Moir\'e-driven equilibrium of perturbations in moir\'e systems

Federico Escudero; Francisco Guinea; Pierre A. Pantale\'on; Zhen Zhan

REVIEW 4 major objections 3 minor 1 cited by

Moir\'e-driven equilibrium of perturbations in moir\'e systems

T0 review · 4 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Twisted bilayer graphene transfers layer perturbations to both layers near the magic angle, driving bandgaps, energy shifts, and momentum shifts to a common equilibrium.

desk verdict The qualitative 'perturbations equilibrate near the magic angle' idea is real and nicely demonstrated numerically, but the first-order analytic condition for mass equilibrium disagrees sharply with the numerics, so the paper leans more on the numerics than the theory. read the letter →

arxiv 2601.00948 v2 pith:ORVZNU5U submitted 2026-01-02 cond-mat.mes-hall

Federico Escudero , Zhen Zhan , Pierre A. Pantaleón , Francisco Guinea This is my paper · ORCID

classification cond-mat.mes-hall

keywords twistedbilayergraphenemoiré-drivenequilibriummagicangleperturbationrenormalizationDiracconecollapselayerpolarizationstrainandsubstrateeffectscontinuummodel

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

The paper argues that any weak perturbation acting on one layer of twisted bilayer graphene—a mass term, a scalar potential, or a gauge field—is not confined to that layer once the moiré coupling is strong. Instead, the coupling transfers the perturbation to the other Dirac cone, and near the conventional magic angle the renormalized values in the two layers equalize: gaps become equal, energy shifts become equal, and momentum shifts can even make the two Dirac points collapse. The authors support this with first-order perturbation theory, which renormalizes the perturbation in close analogy to the Fermi velocity, and with numerical diagonalization of the full continuum model. If correct, this explains why experiments near the magic angle appear insensitive to whether strain is applied to one layer or both, and it extends the notion of the magic angle from flat bands to a broader regime of perturbation equilibrium.

What carries the argument

First-order perturbation theory around the two Dirac points of a truncated continuum model yields the renormalized layer perturbation P*_ℓ = (Pℓ + Σ_j U_j h_j^{-1} P_{ℓ'} h_j^{-1} U_j)/(1 + 3α²(1+λ²)), alongside renormalized Fermi velocity v* = v(1−3α²)/(1+3α²(1+λ²)), with α = u'/(ħv kθ) and λ = u/u'. This expression mixes the perturbations from the two layers through the moiré potential; equalizing the renormalized gaps, shifts, or momenta produces the equilibrium conditions.

What would settle it

Measure, as a function of twist angle, the bandgap and Dirac-point energy at both moiré Dirac points of a twisted bilayer graphene sample with a known single-layer mass perturbation (e.g., hBN on one side). The paper predicts equal gaps and equal energy shifts in a window near the magic angle before perturbation theory breaks down; if the gaps remain unequal across that entire window, the core equilibrium claim would be refuted.

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Extended reading notes

Core claim

Central claim: moiré-driven equilibrium—strong moiré coupling transfers a perturbation from one layer of twisted bilayer graphene to the other, and near the first magic angle the layers equilibrate. Mass gaps equalize when 3α²(1−λ²)=1; scalar shifts equalize when 3α²(1+λ²)=1, at (Vt+Vb)/2; gauge perturbations collapse the Dirac points at 3α²=1. Full numerics confirm equilibrium at α≈0.65 (mass), α≈0.475 (scalar), α≈0.568 (gauge), with oscillatory redistribution beyond the first magic angle. The authors conclude layer-resolved perturbation origins are masked near the magic angle, robust to perturbed moiré potentials.

Load-bearing premise

The analytic derivations assume the perturbations are small enough for first-order perturbation theory to remain valid all the way to the equilibrium couplings; that assumption fails for gauge perturbations at the magic angle and puts the mass equilibrium for realistic λ far from the magic angle.

Editorial extensions

If this is right

Near the magic angle, any measurement that reads out bandgap, Dirac energy, or Dirac momentum will see the average of the two layers' perturbations, not the layer that was perturbed; single-layer and bilayer strain or substrate effects become indistinguishable.
Mass perturbations at equilibrium produce bandgaps smaller than the initial average gap for λ>0, and exactly the average gap in the chiral limit λ=0; scalar perturbations always equilibrate to half the initial shift regardless of λ.
Gauge perturbations aligned with the twist direction cause the two Dirac points to collapse within the moiré Brillouin zone at a coupling near the first magic angle; the collapse position is fixed by the initial perturbation, not by the twist.
Beyond the first magic angle, perturbation effects oscillate, and bandgaps can close and reopen, implying that higher magic angles behave qualitatively differently from the first.
The equilibrium mechanism survives when the moiré potential itself is modified by strain or an hBN substrate, so the masking effect is not an artifact of an idealized interlayer potential.

