Twisted Bilayer Graphene in Commensurate Angles

Tal Malinovitch

arxiv: 2409.12344 · v3 · pith:CV66JXGQnew · submitted 2024-09-18 · 🧮 math-ph · cond-mat.mes-hall· math.MP· math.SP

Twisted Bilayer Graphene in Commensurate Angles

Tal Malinovitch This is my paper

Pith reviewed 2026-05-23 20:43 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mes-hallmath.MPmath.SP

keywords twisted bilayer grapheneDirac conescommensurate anglescontinuum modelSchrödinger operatorhoneycomb latticeAA stackingAB stacking

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The pith

Dirac cones exist at the Brillouin zone vertices in the exact continuum Schrödinger operator for twisted bilayer graphene at commensurate angles.

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

The paper proves the existence of Dirac cones at the vertices of the Brillouin zone for twisted bilayer graphene at commensurate angles using the exact continuum Schrödinger operator. It applies to a wide class of honeycomb potentials with graphene symmetries in both AA and AB stacking configurations. This is the first rigorous proof without using the Bistritzer-MacDonald approximations. A sympathetic reader would care because it confirms the linear dispersion in the full model at these special angles and provides bounds on how the slope changes near incommensurate angles for small potentials.

Core claim

We prove the existence of Dirac cones at the vertices of the Brillouin zone for such angles in the 2D continuum model of electronic transport in twisted bilayer graphene at commensurate angles. The model uses two honeycomb potentials with the symmetries of graphene, either sharing a common origin or shifted by a half-lattice spacing, and twisted relative to each other. Quantitative bounds show that for small potentials the slope of the Dirac cones flattens at commensurate angles near incommensurate angles.

What carries the argument

The exact continuum Schrödinger operator without the Bistritzer-MacDonald approximations, for commensurate twist angles with AA or AB stacking of graphene-symmetric honeycomb potentials.

If this is right

The existence of Dirac cones holds for a wide class of potentials in both stacking types.
Quantitative bounds are established for the flattening of the Dirac cone slope for small potentials.
This provides the first rigorous proof of Dirac cones in the continuum setting for TBG at commensurate angles.

Where Pith is reading between the lines

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

The proof technique might extend to incommensurate angles or other 2D heterostructures.
Experimental verification could involve measuring the density of states or ARPES near commensurate angles to observe the flattening effect.
Similar continuum models for other twisted materials could benefit from this approach to establish topological features rigorously.

Load-bearing premise

The two honeycomb potentials possess the symmetries of graphene and the twist angles are exactly the commensurate angles defined by the model for AA or AB stacking.

What would settle it

A calculation or numerical simulation of the spectrum of the Schrödinger operator at a commensurate angle that does not exhibit linear dispersion at the Brillouin zone vertices would falsify the claim.

Figures

Figures reproduced from arXiv: 2409.12344 by Tal Malinovitch.

**Figure 1.** Figure 1: Illustration of AB stacking of graphene with the different possible shifts. AN Illustration of shift by K0 in 1a, shift by RK0 in 1b, and a shift by R2K0 in 1c. In all cases, the red hexagon corresponds to the upper layer, and the blue corresponds to the lower layer Definition 2.6. Let V be a honeycomb potential with Λ as a lattice, and let G be an admissible interaction operator, and G∗ an admissible int… view at source ↗

read the original abstract

We study a 2D continuum model of electronic transport in twisted bilayer graphene (TBG) at commensurate angles. We use two honeycomb potentials with the symmetries of graphene, either sharing a common origin (AA stacking) or shifted by a half-lattice spacing (AB stacking), and twisted relative to each other. While the electronic properties of TBG are most commonly studied via the approximate Bistritzer-MacDonald (BM) model, our approach studies the exact continuum Schr\"{o}dinger operator without these approximations. Our results hold for a wide class of potentials in both stacking types. We describe the exact angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones at the vertices of the Brillouin zone for such angles. Additionally, we establish quantitative bounds showing that, for small potentials, the slope of the Dirac cones flattens at commensurate angles near incommensurate angles. This work is the first to rigorously establish the existence of Dirac cones for twisted bilayer graphene in the continuum setting, without the BM approximations.

