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arxiv: 2507.17250 · v3 · submitted 2025-07-23 · 🪐 quant-ph · cond-mat.dis-nn· hep-ph· math-ph· math.MP· physics.comp-ph

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

Dinesh Kumar Panda , Colin Benjamin This is my paper

Pith reviewed 2026-05-19 03:23 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nnhep-phmath-phmath.MPphysics.comp-ph
keywords quantum walkstopological phasesflat bandsedge statescyclic graphsDirac conesphase transitions
0
0 comments X

The pith

Step-dependent quantum walks on cyclic graphs produce gapped flat bands, Dirac cones, and protected edge states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that cyclic quantum walks, using only step-dependent operations on simple cycle graphs, generate both gapped and gapless topological phases including topologically nontrivial flat bands and Dirac-cone dispersions. It shows analytically that gap closings in rotation space produce Dirac cones in momentum space and that protected edge states form at interfaces between distinct topological phases. A sympathetic reader would care because the scheme avoids resource-intensive split-step or split-coin protocols, works for both odd and even cycle lengths, and yields states robust against moderate static and dynamic disorder. Flat bands appear exclusively in 4n-site graphs. The approach is presented as a compact platform for noise-resilient quantum information tasks in small-scale systems.

Core claim

We introduce cyclic quantum walks on cyclic graphs and use discrete Fourier transforms to obtain an effective Hamiltonian. This Hamiltonian reveals the conditions for emergence of topological gapped flat bands, shows that gap closings in rotation space imply Dirac cones in momentum space, and demonstrates protected edge states at the interface between distinct topological phases in both odd and even cycle graphs. These edge states remain robust against moderate static and dynamic gate disorder, phase-preserving perturbations, and are independent of initial states.

What carries the argument

The effective Hamiltonian derived via discrete Fourier transform on the cyclic graph, which encodes the topological invariants, phase transitions, and edge-state protection.

If this is right

  • Rotationally symmetric flat bands emerge exclusively in 4n-site graphs.
  • Gap closings in rotation space produce Dirac cones in momentum space.
  • Protected edge states form at interfaces between distinct topological phases for both odd and even cycles.
  • The edge states remain stable under moderate static and dynamic gate disorder and independent of initial state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The resource-efficient protocol could reduce hardware overhead for implementing topological quantum memory or protected state transfer on near-term devices.
  • The difference between odd and even cycle spectra suggests systematic studies of how graph parity controls topological features in other discrete quantum systems.
  • Robustness to disorder implies the edge states might serve as building blocks for compact fault-tolerant components even when perfect isolation from the environment is unavailable.

Load-bearing premise

The discrete Fourier transform applied to the cyclic graph produces an effective Hamiltonian that accurately captures the topological invariants and protects the edge states without further assumptions on physical implementation.

What would settle it

An experiment that measures the energy spectrum of a physical cyclic quantum walk on a 4n-site graph and checks whether gap closings in rotation space produce Dirac cones in momentum space, or whether flat bands appear only for those lengths, would confirm or refute the central claims.

Figures

Figures reproduced from arXiv: 2507.17250 by Colin Benjamin, Dinesh Kumar Panda.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematics of (a) a Dirac cone for CQW, where en [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Energy dispersion vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Energy dispersion vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Probability of the particle at position [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematics of (a) a Dirac cone for CQW, where energy gap closing is linear; (b) two distinct topological phase regimes [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Energy dispersion relation vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Group velocity (Eq. (17) with + sign) vs quasi-momenta [PITH_FULL_IMAGE:figures/full_fig_p014_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. (a) Probability of the particle at position [PITH_FULL_IMAGE:figures/full_fig_p015_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Probability of the particle at position [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. (a) Probability of the particle at position [PITH_FULL_IMAGE:figures/full_fig_p016_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. (a) Edge state at the interface (site 0) between two distinct phases (i.e., with [PITH_FULL_IMAGE:figures/full_fig_p017_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. (a) Edge state at the interface (site 0) between two distinct phases (i.e., with [PITH_FULL_IMAGE:figures/full_fig_p018_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. (a) Absence of edge state due to identical topological phase ( [PITH_FULL_IMAGE:figures/full_fig_p018_19.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces cyclic quantum walks (CQWs) on odd- and even-site cycle graphs, employing step-dependent walks and discrete Fourier transforms to derive an effective Hamiltonian. It analytically establishes conditions for gapped topological flat bands (appearing only in 4n-site graphs), shows that gap closings in rotation space produce Dirac cones in momentum space, and claims to engineer protected edge states at interfaces between distinct topological phases. Numerical simulations are presented to demonstrate robustness of these edge states to moderate static/dynamic gate disorder, phase-preserving perturbations, and independence from initial states.

