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arxiv: 2604.18854 · v1 · submitted 2026-04-20 · ❄️ cond-mat.mes-hall

Landau levels and magneto-optics in 30^circ quasi-periodic twisted bilayer graphene

Masaru Hitomi , Takuto Kawakami , Mikito Koshino This is my paper

Pith reviewed 2026-05-10 03:15 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords twisted bilayer grapheneLandau levelsquasi-periodicmagneto-optics12-fold symmetryvan der Waalsquasicrystal
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The pith

A quasi-band approach calculates Landau levels in 30-degree twisted bilayer graphene by treating them as quantized orbits under 12-fold symmetry.

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

The paper sets out a framework for Landau levels in quasi-periodic twisted bilayer graphene at exactly 30 degrees, a system that lacks translational symmetry yet keeps 12-fold rotational symmetry. The method starts from the zero-field quasi-band Hamiltonian and adds the magnetic field through ordinary momentum substitution, which directly maps the levels onto orbits around pockets in the quasi-band structure. This produces nearly flat bands whose energies change only weakly with field strength and highly degenerate levels that arise because twelve equivalent off-center pockets contribute. The same construction yields the magneto-optical conductivity, where transitions must conserve the angular momentum fixed by the 12-fold symmetry. Readers would care because the resulting spectra are concrete predictions that high-field infrared and terahertz experiments can check.

Core claim

Landau levels in 30-degree twisted bilayer graphene are quantized orbits of the quasi-band pockets and carry two good quantum numbers: the usual Landau-level index and an angular-momentum label set by the underlying 12-fold rotational symmetry. The spectrum contains nearly flat bands with weak magnetic-field dependence together with highly degenerate levels that come from the twelve equivalent off-center pockets. Magneto-optical transitions obey angular-momentum selection rules enforced by the same symmetry.

What carries the argument

Quasi-band formalism that inserts the magnetic field by conventional momentum substitution into the zero-field Hamiltonian, thereby turning Landau levels into quantized orbits of quasi-band pockets.

If this is right

  • Nearly flat bands display only weak dependence on magnetic field strength.
  • Levels exhibit high degeneracy arising from twelve equivalent off-center quasi-band pockets.
  • Optical transitions are restricted by angular-momentum selection rules tied to the 12-fold symmetry.
  • The method supplies an efficient route to bulk magneto-optical spectra for any quasicrystalline van der Waals system with rotational symmetry.

Where Pith is reading between the lines

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

  • The same substitution trick may apply to other rotationally symmetric quasicrystals such as higher-order twisted multilayers.
  • High-field infrared spectra could directly test whether angular momentum remains a conserved quantity in optical transitions.
  • The two-quantum-number classification offers a way to label and track features in related quasicrystalline systems even when translational symmetry is absent.

Load-bearing premise

The magnetic field can be incorporated by ordinary momentum substitution directly into the zero-field quasi-band Hamiltonian even though the system has no translational symmetry.

What would settle it

Magneto-optical conductivity measurements that violate the predicted angular-momentum selection rules, or Landau-level spectra that lack the expected twelve-fold degeneracy, would show the substitution approach fails.

Figures

Figures reproduced from arXiv: 2604.18854 by Masaru Hitomi, Mikito Koshino, Takuto Kawakami.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Lattice structure of 30 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Quasi-band structure projected onto [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Left: Quasi-band structure of 30 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Quasi-band structure (left) and corresponding Lan [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Density plot of the optical absorption spectrum Re [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Enlarged views of the optical absorption spectrum in Fig. 5(c), focusing on Regions C1 and C2. The right panel in [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

