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REVIEW 3 major objections 4 minor 53 references

Modulated Dirac bands and integer hopping ratios in a honeycomb lattice of phenalenyl-tessellation molecules

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper establishes a structural design rule: in a honeycomb lattice of phenalenyl-tessellation molecules, the low-energy Dirac bands are determined solely by the numbers of corner connections between neighboring molecules, so the…

desk verdict Nice integer-ratio effective model for PTM honeycombs, but the abstract overstates 'connections-only' and the gapped case relies on a fitted tau; worth a serious look. read the letter →

arxiv 2506.07819 v1 pith:PHHVXSC7 submitted 2025-06-09 cond-mat.mes-hall

Naoki Morishita , Kenshin Komatsu , Motoharu Kitatani , Koichi Kusakabe This is my paper · ORCID
classification cond-mat.mes-hall
keywords grapheneDiracconesphenalenyl-tessellationmoleculestight-bindingmodelzeromodesFermivelocitynanographenebandgapengineering
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 paper tries to establish a design rule for a family of nanographene lattices: in a honeycomb arrangement of phenalenyl-tessellation molecules (PTMs), the low-energy Dirac bands are governed entirely by how many double-zigzag corners connect each molecule to its neighbors. That rule makes the effective hopping parameters appear as positive integers in fixed ratios, so by choosing the connection counts a designer can decide whether the system is a gapless Dirac metal or a gapped semiconductor. It would matter because it turns a molecular-geometry choice into quantitative electronic design, including tuning the Fermi velocity from near zero up to values comparable to graphene, and it allows vacancy-localized zero modes to coexist with the Dirac electrons. A sympathetic reader would care because this is a parameter-free route, up to one overall scale, to modulated Dirac physics in a carbon-based platform.

What carries the argument

The carrying object is the honeycomb PTM lattice (H-PTM), in which each phenalenyl-tessellation molecule acts as an effective atom and neighboring molecules are joined through double-zigzag corners (DZCs). The identity that does the work is Eq. (8), which maps the many-site bipartite lattice onto the textbook two-site honeycomb tight-binding model with effective hoppings $\tau_i = N_{Di}\tau$. The derivation rests on the zero-sum rule for bipartite zero modes: for the $\sqrt{3}\times\sqrt{3}$ zero-mode wavefunctions on $\alpha$- and $\beta$-PTMs, all inter-PTM coupling terms vanish except those that touch the two sublattices through DZCs, leaving only the connection numbers $N_{Di}$. That is what converts molecular geometry into counting, and it is also what fixes $\tau$ analytically for simple structures through the normalization of the zero-mode amplitudes.

What would settle it

Run a full tight-binding calculation for an H-PTM with connection counts 2:1:1, where the effective model predicts a gapless Dirac band, and for 3:1:1, where it predicts a gap; the central claim fails if the exact spectrum shows a gap in the first case or no gap in the second, or if the gap-closing point shifts away from the effective model's predicted k-point (the M point for 2:1:1).

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Extended reading notes

Core claim

The central claim is that a two-site effective honeycomb model describes the low-energy Dirac bands of an H-PTM lattice exactly enough to decide gap opening and closing. The effective Hamiltonian is $H_{\mathrm{eff}} = \tau$ times the standard honeycomb matrix with entries $N_{D0} + N_{D1} e^{-i\mathbf{k}\cdot\mathbf{a}'_1} + N_{D2} e^{-i\mathbf{k}\cdot\mathbf{a}'_2}$, where $N_{D0}$, $N_{D1}$, $N_{D2}$ are the numbers of double-zigzag-corner connections in the three bond directions and $\tau = 2|\varphi_\alpha||\varphi_\beta|t$ is set by the amplitudes of the $\sqrt{3}\times\sqrt{3}$ zero modes on the two molecular sublattices. Since $\tau$ is just a common scale, the three effective hoppings have positive integer ratios $N_{D0} : N_{D1} : N_{D2}$, and the triangular inequality for those integers decides whether the bands touch in Dirac cones, merge at a critical point, or open a gap. The paper verifies this against full tight-binding spectra for isotropic (1:1:1), critical (2:1:1), and gapped (3:1:1) cases, and shows that vacancies can introduce flat zero modes at $E=0$ without removing the Dirac cone.

