Quantum walks reveal topological flat bands, robust edge states and topological phase transitions in cyclic graphs

Abstract

Topological phases, edge states, and flat bands in synthetic quantum systems are a key resource for topological quantum computing and noise-resilient information processing. We introduce a scheme based on step-dependent quantum walks on cyclic graphs, termed cyclic quantum walks (CQWs), to simulate exotic topological phenomena using discrete Fourier transforms and an effective Hamiltonian. Our approach enables the generation of both gapped and gapless topological phases, including Dirac cone-like energy dispersions, topologically nontrivial flat bands, and protected edge states, all without resorting to resource-intensive split-step or split-coin quantum walk protocols. Odd and even-site cyclic graphs exhibit markedly different spectral characteristics, with rotationally symmetric flat bands emerging exclusively in $4n$-site graphs ($n\in \mathbf{N}$). We analytically establish the conditions for the emergence of topological, gapped flat bands and show that gap closings in rotation space imply the formation of Dirac cones in momentum space. Further, we engineer protected edge states at the interface between distinct topological phases in both odd and even cycle graphs. We numerically demonstrate that the edge states are robust against moderate static and dynamic gate disorder as well as remain stable against phase-preserving perturbations and are independent of initial states. This scheme serves as a resource-efficient and versatile platform to engineer topological phases, transitions, edge states, and flat bands in small-scale quantum systems, opening new avenues for robust quantum memory, protected state transfer, and compact implementations of fault-tolerant quantum technologies.

Figures (21)

• Figure 1: Schematics of (a) a Dirac cone for CQW, where energy gap closing is linear; (b) two distinct topological phase regimes are shown in green and red on a 4-cycle and an edge state (wave peak in blue) is expected to form at the boundary between the phases.
• Figure 2: Energy dispersion vs quasi-momenta $k$ and rotation angle $\theta$ for (a) $N=7$, (b) $N=8$-cycles and (c) $N=1000$ (with Dirac cones). The blue (red) surface refers to the upper (lower) energy band. Winding number $\omega$ vs $\theta$ for (d) $N=7$, (e) $N=8$ and (f) $N=1000$ (continuum limit), for step-dependent ($T=2$) CQW.
• Figure 3: (a) Probability of the particle at position $x=0$ vs time-step $t$ showing chaotic evolution; (b) Absence of edge state due to identical topological phase ($\omega=1$) throughout position space i.e., no boundary; (c) Generation of edge state (persistent over $t$) at the interface (site 0) between two distinct phases (i.e., with $\omega=-1$ and $\omega=+1$), via step-dependent CQW ($T=2$), for 8-cycle.
• Figure 4: Schematics of (a) a Dirac cone for CQW, where energy gap closing is linear; (b) two distinct topological phase regimes are shown in green and red on a 4-cycle and an edge state (wave peak in blue) is expected to form at the boundary between the phases.
• Figure 5: Energy dispersion vs quasi-momenta $k$ and rotation angle $\theta$ for (a) $N=7$, (b) $N=8$-cycles and (c) $N=1000$ (with Dirac cones). The blue (red) surface refers to the upper (lower) energy band.; Winding number $\omega$ vs $\theta$ for (d) $N=7$, (e) $N=8$ and (f) $N=1000$ ($k$ continuum limit), for step-independent ($T=1$) CQW.