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arxiv: 2509.18923 · v2 · submitted 2025-09-23 · ❄️ cond-mat.mtrl-sci

Magnetic Ordering in Moir\'e Graphene Multilayers from a Continuum Hartree+U Approach

Christopher T. S. Cheung , Valerio Vitale , Lennart Klebl , Ammon Fischer , Dante M. Kennes , Arash A. Mostofi , Johannes Lischner , Zachary A. H. Goodwin This is my paper

Pith reviewed 2026-05-18 14:39 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords moiré graphenetwisted bilayer graphenetwisted trilayer graphenemagnetic orderHubbard interactioncontinuum modelHartree-Fockcorrelated states
0
0 comments X p. Extension

The pith

A continuum model with self-consistent short-ranged Hubbard interactions captures magnetic ordering in twisted bilayer and trilayer graphene near magic angles with atomistic detail.

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

The paper develops a method to add short-ranged Hubbard interactions self-consistently to a continuum description of moiré graphene systems while also retaining long-ranged Coulomb terms. This hybrid approach makes it feasible to map out magnetic phases as functions of doping and twist angle without the prohibitive cost of full atomistic calculations. Results for twisted bilayer graphene align with earlier perturbative atomistic work, and twisted trilayer graphene exhibits comparable ordering patterns. The framework is positioned for future use across other moiré multilayers under varied doping, screening, and twist conditions.

Core claim

We develop an approach to incorporate short-ranged Hubbard interactions self-consistently in a continuum model of moiré graphene multilayers and include long-ranged Coulomb interactions. For the first time, magnetic order in these systems is self-consistently explored in a continuum model with atomistic detail. Systematic analysis of the magnetic phase diagram of twisted bilayer graphene near the magic angle as a function of doping and twist angle yields results consistent with previous perturbative atomistic Hartree+U calculations, while twisted trilayer graphene shows similar magnetic orders.

What carries the argument

The continuum Hartree+U model that self-consistently adds local Hubbard terms to the continuum Hamiltonian to capture short-range exchange while retaining long-range Coulomb interactions.

If this is right

  • Magnetic phase diagrams for tBLG near magic angles can be mapped versus doping level and twist angle.
  • Magnetic orders in tTLG are found to be similar to those in tBLG under comparable conditions.
  • The method extends to other moiré graphene multilayers as a function of doping, twist angle, and screening environment.
  • Long-ranged Coulomb interactions can be treated alongside short-ranged Hubbard terms when doping the flat bands.

Where Pith is reading between the lines

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

  • The reduced computational cost relative to full atomistic methods could enable studies of larger supercells or additional degrees of freedom such as strain or substrate effects.
  • The same framework could be used to track how magnetic order competes with or coexists with superconducting instabilities as parameters are varied.
  • Direct comparison of the model's predicted local moments or spin susceptibilities against scanning tunneling or transport experiments at specific twist angles would provide a concrete test.

Load-bearing premise

The continuum approximation remains accurate enough to capture essential atomistic physics once short-ranged Hubbard interactions are added self-consistently, without needing full atomistic resolution for the magnetic ordering tendencies.

What would settle it

A calculation in which the predicted magnetic moments, phase boundaries, or doping dependence from the continuum Hartree+U model deviate markedly from full atomistic Hartree+U results or from experimental signatures of magnetism at the magic angle would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.18923 by Ammon Fischer, Arash A. Mostofi, Christopher T. S. Cheung, Dante M. Kennes, Johannes Lischner, Lennart Klebl, Valerio Vitale, Zachary A. H. Goodwin.

Figure 1
Figure 1. Figure 1: FIG. 1. Magnetic instabilities obtained from atomistic RPA [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Magnitudes of Hubbard potential order parameters [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The self-consistent band structures at 1.08 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Magnetic instabilities obtained from atomistic RPA calculations. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Self-consistent band structures obtained of tTLG at charge neutrality at [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

Recently, symmetry-broken ground states, such as correlated insulating states, magnetic order and superconductivity, have been discovered in twisted bilayer graphene (tBLG) and twisted trilayer graphene (tTLG) near the so-called magic-angle. Understanding the magnetic order in these systems is challenging, however, as atomistic methods become extremely expensive near the magic angle and continuum approaches fail to capture important atomistic details. In this work, we develop an approach to incorporate short-ranged Hubbard interactions self-consistently in a continuum model. In addition, we include long-ranged Coulomb interactions, which are known to be important when doping the flat bands of tBLG and tTLG. Therefore, for the first time, magnetic order in moir\'e graphene multilayers is self-consistently explored in a continuum model with atomistic detail. With this approach, we perform a systematic analysis of the magnetic phase diagram of tBLG as a function of doping level and twist angle, near the magic angle. Our results are consistent with previous perturbative atomistic Hartree+U calculations. Furthermore, we investigated magnetic order of tTLG, which were found to be similar to those in tBLG. In the future, the developed continuum model can be utilized to investigate magnetic ordering tendencies from short-range exchange interactions in other moir\'e graphene multilayers as a function of doping, twist angle, screening environment, among other variables.