Load-bearing premise

The load-bearing premise is that each molecule's low-energy state keeps the uniform $\sqrt{3}\times\sqrt{3}$ zero-mode shape for every wavevector that matters, which is exact only at the gap-closing point; once a gap opens the shape is modulated and the effective hopping must be fitted numerically rather than derived.

Editorial extensions

If this is right

  • The low-energy band character of an H-PTM is fixed by three integers, $N_{D0}$, $N_{D1}$, $N_{D2}$, and one overall scale $\tau$; no other molecular detail matters for whether a gap opens.
  • Choosing integers that violate the triangle inequality opens a band gap without strain, and choosing equality places the Dirac-point merge at a controlled k-point such as the M point.
  • The Fermi velocity can be engineered from near zero (uniaxial lattices with long arms) up to a value comparable to graphene's (short isotropic arms), with explicit $1/L_{\mathrm{arm}}$ scaling laws.
  • Vacancies inside PTM regions create localized zero modes at the Fermi level that coexist with the Dirac cone, giving a built-in coexistence of localized spins and itinerant electrons.
  • Because the hoppings are integer multiples of a common $\tau$, the effective model is parameter-free apart from $\tau$, making the design rules directly checkable by tight-binding calculation.

Reading between the lines

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

  • Editorial inference: the integer-count rule suggests a combinatorial design chart, since every triple $(N_{D0}, N_{D1}, N_{D2})$ realizes a distinct anisotropic honeycomb model, including Dirac-point-merge and gap-opening phases, so one could in principle enumerate all achievable band structures.
  • Editorial inference: the coexistence of vacancy zero modes with a tunable Dirac band points toward systems where localized spins and itinerant electrons interact magnetically, a direction the paper mentions but does not compute.
  • Editorial inference: a testable extension would be to break the symmetry between the alpha and beta molecules, for example by removing atoms from only one type, to see whether the integer-hopping effective model predicts an energy gap at the Dirac point while the zero-sum derivation still holds.
  • Editorial inference: because the effective model depends only on connection counts, small energy shifts on interior carbon atoms that preserve the molecular tiling should leave the Dirac physics unchanged; this insensitivity could be checked by adding random small site-energy variations in tight-binding calculations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes that the low-energy electronic bands of a periodic honeycomb arrangement of phenalenyl-tessellation molecules (PTMs) can be described by a two-site effective honeycomb tight-binding model whose hopping ratios are given by the integers N_D0:N_D1:N_D2, the numbers of inter-PTM double-zigzag-corner connections in the three bond directions. The authors derive this effective model from the √3×√3 zero modes of the constituent PTMs, compare it with full tight-binding calculations for isotropic, critical, and gap-opening connection patterns, discuss Fermi-velocity control as a function of arm length, and show that vacancies introduce localized zero modes that coexist with the Dirac bands.

Significance. If the central claim holds, the paper offers a concrete bottom-up design principle for graphene-based lattices: varying the connection counts between molecular building blocks quantizes the effective hopping ratios, enabling controlled gap opening and Fermi-velocity modulation without large strain. The zero-mode normalization is derived analytically for several isotropic and critical structures, and the qualitative gap-opening/closing criterion is robust because it depends only on the integer ratios, not on the overall scale τ. The coexistence of Dirac bands with vacancy-localized spins extends the potential for correlated-electron devices. However, the claim that the effective model is 'determined only by the connections' is too strong, since the overall hopping scale τ=2|φ_α||φ_β|t depends on the internal zero-mode structure, and in the gap-opening case τ is obtained by numerical fitting rather than analytically. These caveats do not undermine the integer-ratio result itself, which is the paper's central and most valuable contribution.

