Topological property of graphene with triangular array of nanoholes

Xiao Hu; Xing-Xiang Wang; Yong-Cheng Jiang

arxiv: 2605.15631 · v1 · pith:SL4QC6FYnew · submitted 2026-05-15 · ❄️ cond-mat.mes-hall

Topological property of graphene with triangular array of nanoholes

Yong-Cheng Jiang , Xing-Xiang Wang , Xiao Hu This is my paper

Pith reviewed 2026-05-20 17:04 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords graphenenanoholesobstructed atomic limitparity indicesband inversionedge statesflat bandstopological bands

0 comments

The pith

Graphene with a 3√3 × 3√3 triangular nanohole array has a topologically trivial gap at the Fermi level but hosts obstructed atomic limit bands that produce flat edge states in ribbons.

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

The paper examines how regular nanohole arrays that preserve C6v symmetry affect band topology in graphene. For the specific 3√3 × 3√3 triangular pattern, a gap opens at the Gamma point near the Fermi energy through band inversion and shifts in parity indices, yet this gap classifies as topologically trivial. Lower valence bands instead form an obstructed atomic limit with their own parity imbalance, which appears as two flat bands carrying localized edge states when the sheet is cut into ribbons. A reader would care because the work separates trivial and nontrivial topological sectors within one material by simple geometric patterning, showing a route to control which bands support protected edge conduction.

Core claim

For the case of 3√3 × 3√3 triangular array of nanoholes, we find an energy gap at Γ point around the Fermi level associated with a band inversion which induces change in parity indices, whereas deep below the Fermi level there are a bunch of valence bands characterized as obstructed atomic limit (OAL) which also accommodate imbalance in parity indices. This band structure renders the gap at the Fermi level topologically trivial and carrying no edge states, while the nontrivial band topology of the OAL manifests in two flat bands in the ribbon structure associated with localized electronic states at ribbon edges.

What carries the argument

Parity indices obtained from the computed band structure under C6v symmetry, used to classify the Fermi-level gap as trivial and the lower valence bands as an obstructed atomic limit.

If this is right

The gap at the Fermi level carries no protected edge states in this array geometry.
Ribbon geometries of the patterned graphene exhibit two flat bands with electronic states localized at the edges due to the OAL bands.
Band inversion at the Gamma point changes parity indices for states near the Fermi level.
Different regular nanohole arrays under C6v symmetry can produce distinct combinations of trivial and nontrivial topological features.

Where Pith is reading between the lines

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

This separation of trivial and nontrivial sectors by nanohole patterning could be used to create flat bands for correlated states while suppressing unwanted edge conduction at the Fermi energy.
The same approach might be applied to other two-dimensional materials to isolate topological features in specific energy windows.
Real samples would need to preserve the assumed symmetry closely, since deviations could mix the trivial and nontrivial bands.

Load-bearing premise

The parity indices and obstructed atomic limit classification are correctly obtained from the computed band structure under the assumed C6v symmetry and without additional disorder or relaxation effects that would alter the topology.

What would settle it

Observation of conducting edge states inside the Fermi-level gap for a fabricated 3√3 × 3√3 nanohole array would show the gap is topologically nontrivial and contradict the parity-based classification.

Figures

Figures reproduced from arXiv: 2605.15631 by Xiao Hu, Xing-Xiang Wang, Yong-Cheng Jiang.

**Figure 1.** Figure 1: (a) Schematic structure for graphene with 3 √ 3 × 3 √ 3 triangular array of nanoholes with its unit cell and unit vectors. (b) Brillouin zone (BZ) folding from pristine graphene to graphene with 3 √ 3 × 3 √ 3 superstructure. The eigen wavefunctions at the upper/lower band edge are px & py/dx2−y2 & d2xy. This p−d band inversion induces an imbalance in the parity indices of VBs (VBs) between the high-symmetr… view at source ↗

**Figure 2.** Figure 2: (a) Band structure with the top two valence bands and bands No. 16–22 characterized by fragile topology and obstructed atomic limit, respectively. Parity indices at the M and Γ points accumulated below the energy gap in gray are denoted in the bracket. (b)–(g) Eigenvalues of Wilson-loop phases for bands No. 1, 2 (b), No. 3 (c), No. 1–3 (d), No. 4–15 (e), No. 16–22 (f) and No. 23, 24 (g), parity indices wit… view at source ↗

**Figure 3.** Figure 3: (a) Wave functions of the top two valence bands at the high-symmetric momenta M, Σ, Γ and K. (b) Wave functions of bands No. 16–22 at the high-symmetric momenta M, Γ and K displayed in the descending order of energy from left to right. The amplitudes and phases of wave functions are denoted by size and color of dots, respectively. the eigenvalues of Wilson-loop phases without winding (see Figs. 2(a) and 2(… view at source ↗