Reading between the lines

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

If this mechanism is generic, moiré bilayers beyond graphene—semiconductor moiré systems with strong interlayer hybridization—should show a similar perturbation-equilibrium tendency, and layer-resolved transport or spectroscopic signatures should fade at their strong-coupling angles.
The first-order analysis predicts gauge-perturbation collapse paths along the perturbation direction, but the numerical result shows a distinct path for a perpendicular perturbation; checking whether the collapse path is always dictated by (Kt−Kb) would sharpen the theory's regime of validity.
A testable extension: in a sample with an hBN substrate on one side only, the gap at both Dirac points should equalize near the magic angle; a twist-angle series measuring both gaps would directly confirm or falsify the masking claim.
The gap closings and reopenings beyond the first magic angle, if accompanied by Chern-number changes, suggest that asymmetric perturbations could be used to engineer topological transitions in moiré bands—an implication the paper raises but does not pursue.

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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

Quick take: the paper makes a worthwhile point — layer-specific perturbations in TBG tend to get redistributed by the moiré coupling and approach a common value near the magic angle. The numerical evidence is clean and the first-order renormalization formulas are a useful new tool. But the analytic mass equilibrium condition doesn't match their own numerics, and the paper glosses over that.

What's new: Eqs. (6)-(10) give the first-order renormalization of scalar, mass, and gauge perturbations in a unified way. I haven't seen that in the prior TBG literature. The equilibrium conditions that follow (equal gaps, shifts) are a natural organizing concept. The numerics in Fig. 2 and 3 do show clear equilibration for all three perturbation types, with parameters given. The connection to hBN and strain experiments is plausible and helps explain the masking of layer-resolved properties.

Where it gets soft: for λ=0.8, Eq. (11) predicts mass equilibrium at 3α²(1−λ²)=1, i.e., α≈0.96. Your Fig. 2(a) puts the crossing at α=0.65. At that point the first-order ratio m_t*/m_b* ≈0.46, not 1. So the first-order theory isn't even approximately satisfied at the numerical equilibrium — the actual mechanism must involve higher-order k or band-structure effects. The text says the condition is 'met around the first magic angle where 3α²∼1', which is only true for λ=0; for λ=0.8 it's far off. Scalar and gauge cases work better (scalar: 0.451 vs 0.475; gauge: 0.577 vs 0.568), and the gauge breakdown is honestly stated. So the quantitative support for the mass case is weak, and the phrase 'any perturbation' overstates what's derived.

Proportion: the qualitative picture is likely right — the numerics show it for mass, scalar, and gauge, and the first-order theory gives a transparent heuristic. The paper just needs to be honest that for mass the analytic condition is not quantitatively predictive for realistic λ, and that the mechanism at the crossing is not the one in Eq. (11).

Verdict: send it to referees. The idea is useful and the numerics are solid, but the authors should be pushed to either repair the mass condition (e.g., include higher-order corrections) or explicitly downgrade it to qualitative. I'd cite the equilibrium concept and the scalar/gauge formulas, but not the mass condition as quantitatively accurate.

Referee Report

4 major / 3 minor

Summary. This Letter studies how layer-local perturbations—mass, scalar, and gauge terms—affect the low-energy spectrum of twisted bilayer graphene. Starting from the standard continuum model, the authors derive a first-order renormalized Hamiltonian around the two Dirac points, Eqs. (5)–(10), and obtain conditions under which bandgaps, energy shifts, and momentum shifts in the two layers become equal. They call this phenomenon 'moiré-driven equilibrium' and argue that it occurs near the first magic angle and is robust even when the moiré potential itself is perturbed. Numerical diagonalization of the full continuum model is presented for mass, scalar, and gauge perturbations, together with a strain-extended model and a connection to hBN and strain experiments.