major comments (3)
  1. [Abstract, Section II.D, Eq. (8)] The statement that the effective model is 'determined only by the connections between neighboring effective atoms' is not supported by the derivation. Equation (9) defines τ=2|φ_α||φ_β|t, and the normalization factors depend on the internal structure of the PTMs: for the same 1:1:1 connection counts, Section III.A reports τ=2/11t for Fig. 4 but τ=6/√4843t for Fig. 5. Only the ratios τ_0:τ_1:τ_2=N_D0:N_D1:N_D2 are fixed by the connection counts; the overall energy scale is not. The abstract and Section V should be reworded to state that the hopping ratios (not the full model) are determined by the connections.
  2. [Section III.C and Fig. 7] In the gap-opening case, the √3×√3 zero-mode assumption fails because the eigenenergy is not zero, and the value τ=0.0172t is 'numerically determined by averaging the values of |φ_α||φ_β| across all DZCs.' Consequently the quantitative band structure and gap magnitude of the effective model in this section rest on a fitted parameter rather than a first-principles derivation. The qualitative gap-opening prediction is robust because the triangle inequality involves only the integer ratios, but the paper should state this limitation explicitly and temper the claim of a parameter-free validation.
  3. [Section II.D, displayed expansion after Eq. (7)] The replacement of the four terms by the √3-mode expression contains an index error: the first term is written as ψ√3*_A1α H_AB ψ√3_B1α and the third as ψ*_A1β H_AB ψ√3_B1α, whereas the preceding expansion shows that the inter-PTM term is ψ*_A1α H_AB ψ_B1β and the intra-β term is ψ*_A1β H_AB ψ_B1β. The correct simplified expression should read ψ√3*_A1α H_AB ψ√3_B1β + ψ√3*_A1α H_AB ψ_B1α + ψ*_A1β H_AB ψ√3_B1β. As printed, the derivation of Eq. (8) is obscured.
minor comments (4)
  1. [Appendix] The value τ=6/√4863t should read 6/√4843t; with |φ_α|=3/√167 and |φ_β|=1/√29, the product is 6/√(167·29)=6/√4843. The main text uses the correct value.
  2. [Equations (11)-(19)] The fractions are typeset ambiguously (e.g., 'τ= 2 8NPU + 3t' should be τ=2t/(8N_PU+3)); please use clear fraction notation throughout Section IV.A.
  3. [References [34] and [35]] Reference [35] appears to be a duplicate of [34] or an erratum with changed pagination; please verify the citation and list it as an erratum if that is the case.
  4. [Figure 10] In Fig. 10(c), the red-dashed effective-model lines are described as unchanged from Fig. 5(e), but the figure appears to show only the original model; please indicate the effective-model bands more clearly or state that they are omitted from the panel.

Circularity Check

1 steps flagged · score 5.0 of 10

The integer-ratio Dirac-band construction is independent, but the gapped-case validation is partially circular because τ is fitted from the original TB wavefunctions rather than derived.

  1. fitted input called prediction [Section III.C, 'Gap-opening case', paragraph describing Fig. 7]
    "In this case, since the modes on the A-sites in the α-PTM region and on the B-sites in the β-PTM region are modulated from the √3×√3 shape because the eigen energy is not E=0. So, the value of τ=0.0172t in the effective model is numerically determined by averaging the values of |φ_α||φ_β| across all DZCs."