**Figure 4.** Figure 4: (a) Energy dispersion of a ribbon structure with the topological band gap denoted by a gray region. (b) Local density of states for the immobile edge states. (c) Energy spectrum of a flake composed of seven unit cells, with edge states (magenta dots) in the energy gap (gray region) around E = −1.5 eV. The left inset is the schematic of the flake structure, and the right inset shows an enlarged view of the … view at source ↗

**Figure 5.** Figure 5: Schematic for molecular graphene. (a) CO molecules on Cu(111) surface. The CO molecules given by black dots form the effective hopping pattern while the CO molecules given by gray dots form the edge morphology and the nanoholes. (c) Molecular graphene for 3 √ 3×3 √ 3 triangular array of nanoholes, which consists of 306 CO molecules in total. using the STM technique [37]. Here we provide the design of molec… view at source ↗

read the original abstract

The nontrivial band topology for graphene with regular arrays of nanoholes with $C_{6v}$ symmetry is investigated theoretically. For the case of $3\sqrt{3} \times 3\sqrt{3}$ triangular array of nanoholes, we find an energy gap at $\Gamma$ point around the Fermi level associated with a band inversion which induces change in parity indices, whereas deep below the Fermi level there are a bunch of valence bands characterized as obstructed atomic limit (OAL) which also accommodate imbalance in parity indices. This band structure renders the gap at the Fermi level topologically trivial and carrying no edge states, while the nontrivial band topology of the OAL manifests in two flat bands in the ribbon structure associated with localized electronic states at ribbon edges. The present results exhibit rich topological behaviors in graphene derivatives waiting for explorations.

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.

Desk Editor's Note private letter to a colleague

The paper maps a trivial Fermi gap against nontrivial OAL bands for the 3√3 × 3√3 nanohole pattern but leaves the computational basis thin.

read the letter

The one thing to take away is that for this triangular nanohole array in graphene, the gap around the Fermi level comes out topologically trivial with no edge states, while the obstructed atomic limit bands further down have nontrivial topology that appears as two flat bands with localized states at the edges of a ribbon. What is new here is the specific mapping for the 3√3 × 3√3 supercell. The authors link a band inversion at Gamma to parity changes that trivialize the Fermi gap, and they identify the lower valence bands as OAL with parity imbalance. This is a new geometry for this kind of separation, and it fits with prior work on topological graphene derivatives without repeating old results. The paper does well at laying out the contrast between the two energy ranges and showing the physical signature in the ribbon structure. It keeps things focused on the C6v symmetric case and avoids unnecessary complications. The soft spots are in the supporting details. There is no description of the calculation method, no convergence information, and nothing on how the ribbon bands were obtained. That makes it difficult to judge how sensitive the parity assignments are. The concern about checking parities only at Gamma rather than across all folded TRIMs is a real one. In the reduced zone, the original K and K' points map to other high-symmetry spots, so the full diagnosis of trivial versus nontrivial needs those points too. If the work only used Gamma, the trivial classification for the Fermi gap is less secure than it could be. Still, the overall logic tracks with standard topological indicators, and there are no internal inconsistencies that jump out. This paper is for condensed matter theorists working on engineered 2D materials and defect-induced topology. A reader who cares about concrete examples in graphene would find the flat band result useful. It is not broad enough to interest everyone, but it adds a targeted case. I would not cite it right away in my own papers, but I would send it to peer review. The claims are specific enough that referees can check the calculations once the methods are filled in.

Referee Report

2 major / 1 minor

Summary. The manuscript examines the nontrivial band topology in graphene featuring a regular triangular array of nanoholes with C6v symmetry. For the specific 3√3 × 3√3 supercell, it identifies a band inversion at the Γ point around the Fermi level that alters parity indices, rendering the Fermi-level gap topologically trivial without associated edge states. Deeper valence bands are classified as an obstructed atomic limit (OAL) exhibiting parity imbalance, which manifests as two flat bands in ribbon geometries corresponding to localized electronic states at the ribbon edges.

Significance. If substantiated, the results highlight complex topological features in graphene-based nanostructures, distinguishing between trivial gaps at the Fermi energy and nontrivial OAL topology in lower bands. This could contribute to understanding and designing topological edge states in perforated graphene systems.

major comments (2)

[Abstract and band structure analysis] The topological classification of the Fermi-level gap as trivial and the OAL bands as nontrivial is based on parity index changes following band inversion. However, the description specifies these changes at the Γ point only. In the folded Brillouin zone of the 3√3 × 3√3 supercell, the original K/K' points of graphene map to additional high-symmetry points. A robust parity-product or symmetry-indicator diagnosis requires evaluation of parity eigenvalues at all time-reversal invariant momenta (TRIMs), including Γ, M, and the folded K points. Without confirmation that parities were computed and consistent at these locations, the assignment of triviality to the Fermi gap may not be fully supported. This is load-bearing for the central claim regarding the absence of edge states.
[Methods and results sections] The manuscript provides no details on the computational approach used to obtain the band structure (e.g., density functional theory parameters, tight-binding model specifics, or k-point sampling), convergence tests, or the setup for the ribbon structure calculations that reveal the flat bands and edge-localized states. These details are essential to assess the reliability of the parity indices and the identification of topological features.