We develop a theoretical framework for Landau levels in quasi-periodic twisted bilayer graphene at a $30^\circ$ twist angle, a system without translational symmetry but possessing 12-fold rotational symmetry. Using a quasi-band formalism, we incorporate the magnetic field through a conventional momentum substitution in the zero-field Hamiltonian. This approach provides a transparent physical interpretation by directly relating the Landau levels to the quasi-band structure, allowing them to be understood as quantized orbits of quasi-band pockets. By using this method, we reveal distinctive spectral features, including nearly flat bands with weak magnetic-field dependence and highly degenerate levels arising from twelve off-center pockets. The resulting Landau levels are classified by two quantum numbers: the Landau-level index and the angular momentum associated with the underlying quasicrystalline symmetry. We also compute the magneto-optical conductivity and show that optical transitions follow angular-momentum selection rules enforced by the 12-fold symmetry. Our approach provides a symmetry-based and computationally efficient framework for bulk quantum magneto-optics in quasicrystalline van der Waals systems, predicting spectroscopic signatures accessible in high-field infrared and THz experiments.

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

2 major / 2 minor

Summary. The manuscript develops a quasi-band formalism for 30° twisted bilayer graphene (lacking translational symmetry but possessing 12-fold rotational symmetry) and incorporates the magnetic field via direct momentum substitution into the zero-field Hamiltonian. This yields Landau levels interpreted as quantized orbits of quasi-band pockets, classified by Landau index and angular momentum, with nearly flat bands, high degeneracy from twelve off-center pockets, and magneto-optical conductivity obeying angular-momentum selection rules enforced by the quasicrystalline symmetry.

Significance. If rigorously justified, the framework supplies a symmetry-based, computationally efficient route to bulk magneto-optics in quasicrystalline van der Waals systems, directly linking Landau levels to zero-field quasi-band pockets and predicting accessible spectroscopic signatures in high-field infrared/THz experiments. The two-quantum-number classification and selection-rule predictions constitute a clear physical advance over brute-force numerical approaches.

major comments (2)
  1. [Method (quasi-band formalism and magnetic-field incorporation)] The central methodological step—direct substitution of the vector potential into the zero-field quasi-band Hamiltonian—is load-bearing for all subsequent claims (Landau-level classification, degeneracy counting, and optical selection rules). Because the underlying system lacks Bloch periodicity, an explicit derivation or projection argument is required to confirm that this substitution reproduces the correct magnetic translation algebra and does not omit Hofstadter-like fractal splittings; without such justification or a benchmark against the zero-field limit, the resulting level structure remains unverified.
  2. [Landau-level spectrum and classification] The reported high degeneracy arising from twelve off-center pockets and the angular-momentum quantum number are central to the spectroscopic predictions, yet no explicit counting of states or matrix-element derivation is supplied that demonstrates how the 12-fold symmetry commutes with the magnetic Hamiltonian after substitution. A concrete check (e.g., degeneracy lifting under small symmetry-breaking perturbations) would be needed to substantiate the claim.
minor comments (2)
  1. [Magneto-optical conductivity] The abstract states that optical transitions follow angular-momentum selection rules, but the corresponding conductivity formula or matrix-element expression is not referenced; adding an equation number would improve traceability.
  2. [Computational implementation] Numerical details (basis size, cutoff for quasi-band construction, field-strength range) are absent from the provided text; their inclusion would allow readers to assess convergence of the reported flat-band and degeneracy features.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have carefully considered the major comments and provide point-by-point responses below, outlining the revisions we will implement to address the concerns.

read point-by-point responses

  1. Referee: The central methodological step—direct substitution of the vector potential into the zero-field quasi-band Hamiltonian—is load-bearing for all subsequent claims (Landau-level classification, degeneracy counting, and optical selection rules). Because the underlying system lacks Bloch periodicity, an explicit derivation or projection argument is required to confirm that this substitution reproduces the correct magnetic translation algebra and does not omit Hofstadter-like fractal splittings; without such justification or a benchmark against the zero-field limit, the resulting level structure remains unverified.

