$ for different system sizes for $W=6$, and for $V=0$ (noninteracting). For $V=0$ $\overline{⟨⟨X^2⟩⟩}$ saturates at time of the order one and with system size. For $V\ne 0$, $\overline{⟨⟨X^2⟩⟩}\sim \log (t)$.

Figure 5

$T_{\min }$ for different system size and in two different phases. For $W=1.5$ extended phase, it grows algebraically. In the localized phase ($W=6$) the time to entangled two separated region of the systems grows exponentially with their distance.