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.

Figures (18)

• Figure 1: Schematics for (a) symmetric gapped flat-bands (red, blue) and gapless flat-bands (green); (b) creating a phase boundary between site 0 and the remaining sites in a 7-site cycle, where an edge-state (blue peak, which is robust and remains nearly unchanged over time), is expected to appear via SCSS-CQW dynamics. Note: The single-step evolution operator of SCSS-CQW is a composite sequence, $\hat{U}_{evo} = \hat{S}_{+} \hat{C}_\gamma \hat{S}_{-} \hat{C}_\gamma$, involving alternating two conditional shifts and coin rotations, rather than a single homogeneous coin–shift operation, $\hat{U}_{evo} = \hat{S} \hat{C}_\gamma$.
• Figure 2: Dependence of quasienergy $E(k)$ on momentum $k$ and coin angle ($\gamma$) for: (a) $7$-cycle; (b) $8$-cycle; (c) $1000$-cycle. The red (blue) curves represent the upper (lower) quasienergy branches. (d)–(f) The corresponding winding number $\omega$ as a function of $\gamma$ for the $7$-cycle, $8$-cycle, and $1000$-cycle, respectively, obtained using the SI SCSS-CQW protocol with $D=1$. A fractional winding number of $-\frac{1}{2}$ is observed, with no phase transition.
• Figure 3: Dependence of quasienergy $E(k)$ on momentum $k$ and coin angle ($\gamma$) for: (a) $7$-cycle; (b) $8$-cycle; (c) $1000$-cycle. The red (blue) curves represent the upper (lower) quasienergy branches. (d)–(f) The corresponding winding number $\omega$ as a function of $\gamma$ for the $7$-cycle, $8$-cycle, and $1000$-cycle, respectively, obtained using the SD SCSS-CQW protocol with $D=2$. Fractional winding numbers $\pm \frac{1}{2}$ are observed, with a phase transition occurring at $\gamma=\pi$.
• Figure 4: (a) Spatial probability $P(0)$ of the walker at site $0$ as a function of discrete time steps $t$ for rotation angle $\gamma=\frac{3\pi}{4}$. (b) A uniform topological phase ($\omega=-\frac{1}{2}$ for $\gamma=\frac{3\pi}{4}$) produces no boundary and therefore no edge state, as reflected in the probability profile $P(x)$ versus $x$ at different $t$. (c) Introducing an interface between two distinct topological phases ($\omega=\frac{1}{2}$ for $\gamma=\frac{6\pi}{4}$ and $\omega=-\frac{1}{2}$ for $\gamma=\frac{3\pi}{4}$) results in a robust edge state localized at site $x=0$, persisting over time, as demonstrated using the SD SCSS-CQW protocol ($D=2$) on a $7$-site cyclic graph.
• Figure 5: Edge-state survival probability $P(x=0)$ vs time step $t$ using SD SCSS-CQW ($D=2$) on a $7$-cycle graph, without any disorder. The numerical data exhibit fluctuations around a constant mean value 0.85, and are well captured by a constant fit (red), demonstrating long-time saturation and the stability of the localized edge state.