Robust Floquet Topological Phases and Anomalous $π$-Modes in Quasiperiodic Quantum Walks

TL;DR

The paper investigates one-dimensional discrete-time quantum walks with Fibonacci-modulated coin rotations to understand how quasiperiodicity affects Floquet topological phases. It uses Fibonacci rational approximants and long-time averaging of the mean chiral displacement (MCD) to extract a bulk winding invariant, revealing a fractal phase diagram with a robust topological region characterized by $\nu=-1$. It further shows edge states residing at both $E=0$ and $E=\pi$ with identical localization lengths, demonstrating that topological protection persists despite a Cantor-like bulk spectrum due to chiral symmetry. The work highlights that Floquet topology survives strong aperiodic disorder, offering a concrete route to observing non-equilibrium fractal topological phases in photonic experiments and informing disorder-immune topological photonics in aperiodic media.

Abstract

We uncover the global topological phase diagram of one-dimensional discrete-time quantum walks driven by Fibonacci-modulated coin parameters. Utilizing the mean chiral displacement (MCD) as dynamical probe, we identify robust topological phases defined by a strictly quantized winding number $ν=-1$ and exponentially localized edge states. Crucially, we discover that these topological edge modes emerges not only at zero energy but also at the quasienergy zone boundary $E=π$, exhibiting identical localization robustness despite the fractal nature of the bulk spectrum. These results demonstrate that Floquet topological protection remains intact amidst quasiperiodic disorder, offering a concrete route to observing exotic non-equilibrium phases in photonic experiments.

Figures (5)

• Figure 1: (Color online) Global topological phase diagram. The winding number $\nu$, extracted from the mean chiral displacement (MCD), is mapped in the $(\theta_A, \theta_B)$ parameter plane for $N=F_{18}=2584$. Yellow regions indicate the topological phase ($\nu\approx -1$), while dark regions correspond to metallic or gapless regimes. The color scale is saturated at $-12$ for visibility; actual MCD values in the metallic regions reach $\sim -1200$.
• Figure 2: (Color online) Topological phase transition and quantization. (a) The topological winding number $\nu$, defined as twice the long-time averaged mean chiral displacement (MCD), is plotted against $\theta_A$ along the cut $\theta_B = 0$. Robust plateaus are observed in the topological regions (centered around $\theta_A\approx \pm\pi/2$), whereas the metallic regions exhibit large fluctuations or asymptotic divergence, signifying the ballistic nature of the wave packet. (b) Magnified view of the topological regime ($\pi/4 \leq \theta_A \leq 3\pi/4$), confirming a quantized plateau at $\nu=-1$.
• Figure 3: (Color online) Topological edge states and their localization properties calculated at $\theta_A=1$ and $\theta_B=0$. (a) Quasienergy spectrum. Bulk states are shown in black dots. The isolated zero-energy modes (solid royal blue circles) and $\pi$-energy modes (open red triangles) are clearly visible within the gaps. (b) Spatial probability distributions of the edge modes shown in (a). The zero mode (blue circles) and the $\pi$ mode (red triangles) exhibit identical localization lengths, despite their distinct microscopic oscillation patterns. The $\pi$ mode is spatially mirrored for comparison. The gray dashed line indicates the exponential fit with $\xi \approx 0.66$, shifted vertically for clarity.
• Figure S1: (Color online) Extended dynamical phase diagram utilizing the mean chiral displacement (MCD) under periodic boundary conditions. The data were computed for a system size of $N=F_{15}=610$ with a long-time average over $T=2450$ steps. The topological phase (bright yellow region, $\nu\approx -1$) remains strictly quantized and featureless, demonstrating its robustness. In contrast, the metallic and critical phases exhibit intricate self-similar patterns reminiscent of the Hofstadter butterfly. These complex structures arise from the interference of wave packets wrapping around the system boundaries, visually highlighting the fractal nature of the bulk dynamics.
• Figure S2: (Color online) Quasienergy spectrum as a function of $\theta_A$ for fixed $\theta_B=0$. The bulk bands (black dots) exhibit a fractal structure characteristic of the Fibonacci modulation. Topological edge states are clearly resolved within the gaps centered at $E=0$ (blue dots) and $E=\pi$ (red dots). Note that due to the $2\pi$-periodicity of the quasienergy, states at the lower boundary $E=-\pi$ are mapped to $E=+\pi$. As $\theta_A$ deviates from the deep topological limit ($\pm\pi/2$), the energy gaps narrow, leading to an increase in the localization length of the edge modes.