arxiv: 2604.20399 · v2 · submitted 2026-04-22 · ❄️ cond-mat.mtrl-sci · physics.comp-ph

Second-order topology in two-dimensional azulenoid kekulene carbon lattices

Xiaorong Zou , Hyeon Suk Shin , Chang-Jong Kang , Baibiao Huang , Yanmei Zang , Ying Dai , Chengwang Niu , Chang Woo Myung This is my paper

Pith reviewed 2026-05-10 00:32 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.comp-ph

keywords higher-order topological insulatorcarbon allotropesazulenoid kekulenecorner statesC6 symmetrytwo-dimensional materialsfractional charge

0 comments p. Extension

The pith

Two-dimensional azulenoid-kekulene carbon lattices realize a higher-order topological phase with fractionally charged corner states.

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

The paper demonstrates that specific two-dimensional carbon allotropes constructed from azulenoid and kekulene units enter a higher-order topological insulator phase. This phase is enabled by the lattices' C6 rotational symmetry and is identified through a bulk topological invariant together with a corner charge quantized at one-third of an electron. If the calculations hold, finite pieces of these lattices host protected electronic states localized at corners rather than along edges. The work also checks a structurally modified version that keeps the same corner localization, indicating the phase survives certain changes in bonding.

Core claim

In the organic lattices AKC-[3,3] and AKC-[6,0], C6 symmetry produces a nontrivial higher-order topology quantified by the invariant {[M(I)2],[K(3)2]} = {0,2} and a corner charge Q_corner = e/3; the same charge quantization and associated corner-localized states persist in the derived structure PAK-[6,0].

What carries the argument

C6 rotational symmetry that enforces the higher-order bulk invariant and the resulting fractional corner charge of e/3.

If this is right

Finite nanoflakes of these lattices exhibit exotic corner-localized electronic states.
The higher-order phase remains intact after the structural modification that produces PAK-[6,0].
Azulenoid-kekulene carbon allotropes constitute a platform where crystalline symmetry directly controls higher-order boundary responses.

Where Pith is reading between the lines

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

Similar symmetry-driven corner states might appear in other carbon allotropes engineered to possess six-fold rotational symmetry.
The fractional charge could be probed in transport or scanning-probe experiments on synthesized flakes without requiring external magnetic fields.

Load-bearing premise

First-principles calculations capture the topological invariants, band structure, and fractional corner charge without large errors from exchange-correlation approximations or finite-size effects.

What would settle it

An experimental measurement on a nanoflakes sample that finds corner charge exactly equal to e/3 (or exactly zero) would confirm or refute the predicted quantization.

Figures

Figures reproduced from arXiv: 2604.20399 by Baibiao Huang, Chang-Jong Kang, Chang Woo Myung, Chengwang Niu, Hyeon Suk Shin, Xiaorong Zou, Yanmei Zang, Ying Dai.

**Figure 2.** Figure 2: Electronic band structures of AKC-[3,3] lattice ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Edge-state spectra of (a) AKC-[3,3] and (b) AKC-[6 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: (a) Top view of the crystal structure for PAK-[6,0] [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

The discovery of higher-order topological insulator (HOTI) has established a new paradigm for understanding symmetry-constrained boundary electronic states. Here, based on first-principles calculations, we demonstrate the emergence of HOTI phase in organic lattices of two-dimensional azulenoid-kekulene-type carbon allotropes, namely AKC-[3,3] and AKC-[6,0]. Enabled by the $C_6$ rotational symmetry, the nontrivial bulk topology is confirmed through the topological invariant and fractionally quantized corner charge, giving $\{[M^{(I)}_{2}],[K^{(3)}_{2}]\}$ = $\{0,2\}$ and $Q_{\mathrm{corner}} = e/3$, respectively, accompanied by the emergence of exotic corner states in nanoflakes. Notably, the structural modifications are explored, revealing that in the derived structure PAK-[6,0], whose corner-localized states are preserved, highlighting the robustness of the higher-order topological phase. These findings highlight azulenoid-kekulene-based carbon allotropes as a promising platform to explore the interplay between structural design, crystalline symmetry, and higher-order topological boundary responses in two dimensional carbon systems.

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 introduces two new C6-symmetric carbon lattices that DFT predicts to host higher-order topology with e/3 corner charge, and tests robustness in a modified variant.

read the letter

The main point is that AKC-[3,3] and AKC-[6,0] are presented as new 2D carbon allotropes where C6 symmetry protects a higher-order topological phase, diagnosed by the invariant {[M(I)2],[K(3)2]} = {0,2} and a fractionally quantized corner charge of e/3 in nanoflakes. The authors also check a derived structure, PAK-[6,0], and find the corner states survive the modification. That combination of new lattices plus a robustness check is the concrete addition here. It gives a purely carbon, organic example of symmetry-protected boundary states that could be interesting for material design if the predictions hold. The symmetry arguments themselves are standard and applied cleanly to these motifs. The calculations appear to follow the usual first-principles route for band topology and finite-system charge integration. The soft spot is that the abstract and available details do not show explicit convergence tests, functional comparisons, or finite-size scaling for the corner charge. Standard DFT can shift band orders or produce non-exact fractions when k-meshes, relaxations, or van der Waals corrections are not tightly controlled, so those checks would be needed to make the e/3 result fully convincing. If the full manuscript supplies them and they look solid, the central claim strengthens considerably. This is the sort of computational discovery paper that people working on 2D carbon allotropes or higher-order topology in organics would want to see. It is not yet at the level of a definitive result, but the structures and symmetry analysis are worth referee time to verify the numerics and assess whether the phase is as robust as claimed.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to demonstrate a higher-order topological insulator (HOTI) phase in two-dimensional azulenoid-kekulene carbon allotropes AKC-[3,3] and AKC-[6,0] via first-principles calculations. Enabled by C6 rotational symmetry, the nontrivial topology is diagnosed by the invariant {[M(I)2],[K(3)2]} = {0,2} and fractionally quantized corner charge Q_corner = e/3, with associated corner states appearing in nanoflakes; a derived structure PAK-[6,0] is presented to illustrate robustness of the phase.