    Authors: We acknowledge the need for a more rigorous justification of the momentum substitution in the quasi-periodic context. In the revised manuscript, we will add a dedicated section deriving the effective magnetic Hamiltonian from the quasi-band model, demonstrating that it preserves the magnetic translation algebra consistent with the 12-fold symmetry. We will also include a benchmark calculation showing that as the magnetic field approaches zero, the Landau levels recover the original quasi-band dispersion without additional splittings. This addresses the concern regarding potential omission of Hofstadter-like features, which in this approximation are not present due to the lack of translational periodicity. revision: yes

  2. Referee: The reported high degeneracy arising from twelve off-center pockets and the angular-momentum quantum number are central to the spectroscopic predictions, yet no explicit counting of states or matrix-element derivation is supplied that demonstrates how the 12-fold symmetry commutes with the magnetic Hamiltonian after substitution. A concrete check (e.g., degeneracy lifting under small symmetry-breaking perturbations) would be needed to substantiate the claim.

    Authors: The 12-fold rotational symmetry is preserved in our formulation because the vector potential substitution is performed in a symmetric gauge that respects the quasicrystalline symmetry. The degeneracy is a direct consequence of the twelve equivalent pockets being mapped onto each other by rotations. In the revision, we will provide an explicit group-theoretic counting of the degenerate states and derive the optical matrix elements under the angular momentum selection rules. Furthermore, we will add a numerical test applying a small symmetry-breaking perturbation to show the expected lifting of degeneracy, confirming the symmetry protection. revision: yes

Circularity Check

0 steps flagged

No circularity: standard minimal coupling applied to independently defined quasi-band Hamiltonian

full rationale

The derivation begins from a zero-field quasi-band Hamiltonian constructed via 12-fold rotational symmetry for the 30° twisted bilayer, then applies conventional momentum substitution to incorporate the vector potential. This produces Landau levels classified by index and angular momentum, followed by magneto-optical conductivity via angular-momentum selection rules. No step reduces a prediction to a fitted parameter by construction, no self-citation chain supplies the load-bearing uniqueness or ansatz, and the quasi-band structure is not defined in terms of the magnetic-field results. The framework is self-contained as a direct computational mapping from the symmetry-adapted zero-field model.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the validity of quasi-band formalism for aperiodic systems and the applicability of standard magnetic substitution without translational invariance; no explicit free parameters or new entities are mentioned.

axioms (2)
  • domain assumption Quasi-band formalism remains valid when translational symmetry is absent but rotational symmetry is present.
    Invoked to justify the entire substitution approach for the 30° twist.
  • domain assumption Magnetic field incorporation via momentum substitution in the zero-field Hamiltonian captures the Landau level structure.
    Central to relating levels to quasi-band pockets.

pith-pipeline@v0.9.0 · 5499 in / 1363 out tokens · 34443 ms · 2026-05-10T03:15:34.093556+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

221 extracted references · 221 canonical work pages

  1. [1]

    Physical Review Letters , volume =

    Graphene Bilayer with a Twist: Electronic Structure , author =. Physical Review Letters , volume =. 2007 , month =

    work page 2007

  2. [2]

    Physical Review B , volume =

    Commensuration and interlayer coherence in twisted bilayer graphene , author =. Physical Review B , volume =. 2010 , month =

    work page 2010

  3. [3]

    Nano Letters , volume=

    Localization of Dirac electrons in rotated graphene bilayers , author=. Nano Letters , volume=. 2010 , publisher=

    work page 2010

  4. [4]

    Physical Review B , volume=

    Electronic structure of turbostratic graphene , author=. Physical Review B , volume=. 2010 , publisher=

    work page 2010

  5. [5]

    Physical Review B , volume =

    Flat bands in slightly twisted bilayer graphene: Tight-binding calculations , author =. Physical Review B , volume =. 2010 , month =

    work page 2010

  6. [6]

    Proceedings of the National Academy of Sciences , volume=

    Moire bands in twisted double-layer graphene , author=. Proceedings of the National Academy of Sciences , volume=. 2011 , publisher=

    work page 2011

  7. [7]

    Physical Review B , volume =

    Local sublattice-symmetry breaking in rotationally faulted multilayer graphene , author =. Physical Review B , volume =. 2011 , month =

    work page 2011

  8. [8]

