Fractional Topological Phases, Flat Bands, and Robust Edge States on Finite Cyclic Graphs via Single-Coin Split-Step Quantum Walks

Abstract: We report the first realization of a fractional topological phase in a fully unitary, noninteracting discrete-time quantum walk implemented on finite cyclic graphs. Using a single-coin split-step cyclic quantum walk (SCSS-CQW), we uncover topological phenomena that are inaccessible within conventional cyclic quantum-walk dynamics. The protocol enables controlled engineering of quasienergy spectra, flat bands, and topological phase transitions through the step-dependency parameter and coin-rotation angle. We show that cyclic graphs with even and odd numbers of sites exhibit qualitatively different band structures, while rotational flat bands arise exclusively in $4n$-site cycles; a general analytic condition for their emergence is derived. The SCSS-CQW produces fractional winding numbers $\pm \frac{1}{2}$ (Zak phases $\pm \fracπ{2}$), in sharp contrast with the integer invariants of standard quantum walks. These fractional invariants lead to an unconventional bulk--boundary correspondence and support edge states beyond the usual integer topological classification. In the step-dependent protocol, transitions between distinct fractional winding sectors generate robust edge modes. Numerical simulations show that these states remain stable in the presence of both dynamic and static coin disorder as well as phase-preserving perturbations, while survival-probability analysis demonstrates their long-time persistence. Requiring only a constant number of detectors independent of the evolution time, the proposed scheme offers a minimal-resource and experimentally accessible platform for realizing fractional topology, flat bands, and protected edge states in small-scale synthetic quantum systems.