Delocalized glassy dynamics and many-body localization

We analyze the unusual slow dynamics that emerges in the bad metal delocalized phase preceding the many-body localization transition by using single-particle Anderson localization on the Bethe lattice as a toy model of many-body dynamics in Fock space. We probe the dynamical evolution by measuring observables such as the imbalance and equilibrium correlation functions, which display slow dynamics and power laws strikingly similar to those observed in recent simulations and experiments. We relate this unusual behavior to the nonergodic spectral statistics found on Bethe lattices. We discuss different scenarios, such as a true intermediate phase which persists in the thermodynamic limit versus a glassy regime established on finite but very large time and length scales only, and their implications for real-space dynamical properties. In the latter, slow dynamics and power laws extend on a very large time window but are eventually cut off on a time scale that diverges at the many-body localization transition.

Figure 1

Imbalance $I\left(t\right)$ as a function of time for different disorder strengths (log-log plot). Dashed straight lines highlight the apparent power-law behavior $I\left(t\right)\sim t^{-\beta }$.

Figure 2

Equilibrium correlation function $C\left(t\right)$ as a function of time for different disorder strengths (log-log plot). Dashed straight lines highlight the apparent power-law behavior $C\left(t\right)\sim t^{-\alpha }$.

Figure 3

Log-log plot of $K_2\left(E\right)$ for different system sizes $M=2^n$ ($n=11,12,14,15$). The straight dashed line highlights the apparent power-law behavior $K_2\left(E\right)\sim E^{-\mu }$. Inset: Power-law exponents $\alpha ,\beta ,\mu$ observed numerically as a function of $W$. For smaller values of $W$ our precision on the exponents decreases since we can track the power laws for fewer decades. The difference found for $W=11,12$ is explained in the Supplemental Material [37].