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

arXiv:2603.07701v2 Announce Type: replace-cross 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 $\pm \frac{\pi}{2}$), in sharp contrast with the integer invariants in standard cyclic quantum walks. These fractional invariants lead to an unconventional bulk-boundary correspondence and support protected {zero energy} edge states beyond the usual integer topological classification. In the step-dependent protocol, transitions between distinct fractional winding sectors generate robust edge modes {pinned at zero quasienergy}. 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 and fewer operators, the proposed scheme offers a minimal-resource and experimentally accessible platform for realizing fractional topology, flat bands, and protected {zero energy} edge states in small-scale synthetic quantum systems.

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 $\pm \frac{\pi}{2}$), in sharp contrast with the integer invariants in standard cyclic quantum walks. These fractional invariants lead to an unconventional bulk-boundary correspondence and support protected {zero energy} edge states beyond the usual integer topological classification. In the step-dependent protocol, transitions between distinct fractional winding sectors generate robust edge modes {pinned at zero quasienergy}. 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 and fewer operators, the proposed scheme offers a minimal-resource and experimentally accessible platform for realizing fractional topology, flat bands, and protected {zero energy} edge states in small-scale synthetic quantum systems.

Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Applied Physics (physics.app-ph); Quantum Physics (quant-ph); Computation (stat.CO)

arXiv:2603.07701 [cond-mat.str-el]

(or arXiv:2603.07701v2 [cond-mat.str-el] for this version)

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

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