    With the gap open there is no E=0 eigenstate, so the √3×√3 zero-mode ansatz that produced Eq. (8) breaks down; the paper therefore sets τ=0.0172t by averaging |φ_α||φ_β| across all DZCs using the very same original TB wavefunctions whose bands are then compared with the effective model in Fig. 7(e). The quantitative agreement of the gapped bands is consequently imposed by this fitted value rather than independently derived, and the abstract's phrase 'determined only by the connections' cannot hold for the overall energy scale in this case. What remains non-circular is the qualitative gap/no-gap statement, because it depends only on the integer ratio τ0:τ1:τ2=3:1:1 through the triangle inequality and the common factor τ cancels.

full rationale

The central construction of Eq. (8) is not circular: the integer ratios N_D0:N_D1:N_D2 follow from explicitly counting the 2N_Di inter-PTM DZC bonds in the E=0 √3×√3 zero-mode subspace, and the common prefactor τ=2|φ_α||φ_β|t is computed analytically from normalized zero-mode wavefunctions in the gapless and critical examples (Appendix, Fig. A.1). The abstract's phrase 'determined only by the connections' is best read as applying to the integer hopping ratios, since τ itself depends on internal PTM normalization. The one genuinely circular element is the gap-opening case of Sec. III.C: with no zero-energy eigenstate, the √3×√3 assumption fails, and τ=0.0172t is numerically determined by averaging |φ_α||φ_β| across all DZCs from the same original TB wavefunctions that are then compared with the effective model in Fig. 7(e). That particular quantitative agreement is therefore imposed by the fit rather than independently predicted. The qualitative gap-opening prediction survives because it relies only on τ0=3τ, τ1=τ, τ2=τ and the triangle inequality, in which τ cancels. No load-bearing uniqueness theorem is invoked from the authors' prior work; the zero-sum rule is supported by external references [38–41] and by explicit analytical solutions in the Appendix. The derivation in Sec. II.D also contains an apparent index slip in the displayed expansion (first term written with ψ^√3_B1α instead of ψ^√3_B1β), but this is an expositional flaw rather than an additional circular step. Overall, the integer-ratio and gap/no-gap claims are independent; only the gapped quantitative comparison is partially circular.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The effective atoms are an abstraction of PTMs, not new physical entities. The only fitted number is tau in the gap-opening case; the integer ratios are constructed from connection counts. The zero-sum rule and single-mode projection are the load-bearing assumptions.

free parameters (1)
  • tau (gap-opening case) = 0.0172 t
    In Section III.C, tau is numerically determined by averaging |phi_alpha||phi_beta| across DZCs because the zero-mode shape is modulated when the gap opens. This is a fitted value used to compare the effective model with the original TB bands.
assumptions (3)
  • standard math Zero-sum rule for zero-energy eigenstates of bipartite lattices
    Used in Section II.D to impose the sqrt(3)xsqrt(3) shape on the zero-mode wavefunctions. This is a standard property of tight-binding zero modes.
  • domain assumption Low-energy subspace is spanned by one zero-mode-like eigenvector pair per unit cell
    Sections II.C-II.D project onto the lowest eigenvector pair and drop all off-diagonal couplings in the transformed basis. This assumes no other low-energy states mix with the sqrt(3)xsqrt(3) mode.
  • domain assumption Inter-PTM coupling occurs only through double-zigzag corner bonds
    In the derivation of Eq. (8), only DZC connections contribute to the effective hopping; other inter-molecular contacts are neglected.

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Pith. "Pith review of Modulated Dirac bands and integer hopping ratios in a honeycomb lattice of phenalenyl-tessellation molecules." pith.science (2026). https://pith.science/paper/PHHVXSC7

@misc{pith2026250607819,
  author       = {Pith},
  title        = {Pith review of: Modulated Dirac bands and integer hopping ratios in a honeycomb lattice of phenalenyl-tessellation molecules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHHVXSC7}},
  note         = {Machine review of arXiv:2506.07819}
}
abstract