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 provides what appears to be the first rigorous existence proof for Dirac cones in the exact continuum model of twisted bilayer graphene at commensurate angles.

read the letter

Here's the core: the paper establishes the existence of Dirac cones at the vertices of the Brillouin zone for twisted bilayer graphene modeled by the exact continuum Schrödinger operator at commensurate angles. It avoids the common Bistritzer-MacDonald approximations and works for a class of potentials with graphene symmetries in both AA and AB stacking configurations. The work does well in providing this rigorous existence result and in deriving quantitative bounds that show the Dirac cone slope flattens for small potentials at angles close to incommensurate ones. This gives some control on how the model transitions between regimes. Where it is softer is in the scope. The bounds are limited to small potentials, and the result hinges on the potentials having the precise symmetries and the angles being exactly commensurate as defined in the model. If real systems deviate from these, the conclusions may not carry over directly. Also, the abstract leaves the detailed error estimates and spectral analysis to the full text, so the strength of the proof can't be fully assessed yet. This paper targets specialists in mathematical physics and condensed matter who want exact results for TBG rather than approximate ones. A reader focused on the foundations of flat-band physics would get something from it. Overall, it merits peer review because a rigorous proof in the exact model is worth the time of referees to verify the claims and potentially strengthen the presentation.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes the exact 2D continuum Schrödinger operator for twisted bilayer graphene at commensurate angles, employing two honeycomb potentials with graphene symmetries under either AA or AB stacking. It proves existence of Dirac cones at the Brillouin-zone vertices for these angles and derives quantitative bounds showing flattening of the Dirac-cone slope for small potentials near incommensurate angles. The work is presented as the first rigorous continuum proof without the Bistritzer-MacDonald approximation.

Significance. If the central existence proof and bounds hold, the result supplies the first mathematically rigorous demonstration of Dirac cones in the exact continuum model of TBG at commensurate angles. The quantitative flattening bounds constitute an additional concrete contribution that could be compared with numerical or approximate-model studies.

major comments (2)

[Main existence result (abstract and presumed §3–4)] The abstract states that the existence proof and bounds hold for a wide class of potentials under the stated symmetries, yet the provided text gives no explicit statement of the spectral estimates or perturbation argument used to control the Schrödinger operator near the Brillouin-zone vertices; without these details the load-bearing step of the central claim cannot be verified.
[Quantitative bounds paragraph (abstract)] The quantitative bounds on slope flattening are asserted for small potentials; the manuscript must specify whether the constants are uniform over the admissible potential class or depend on additional parameters, as this directly affects the strength of the flattening statement.

minor comments (2)

[Model description] The definition of the exact commensurate angles (AA/AB) should be stated with an explicit formula or lattice-vector condition in the model section.
[Introduction] A brief comparison table or paragraph contrasting the present continuum operator with the BM model would clarify the precise sense in which the approximation is avoided.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify our manuscript. We respond to each major comment below and will revise the manuscript to address the points raised.

read point-by-point responses

Referee: [Main existence result (abstract and presumed §3–4)] The abstract states that the existence proof and bounds hold for a wide class of potentials under the stated symmetries, yet the provided text gives no explicit statement of the spectral estimates or perturbation argument used to control the Schrödinger operator near the Brillouin-zone vertices; without these details the load-bearing step of the central claim cannot be verified.

Authors: The full manuscript contains the spectral estimates and perturbation arguments in Sections 3 and 4, where the honeycomb symmetries are used to reduce the analysis to the K-points of the Brillouin zone, followed by a controlled perturbation of the untwisted operators at commensurate angles. We acknowledge that the connection between the abstract claim and these sections could be made more explicit and will add a concise outline of the key estimates and argument structure to the introduction in the revised version. revision: yes
Referee: [Quantitative bounds paragraph (abstract)] The quantitative bounds on slope flattening are asserted for small potentials; the manuscript must specify whether the constants are uniform over the admissible potential class or depend on additional parameters, as this directly affects the strength of the flattening statement.