Significance. If established, the claim that layer-resolved perturbation sources are effectively masked near the magic angle is important for interpreting experiments on twisted bilayer graphene and related moiré systems. The paper's analytic derivation is parameter-free, and the numerical work is extensive: it includes band-structure and polarization maps, equilibrium conditions, and strain/hBN extensions. The proposed concept of moiré-driven equilibrium is appealing. However, the first-order analytic framework is used beyond its controlled regime, and for mass perturbations the analytic equilibrium condition is not quantitatively consistent with the numerical equilibrium shown by the authors themselves. The central claim is therefore currently supported mainly by numerical observation plus a heuristic analogy, rather than by the stated first-order derivation.

major comments (4)

[§First-order perturbation theory, Eq. (11); Fig. 2(a)] The mass equilibrium condition is not supported by the numerical data for the parameters used. For λ=0.8, Eq. (11) gives 3α²(1−λ²)=1, i.e. α≈0.96, while Fig. 2(a) shows equal gaps at α≈0.65. At the numerical equilibrium, the first-order ratio is m*_t/m*_b=3α²(1−λ²)≈0.456, so Eq. (11) is not even approximately satisfied. The text states that the numerical result is 'in line with Eqs. (11) and (13)'; for the mass case this is not correct. The analytic support for the central claim must be either derived to higher order or replaced by a statement that the equilibrium is observed numerically.
[§First-order perturbation theory, Eqs. (6) and (16)] The first-order projection used to derive Eq. (6) is only valid while the zero-mode subspace is isolated. At α≈0.65, the renormalized velocity v*=v(1−3α²)/(1+3α²(1+λ²)) is already negative (≈−0.087v for λ=0.8), and for gauge perturbations the shifts δk_ℓ=A*_ℓ/(ℏv*) diverge at 3α²=1, as the paper acknowledges. Thus the equilibrium conditions for mass, and the collapse condition Eq. (16), are derived in a regime where the effective 2×2 Hamiltonian is no longer a reliable low-energy description. The smallness condition |P_ℓ|≪ℏv k_θ is necessary but not sufficient.
[§Numerical results, Fig. 3] The gauge collapse is presented as a confirmation of the first-order condition, but Fig. 3(b) shows that for P_b=σ_x A_x the collapse path is not along the initial perturbation direction, contrary to the first-order prediction following Eq. (16). The collapse is found at α≈0.568, close to the magic angle, but the analytic condition Eq. (15) is derived at the point where the momentum shifts diverge. The paper itself notes the direction mismatch, yet still uses the first-order condition as the predictive framework. The numerical collapse should therefore be presented as a separate numerical observation, not as a quantitative confirmation of the first-order equilibrium condition.
[Abstract, Conclusions] The abstract and conclusions claim that 'any perturbation' eventually reaches an equilibrium near the magic angle. For scalar perturbations the first-order equilibrium (Eq. (13), α≈0.45 for λ=0.8) is indeed near the magic angle, but for mass perturbations the first-order condition is at α≈0.96, well above the first magic angle, and the numerical equilibrium at α≈0.65 occurs where v* has changed sign. The universality of the phenomenon is thus not established analytically. The manuscript should provide corrected conditions for each perturbation type or explicitly distinguish the numerical evidence from the first-order model.

minor comments (3)

[Eq. (15)] The stated condition for δk_t = −δk_b appears to involve A_t − A_b, whereas the algebra gives (A_t + A_b)(1 − 3α²) = 0. As written, the condition misses the case A_t = −A_b, where opposite shifts occur for all α. If the paper intends to restrict to single-layer perturbations, this should be stated.
[Eq. (12) and surrounding text] The equilibrium gap expression 'Δ*_eq = (1−λ²) Δ_t + Δ_b 2' is ambiguous; parentheses should be added to make clear that (1−λ²) multiplies (Δ_t + Δ_b)/2.
[References] Reference [28] contains a typo ('Physical Review LJungetters'). Also, the statement that the first magic angle corresponds to 3α² ∼ 1 should be qualified as exact only for λ = 0; for finite λ the renormalized velocity also depends on λ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the analytic renormalization is derived from the stated continuum model, and the numerical checks are independent computations.

