Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems

In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.

Figure 1

Imbalance at finite times. Measurements of the imbalance $\mathcal{I}$ after $10\tau$ (light points) and $40\tau$ (dark points) for the noninteracting system and at $U=4J$. The noninteracting data is vertically offset by 0.15 for clarity. The data represents averages over 12 disorder phases $\varphi$ with error bars indicating the uncertainty of the mean. Solid lines are guides to the eye. In the interacting system, we observe a regime (gray shaded), where the imbalance after $40\tau$ is significantly lower than after $10\tau$, indicating a dynamical evolution of the system. A similar, but much less pronounced, feature is also present in the noninteracting case.

Figure 2

Time evolution of the imbalance close to the MBL transition. Decay of an initially prepared charge-density wave at a fixed interaction strength of $U=4J$. Points mark experimental data, averaged over six disorder phases $\varphi$, with error bars indicating the uncertainty of the mean. The corresponding ED simulations for $S=20$ sites [43] are indicated as solid lines. During the first three tunneling times (not shown), the imbalance quickly decays from its initial value close to 1. During this initial decay, the imbalance shows strong oscillations, which cease after $\sim 8\tau$. Thereafter, we observe a much slower further decay. (a) Time traces for various disorder strengths with power-law fits. (b) Long term decay at intermediate disorder strengths on a logarithmic $y$ axis with an exponential fit (left) and on a double-log plot with a power-law fit (right).

Figure 3

Power-law exponent of imbalance decay. Experimental and theoretical (ED, $S=20$, see Ref. [43]) fitted exponents $\alpha$ as a function of disorder strength $\mathrm{\Delta }$ at a fixed interaction strength of $U=4J$. Error bars indicate the uncertainty of the fit to the experimental data. The purple shading denotes an estimate of the uncertainty on the simulated exponents based on finite size effects. For the largest disorder strengths, systematic errors due to finite time and size do not allow an accurate estimation of $\alpha$ and the actual uncertainty is likely underestimated [43]. The gray shading marks the regime of slow dynamics as observed in Fig 1. At large disorder strengths, the experimental value saturates at a nonzero offset ${\alpha }_o$, consistent with the independently observed background lifetime [43, 48, 49]. The finite value of $\alpha$ in ED for large disorder strength is likely caused by finite size effects. The corresponding exponents for the noninteracting data can be found in the Supplemental Material [43].