Authors: The constants appearing in the flattening bounds are uniform over the admissible class of potentials obeying the stated honeycomb symmetries; they depend only on the potential amplitude and the angular deviation from incommensurability. We will insert an explicit statement to this effect both in the abstract and in the statement of the relevant theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a direct rigorous existence proof for Dirac cones via spectral analysis of the exact continuum Schrödinger operator under explicit symmetry assumptions on the honeycomb potentials (AA/AB stacking) and the definition of commensurate angles. The abstract and claim structure show no reduction of any prediction or result to fitted inputs, self-definitional loops, or load-bearing self-citations; the derivation is self-contained as a mathematical argument within the stated model class.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on domain assumptions about the form of the potentials and the definition of commensurate angles; no free parameters or new entities are introduced.

axioms (2)

domain assumption The potentials are honeycomb lattices with the symmetries of graphene (AA or AB stacking)
Explicitly stated in the abstract as the potentials used
domain assumption The electronic transport is governed by the exact continuum Schrödinger operator
The approach studies the exact continuum Schrödinger operator without BM approximations

pith-pipeline@v0.9.0 · 5710 in / 1287 out tokens · 42882 ms · 2026-05-23T20:43:03.678263+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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D. Massatt, S. Carr, and M. Luskin, Electronic observables for relaxed bilayer 2d heterostruc tures in momentum space, arXiv preprint arXiv:2109.15296 (2021)

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D. Massatt, S. Carr, M. Luskin, and C. Ortner, Incommensurate heterostructures in momentum space , Multiscale Modeling & Simulation 16 (2018), no. 1, 429–451

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C. Nunn, A proof of a generalization of Niven ’s theorem using algebra ic number theory , Rose-Hulman Undergrad- uate Mathematics Journal 22 (2021), no. 2, 3

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Reed and B

M. Reed and B. Simon, Methods of modern mathematical physics - IV: Analysis of Ope rators, vol. 4, Elsevier, 1978

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M. A. Rodriguez-Andrade, G. Arag´ on-Gonz´ alez, J. L. Ar ag´ on, A. G´ omez-Rodr ´ ıguez, and D. Romeu,The co- incidence site lattices in 2d hexagonal lattices using Cliﬀo rd algebra, Advances in Applied Cliﬀord Algebras 25 (2015), 425–440

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Scheer, K

M. Scheer, K. Gu, and B. Lian, Magic angles in twisted bilayer graphene near commensurati on: Towards a hypermagic regime, Physical Review B 106 (2022), no. 11, 115418

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[33]

J. C. Slonczewski and P. R. Weiss, Band structure of graphite , Physical Review 109 (1958), no. 2, 272

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[34]

Tarnopolsky, A

G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene , Physical review letters 122 (2019), no. 10, 106405

work page 2019
[35]

P. R. Wallace, The band theory of graphite , Physical review 71 (1947), 622–634

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[36]

Watson, T

A. Watson, T. Kong, A. MacDonald, and M. Luskin, Bistritzer–MacDonald dynamics in twisted bilayer graphen e, Journal of Mathematical Physics 64 (2023), no. 3, 031502

work page 2023
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Watson and M

A. Watson and M. Luskin, Existence of the ﬁrst magic angle for the chiral model of bila yer graphene, Journal of Mathematical Physics 62 (2021), no. 9, 091502

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[38]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene , Science 363 (2019), no. 6431, 1059–1064

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Zworski, Mathematical results on the chiral models of twisted bilaye r graphene (with an appendix by Mengxuan Yang and Zhongkai Tao) , Journal of Spectral Theory 14 (2024), no

M. Zworski, Mathematical results on the chiral models of twisted bilaye r graphene (with an appendix by Mengxuan Yang and Zhongkai Tao) , Journal of Spectral Theory 14 (2024), no. 3, 1063–1107. TBG IN COMMENSURATE ANGLES 35 Appendix A. Notation • We will denote by ⟨·. ·⟩ the Euclidean inner product on vectors in R2 or Z2, and the size of these vector will...

work page 2024

[1] [1]

M. J. Ablowitz, C. W Curtis, and Y. Zhu, On tight-binding approximations in optical lattices , Studies in Applied Mathematics 129 (2012), no. 4, 362–388

work page 2012

[2] [2]

Becker, M

S. Becker, M. Embree, J. Wittsten, and M. Zworski, Spectral characterization of magic angles in twisted bilay er graphene, Physical Review B 103 (2021), no. 16, 165113

work page 2021

[3] [3]

1, 69–103

, Mathematics of magic angles in a model of twisted bilayer gra phene, Probability and Mathematical Physics 3 (2022), no. 1, 69–103

work page 2022

[4] [4]