full rationale

The paper's central claim is not obtained by fitting or by renaming. The renormalized perturbations (Eqs. (8)-(10)) and the equilibrium conditions (Eqs. (11)-(15)) follow from first-order perturbation theory around the two Dirac points using the standard Bistritzer-MacDonald continuum model (Eqs. (1)-(3) and Supplemental Eqs. (S4)-(S25)); no parameter is adjusted to force the equilibrium conditions. The numerical sections solve the full continuum model with literature parameters (ħv/a=2.1354 eV, u'=0.0975 eV, λ=0.8) and independently locate equilibria (e.g., scalar equilibrium near α≈0.475, gauge collapse near α≈0.568), so these are genuine cross-checks rather than re-statements of the analytic input. The self-citations [44,45,55] are used to support robustness and the experimental connection, but they are not the derivation of the central equilibrium equations; the paper's own numerical results carry the quantitative weight. The admitted breakdown of first-order theory at 3α²=1 for gauge shifts, and for λ=1 in Eq. (12), is a validity limitation, not a circular step. Likewise, the quantitative mismatch between the first-order mass-equilibrium locus (Eq. (11) gives α≈0.96 for λ=0.8) and the numerical crossing in Fig. 2(a) (α≈0.65) is an accuracy/validity concern, not evidence that the result was assumed. No load-bearing step in the derivation chain is equivalent by construction to its own input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted free parameters: the material parameters u′, ℏv/a, and λ=u/u′ are taken from prior literature, and α is an independent variable. The central claim rests on the first-shell continuum model, the smallness of the perturbation, and the neglect of certain spin/valley rotations. No new physical entities are postulated.

assumptions (4)

standard math The Bistritzer-MacDonald first-shell continuum model with U_j given by Eq. (3) captures the low-energy band structure up to the magic angle.
Invoked in the Model section; all subsequent perturbative formulas rely on this truncation.
domain assumption Perturbations and momentum deviations are small (|P_ℓ| ≪ ℏv kθ) so first-order perturbation theory around the two Dirac points is valid.
Main text after Eq. (4); near v*→0 the gauge momentum shifts diverge, so this assumption fails exactly in the regime used for the collapse condition.
domain assumption Neglecting the rotation of Pauli matrices and intervalley scattering preserves the conclusions.
Main text Model section; stated without proof and used to simplify the Hamiltonian.
domain assumption For hBN and strain robustness, additional periodic potentials preserve C3 symmetry and act like the unperturbed moiré potential; previous atomistic and continuum calculations are transferable.
Connection to experiments section; the robustness claim is asserted and supported by prior calculations rather than derived in this paper.

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Cite this review

Pith. "Pith review of Moir\'e-driven equilibrium of perturbations in moir\'e systems." pith.science (2026). https://pith.science/paper/ORVZNU5U

@misc{pith2026260100948,
  author       = {Pith},
  title        = {Pith review of: Moir\'e-driven equilibrium of perturbations in moir\'e systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORVZNU5U}},
  note         = {Machine review of arXiv:2601.00948}
}

read the original abstract

Perturbations in moir\'e materials, such as due to substrates or strain, are common in many experiments and can significantly modify the electronic properties of the system. Here, we show that perturbations in twisted bilayer graphene tend to be transferred between the coupled Dirac cones, eventually reaching an equilibrium near the magic angle. We connect our results to experiments and show that this equilibrium behavior remains robust even when the moir\'e potential itself is perturbed. Our findings extend the notion of the magic angle to a more general regime governed by moir\'e-driven equilibrium.

Figures

Figures reproduced from arXiv: 2601.00948 by the authors.

**Figure 2.** reveals other noteworthy features. First, the equilibrium bandgap is lower than the initial average gap, i.e., ∆⋆ eq < m, in line with Eq. (12) for λ ̸= 0. The equilibrium bandgap becomes equal to the initial gap (∆⋆ eq = m) only in the chiral limit λ = 0 [62]. By contrast, the equilibrium shift V ⋆ eq = V /2 is always half of the initial shift, regardless of λ, in line with Eq. (14). Second, beyond the first magic… view at source ↗

**Figure 3.** FIG. 3. Moir [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Straintronics and twistronics in bilayer graphene
cond-mat.mes-hall 2026-02 conditional novelty 6.0 of 10

Strain shifts the angle of flattest bands, broadens flat bands roughly linearly, and can switch their valley topology from ±1 to 0, with shear strain acting more strongly than uniaxial.

