Topological carbon materials: a new perspective

Marvin L. Cohen; Shengbai Zhang; Xiaohong Yan; Yuanping Chen; Yuee Xie

REVIEW 3 major objections 5 minor 14 references

Topological carbon materials: a new perspective

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Carbon allotropes realize a spinless topological class, distinct from heavy-element materials, because their spin-orbit coupling is negligible.

desk verdict A useful review of spinless topological carbon, but the CHC experimental anchor is disputed and an uncited higher-order TI claim needs fixing. read the letter →

arxiv 1908.10108 v1 pith:P4CJQN5T submitted 2019-08-27 cond-mat.mtrl-sci cond-mat.mes-hall

Yuanping Chen , Yuee Xie , Xiaohong Yan , Marvin L. Cohen , Shengbai Zhang This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.mes-hall

keywords topologicalsemimetalcarbonallotropespinlesstime-reversalsymmetrynodalringWeylsurfacetriplepointnexusnetworkorbitalfrustration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This review argues that carbon allotropes are a distinct arena for topological physics because carbon's spin-orbit coupling is negligibly small. Treating electron spin as a dummy variable makes time reversal satisfy $T^2 = 1$ instead of $T^2 = -1$, so the topological classifications possible in carbon differ fundamentally from those in heavy-element materials. The paper catalogs predicted topological phases across carbon structures—Weyl-like points and loops, nodal rings of three types, Weyl surfaces, triple points, and nexus networks—and traces them all to the same orbital origin: the $p$ orbitals of sp$^2$ carbon behaving like rank-1 tensors. If these predictions hold, carbon offers unusually clean topological electronic states, with a single $p$ orbital dominating the gapless physics over a nearly 10 eV window.

What carries the argument

The central machinery is the spinless time-reversal symmetry with $T^2 = 1$, which changes how band crossings are classified, together with the orbital physics of carbon's $p$ electrons, which behave as rank-1 tensor (vector) degrees of freedom on frustrated lattices. The review uses $\mathbf{k}\cdot\mathbf{p}$ models built from symmetry representations—including Gell-Mann matrices for isospin-1 triplet fermions and 3×3 Hamiltonians for nexus networks—to capture the band crossings, and shows how each topological object, be it a Weyl-like point, nodal ring, Weyl surface, triple point, or nexus network, is protected by mirror, screw, or sublattice symmetries.

What would settle it

Angle-resolved photoemission on a synthesized carbon honeycomb or graphene network should find the predicted Fermi arcs, drumhead states, or nodal-ring crossings at the stated energies and momenta; a null result, or a beyond-DFT calculation that gaps the crossings, would overturn the spinless classification for that material.

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Extended reading notes

Core claim

The central claim is that topological semimetallic phases in carbon allotropes form a new class, distinct from conventional spin-orbit-coupled topological materials, because the spin degree of freedom is effectively frozen out. With negligible spin-orbit coupling, time reversal obeys $T^2 = 1$ (spinless) rather than $T^2 = -1$, so the band crossings in graphene-based networks are not Dirac or Weyl points in the usual spinful sense but their spinless analogues—Weyl-like points, nodal rings and loops, nodal surfaces, triple points, and nexus networks—protected by spatial and sublattice symmetries rather than by Kramers degeneracy. The review supports this by surveying first-principles band structures and $\mathbf{k}\cdot\mathbf{p}$ models of a series of 3D carbon allotropes (interpenetrating graphene networks, quadrilateral graphene networks, pentagon carbon, carbon honeycombs) and showing that each predicted phase arises from the same orbital physics: $p$ orbitals on sp$^2$-hybridized atoms acting as tensor degrees of freedom in frustrated lattices. It also extends the same classification to other light-element materials such as boron.

Load-bearing premise

The whole catalog of 3D spinless topological phases depends on the predicted carbon allotropes actually existing as stable, synthesizable materials with the band structures that density-functional theory gives them; if the structures are not realized, or if the one claimed synthesis (carbon honeycomb) is misidentified, the spinless class has no confirmed host.