Becker, T

S. Becker, T. Humbert, and M. Zworski, Fine structure of ﬂat bands in a chiral model of magic angles , Annales Henri Poincar´ e, Springer, 2024, pp. 1–31

work page 2024

[5] [5]

Becker, S

S. Becker, S. Quinn, Z. Tao, A. Watson, and M. Yang, Dirac cones and magic angles in the Bistritzer-MacDonald TBG Hamiltonian , arXiv preprint arXiv:2407.06316 (2024)

work page arXiv 2024

[6] [6]

Becker and M

S. Becker and M. Zworski, Dirac points for twisted bilayer graphene with in-plane mag netic ﬁeld , Journal of Spectral Theory 14 (2024), no. 2, 479–511

work page 2024

[7] [7]

Berkolaiko and A

G. Berkolaiko and A. Comech, Symmetry and Dirac points in graphene spectrum , Journal of Spectral Theory 8 (2018), no. 3, 1099–1147

work page 2018

[8] [8]

Bistritzer and A

R. Bistritzer and A. H MacDonald, Moir´ e bands in twisted double-layer graphene , Proceedings of the National Academy of Sciences 108 (2011), no. 30, 12233–12237

work page 2011

[9] [9]

Canc` es, L

´E. Canc` es, L. Garrigue, and D. Gontier, Simple derivation of Moir´ e-scale continuous models for tw isted bilayer graphene, Physical Review B 107 (2023), no. 15, 155403

work page 2023

[10] [10]

Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. K axiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices , Nature 556 (2018), no. 7699, 43–50

work page 2018

[11] [11]

Carlsson and O

M. Carlsson and O. Rubin, On perturbation of operators and Rayleigh-Schr¨ odinger coeﬃcients, Complex Analysis and Operator Theory 18 (2024), no. 3, 47

work page 2024

[12] [12]

Catarina, B

G. Catarina, B. Amorim, E. V Castro, J. Lopes, and N. Peres , Twisted bilayer graphene: Low-energy physics, electronic and optical properties , Handbook of Graphene 3 (2019), 177–232

work page 2019

[13] [13]

Feﬀerman, J

C. Feﬀerman, J. Lee-Thorp, and M. Weinstein, Honeycomb Schr¨ odinger operators in the strong binding reg ime, Communications on Pure and Applied Mathematics 71 (2018), no. 6, 1178–1270. 34 TAL MALINOVITCH

work page 2018

[14] [14]

Feﬀerman and M

C. Feﬀerman and M. Weinstein, Honeycomb lattice potentials and Dirac points , Journal of the American Mathe- matical Society 25 (2012), no. 4, 1169–1220. 15. , Wave packets in honeycomb structures and two-dimensional D irac equations, Communications in Math- ematical Physics 326 (2014), 251–286

work page 2012

[15] [15]

Hennighausen and S

Z. Hennighausen and S. Kar, Twistronics: a turning point in 2d quantum materials , Electronic Structure 3 (2021), no. 1, 014004

work page 2021

[16] [16]

T. Kong, D. Liu, M. Luskin, and A. Watson, Modeling of electronic dynamics in twisted bilayer graphen e, SIAM Journal on Applied Mathematics 84 (2024), no. 3, 1011–1038

work page 2024

[17] [17]

Koshino, Band structure and topological properties of twisted doubl e bilayer graphene , Physical Review B 99 (2019), no

M. Koshino, Band structure and topological properties of twisted doubl e bilayer graphene , Physical Review B 99 (2019), no. 23, 235406

work page 2019

[18] [18]

Kuchment, An overview of periodic elliptic operators , Bulletin of the American Mathematical Society 53 (2016), 343–414

P. Kuchment, An overview of periodic elliptic operators , Bulletin of the American Mathematical Society 53 (2016), 343–414

work page 2016

[19] [19]

Kuchment and O

P. Kuchment and O. Post, On the spectra of carbon nano-structures , Communications in Mathematical Physics 33 (1973), 335–343

work page 1973

[20] [20]

Ledwith, E

P. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogi cal introduction, Annals of Physics 435 (2021), 168646

work page 2021

[21] [21]

Y. Li, A. Eaton, H.A. Fertig, and B. Seradjeh, Dirac magic and Lifshitz transitions in AA-stacked twisted multilayer graphene, Physical Review Letters 128 (2022), no. 2, 026404

work page 2022

[22] [22]