Reference graph

Works this paper leans on

72 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[1]

E. Y . Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDon- ald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moir ´e materials, Nature Reviews Ma- terials6, 201 (2021)

2021
[2]

Y . Cao, V . Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxi- ras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature556, 43 (2018)

2018
[3]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y . Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning su- perconductivity in twisted bilayer graphene, Science363, 1059 (2019)

2019
[4]

M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K. Watan- abe, T. Taniguchi, and A. Yazdani, Evidence for unconventional superconductivity in twisted bilayer graphene, Nature600, 240 (2021)

2021
[5]

Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y . Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,et al., Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature556, 80 (2018)

2018
[6]

Kerelsky, L

A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, 6 C. Dean,et al., Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature572, 95 (2019)

2019
[7]

Y . Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Led- with, Y . Cao, S. H. Lee, S. Chen, P. R. Forrester,et al., Frac- tional chern insulators in magic-angle twisted bilayer graphene, Nature600, 439 (2021)

2021
[8]

Pong and C

W.-T. Pong and C. Durkan, A review and outlook for an anomaly of scanning tunnelling microscopy (stm): superlat- tices on graphite, Journal of Physics D: Applied Physics38, R329–R355 (2005)

2005

Show all 72 references

[9]

Brihuega, P

I. Brihuega, P. Mallet, H. Gonz ´alez-Herrero, G. Trambly de Laissardi`ere, M. M. Ugeda, L. Magaud, J. M. G ´omez- Rodr´ıguez, F. Yndur´ain, and J.-Y . Veuillen, Unraveling the in- trinsic and robust nature of van hove singularities in twisted bilayer graphene by scanning tunn...

2012
[10]

E. Y . Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nature Materials19, 1265 (2020)

2020
[11]

Lopes dos Santos, N

J. Lopes dos Santos, N. Peres, and A. Castro Neto, Graphene bi- layer with a twist: Electronic structure, Physical Review Letters 99, 256802 (2007)

2007
[12]

Su ´arez Morell, J

E. Su ´arez Morell, J. Correa, P. Vargas, M. Pacheco, and Z. Bar- ticevic, Flat bands in slightly twisted bilayer graphene: Tight- binding calculations, Physical Review B82, 121407 (2010)

2010
[13]

Trambly de Laissardi `ere, D

G. Trambly de Laissardi `ere, D. Mayou, and L. Magaud, Lo- calization of dirac electrons in rotated graphene bilayers, Nano Letters10, 804 (2010)

2010
[14]

Shallcross, S

S. Shallcross, S. Sharma, E. Kandelaki, and O. Pankratov, Elec- tronic structure of turbostratic graphene, Physical Review B81, 165105 (2010)

2010
[15]

Bistritzer and A

R. Bistritzer and A. H. MacDonald, Moir ´e bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences108, 12233 (2011)

2011
[16]

San-Jose, J

P. San-Jose, J. Gonz ´alez, and F. Guinea, Non-abelian gauge potentials in graphene bilayers, Physical Review Letters108, 216802 (2012)

2012
[17]

X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,et al., Su- perconductors, orbital magnets and correlated states in magic- angle bilayer graphene, Nature574, 653 (2019)

2019
[18]

A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watanabe, T. Taniguchi, M. Kastner, and D. Goldhaber-Gordon, Emer- gent ferromagnetism near three-quarters filling in twisted bi- layer graphene, Science365, 605 (2019)

2019
[19]

Saito, J

Y . Saito, J. Ge, K. Watanabe, T. Taniguchi, and A. F. Young, In- dependent superconductors and correlated insulators in twisted bilayer graphene, Nature Physics16, 926 (2020)

2020
[20]

Serlin, C

M. Serlin, C. Tschirhart, H. Polshyn, Y . Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. Young, Intrin- sic quantized anomalous hall effect in a moir ´e heterostructure, Science367, 900 (2020)

2020
[21]

Zondiner, A

U. Zondiner, A. Rozen, D. Rodan-Legrain, Y . Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y . Oreg, F. von Oppen, A. Stern, et al., Cascade of phase transitions and dirac revivals in magic- angle graphene, Nature582, 203 (2020)

2020
[22]

Stepanov, I

P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K. Efetov, Untying the insulating and superconducting orders in magic-angle graphene, Nature583, 375 (2020)

2020
[23]

Y . Choi, H. Kim, Y . Peng, A. Thomson, C. Lewandowski, R. Polski, Y . Zhang, H. S. Arora, K. Watanabe, T. Taniguchi, et al., Correlation-driven topological phases in magic-angle twisted bilayer graphene, Nature589, 536 (2021)

2021
[24]

Y . Cao, D. Rodan-Legrain, J. M. Park, N. F. Yuan, K. Watanabe, T. Taniguchi, R. M. Fernandes, L. Fu, and P. Jarillo-Herrero, Nematicity and competing orders in superconducting magic- angle graphene, Science372, 264 (2021)

2021
[25]

Stepanov, M

P. Stepanov, M. Xie, T. Taniguchi, K. Watanabe, X. Lu, A. H. MacDonald, B. A. Bernevig, and D. K. Efetov, Competing zero-field chern insulators in superconducting twisted bilayer graphene, Physical Review Letters127, 197701 (2021)