A family of nanographene molecules called phenalenyl-tessellation molecules (PTMs) exhibits two types of zero modes: a $\sqrt{3} \times \sqrt{3}$ type that spreads over the entire molecule and a vacancy-localized type. A periodic system of PTMs is expected to have low-energy bands that strongly reflect the properties of the zero modes of PTMs as effective atoms. In this study, we show that the low-energy Dirac bands in a class of honeycomb PTMs (H-PTM) can be represented by an effective honeycomb model which is determined only by the connections between neighboring effective atoms.The hopping parameters of H-PTM in each direction take positive integer ratios according to the connection order between two PTMs.By structurally designing each PTM, we can change the connection order of the PTMs and hence modulate the energy gap and the Fermi velocity of the Dirac band of the H-PTM. Moreover, we confirm that Dirac bands coexist with vacancy-localized zero modes in the H-PTM with vacancies.The result indicates that the nanographene structure arranging PTMs as effective atoms extends material design freedom that effectively generates a modulated Dirac electron system with coexisting localized electron spins for graphene-based electronic and quantum devices.

Figures

Figures reproduced from arXiv: 2506.07819 by the authors.

Figure 1
Figure 1. FIG. 1. A honeycomb lattice with an A-site and a B-site. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) An [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Eigenvectors at the K-point on (a) the A-sites [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Results of an isotropic case: (a) The original H–PTM; (b) effective model with [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Another isotropic case with [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. A critical-parameter case with [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. An anisotropic case with [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. H–PTMs composed of a single row of PUs: (a) an isotropic case with [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Arm-length [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Introduction of a vacancy in a [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]

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Works this paper leans on

53 extracted references · 48 canonical work pages

  1. [1]

    K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric field effect in atomically thin carbon films, Science306, 666 (2004)

    2004

  2. [2]

    Because the effective two-site system corresponding to Eq

    Denoting the effective hopping parameters in the three directions asτ 0 =N D0τ,τ 1 =N D1τandτ 2 =N D2τ, we find thatτ 0,τ 1 andτ 2 have positive integer ratios. Because the effective two-site system corresponding to Eq. (8) is bipartite, the two eigenvalues are symmetric with respect to the Fermi level and take values of±E eff(⃗k). E. Band gap and F ermi ...

    work page

  3. [3]

    A. K. Geim and K. S. Novoselov, The rise of graphene, Nature Materials6, 183 (2007)

    2007

  4. [4]

    K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, A roadmap for graphene, Nature490, 192 (2012)

    work page 2012

  5. [5]

    K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, 22 FIG. A.1. The analytical solutions of the wavefunctions for the system shown in (a) Fig. 4 (A- sites, K-point), (b) Fig. 6 (A-sites, M-point), (c) Fig. 5 (A-sites, K-point) and (d) Fig. 5 (B-sites, K-point). 23 S. V. Dubonos, and A. A. Firsov, Two-dimensional gas ...

    work page 2005

  6. [6]

    V. B. Mbayachi, E. Ndayiragije, T. Sammani, S. Taj, E. R. Mbuta, and A. ullah khan, Graphene synthesis, characterization and its applications: A review, Results in Chemistry3, 100163 (2021)

    work page 2021

  7. [7]

    Kireev, S

    D. Kireev, S. K. Ameri, A. Nederveld, J. Kampfe, H. Jang, N. Lu, and D. Akinwande, Fabri- cation, characterization and applications of graphene electronic tattoos, Nature Protocols16, 2395 (2021)

    work page 2021

  8. [8]

    E. L. Wolf, Applications of Graphene: An Overview (Springer International Publishing, 2014)

    work page 2014

Show all 53 references
  1. [9]

    Nakada, M

    K. Nakada, M. Fujita, G. Dresselhaus, and M. S. Dresselhaus, Edge state in graphene ribbons: Nanometer size effect and edge shape dependence, Phys. Rev. B54, 17954 (1996)

    1996

  2. [10]

    O. V. Yazyev, R. B. Capaz, and S. G. Louie, Theory of magnetic edge states in chiral graphene nanoribbons, Phys. Rev. B84, 115406 (2011)

    2011

  3. [11]

    Similarly, the analytical solution on the A-sites at the M-point of the system in Fig

    Therefore, τ= 2|φ α||φβ|t= 2/11t. Similarly, the analytical solution on the A-sites at the M-point of the system in Fig. 6 is also shown in Fig. A.1 (b). In this case,|φ α|=|φ β|= 3/ √ 838 andτ= 2|φ α||φβ|t= 9/419t. The analytical solutions on the A-sites and the B-sites of th...