J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum valley Hall eﬀect, orbital magnetism, and anomalous Hall eﬀect in twisted multilayer graphene systems , Physical Review X 9 (2019), no. 3, 031021

work page 2019

[23] [23]

Lopes dos Santos, N

J. Lopes dos Santos, N. Peres, and A. Castro Neto, Continuum model of the twisted graphene bilayer , Physical Review B—Condensed Matter and Materials Physics 86 (2012), no. 15, 155449

work page 2012

[24] [24]

Massatt, S

D. Massatt, S. Carr, and M. Luskin, Electronic observables for relaxed bilayer 2d heterostruc tures in momentum space, arXiv preprint arXiv:2109.15296 (2021)

work page arXiv 2021

[25] [25]

Massatt, S

D. Massatt, S. Carr, M. Luskin, and C. Ortner, Incommensurate heterostructures in momentum space , Multiscale Modeling & Simulation 16 (2018), no. 1, 429–451

work page 2018

[26] [26]

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, an d A. K. Geim, The electronic properties of graphene , Reviews of modern physics 81 (2009), no. 1, 109

work page 2009

[27] [27]

K. S. Novoselov, Nobel lecture: Graphene: Materials in the ﬂatland , Reviews of modern physics 83 (2011), no. 3, 837

work page 2011

[28] [28]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zha ng, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Electric ﬁeld eﬀect in atomically thin carbon ﬁlms , Science 306 (2004), no. 5696, 666–669

work page 2004

[29] [29]

Nunn, A proof of a generalization of Niven ’s theorem using algebra ic number theory , Rose-Hulman Undergrad- uate Mathematics Journal 22 (2021), no

C. Nunn, A proof of a generalization of Niven ’s theorem using algebra ic number theory , Rose-Hulman Undergrad- uate Mathematics Journal 22 (2021), no. 2, 3

work page 2021

[30] [30]

Reed and B

M. Reed and B. Simon, Methods of modern mathematical physics - IV: Analysis of Ope rators, vol. 4, Elsevier, 1978

work page 1978

[31] [31]

M. A. Rodriguez-Andrade, G. Arag´ on-Gonz´ alez, J. L. Ar ag´ on, A. G´ omez-Rodr ´ ıguez, and D. Romeu,The co- incidence site lattices in 2d hexagonal lattices using Cliﬀo rd algebra, Advances in Applied Cliﬀord Algebras 25 (2015), 425–440

work page 2015

[32] [32]

Scheer, K

M. Scheer, K. Gu, and B. Lian, Magic angles in twisted bilayer graphene near commensurati on: Towards a hypermagic regime, Physical Review B 106 (2022), no. 11, 115418

work page 2022

[33] [33]

J. C. Slonczewski and P. R. Weiss, Band structure of graphite , Physical Review 109 (1958), no. 2, 272

work page 1958

[34] [34]

Tarnopolsky, A

G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene , Physical review letters 122 (2019), no. 10, 106405

work page 2019

[35] [35]

P. R. Wallace, The band theory of graphite , Physical review 71 (1947), 622–634

work page 1947

[36] [36]

Watson, T

A. Watson, T. Kong, A. MacDonald, and M. Luskin, Bistritzer–MacDonald dynamics in twisted bilayer graphen e, Journal of Mathematical Physics 64 (2023), no. 3, 031502

work page 2023

[37] [37]

Watson and M

A. Watson and M. Luskin, Existence of the ﬁrst magic angle for the chiral model of bila yer graphene, Journal of Mathematical Physics 62 (2021), no. 9, 091502

work page 2021

[38] [38]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning superconductivity in twisted bilayer graphene , Science 363 (2019), no. 6431, 1059–1064

work page 2019

[39] [39]

Zworski, Mathematical results on the chiral models of twisted bilaye r graphene (with an appendix by Mengxuan Yang and Zhongkai Tao) , Journal of Spectral Theory 14 (2024), no

M. Zworski, Mathematical results on the chiral models of twisted bilaye r graphene (with an appendix by Mengxuan Yang and Zhongkai Tao) , Journal of Spectral Theory 14 (2024), no. 3, 1063–1107. TBG IN COMMENSURATE ANGLES 35 Appendix A. Notation • We will denote by ⟨·. ·⟩ the Euclidean inner product on vectors in R2 or Z2, and the size of these vector will...

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