Editorial extensions

If this is right

Spinless carbon allotropes should be classified by the BDI class with a local $Z_2$ invariant, not by the standard spinful topological periodic table.
Surface probes should find Fermi arcs connecting Weyl-like points, drumhead states inside nodal rings, and corner states in graphdiyne nanodisks.
The same orbital mechanism predicts that other light-element materials, including the boron allotropes reviewed, belong to the same spinless topological class.
If the graphdiyne identification is correct, carbon would be the first material to host both a topological insulator and a second-order topological insulator.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A finite but small spin-orbit coupling will eventually split each spinless crossing into spinful counterparts or open a gap; the paper does not quantify the threshold at which the spinless classification breaks down, so a tight-binding study of interpenetrating graphene networks with variable spin-orbit coupling would directly test the regime of validity.
The spinless classification likely applies to other ultralight-element crystals (silicon, boron, boron nitride) with negligible spin-orbit coupling, making a systematic search for spinless Weyl and nodal-chain phases a natural extension.
The claimed nearly 10 eV clean window around the Fermi level, if accurate, makes carbon networks unusually promising for isolating topological bands in transport experiments; this is an inference beyond the paper's explicit claims.
Should the carbon honeycomb synthesis be independently confirmed, the nexus-network phase would be the first materialized spinless nexus phase; otherwise the catalog remains dependent on future synthesis.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You asked about arXiv:1908.10108. Short version: it's a competent, genuinely useful review of topological carbon allotropes, but two things need attention before I'd trust it as a reference. The only experimentally realized 3D network it leans on (carbon honeycomb) has a disputed identification that the review never confronts, and one of its headline claims — graphynes as the first second-order topological insulators — is made without a citation.

What's good: the review does a real service by gathering the scattered literature on carbon-based semimetals and organizing it around a clear physical idea: because SOC in carbon is negligible, spin is a dummy variable, so time reversal squares to +1 and the topological classifications are the spinless ones. That perspective is correct and is stated more cleanly here than in the original papers. The reproduced k.p and tight-binding models (Eqs 3-8) are internally consistent and match the DFT results they are drawn from. The sections on IGN, pentagon carbon, and the nexus-network construction are helpful entry points. Self-citation is present, but that's expected in a review by a group that did much of this work; it didn't bother me.

Now the soft spots. The CHC issue is not minor. Section 2.3 presents Krainyukova and Zubarev's synthesis as if it settled the matter, and Section 4.4 uses CHC-1 as the physical host for the nexus network. But Ref. 128 (Fthenakis) argues the all-sp2 CHC structure is unstable due to dangling p-orbitals, so the deposited films may not be the predicted structure. The review cites Ref. 128 in a list but never addresses the objection. That's a gap a reader needs to know about. It doesn't kill the central thesis — the spinless classification and most of the predicted phases are theoretical — but it does mean the only experimental 3D anchor is shaky.

The second-order TI claim in Section V(4) is a separate, smaller problem. It says 'it has been proposed' that graphynes are the first second-order TIs, with no citation, and then leans on that to say the first second-order TI was born in carbon. That should either be cited properly or dropped. Also, the twisted-bilayer-graphene analogy is explicitly speculative; fine, but it reads as padding.

Who is this for? Someone looking for a survey of topological carbon, or a lecturer building a module on spinless topology. It deserves a serious referee, but the authors should be asked to fix the CHC discussion and the missing citation before it goes out. My verdict: use it with care.

Referee Report

3 major / 5 minor

Summary. The paper is a review article arguing that carbon allotropes host topological semimetals that are fundamentally distinct from conventional spin-orbit-coupling-based topological materials because of the negligible spin-orbit coupling in carbon. It surveys carbon structures from 1D to 3D, reviews the classification of nodal points, nodal lines, and nodal surfaces, and then catalogs predicted topological phases in 3D carbon networks (Weyl-like loops and points, nodal rings, nodal surfaces, triple points, and nexus networks), with an extension to boron. The article also comments on twisted graphene and on graphynes as possible second-order topological insulators.