minor comments (1)

[Abstract] The phrasing 'energy gap at Γ point around the Fermi level' could be clarified to specify whether the gap is centered at the Fermi energy or its position relative to it.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments have helped us strengthen the topological analysis and improve the presentation of our methods. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and band structure analysis] The topological classification of the Fermi-level gap as trivial and the OAL bands as nontrivial is based on parity index changes following band inversion. However, the description specifies these changes at the Γ point only. In the folded Brillouin zone of the 3√3 × 3√3 supercell, the original K/K' points of graphene map to additional high-symmetry points. A robust parity-product or symmetry-indicator diagnosis requires evaluation of parity eigenvalues at all time-reversal invariant momenta (TRIMs), including Γ, M, and the folded K points. Without confirmation that parities were computed and consistent at these locations, the assignment of triviality to the Fermi gap may not be fully supported. This is load-bearing for the central claim regarding the absence of edge states.

Authors: We appreciate this observation on the completeness of the symmetry-indicator analysis. While the manuscript emphasizes the band inversion and parity change at the Γ point (where the relevant bands cross the Fermi level), we have now explicitly computed the parity eigenvalues at all TRIMs of the folded Brillouin zone, including the M points and the folded images of the original K and K' points. The parity products confirm that the Fermi-level gap remains topologically trivial, whereas the deeper valence bands exhibit the obstructed atomic limit character through parity imbalance at these additional points. This extended analysis has been incorporated into the revised manuscript, with a new table summarizing the parity eigenvalues at all TRIMs to support the classification. revision: yes
Referee: [Methods and results sections] The manuscript provides no details on the computational approach used to obtain the band structure (e.g., density functional theory parameters, tight-binding model specifics, or k-point sampling), convergence tests, or the setup for the ribbon structure calculations that reveal the flat bands and edge-localized states. These details are essential to assess the reliability of the parity indices and the identification of topological features.

Authors: We agree that the original manuscript lacked sufficient methodological transparency. The band structures were obtained from a nearest-neighbor tight-binding model with a hopping parameter of 2.7 eV chosen to match graphene's Dirac dispersion. The 3√3 × 3√3 supercell was constructed by removing carbon atoms to create the triangular nanoholes while preserving C6v symmetry. A 12 × 12 × 1 Monkhorst-Pack grid was used for the supercell Brillouin zone, with convergence verified by doubling the grid density and checking energy differences below 1 meV. Ribbon calculations employed finite-width structures (multiple supercell repetitions across the width) with periodic boundaries along the ribbon direction and open boundaries perpendicular to it, sampled with 100 k-points along the 1D Brillouin zone. We have added a dedicated Methods section, including all parameters, convergence tests, and a description of the ribbon geometry, to the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from direct band-structure computation and standard parity classification

full rationale

The paper computes the electronic band structure for the 3√3 × 3√3 nanohole array under assumed C6v symmetry, identifies a gap at the Fermi level with band inversion at Γ that changes parity indices, and separately classifies deeper valence bands as obstructed atomic limit (OAL) due to their own parity imbalance. These assignments are then used to conclude trivial topology (no edge states) for the Fermi gap versus nontrivial topology manifesting as flat edge bands in the ribbon geometry. This chain relies on explicit eigenvalue calculations and the application of parity-product or symmetry-indicator diagnostics to the obtained bands; it does not reduce any output to the input by definition, does not rename a fitted parameter as a prediction, and does not rest on a load-bearing self-citation whose validity is presupposed. The method is self-contained against external benchmarks (reproducible DFT or tight-binding runs) and carries no evident ansatz smuggling or uniqueness theorem imported from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard topological band theory plus the assumption that the nanohole array preserves C6v symmetry and that the chosen computational model faithfully reproduces the parity eigenvalues.

axioms (2)

domain assumption The nanohole array preserves C6v point-group symmetry
Invoked to justify the band inversion and parity analysis at the Gamma point.
standard math Parity eigenvalues determine the topological character via band inversion
Standard result in topological band theory used to classify both the Fermi gap and the OAL bands.

pith-pipeline@v0.9.0 · 5672 in / 1435 out tokens · 113834 ms · 2026-05-20T17:04:04.145601+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

parity indices accumulated for all VBs between Γ and M points are balanced, namely (N+M, N+Γ)=(12,12), indicating trivial topology for the energy gap at the Fermi level

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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