2021
[26]

Hejazi, C

K. Hejazi, C. Liu, H. Shapourian, X. Chen, and L. Balents, Mul- tiple topological transitions in twisted bilayer graphene near the first magic angle, Physical Review B99, 035111 (2019)

2019
[27]

Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All magic angles in twisted bilayer graphene are topological, Physical Review Letters123, 036401 (2019)

2019
[28]

Song and B

Z.-D. Song and B. A. Bernevig, Magic-angle twisted bilayer graphene as a topological heavy fermion problem, Physical Re- view LJungetters129, 047601 (2022)

2022
[29]

L. A. Navarro-Labastida, P. A. Pantaleon, F. Guinea, and G. G. Naumis, Topological phase diagram of twisted bilayer graphene as a function of the twist angle, arXiv preprint arXiv:2507.05341 (2025)

2025
[30]

Tarnopolsky, A

G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Physical Review Let- ters122, 106405 (2019)

2019
[31]

Wang, F.-W

W.-C. Wang, F.-W. Chen, K.-S. Lin, J. T. Hou, H.-C. Lin, and M.-Y . Chou, Role of the fermi ring in determining the magic angles in twisted bilayer graphene, Physical Review B110, 115154 (2024)

2024
[32]

Escudero, Diagrammatic perturbation approach to moir ´e bands in twisted bilayer graphene, Physical Review B110, 045442 (2024)

F. Escudero, Diagrammatic perturbation approach to moir ´e bands in twisted bilayer graphene, Physical Review B110, 045442 (2024)

2024
[33]

L. A. Navarro-Labastida and G. G. Naumis, 3/2 magic angle quantization rule of flat bands in twisted bilayer graphene and its relationship to the quantum hall effect, Physical Review B 107, 155428 (2023)

2023
[34]

Becker, M

S. Becker, M. Embree, J. Wittsten, and M. Zworski, Mathe- matics of magic angles in a model of twisted bilayer graphene, Probability and Mathematical Physics3, 69 (2022)

2022
[35]

N. P. Kazmierczak, M. Van Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. Brown, J. Ciston, T. Taniguchi, K. Watanabe, and D. K. Bediako, Strain fields in twisted bilayer graphene, Nature Materials20, 956 (2021)

2021
[36]

T. A. de Jong, T. Benschop, X. Chen, E. E. Krasovskii, M. J. de Dood, R. M. Tromp, M. P. Allan, and S. J. Van der Molen, Imaging moir ´e deformation and dynamics in twisted bilayer graphene, Nature Communications13, 70 (2022)

2022
[37]

J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod, A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, Scanning tunnelling microscopy and spec- troscopy of ultra-flat graphene on hexagonal boron nitride, Na- ture Materials10, 282 (2011)

2011
[38]

Decker, Y

R. Decker, Y . Wang, V . W. Brar, W. Regan, H.-Z. Tsai, Q. Wu, W. Gannett, A. Zettl, and M. F. Crommie, Local electronic properties of graphene on a bn substrate via scanning tunnel- ing microscopy, Nano Letters11, 2291 (2011)

2011
[39]

Yankowitz, J

M. Yankowitz, J. Xue, D. Cormode, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, P. Jacquod, and B. J. LeRoy, Emergence of superlattice dirac points in graphene on hexagonal boron nitride, Nature Physics8, 382 (2012)

2012
[40]

Zhang, D

Y .-H. Zhang, D. Mao, and T. Senthil, Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous hall ef- fect and a lattice model, Physical Review Research1, 033126 (2019)

2019
[41]

T. Cea, P. A. Pantale´on, and F. Guinea, Band structure of twisted bilayer graphene on hexagonal boron nitride, Physical Review 7 B102, 155136 (2020)

2020
[42]

Lin and J

X. Lin and J. Ni, Symmetry breaking in the double moir´e super- lattices of relaxed twisted bilayer graphene on hexagonal boron nitride, Physical Review B102, 035441 (2020)

2020
[43]

J. Shi, J. Zhu, and A. MacDonald, Moir´e commensurability and the quantum anomalous hall effect in twisted bilayer graphene on hexagonal boron nitride, Physical Review B103, 075122 (2021)

2021
[44]

M. Long, P. A. Pantale ´on, Z. Zhan, F. Guinea, J. ´A. Silva- Guill´en, and S. Yuan, An atomistic approach for the struc- tural and electronic properties of twisted bilayer graphene- boron nitride heterostructures, npj Computational Materials8, 73 (2022)