Significance. If the central thesis holds, the review provides a valuable unifying perspective: light-element carbon allotropes form a distinct spinless topological class with clean single-p-orbital physics, broadening the search for topological semimetals beyond heavy-element compounds. The manuscript's strengths include internally consistent tight-binding and k·p models (Eqs. 3-8) that match the cited DFT results, a comprehensive catalog of topological elements in carbon networks, and an explicit statement of the spinless time-reversal classification (Section V(1)). The significance is conditional, however, because most 3D carbon hosts are unsynthesized and the only experimentally claimed 3D network, carbon honeycomb (CHC), is disputed in the very reference the review cites but does not engage.

major comments (3)

[§2.3 and §4.4 (Fig. 19)] The review presents carbon honeycomb (CHC) as experimentally realized and uses it as the physical host for the nexus-network phase in §4.4. At the same time it cites Ref. 128 without addressing that work's core objection: the proposed all-sp2 CHC structures are unstable because of dangling p-orbitals, so the vacuum-deposited films may not be the predicted CHC. If this objection stands, the only synthesized 3D host for the spinless topological phases vanishes, and the nexus-network predictions lack an experimentally realized material. The authors should either rebut Ref. 128 or explicitly qualify all CHC-based predictions as conditional on the structure identification.
[§V(4)] The claim that 'some of the graphynes discussed earlier are in fact the first second-order TIs' and that graphdiyne would be 'the first example of second-order TIs to be experimentally synthesized' is made without any supporting citation, derivation, or symmetry analysis. As this is presented as a major highlight of the review, the authors must provide a reference to the original prediction and a brief justification; otherwise the claim should be removed.
[§V(1)] The central classification statement—that treating spin as a dummy variable yields a T^2=1 fundamental time reversal and that phases in light-element materials are 'fundamentally distinct' from SOC systems—should be qualified. Real carbon has small but finite spin-orbit coupling, and the spinless classification is an idealization. The review should state the limits of this approximation and discuss how a nonzero SOC would affect the protection of the cataloged nodal features (e.g., the Weyl-like points and nodal surfaces in §4.4), rather than presenting the spinless class as the physical classification of carbon.

minor comments (5)

[§4.4, Eq. (4)] Equation (4), which defines the k·p Hamiltonian for the armchair graphene networks, is not displayed in the manuscript; only the parameter list A1, B1, A2, B2, C is given. The Hamiltonian should be restored so that the three types of nodal rings can be traced to the model.
[§4.4 (CHC discussion)] Near Fig. 19, 'couture map' should read 'contour map'.
[§2.3 and §4.4] The names 'Mackay crystals' and 'Mackey-Terrones crystal' are inconsistent; the spelling and the correspondence to reference 163 should be checked.
[§3.3] The word 'planner' should be 'planar' in the discussion of nodal surfaces.
[§2.2 (Fig. 2)] The text says rolling graphene produces a carbon nanotube shown in Fig. 2(b) and wrapping produces a fullerene shown in Fig. 2(c), but the figure caption labels Fig. 2(b) as a fullerene and Fig. 2(c) as a carbon nanotube; the text and caption should be reconciled.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the review's claims rest on DFT band structures, explicit symmetry arguments, and external benchmarks; self-citations are non-load-bearing pointers to prior computational work.

full rationale

This is a review article, not a new derivation. The load-bearing statements either summarize published DFT/band-structure results (including external works, e.g., Refs 42-47 and 163) or follow from explicitly stated symmetry/approximation arguments, such as the spinless T^2=1 classification in Sec. V(1) and the k.p models in Sec. 4.4. Equations (4), (5), and (8) are k.p or tight-binding models whose parameters are fitted to, or constrained by, DFT band structures; the paper uses them to classify or reproduce crossings already present in the DFT data, not to fit a parameter and then report it as an independent prediction. Self-citations such as Ref. 321 for nexus networks point to the authors' prior computational studies but are not invoked as a uniqueness theorem forbidding alternatives; the central claim that carbon hosts spinless topological semimetals is independently supported by external computations and by the explicitly stated negligible-SOC approximation. The review's own limitation in Sec. V(2), that most 3D carbon structures are unsynthesized, and the disputed CHC identification noted in Sec. 2.3 (Ref. 128) are material-realization/correctness risks, not circularity. No equation or claim reduces to its own input by construction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 2 invented entities