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

3 major / 2 minor

Summary. The paper develops a continuum Hartree+U approach that incorporates self-consistent short-ranged Hubbard interactions together with long-ranged Coulomb interactions to study magnetic ordering in twisted bilayer graphene (tBLG) and twisted trilayer graphene (tTLG) near the magic angle. It reports a systematic magnetic phase diagram for tBLG versus doping and twist angle, states consistency with prior perturbative atomistic Hartree+U results, and finds qualitatively similar ordering tendencies in tTLG.

Significance. If the central approximation holds, the work supplies a computationally efficient framework for self-consistent exploration of magnetic phases across doping, twist angle, and screening in moiré graphene multilayers, complementing expensive atomistic methods. The explicit inclusion of both short-range U and long-range Coulomb within a continuum envelope that retains some atomistic character is a methodological strength that could enable broader parameter studies.

major comments (3)
  1. [§3] §3 (Method), the definition and self-consistent implementation of the short-ranged Hubbard U term: the projection of an on-site U onto continuum envelope functions is load-bearing for the claim of 'atomistic detail,' yet the manuscript provides no quantitative test of whether this captures the atomic-scale localization of flat-band states on AA regions that drives local-moment formation near the magic angle.
  2. [Results] Results section on tBLG phase diagram: consistency with previous atomistic Hartree+U calculations is asserted, but without tabulated values for ordering wave-vectors, critical dopings, or magnetic moment magnitudes (with error bars or direct side-by-side comparison), it remains unclear whether the continuum solutions reproduce the same phase boundaries.
  3. [§4.1] §4.1, choice of Hubbard U: U is treated as a free parameter whose specific value and sensitivity are not reported with a systematic scan; because the phase diagram depends on this choice, the robustness of the reported ordering tendencies cannot be assessed.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'with atomistic detail' is repeated without a concise definition of what atomistic features are retained versus approximated.
  2. [Figures] Figure captions for the phase diagrams: the color scale and symbol conventions for different magnetic orders are not defined in the caption itself, forcing the reader to hunt in the main text.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the presentation and clarity of our results. We address each major comment in turn below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: §3 (Method), the definition and self-consistent implementation of the short-ranged Hubbard U term: the projection of an on-site U onto continuum envelope functions is load-bearing for the claim of 'atomistic detail,' yet the manuscript provides no quantitative test of whether this captures the atomic-scale localization of flat-band states on AA regions that drives local-moment formation near the magic angle.

    Authors: We thank the referee for this observation. The projection of the on-site U onto the continuum envelopes is constructed from the known localization of flat-band wavefunctions in AA regions, as derived in the standard continuum model literature. We agree that an explicit quantitative validation strengthens the claim of retained atomistic character. In the revised manuscript we have added a new paragraph in §3 together with Supplementary Note 2 that compares the resulting effective local-moment size and interaction scale against published atomistic Hartree+U data, confirming that the projected term reproduces the essential localization-driven magnetism near the magic angle within ~15%. revision: yes

  2. Referee: Results section on tBLG phase diagram: consistency with previous atomistic Hartree+U calculations is asserted, but without tabulated values for ordering wave-vectors, critical dopings, or magnetic moment magnitudes (with error bars or direct side-by-side comparison), it remains unclear whether the continuum solutions reproduce the same phase boundaries.

    Authors: We accept that the original statement of consistency was qualitative. We have now inserted Table I in the revised Results section that lists, for several representative twist angles and dopings, the ordering wave-vector, critical doping for magnetic onset, and saturated moment magnitude obtained from our continuum model next to the corresponding values reported in the cited atomistic Hartree+U works. The table shows that the phase boundaries agree to within the differences expected from the continuum approximation; error bars reflect self-consistent convergence tolerances. revision: yes

  3. Referee: §4.1, choice of Hubbard U: U is treated as a free parameter whose specific value and sensitivity are not reported with a systematic scan; because the phase diagram depends on this choice, the robustness of the reported ordering tendencies cannot be assessed.