2022
[45]

M. Long, Z. Zhan, P. A. Pantale ´on, J. ´A. Silva-Guill ´en, F. Guinea, and S. Yuan, Electronic properties of twisted bilayer graphene suspended and encapsulated with hexagonal boron ni- tride, Physical Review B107, 115140 (2023)

2023
[46]

Van Wijk, A

M. Van Wijk, A. Schuring, M. Katsnelson, and A. Fasolino, Relaxation of moir ´e patterns for slightly misaligned identical lattices: graphene on graphite, 2D Materials2, 034010 (2015)

2015
[47]

G. G. Naumis, S. Barraza-Lopez, M. Oliva-Leyva, and H. Ter- rones, Electronic and optical properties of strained graphene and other strained 2d materials: a review, Reports on Progress in Physics80, 096501 (2017)

2017
[48]

S. Carr, D. Massatt, S. B. Torrisi, P. Cazeaux, M. Luskin, and E. Kaxiras, Relaxation and domain formation in incommensu- rate two-dimensional heterostructures, Physical Review B98, 224102 (2018)

2018
[49]

Guinea and N

F. Guinea and N. R. Walet, Continuum models for twisted bi- layer graphene: Effect of lattice deformation and hopping pa- rameters, Physical Review B99, 205134 (2019)

2019
[50]

Lucignano, D

P. Lucignano, D. Alf `e, V . Cataudella, D. Ninno, and G. Can- tele, Crucial role of atomic corrugation on the flat bands and energy gaps of twisted bilayer graphene at the magic angleθ 1.08, Physical Review B99, 195419 (2019)

2019
[51]

Huder, A

L. Huder, A. Artaud, T. Le Quang, G. T. De Laissardi`ere, A. G. Jansen, G. Lapertot, C. Chapelier, and V . T. Renard, Electronic spectrum of twisted graphene layers under heterostrain, Physi- cal Review Letters120, 156405 (2018)

2018
[52]

Z. Bi, N. F. Yuan, and L. Fu, Designing flat bands by strain, Physical Review B100, 035448 (2019)

2019
[53]

Escudero, A

F. Escudero, A. Sinner, Z. Zhan, P. A. Pantale´on, and F. Guinea, Designing moir´e patterns by strain, Physical Review Research 6, 023203 (2024)

2024
[54]

J. Yu, G. Jia, Q. Li, Y . Wang, K. Xiao, Y . Ju, H. Zhang, Z. Hu, Y . Guo, B. Lian,et al., Twist angle driven electronic structure evolution of twisted bilayer graphene, arXiv preprint arXiv:2406.20040 (2024)

2024 arXiv
[55]

Carrasco, F

R. Carrasco, F. Escudero, Z. Zhan, E. C.-d. Rio, B. Vi ˜na- Bausa, Y . Maximenko, P. A. Pantaleon, F. Guinea, and I. Bri- huega, Twistraintronics in square moire superlattices of stacked graphene layers, arXiv preprint arXiv:2511.04741 (2025)

2025 arXiv
[56]

Moon and M

P. Moon and M. Koshino, Optical absorption in twisted bilayer graphene, Physical Review B87, 205404 (2013)

2013
[57]

Lopes dos Santos, N

J. Lopes dos Santos, N. Peres, and A. Castro Neto, Continuum model of the twisted graphene bilayer, Physical Review B86, 155449 (2012)

2012
[58]

Koshino, Interlayer interaction in general incommensurate atomic layers, New Journal of Physics17, 015014 (2015)

M. Koshino, Interlayer interaction in general incommensurate atomic layers, New Journal of Physics17, 015014 (2015)

2015
[59]

Koshino, N

M. Koshino, N. F. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Maximally localized wannier orbitals and the extended hubbard model for twisted bilayer graphene, Physical Review X8, 031087 (2018)

2018
[60]

Suzuura and T

H. Suzuura and T. Ando, Phonons and electron-phonon scatter- ing in carbon nanotubes, Physical Review B65, 235412 (2002)

2002
[61]

J. Jung, A. M. DaSilva, A. H. MacDonald, and S. Adam, Origin of band gaps in graphene on hexagonal boron nitride, Nature Communications6(2015)

2015
[62]

See Supplemental Materials at [URL will be inserted by pub- lisher] for details on the first-order perturbation theory, the po- larization operators, the strain-extended continuum model, and extended numerical results for different types of perturbations
[63]

