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

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.