Localization Counteracts Decoherence in Noisy Floquet Topological Chains

The topological phases of periodically driven, or Floquet systems, rely on a perfectly periodic modulation of system parameters in time. Even the smallest deviation from periodicity leads to decoherence, causing the boundary (end) states to leak into the system’s bulk. Here, we show that in one dimension this decay of topologically protected end states depends fundamentally on the nature of the bulk states: a dispersive bulk results in an exponential decay, while a localized bulk slows the decay down to a diffusive process. The localization can be due to disorder, which remarkably counteracts decoherence even when it breaks the symmetry responsible for the topological protection. We derive this result analytically, using a novel, discrete-time Floquet-Lindblad formalism and confirm our findings with the help of numerical simulations. Our results are particularly relevant for experiments, where disorder can be tailored to protect Floquet topological phases from decoherence.

Figure 1

Floquet chain with timing noise. (a) Driving protocol: $H_i$, $i=1,\dots ,4$, is active for time interval $T_i=T/4$, enabling hopping along bonds as indicated by the colors. Sites are labeled according to their doublet $j$ (gray dashed ellipses) and their sublattice $s$ (empty or full circles). In a finite ladder of length $L$, two end sites marked by dashed circles have no doublet partner, $(0,-)$ and $(L,+)$, and host topologically protected end states. (b) Timing noise corresponds to fluctuations of the durations $T_i$ by random uncorrelated amounts ${\tau }_i$. (c) For a dispersive bulk, excitations out of the end state $|e⟩$ due to the timing noise are carried away with a finite group velocity $v_g$, leading to an exponential decay. (d) For a localized bulk, excitations perform a random walk between localized states, resulting in a diffusive decay.

Figure 2

End state survival probability $s=\overline{|⟨e|{\psi }_n⟩{|}^2}$ with driving cycles $n$. A clean, dispersive system (blue) leads to exponential decay with finite-size saturation at long times. A localized bulk shows diffusive decay at long times, both for a clean system at resonant driving (yellow) and a disordered system with on-site disorder strength $V$ (orange and green). For comparison, the black solid (dashed) line indicates purely exponential decay with rate $2J^2{\tau }^2$ (diffusive decay $\sim t^{-1/2}$ where $t=nT$). The blue (yellow) curve is averaged over 2000 (5000) noise realizations, while the green and orange curves are averaged over 4000 simultaneous disorder-noise configurations.