Significance. If the analytical conditions and interface construction hold, the work offers a resource-efficient platform for realizing flat bands, Dirac cones, and robust edge states in small-scale quantum systems without split-step protocols. The explicit distinction between odd/even cycles and the numerical robustness tests against disorder constitute concrete strengths that could support applications in protected state transfer and compact topological quantum devices.

major comments (1)
  1. [abstract and edge-state engineering section] The central claim that protected edge states form at interfaces between distinct topological phases (as stated in the abstract and the section on engineering edge states) rests on bulk invariants computed from the translationally invariant effective Hamiltonian obtained via DFT. Because an interface explicitly breaks translational invariance, the manuscript does not demonstrate that the bulk-boundary correspondence continues to hold; no explicit interface Hamiltonian, gap-closing analysis at the junction, or proof of localization is provided to confirm topological protection.
minor comments (1)
  1. [introduction] Notation for the effective Hamiltonian and the rotation-space versus momentum-space spectra should be introduced with explicit equations early in the text to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the major comment below and describe the revisions we intend to implement.

read point-by-point responses

  1. Referee: [abstract and edge-state engineering section] The central claim that protected edge states form at interfaces between distinct topological phases (as stated in the abstract and the section on engineering edge states) rests on bulk invariants computed from the translationally invariant effective Hamiltonian obtained via DFT. Because an interface explicitly breaks translational invariance, the manuscript does not demonstrate that the bulk-boundary correspondence continues to hold; no explicit interface Hamiltonian, gap-closing analysis at the junction, or proof of localization is provided to confirm topological protection.

    Authors: We agree that the interface construction breaks translational invariance and that the manuscript invokes bulk topological invariants without supplying an explicit interface Hamiltonian or a direct analytical demonstration that the bulk-boundary correspondence holds at the junction. The current numerical results show localized states whose robustness to disorder is consistent with topological protection, yet they do not constitute a proof of localization. In the revised manuscript we will add an explicit description of the interface Hamiltonian, a gap-closing analysis across the phase boundary, and supplementary numerical diagnostics of localization length to substantiate the topological character of the observed edge states. revision: yes

Circularity Check

0 steps flagged

Derivation from DFT on cyclic graphs is self-contained with no reduction to inputs

full rationale

The paper derives an effective Hamiltonian via discrete Fourier transform applied to the translationally invariant cyclic-graph evolution operator, then analytically obtains conditions for gapped flat bands and gap-closing implications for Dirac cones directly from the resulting momentum-space spectrum. Edge-state claims are supported by separate numerical simulations at engineered interfaces. No parameters are fitted to the target topological quantities, no self-citations carry the central load, and the steps do not reduce by construction to the inputs; the results follow from the graph structure and walk protocol under the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides insufficient detail to enumerate specific free parameters, axioms, or invented entities; standard quantum mechanics and graph theory are implicitly used but not itemized.

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Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages · 1 internal anchor

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    Analytical results on energy dispersion, effective mass and group velocity As discussed in the main text page 2, part on”Model”, a cyclic quantum walk (CQW) describes the propagation of the spatial distribution of a single quantum particle (e.g., electron or photon) on anN-cycle graph, i.e., onNsites of a cyclic graph (e.g., atomic sites or position or or...

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    whenever the group velocity (v gr) is zero, i.e., atθ= (2n+ 1) π T , n∈Z + ∪ {0}(Eq. (23)). We note that gapless flat-bands are not possible in CQW, as it would require cos T θ 2 ̸= 0 which differs from the condition of flat band formation in CQW systems. (a) (b) (c) FIG. 13. Group velocity (Eq. (17) with + sign) vs quasi-momentakand rotation angleθwith (...

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