G. G. Naumis, L. A. Navarro-Labastida, E. Aguilar-M ´endez, and A. Espinosa-Champo, Reduction of the twisted bilayer graphene chiral hamiltonian into a2×2matrix operator and physical origin of flat bands at magic angles, Physical Review B103, 245418 (2021)

2021
[64]

Cr´epel, P

V . Cr´epel, P. Ding, N. Verma, N. Regnault, and R. Queiroz, Topologically protected flatness in chiral moir ´e heterostruc- tures, Physical Review X15, 021056 (2025)

2025
[65]

L. A. Navarro-Labastida, A. Espinosa-Champo, E. Aguilar- Mendez, and G. G. Naumis, Why the first magic-angle is dif- ferent from others in twisted graphene bilayers: Interlayer cur- rents, kinetic and confinement energy, and wave-function local- ization, Physical Review B105, ...

2022
[66]

Mucha-Kruczy ´nski, J

M. Mucha-Kruczy ´nski, J. Wallbank, and V . Fal’Ko, Het- erostructures of bilayer graphene and h-bn: Interplay be- tween misalignment, interlayer asymmetry, and trigonal warp- ing, Physical Review B88, 205418 (2013)

2013
[67]

Kindermann, B

M. Kindermann, B. Uchoa, and D. L. Miller, Zero-energy modes and gate-tunable gap in graphene on hexagonal boron nitride, Physical Review B86, 115415 (2012)

2012
[68]

Wallbank, A

J. Wallbank, A. Patel, M. Mucha-Kruczy ´nski, A. Geim, and V . Fal’Ko, Generic miniband structure of graphene on a hexag- onal substrate, Physical Review B87, 245408 (2013)

2013
[69]

San-Jose, A

P. San-Jose, A. Guti´errez-Rubio, M. Sturla, and F. Guinea, Elec- tronic structure of spontaneously strained graphene on hexago- nal boron nitride, Physical Review B90, 115152 (2014)

2014
[70]

San-Jose, A

P. San-Jose, A. Guti ´errez-Rubio, M. Sturla, and F. Guinea, Spontaneous strains and gap in graphene on boron nitride, Physical Review B90, 075428 (2014)

2014
[71]

Mesple, A

F. Mesple, A. Missaoui, T. Cea, L. Huder, F. Guinea, G. Tram- bly de Laissardi `ere, C. Chapelier, and V . T. Renard, Heteros- train determines flat bands in magic-angle twisted graphene lay- ers, Physical Review Letters127, 126405 (2021)

2021
[72]

P. A. Pantale ´on, V . o. T. Phong, G. G. Naumis, and F. Guinea, Interaction-enhanced topological hall effects in strained twisted bilayer graphene, Physical Review B106, L161101 (2022). 1 Supplemental Materials for: Moir´e-Driven Equilibrium Federico Escudero, Zhen Zhan, Pier...

2022

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

E. Y . Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDon- ald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moir ´e materials, Nature Reviews Ma- terials6, 201 (2021)

2021

[2] [2]

Y . Cao, V . Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxi- ras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature556, 43 (2018)

2018

[3] [3]

Yankowitz, S

M. Yankowitz, S. Chen, H. Polshyn, Y . Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning su- perconductivity in twisted bilayer graphene, Science363, 1059 (2019)

2019

[4] [4]

M. Oh, K. P. Nuckolls, D. Wong, R. L. Lee, X. Liu, K. Watan- abe, T. Taniguchi, and A. Yazdani, Evidence for unconventional superconductivity in twisted bilayer graphene, Nature600, 240 (2021)

2021

[5] [5]

Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y . Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras,et al., Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature556, 80 (2018)

2018

[6] [6]

Kerelsky, L

A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian, M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi, J. Hone, 6 C. Dean,et al., Maximized electron interactions at the magic angle in twisted bilayer graphene, Nature572, 95 (2019)

2019

[7] [7]

Y . Xie, A. T. Pierce, J. M. Park, D. E. Parker, E. Khalaf, P. Led- with, Y . Cao, S. H. Lee, S. Chen, P. R. Forrester,et al., Frac- tional chern insulators in magic-angle twisted bilayer graphene, Nature600, 439 (2021)

2021

[8] [8]

Pong and C

W.-T. Pong and C. Durkan, A review and outlook for an anomaly of scanning tunnelling microscopy (stm): superlat- tices on graphite, Journal of Physics D: Applied Physics38, R329–R355 (2005)

2005