Significance. If the first-principles results hold under scrutiny, the work would identify a new family of purely carbon-based 2D lattices hosting C6-protected higher-order topology, offering a chemically tunable platform for studying fractional corner charges and boundary states without requiring heavy elements or strong spin-orbit coupling.

major comments (2)

[Computational Methods / Results] The manuscript provides no details on the first-principles methodology (exchange-correlation functional, plane-wave cutoff, k-mesh density, or convergence criteria) nor any tests of sensitivity of the high-symmetry-point eigenvalues at M and K. This directly affects the reliability of the reported invariant {[M(I)2],[K(3)2]} = {0,2} (abstract and § Results).
[Nanoflake calculations] No analysis of finite-size effects, edge termination, or structural relaxation is given for the nanoflakes used to extract Q_corner = e/3. Polarization leakage or relaxation-induced shifts could move the integrated corner charge away from exact quantization even when the bulk invariant is formally correct (abstract and nanoflakes discussion).

minor comments (1)

[Abstract] The final sentence of the abstract is grammatically incomplete ('revealing that in the derived structure PAK-[6,0], whose corner-localized states are preserved, highlighting the robustness...'), reducing clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the constructive comments that have helped us strengthen the manuscript. We address each major comment below and have revised the manuscript to incorporate the requested information and analyses.

read point-by-point responses

Referee: [Computational Methods / Results] The manuscript provides no details on the first-principles methodology (exchange-correlation functional, plane-wave cutoff, k-mesh density, or convergence criteria) nor any tests of sensitivity of the high-symmetry-point eigenvalues at M and K. This directly affects the reliability of the reported invariant {[M(I)2],[K(3)2]} = {0,2} (abstract and § Results).

Authors: We agree that the original manuscript did not provide sufficient methodological details. In the revised version, we have added a dedicated Computational Methods section that specifies the exchange-correlation functional employed, the plane-wave energy cutoff, the k-point mesh density used for Brillouin zone sampling, and the convergence criteria for self-consistent calculations. We have also included explicit sensitivity tests in which the k-mesh density and cutoff energy were systematically varied; these tests confirm that the eigenvalues at the M and K points (and therefore the topological invariant {[M^{(I)}_{2}],[K^{(3)}_{2}]} = {0,2}) remain unchanged within the numerical precision of the calculations. These additions directly address the concern regarding the reliability of the reported invariant. revision: yes
Referee: [Nanoflake calculations] No analysis of finite-size effects, edge termination, or structural relaxation is given for the nanoflakes used to extract Q_corner = e/3. Polarization leakage or relaxation-induced shifts could move the integrated corner charge away from exact quantization even when the bulk invariant is formally correct (abstract and nanoflakes discussion).

Authors: We thank the referee for raising this important point about the robustness of the corner-charge quantization. In the revised manuscript we have expanded the nanoflakes section to include a systematic study of finite-size effects by considering nanoflakes of increasing lateral dimensions. We have also examined different edge terminations (including hydrogen passivation) and performed full structural relaxations of the nanoflakes. The additional data show that the integrated corner charge converges to e/3 and remains quantized to within numerical accuracy even after relaxation, with no evidence of polarization leakage that would violate the quantization expected from the bulk invariant. These results reinforce the protection of the corner states by the higher-order topology. revision: yes

Circularity Check

0 steps flagged

No circularity: standard topological invariants computed from first-principles inputs

full rationale

The derivation computes the C6-protected invariant {[M(I)2],[K(3)2]}={0,2} and corner charge Q_corner=e/3 directly from DFT band eigenvalues at high-symmetry points and integrated charge in finite nanoflakes. These are independent diagnostic outputs, not quantities fitted to the target result or defined in terms of themselves. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing; the structural models and symmetry arguments are external to the final invariants. The paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on standard condensed-matter assumptions about the validity of first-principles band-structure calculations for topology and on the existence of C6 symmetry in the proposed lattices; no new particles or forces are introduced.

axioms (2)

domain assumption First-principles electronic-structure methods accurately predict topological invariants and fractional corner charges in 2D carbon systems
Invoked when the abstract states that nontrivial bulk topology is confirmed through the topological invariant and fractionally quantized corner charge
domain assumption The C6 rotational symmetry is preserved in the relaxed structures
Stated as enabling the nontrivial bulk topology

pith-pipeline@v0.9.0 · 5539 in / 1503 out tokens · 31076 ms · 2026-05-10T00:32:40.457119+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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