  4. [12]

    Fujita, K

    M. Fujita, K. Wakabayashi, K. Nakada, and K. Kusakabe, Peculiar localized state at zigzag graphite edge, Journal of the Physical Society of Japan65, 1920 (1996)

    1996

  5. [13]

    C. Ma, Z. Xiao, A. A. Puretzky, H. Wang, A. Mohsin, J. Huang, L. Liang, Y. Luo, B. J. Lawrie, G. Gu, W. Lu, K. Hong, J. Bernholc, and A.-P. Li, Engineering edge states of graphene nanoribbons for narrow-band photoluminescence, ACS Nano14, 5090 (2020), pMID: 32283017

    2020

  6. [14]

    Ziatdinov, H

    M. Ziatdinov, H. Lim, S. Fujii, K. Kusakabe, M. Kiguchi, T. Enoki, and Y. Kim, Chemically induced topological zero mode at graphene armchair edges, Physical Chemistry Chemical Physics19, 5145 (2017)

    2017

  7. [15]

    Kusakabe, K

    K. Kusakabe, K. Nishiguchi, S. Teranishi, and Y. Wicaksono, Edge states caused by shift of dirac points at the armchair edge of distorted nanographene, Journal of the Physical Society of Japan87, 084706 (2018)

    2018

  8. [16]

    V. M. Pereira, J. M. B. Lopes dos Santos, and A. H. Castro Neto, Modeling disorder in graphene, Phys. Rev. B77, 115109 (2008)

    2008

  9. [17]

    H. Wang, H. S. Wang, C. Ma, L. Chen, C. Jiang, C. Chen, X. Xie, A.-P. Li, and X. Wang, Graphene nanoribbons for quantum electronics, Nature Reviews Physics3, 791 (2021). 24

    2021

  10. [18]

    V. M. Pereira, F. Guinea, J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Disorder induced localized states in graphene, Phys. Rev. Lett.96, 036801 (2006), erratum: Phys. Rev. Lett.98(2007) 259902

    2006

  11. [19]

    Morishita, G

    N. Morishita, G. K. Sunnardianto, S. Miyao, and K. Kusakabe, Theoretical analysis of pseudo- degenerate zero-energy modes in vacancy-centered hexagonal armchair nanographene, Journal of the Physical Society of Japan85, 084703 (2016)

    2016

  12. [20]

    P. O. Lehtinen, A. S. Foster, Y. Ma, A. V. Krasheninnikov, and R. M. Nieminen, Irradiation- induced magnetism in graphite: A density functional study, Phys. Rev. Lett.93, 187202 (2004)

    2004

  13. [21]

    Ziatdinov, S

    M. Ziatdinov, S. Fujii, K. Kusakabe, M. Kiguchi, T. Mori, and T. Enoki, Direct imaging of monovacancy-hydrogen complexes in a single graphitic layer, Phys. Rev. B89, 155405 (2014)

    2014

  14. [22]

    Morishita, Y

    N. Morishita, Y. Oishi, T. Yamaguchi, and K. Kusakabe, S = 1 antiferromagnetic electron- spin systems on hydrogenated phenalenyl-tessellation molecules for material-based quantum- computation resources, Applied Physics Express14, 121005 (2021)

    2021

  15. [23]

    Morishita and K

    N. Morishita and K. Kusakabe, Counting the zero modes and magnetic moment by topology of a phenalenyl-tessellation molecule with vacancies, Journal of the Physical Society of Japan 88, 124707 (2019)