This paper is a review, so the free parameters listed are technical coefficients from previously published k.p and tight-binding models that the review reproduces to explain topological phases. They are not newly fitted here, but the central synthesis relies on their validity. The axioms are the domain assumptions inherited from the cited DFT literature, especially the neglect of spin-orbit coupling and the presumption of synthesizability.

free parameters (4)

Tight-binding hoppings t13, t14, t23, t24 in IGN model (Eq. 3) = not specified in review (from DFT fit in Ref. 36)
Zero-energy condition cos(kc c/2) = sqrt(t13 t24/(4 t14 t23)) depends on these fitted hopping amplitudes; no values or fit details are given in this review.
k.p parameters A1, B1, A2, B2, C in Eq. (4) = not listed
Parameters 'obtained by fitting to the DFT results' (Sec. 4.4, AGNW nodal rings). The classification of type-I/II/III rings depends on their signs.
k.p coefficients A, B1, B2, C, D1, D2, Delta in Eq. (5) = not listed
Determined by fitting DFT results for pentagon carbon; the triple-point transition is set by Delta=0 and Kc = sqrt(-Delta/B1).
Constants A1,2, B1,2, C1,2, D, alpha, beta in Eq. (8) = not listed
Real constants in the CHC-1 nexus-network k.p model; the nexus phase requires alpha != 0 and beta != 0.

assumptions (4)

domain assumption Spin-orbit coupling in carbon is negligible, so spin can be treated as a dummy variable and the time-reversal operator satisfies T^2=+1.
Explicitly stated in Section V(1) and used throughout Sec. 4.4 for Weyl-like points, triple points, and nodal surfaces; if SOC were significant, these phases would gap out or change class.
domain assumption DFT band structures for the proposed allotropes are accurate enough to establish band crossings and topological invariants.
All topological phase assignments in Sec. 4.4 rest on first-principles calculations from cited references; no experimental verification exists for most structures.
domain assumption The predicted carbon allotropes are structurally stable or synthesizable.
The review acknowledges (Sec. V(2)) that most 3D carbon structures have not been synthesized; the material realization of the topological phases depends on this.
standard math Standard topological band theory (Berry phase, Chern numbers, bulk-boundary correspondence) applies to spinless systems.
Used for Weyl monopoles, Fermi arcs, Z2 invariants, and BDI classification in Secs. 3 and 4.

invented entities (2)

Isospin-1 triplet fermion (spin-1 Gell-Mann quasiparticle)
purpose: Describes the triple point at Gamma in pentagon carbon networks with threefold degeneracy.
Presented as a new quasiparticle in Sec. 4.4, but the host structure has not been synthesized and no ARPES or transport signature is provided.
Nexus network (winding connectivity of nodal lines between triple points)
purpose: Describes the connectivity of nodal lines in CHC-1 carbon honeycomb.
A topological connectivity pattern predicted by a k.p model (Eq. 8) and DFT; CHC-1 has been experimentally claimed but the topological nexus network has not been directly observed.

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Cite this review

Pith. "Pith review of Topological carbon materials: a new perspective." pith.science (2026). https://pith.science/paper/P4CJQN5T

@misc{pith2026190810108,
  author       = {Pith},
  title        = {Pith review of: Topological carbon materials: a new perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4CJQN5T}},
  note         = {Machine review of arXiv:1908.10108}
}

read the original abstract

Carbon has numerous one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) allotropic structures. The study of carbon materials has been a major focus of material science and condensed matter physics. Previous studies have identified different classes of topological semimetallic carbon allotropes with different topological phases. In this review, we first give a brief summary of the development of carbon allotropes from 1D to 3D. Next, we discuss topological properties of carbon materials and their physical origin. Then, we consider possible expansion of the topological study of carbon materials to other light-element materials such as boron. Finally, we present future prospects in pursue of topological physics within carbon allotropes.

