Quantum revivals and many-body localization

We show that the magnetization of a single “qubit” spin weakly coupled to an otherwise isolated disordered spin chain exhibits periodic revivals in the localized regime, and retains an imprint of its initial magnetization at infinite time. We demonstrate that the revival rate is strongly suppressed upon adding interactions after a time scale corresponding to the onset of the dephasing that distinguishes many-body localized phases from Anderson insulators. In contrast, the ergodic phase acts as a bath for the qubit, with no revivals visible on the time scales studied. The suppression of quantum revivals of local observables provides a quantitative, experimentally observable alternative to entanglement growth as a measure of the “nonergodic but dephasing” nature of many-body localized systems.

Figure 1

Quench protocol and magnetization dynamics. (a) We consider the postquench dynamics of a two-level “qubit” $\mathbf{S}$ coupled to a one-dimensional chain of atoms with strength $\lambda$ [see Eq. (1)]. (b) Time series for a single instance of disorder, and influence of the interactions strength $J_z=0.0,0.025$, and 0.05 on the revivals. (c) Averaged time evolution in the ergodic and MBL phases. Observe that the disorder-averaged magnetization $\overline{\langle S^z\left(t\right)\rangle }$ is only slightly diminished with little or no finite-size scaling in the MBL phase, in contrast to the ergodic phase where the magnetization is significantly lower at long times, scaling to zero as $L\to \infty$. Note that the weak oscillations in the MBL phase scale with ${\lambda }^{-1}$, and were found to be apparently independent of any localization physics.

Figure 2

Phase diagram of the disordered XXZ chain, obtained by finite-size-scaling analysis of infinite-time qubit magnetization $\overline{S_{\infty }^z}$. Inset: Sample finite-size scaling for two representative cuts, shown. The dashed extrapolations are shown to guide the eye; see [37] for a detailed finite-size analysis.

Figure 3

Quantum revivals. Disorder-averaged revival rate $\overline{\mathcal{N}\left(T\right)}/T$ as a function of total time, $T$. Upon adding interactions of strength $J_z$, revivals are suppressed beyond $T^{*}\sim J_z^{-1}$. Inset: The same data collapses onto a universal curve when plotted against $J_{z}T$, with ${\mathcal{N}}_0\left(T\right)={\left.\mathcal{N}\left(T\right)\right|}_{J_z=0}$.