    Physical Review B , volume =

    Effects of electrostatic fields and charge doping on the linear bands in twisted graphene bilayers , author =. Physical Review B , volume =. 2011 , month =

    work page 2011

  9. [9]

    Physical Review B , volume =

    Continuum model of the twisted graphene bilayer , author =. Physical Review B , volume =. 2012 , month =

    work page 2012

  10. [10]

    Physical Review B , volume =

    Energy spectrum and quantum Hall effect in twisted bilayer graphene , author =. Physical Review B , volume =. 2012 , month =

    work page 2012

  11. [11]

    Physical Review B , volume =

    Numerical studies of confined states in rotated bilayers of graphene , author =. Physical Review B , volume =. 2012 , month =

    work page 2012

  12. [12]

    Physical Review B , volume =

    Optical absorption in twisted bilayer graphene , author =. Physical Review B , volume =. 2013 , month =

    work page 2013

  13. [13]

    Nature , volume=

    Unconventional superconductivity in magic-angle graphene superlattices , author=. Nature , volume=. 2018 , publisher=

    work page 2018

  14. [14]

    Science , volume=

    Tuning superconductivity in twisted bilayer graphene , author=. Science , volume=. 2019 , publisher=

    work page 2019

  15. [15]

    Nature , volume=

    Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene , author=. Nature , volume=. 2019 , publisher=

    work page 2019

  16. [16]

    Nature , volume=

    Correlated insulator behaviour at half-filling in magic-angle graphene superlattices , author=. Nature , volume=. 2018 , publisher=

    work page 2018

  17. [17]

    Science , volume=

    Stacking-engineered ferroelectricity in bilayer boron nitride , author=. Science , volume=. 2021 , publisher=

    work page 2021

  18. [18]

    Nature Communications , volume=

    Charge-polarized interfacial superlattices in marginally twisted hexagonal boron nitride , author=. Nature Communications , volume=. 2021 , publisher=

    work page 2021

  19. [19]

    Nanoscale lattice dynamics in hexagonal boron nitride moir

    Moore, SL and Ciccarino, CJ and Halbertal, Dorri and McGilly, LJ and Finney, NR and Yao, Kaiyuan and Shao, Yinming and Ni, Guangxin and Sternbach, Aaron and Telford, EJ and others , journal=. Nanoscale lattice dynamics in hexagonal boron nitride moir. 2021 , publisher=

    work page 2021

  20. [20]

    Nano Letters , volume=

    Local electronic properties of graphene on a BN substrate via scanning tunneling microscopy , author=. Nano Letters , volume=. 2011 , publisher=

    work page 2011

  21. [21]

    Nature materials , volume=

    Scanning tunnelling microscopy and spectroscopy of ultra-flat graphene on hexagonal boron nitride , author=. Nature materials , volume=. 2011 , publisher=

    work page 2011

  22. [22]

    Nature physics , volume=

    Emergence of superlattice Dirac points in graphene on hexagonal boron nitride , author=. Nature physics , volume=. 2012 , publisher=

    work page 2012

  23. [23]

    Nature Reviews Physics , volume=

    van der Waals heterostructures combining graphene and hexagonal boron nitride , author=. Nature Reviews Physics , volume=. 2019 , publisher=

    work page 2019

  24. [24]

    Physical Review B , volume =

    Zero-energy modes and gate-tunable gap in graphene on hexagonal boron nitride , author =. Physical Review B , volume =. 2012 , month =

    work page 2012

  25. [25]

    Physical Review B , volume =

    Generic miniband structure of graphene on a hexagonal substrate , author =. Physical Review B , volume =. 2013 , month =

    work page 2013

  26. [26]

    Physical Review Letters , volume=

    Electron interactions and gap opening in graphene superlattices , author=. Physical Review Letters , volume=. 2013 , publisher=

    work page 2013

  27. [27]

    Electronic properties of graphene/hexagonal-boron-nitride moir

    Moon, Pilkyung and Koshino, Mikito , journal=. Electronic properties of graphene/hexagonal-boron-nitride moir. 2014 , publisher=

    work page 2014

  28. [28]