Figures

Figures reproduced from arXiv: 1908.10108 by the authors.

**Figure 14.** (a) Band structure of the Kagome graphene in Fig. 4(a). (b) [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 11 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

1 Falcao, E. H. & Wudl, F. Carbon allotropes: beyond graphite and diamond. Journal of Chemical Technology & Biotechnology: International Research in Process, Environmental & Clean Technology 82, 524-531 (2007). 2 Zhang, Q., Huang, J. Q., Qian, W. Z., Zhang, Y . Y . & Wei, F. The road for nanomaterials industry: A review of carbon nanotube production, post...

arXiv 2007

[2] [2]

176 Liu, Z

Nature materials 13, 677 (2014). 176 Liu, Z. et al. Discovery of a three-dimensional topological Dirac semimetal, Na3Bi. Science 343, 864-867 (2014). 177 Young, S. M. et al. Dirac semimetal in three dimensions. Physical review letters 108, 140405 (2012). 178 Yang, H. et al. Prediction of triple point fermions in simple half -Heusler topological insulators...

work page 2014

[3] [3]

202 Yan, Z

Physical review letters 118, 176402 (2017). 202 Yan, Z. et al. Nodal-link semimetals. Physical Review B 96, 041103 (2017). 203 Chen, C. et al. Weyl-loop half-metal in Li 3 (FeO

work page 2017

[4] [4]

185 Huang, H., Liu, J., Vanderbilt, D

Nature communications 7, 10556 (2016). 185 Huang, H., Liu, J., Vanderbilt, D. & Duan, W. Topological nodal-line semimetals in alkaline-earth 45 stannides, germanides, and silicides. Physical Review B 93, 201114 (2016). 186 Xu, Q., Yu, R., Fang, Z., Dai, X. & Weng, H. Topological nodal line semimetals in the CaP 3 family of materials. Physical Review B 95,...

work page 2016

[5] [6]

204 Xiao, M., Sun, X

Physical Review B 99, 075131 (2019). 204 Xiao, M., Sun, X. -Q. & Fa n, S. Nodal chain semimetal in geometrically frustrated systems. Physical Review B 99, 094206 (2019). 205 Huang, H., Jiang, W., Jin, K.-H. & Liu, F. Tunable topological semimetal states with ultraflat nodal rings in strained YN. Physical Review B 98, 045131 (2018). 206 Shao, D.-F., Zhang,...

work page 2019

[6] [7]

216 Yu, W

Physical Review Materials 2, 014202 (2018). 216 Yu, W. C. et al. Nonsymmorphic cubic Dirac point and crossed nodal rings across the ferroelectric phase transition in LiOsO

work page 2018

[7] [8]

124 Umemoto, K., Wentzcovitch, R

Physical review letters 107, 215502 (2011). 124 Umemoto, K., Wentzcovitch, R. M., Saito, S. & Miyake, T. Body -Centered Tetragonal C 4: A Viable s p 3 Carbon Allotrope. Physical review letters 104, 125504 (2010). 125 Chen, X.-Q., Niu, H., Franchin i, C., Li, D. & Li, Y . Hardness of T-carbon: Density functional theory calculations. Physical Review B 84, 1...

work page 2011

[8] [9]

217 Lian, J

Physical Review Materials 2, 051201 (2018). 217 Lian, J. et al. Multi-loop node line states in ternary MgSrSi -type crystals. npj Computational Materials 5, 10 (2019). 218 Cai, J., Xie, Y ., Chang, P.-Y ., Kim, H.-S. & Chen, Y .-P. Nodal-chain network, intersecting nodal rings and triple points coexisting in nonsymmorphic Ba3Si4. Physical Chemistry Chemic...

work page 2018