    2019

  16. [24]

    Morishita and K

    N. Morishita and K. Kusakabe, Zero-energy modes in a super-chiral nanographene network of phenalenyl-tessellation molecules, Physics Letters A408, 127462 (2021)

    2021

  17. [25]

    Ortiz, Olympicene radicals as building blocks of two-dimensional anisotropic networks, Phys

    R. Ortiz, Olympicene radicals as building blocks of two-dimensional anisotropic networks, Phys. Rev. B108, 115113 (2023)

    2023

  18. [26]

    Tamaki, T

    G. Tamaki, T. Kawakami, and M. Koshino, Topological junction states and their crystalline network in systems with chiral symmetry: Application to graphene nanoribbons, Phys. Rev. B101, 205311 (2020)

    2020

  19. [27]

    Calogero, N

    G. Calogero, N. R. Papior, B. Kretz, A. Garcia-Lekue, T. Frederiksen, and M. Brandbyge, Electron transport in nanoporous graphene: Probing the talbot effect, Nano Letters19, 576 (2019)

    2019

  20. [28]

    Hasegawa, R

    Y. Hasegawa, R. Konno, H. Nakano, and M. Kohmoto, Zero modes of tight-binding electrons on the honeycomb lattice, Phys. Rev. B74, 033413 (2006)

    2006

  21. [29]

    For systems with arms consisting of a single row of PUs, such as shown in Fig

    Therefore, τ= 2|φ α||φβ|t= 6/ √ 4863t. For systems with arms consisting of a single row of PUs, such as shown in Fig. 3 and Fig. 8, the analytical solution can be easily obtained by simply considering a wavefunction that spreads with uniform amplitude on the arms, based on Fig...

  22. [30]

    Ortiz, G

    R. Ortiz, G. Catarina, and J. Fern´ andez-Rossier, Theory of triangulene two-dimensional crys- tals, 2D Materials10, 015015 (2022). 25

    2022

  23. [31]

    Madail, R

    L. Madail, R. G. Dias, and J. Fern´ andez-Rossier, Exotic edge states ofC 3 high-fold fermions in honeycomb lattices, Phys. Rev. Res.6, 043262 (2024)

    2024

  24. [32]

    Oliva-Leyva and G

    M. Oliva-Leyva and G. G. Naumis, Effective dirac hamiltonian for anisotropic honeycomb lattices: Optical properties, Phys. Rev. B93, 035439 (2016)

    2016

  25. [33]

    Kishigi, K

    K. Kishigi, K. Ueno, E. Miyamoto, and Y. Hasegawa, Dirac cones on the generalized honey- comb lattice, Journal of Physics: Conference Series334, 012047 (2011)

    2011

  26. [34]

    V. M. Pereira and A. H. Castro Neto, Strain engineering of graphene’s electronic structure, Phys. Rev. Lett.103, 046801 (2009)

    2009

  27. [35]

    D. G. Papageorgiou, I. A. Kinloch, and R. J. Young, Mechanical properties of graphene and graphene-based nanocomposites, Progress in Materials Science90, 75 (2017)

    2017

  28. [36]

    V. M. Pereira, A. H. Castro Neto, and N. M. R. Peres, Tight-binding approach to uniaxial strain in graphene, Phys. Rev. B80, 045401 (2009)

    2009

  29. [37]

    Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening, ACS Nano2, 2301 (2008), pMID: 19206396

    2008

  30. [38]

    Z. H. Ni, T. Yu, Y. H. Lu, Y. Y. Wang, Y. P. Feng, and Z. X. Shen, Uniaxial strain on graphene: Raman spectroscopy study and band-gap opening, ACS Nano3, 483 (2009), pMID: 19583275

    2009

  31. [39]