    Nature , volume=

    Cloning of Dirac fermions in graphene superlattices , author=. Nature , volume=. 2013 , publisher=

    work page 2013

  29. [29]

    Nature , volume=

    Hofstadter’s butterfly and the fractal quantum Hall effect in moir\'e superlattices , author=. Nature , volume=. 2013 , publisher=

    work page 2013

  30. [30]

    Science , volume=

    Massive Dirac fermions and Hofstadter butterfly in a van der Waals heterostructure , author=. Science , volume=. 2013 , publisher=

    work page 2013

  31. [31]

    Science , volume=

    Evidence for a fractional fractal quantum Hall effect in graphene superlattices , author=. Science , volume=. 2015 , publisher=

    work page 2015

  32. [32]

    Nature physics , volume=

    Hierarchy of Hofstadter states and replica quantum Hall ferromagnetism in graphene superlattices , author=. Nature physics , volume=. 2014 , publisher=

    work page 2014

  33. [33]

    Science , volume=

    High-temperature quantum oscillations caused by recurring Bloch states in graphene superlattices , author=. Science , volume=. 2017 , publisher=

    work page 2017

  34. [34]

    Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene , author=. Nat. Commun. , volume=. 2019 , publisher=

    work page 2019

  35. [35]

    Nature , volume=

    Tunable spin-polarized correlated states in twisted double bilayer graphene , author=. Nature , volume=. 2020 , publisher=

    work page 2020

  36. [36]

    Nature Physics , volume=

    Correlated states in twisted double bilayer graphene , author=. Nature Physics , volume=. 2020 , publisher=

    work page 2020

  37. [37]

    Nature Physics , volume=

    Symmetry breaking in twisted double bilayer graphene , author=. Nature Physics , volume=. 2021 , publisher=

    work page 2021

  38. [38]

    Nature , volume=

    Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene , author=. Nature , volume=. 2021 , publisher=

    work page 2021

  39. [39]

    Nature Communications , volume=

    Observation of long-lived interlayer excitons in monolayer MoSe2--WSe2 heterostructures , author=. Nature Communications , volume=. 2015 , publisher=

    work page 2015

  40. [40]

    Physical Review Letters , volume =

    Hubbard Model Physics in Transition Metal Dichalcogenide Moir\'e Bands , author =. Physical Review Letters , volume =. 2018 , month =

    work page 2018

  41. [41]

    Physical Review Letters , volume=

    Topological insulators in twisted transition metal dichalcogenide homobilayers , author=. Physical Review Letters , volume=. 2019 , publisher=

    work page 2019

  42. [42]

    Signatures of moir

    Seyler, Kyle L and Rivera, Pasqual and Yu, Hongyi and Wilson, Nathan P and Ray, Essance L and Mandrus, David G and Yan, Jiaqiang and Yao, Wang and Xu, Xiaodong , journal=. Signatures of moir. 2019 , publisher=

    work page 2019

  43. [43]

    Evidence for moir

    Tran, Kha and Moody, Galan and Wu, Fengcheng and Lu, Xiaobo and Choi, Junho and Kim, Kyounghwan and Rai, Amritesh and Sanchez, Daniel A and Quan, Jiamin and Singh, Akshay and others , journal=. Evidence for moir. 2019 , publisher=

    work page 2019

  44. [44]

    Nano Letters , volume=

    Giant valley-Zeeman splitting from spin-singlet and spin-triplet interlayer excitons in WSe2/MoSe2 heterostructure , author=. Nano Letters , volume=. 2019 , publisher=

    work page 2019

  45. [45]

    Nature materials , volume=

    Correlated electronic phases in twisted bilayer transition metal dichalcogenides , author=. Nature materials , volume=. 2020 , publisher=

    work page 2020

  46. [46]

