Localization persisting under aperiodic driving

Localization may survive in periodically driven (Floquet) quantum systems, but is generally unstable for aperiodic drives. In this Letter, we identify a hidden conservation law originating from a chiral symmetry in a disordered spin-$\frac{1}{2}$ XX chain. This protects indefinitely long-lived localization for general—even aperiodic—drives. Therefore, rather counterintuitively, adding further potential disorder which spoils the conservation law delocalizes the system, via a controllable parametrically long-lived prethermal regime. This provides an example of persistent single-particle “localization without eigenstates.”

Figure 1

(a) Average PR for eigenstates of the single-particle time evolution operator $U_m$ at time $t=2^{m}T$. Results are calculated for different system sizes and averaged over 100 disorder realization. The PR remains low in the absence of potential disorder, whereas a finite value $h_{\text{max}}=0.51$ results in delocalization. The inset shows the size dependence of $\overline{R}/L$ at $t\approx 10$ and the scaling $1/L$ is obtained to confirm the localization for short time. (b) PR for the static Hamiltonian, Floquet operator, and TM driving after sufficiently long time $t\approx {10}^{12}$ computed for $L=200$. Potential disorder leads to stronger localization in static and Floquet systems, while any nonvanishing $h_{\text{max}}$ delocalizes the system driven quasiperiodically. We use $J_0=1, J_{\text{max}}=1.6\pi , \delta h=1, h_{\text{max}}=0.51, 1/T=10$ for the simulation.

Figure 2

(a) Entanglement dynamics for different driving rates, with or without potential disorder, averaged over 100 disorder realization and different initial states. Steady state can be delocalized by a nonvanishing potential disorder and thermalized. (b) System size dependence of the entanglement entropy. Prethermal (red dots) and the final plateau (purple squares) exhibit an area- and volume-law scaling, respectively. We use parameters $J_0=1, J_{\text{max}}=1.6\pi , \delta h=1, h_{\text{max}}=0.51, L=400$ for the simulation.

Figure 3

(a) Algebraic dependence of ${\tau }_S$ on chiral symmetry-breaking perturbation $\delta$ with parameter $1/T=1, L=1400$. Dashed black lines correspond to the algebraic fit ${\delta }^{-2.6}$. (b) Delocalization time $\sqrt{ln{\tau }_S}$ vs driving rate $ln1/T$ with parameter $h_{\text{max}}=1.1, J_{\text{nnn}}=0$. The numerical data fit well with a straight line of slope $\sqrt{C}\approx 1$, indicating ${\tau }_S$ behaves as ${\tau }_S\sim e^{C{[ln(T^{-1}/g)]}^2}$. Both figures use $J_0=1, J_{\text{max}}=1.6\pi , \delta h=1$ and 100 disorder average.