    Komatsu, N

    K. Komatsu, N. Morishita, M. Kitatani, and K. Kusakabe, Designing a polymerized phenalenyl tessellation molecule to realize a super-honeycomb antiferromagnetic s = 3/2 spin system, Journal of the Physical Society of Japan94, 044704 (2025)

    2025

  32. [40]

    A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys.81, 109 (2009)

    2009

  33. [41]

    W. T. Borden and E. R. Davidson, Effects of electron repulsion in conjugated hydrocarbon diradicals, Journal of the American Chemical Society99, 4587 (1977)

    1977

  34. [42]

    G. G. Cash and J. R. Dias, Computation, properties and resonance topology of benzenoid monoradicals and polyradicals and the eigenvectors belonging to their zero eigenvalues, Journal of Mathematical Chemistry30, 429 (2001)

    2001

  35. [43]

    R. F. Langler, How to recognize disjoint hydrocarbons: A h¨ uckel theory topic, The Chemical Educator7, 1 (2002). 26

    2002

  36. [44]

    Orimoto and Y

    Y. Orimoto and Y. Aoki, Analytical method for predicting ferromagnetic properties of benzyl- radical polymers based on NBMO theory, Journal of Chemical Theory and Computation2, 786 (2006)

    2006

  37. [45]

    Wakabayashi, M

    K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Electronic and magnetic properties of nanographite ribbons, Phys. Rev. B59, 8271 (1999)

    1999

  38. [46]

    Enoki and T

    T. Enoki and T. Ando, Physics and chemistry of graphene: graphene to nanographene (CRC Press, 2019)

    2019

  39. [47]

    Kimouche, M

    A. Kimouche, M. M. Ervasti, R. Drost, S. Halonen, A. Harju, P. M. Joensuu, J. Sainio, and P. Liljeroth, Ultra-narrow metallic armchair graphene nanoribbons, Nat. Commun.6, 10177 (2015)

    2015

  40. [48]

    R. D. McCurdy, A. Delgado, J. Jiang, J. Zhu, E. C. H. Wen, R. E. Blackwell, G. C. Veber, S. Wang, S. G. Louie, and F. R. Fischer, Engineering robust metallic zero-mode states in olympicene graphene nanoribbons, Journal of the American Chemical Society145, 15162 (2023)

    2023

  41. [49]

    P. H. Jacobse, R. D. McCurdy, J. Jiang, D. J. Rizzo, G. Veber, P. Butler, R. Zuzak, S. G. Louie, F. R. Fischer, and M. F. Crommie, Bottom-up assembly of nanoporous graphene with emergent electronic states, Journal of the American Chemical Society142, 13507 (2020)

    2020

  42. [50]

    Wakabayashi, K.-i

    K. Wakabayashi, K.-i. Sasaki, T. Nakanishi, and T. Enoki, Electronic states of graphene nanoribbons and analytical solutions, Science and Technology of Advanced Materials11, 054504 (2010)

    2010

  43. [51]

    Gonzalez-Herrero, J

    H. Gonzalez-Herrero, J. M. Gomez-Rodriguez, P. Mallet, M. Moaied, J. J. Palacios, C. Sal- gado, M. M. Ugeda, J.-Y. Veuillen, F. Yndurain, and I. Brihuega, Atomic-scale control of graphene magnetism by using hydrogen atoms, Science352, 437 (2016)

    2016

  44. [52]

    Kusakabe and H

    K. Kusakabe and H. Aoki, Ferromagnetic spin-wave theory in the multiband hubbard model having a flat band, Phys. Rev. Lett.72, 144 (1994)

    1994

  45. [53]

    Naoki, Morishita and Kenshin, Komatsu and Motoharu, Kitatani and Koichi Kusakabe, Tight- binding calculation files of modulated dirac bands and integer hopping ratios in a honey- comb lattice of phenalenyl-tessellation molecules, materials Cloud Archive 2025.81 (2025), https:/...

    2025 doi

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