    Strongly correlated electrons and hybrid excitons in a moir

    Shimazaki, Yuya and Schwartz, Ido and Watanabe, Kenji and Taniguchi, Takashi and Kroner, Martin and Imamo. Strongly correlated electrons and hybrid excitons in a moir. Nature , volume=. 2020 , publisher=

    work page 2020

  47. [47]

    Nature , volume=

    Quantum criticality in twisted transition metal dichalcogenides , author=. Nature , volume=. 2021 , publisher=

    work page 2021

  48. [48]

    Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moir

    Miao, Shengnan and Wang, Tianmeng and Huang, Xiong and Chen, Dongxue and Lian, Zhen and Wang, Chong and Blei, Mark and Taniguchi, Takashi and Watanabe, Kenji and Tongay, Sefaattin and others , journal=. Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moir. 2021 , publisher=

    work page 2021

  49. [49]

    Skyrmions in the Moir

    Tong, Qingjun and Liu, Fei and Xiao, Jiang and Yao, Wang , journal=. Skyrmions in the Moir. 2018 , publisher=

    work page 2018

  50. [50]

    Physical Review Letters , volume=

    Stacking domain wall magnons in twisted van der Waals magnets , author=. Physical Review Letters , volume=. 2020 , publisher=

    work page 2020

  51. [51]

    Noncollinear phases in moir

    Hejazi, Kasra and Luo, Zhu-Xi and Balents, Leon , journal=. Noncollinear phases in moir. 2020 , publisher=

    work page 2020

  52. [52]

    Physical Review B , volume=

    Heterobilayer moir\'e magnets: Moir\'e skyrmions and commensurate-incommensurate transitions , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  53. [53]

    Physical Review B , volume=

    Skyrmions in twisted van der Waals magnets , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  54. [54]

    Physical Review Research , volume=

    Magnetization textures in twisted bilayer Cr X 3 (X= Br, I) , author=. Physical Review Research , volume=. 2021 , publisher=

    work page 2021

  55. [55]

    Nature Physics , volume=

    Twist engineering of the two-dimensional magnetism in double bilayer chromium triiodide homostructures , author=. Nature Physics , volume=. 2022 , publisher=

    work page 2022

  56. [56]

    One-dimensional flat bands in twisted bilayer germanium selenide , author=. Nat. Commun. , volume=. 2020 , publisher=

    work page 2020

  57. [57]

    Physical Review B , volume=

    Effective continuum model of twisted bilayer GeSe and origin of the emerging one-dimensional mode , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  58. [58]

    Journal of Physical Chemistry Letter , volume=

    Emergence and Tuning of Multiple Flat Bands in Twisted Bilayer -Graphyne , author=. Journal of Physical Chemistry Letter , volume=. 2021 , publisher=

    work page 2021

  59. [59]

    Science , volume =

    Dirac electrons in a dodecagonal graphene quasicrystal , author =. Science , volume =. 2018 , doi =

    work page 2018

  60. [60]

    Proceedings of the National Academy of Sciences , volume =

    Quasicrystalline 30 twisted bilayer graphene as an incommensurate superlattice with strong interlayer coupling , author =. Proceedings of the National Academy of Sciences , volume =. 2018 , doi =

    work page 2018

  61. [61]

    2D Materials , volume=

    Scanning tunneling microscopy study of the quasicrystalline 30 twisted bilayer graphene , author=. 2D Materials , volume=. 2019 , publisher=

    work page 2019

  62. [62]

    Nano Letters , volume =

    30-Twisted Bilayer Graphene Quasicrystals from Chemical Vapor Deposition , author =. Nano Letters , volume =. 2020 , doi =

    work page 2020

  63. [63]

    ACS nano , volume=

    Interlayer decoupling in 30 twisted bilayer graphene quasicrystal , author=. ACS nano , volume=. 2020 , publisher=

    work page 2020

  64. [64]

    ACS nano , volume=

    Ultrafast unbalanced electron distributions in quasicrystalline 30 twisted bilayer graphene , author=. ACS nano , volume=. 2019 , publisher=

    work page 2019

  65. [65]

    2D Materials , volume=

    Synchronous growth of 30-twisted bilayer graphene domains with millimeter scale , author=. 2D Materials , volume=. 2021 , publisher=

    work page 2021

  66. [66]

    Physical Review B , volume=

    Atomic arrangements of quasicrystal bilayer graphene: Interlayer distance expansion , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  67. [67]

    Emergent moir

    Li, Jingwei and Bao, Kejie and Sun, Honglin and Yan, Xingxu and Huang, Ting and Zhang, Qicheng and Zhou, Yaoqiang and Huang, Yifeng and Liu, Zhenjing and Das, Paul Masih and others , journal=. Emergent moir. 2025 , publisher=

    work page 2025

  68. [68]

    Quasicrystalline electronic states in

    Moon, Pilkyung and Koshino, Mikito and Son, Young-Woo , journal =. Quasicrystalline electronic states in. 2019 , month =. doi:10.1103/PhysRevB.99.165430 , url =

    work page doi:10.1103/physrevb.99.165430 2019

  69. [69]

    npj Computational Materials , volume=

    Dodecagonal bilayer graphene quasicrystal and its approximants , author=. npj Computational Materials , volume=. 2019 , publisher=

    work page 2019

  70. [70]

    Superlubricity in quasicrystalline twisted bilayer graphene , author =. Phys. Rev. B , volume =. 2016 , month =. doi:10.1103/PhysRevB.93.201404 , url =

    work page doi:10.1103/physrevb.93.201404 2016

  71. [71]

    Physical Review B , volume=

    Pressure and electric field dependence of quasicrystalline electronic states in 30 twisted bilayer graphene , author=. Physical Review B , volume=. 2020 , publisher=

    work page 2020

  72. [72]

    Physical Review B , volume=

    Electronic structure of 30 twisted double bilayer graphene , author=. Physical Review B , volume=. 2020 , publisher=

    work page 2020

  73. [73]

    Physical Review B , volume=

    Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in the strong coupling limit , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  74. [74]

    Ghadimi and B.-J

    Quasiperiodic Pairing in Graphene Quasicrystals , volume =. Nano Letters , author =. 2025 , note =. doi:10.1021/acs.nanolett.4c04386 , number =

    work page doi:10.1021/acs.nanolett.4c04386 2025

  75. [75]

    Physical Review B , volume=

    High-energy Landau levels in graphene beyond nearest-neighbor hopping processes: Corrections to the effective Dirac Hamiltonian , author=. Physical Review B , volume=. 2022 , publisher=

    work page 2022

  76. [76]

    Frontiers in Carbon , author =

    Quasicrystalline 30 twisted bilayer graphene fractal patterns and electronic localization properties , volume =. Frontiers in Carbon , author =. 2024 , note =. doi:10.3389/frcrb.2024.1496179 , urldate =

    work page doi:10.3389/frcrb.2024.1496179 2024

  77. [77]

    Physical Review B , author =

    Interlayer hybridization in graphene quasicrystal and other bilayer graphene systems , volume =. Physical Review B , author =. 2022 , pages =. doi:10.1103/PhysRevB.105.125403 , number =

    work page doi:10.1103/physrevb.105.125403 2022

  78. [78]

    Physical Review B , author =

    Emergent localization in dodecagonal bilayer quasicrystals , volume =. Physical Review B , author =. 2019 , pages =. doi:10.1103/PhysRevB.99.245401 , number =

    work page doi:10.1103/physrevb.99.245401 2019

  79. [79]

    Physical Review B , volume=

    Quasicrystalline electronic states in twisted bilayers and the effects of interlayer and sublattice symmetries , author=. Physical Review B , volume=. 2021 , publisher=

    work page 2021

  80. [80]

    Physical Review B , author =

    Theory of quantum oscillations in quasicrystals: Quantizing spiral Fermi surfaces , volume =. Physical Review B , author =. doi:10.1103/PhysRevB.100.081405 , number =

    work page doi:10.1103/physrevb.100.081405

Showing first 80 references.