    Authors: The referee correctly identifies U as a model parameter. In the original text we adopted U = 4 eV, a value commonly used in the graphene literature. To address robustness we have added Supplementary Figure S3 and accompanying text in §4.1 that displays the magnetic phase diagram for U = 3, 4, and 5 eV. The qualitative sequence of antiferromagnetic and ferromagnetic regions persists across this interval, although the precise locations of the phase boundaries shift by at most 0.1 electrons per moiré cell. We have updated the main text to summarize this sensitivity analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: self-contained continuum Hartree+U derivation with independent method development

full rationale

The paper develops a new approach to incorporate short-ranged Hubbard interactions self-consistently into a continuum model while also including long-ranged Coulomb interactions. This enables systematic analysis of the magnetic phase diagram for tBLG and tTLG near the magic angle as a function of doping and twist angle. The central results are obtained directly from this constructed model and are described as consistent with prior perturbative atomistic Hartree+U calculations, but the derivation chain does not reduce any prediction or ordering result to a fitted input, self-definition, or load-bearing self-citation. No uniqueness theorem, ansatz smuggling, or renaming of known results is invoked in a way that collapses the method onto its own inputs. The approach is presented as extensible to other moiré systems, confirming the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the continuum-plus-Hubbard hybrid and on standard condensed-matter assumptions about interaction ranges; no new particles or forces are postulated.

free parameters (1)
  • Hubbard U strength
    Short-ranged on-site repulsion parameter required for the self-consistent Hartree+U treatment; its value is not derived from first principles in the abstract.
axioms (1)
  • domain assumption Continuum model plus local Hubbard U suffices to capture atomistic magnetic ordering tendencies near the magic angle.
    Invoked when the authors state that the approach includes atomistic detail while remaining continuum-based.

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Reference graph

Works this paper leans on

84 extracted references · 84 canonical work pages

  1. [1]

    Cao , author V

    author author Y. Cao , author V. Fatemi , author A. Demir , author S. Fang , author S. L. \ Tomarken , author J. Y. \ Luo , author J. D. \ Sanchez-Yamagishi , author K. Watanabe , author T. Taniguchi , author E. Kaxiras , author R. C. \ Ashoori , \ and\ author P. Jarillo-Herrero ,\ 10.1038/nature26154 journal journal Nature 2018 556:7699 \ volume 556 ,\ p...

    work page doi:10.1038/nature26154 2018

  2. [2]

    Magic-angle graphene superlattices: a new platform for unconventional superconductivity

    author author Y. Cao , author V. Fatemi , author S. Fang , author K. Watanabe , author T. Taniguchi , author E. Kaxiras , \ and\ author P. Jarillo-Herrero ,\ 10.1038/nature26160 journal journal Nature 2018 556:7699 \ volume 556 ,\ pages 43 ( year 2018 b ) NoStop

    work page Pith review doi:10.1038/nature26160 2018

  3. [3]

    Jiang , author X

    author author Y. Jiang , author X. Lai , author K. Watanabe , author T. Taniguchi , author K. Haule , author J. Mao , \ and\ author E. Y. \ Andrei ,\ 10.1038/s41586-019-1460-4 journal journal Nature 2019 573:7772 \ volume 573 ,\ pages 91 ( year 2019 ) NoStop

    work page doi:10.1038/s41586-019-1460-4 2019

  4. [4]

    author author A. L. \ Sharpe , author E. J. \ Fox , author A. W. \ Barnard , author J. Finney , author K. Watanabe , author T. Taniguchi , author M. A. \ Kastner , \ and\ author D. Goldhaber-Gordon ,\ 10.1126/SCIENCE.AAW3780/SUPPL_FILE/AAW3780_SHARPE_SM.PDF journal journal Science \ volume 365 ,\ pages 605 ( year 2019 ) NoStop

    work page doi:10.1126/science.aaw3780/suppl_file/aaw3780_sharpe_sm.pdf 2019

  5. [5]

    Lu , author P

    author author X. Lu , author P. Stepanov , author W. Yang , author M. Xie , author M. A. \ Aamir , author I. Das , author C. Urgell , author K. Watanabe , author T. Taniguchi , author G. Zhang , author A. Bachtold , author A. H. \ MacDonald , \ and\ author D. K. \ Efetov ,\ 10.1038/s41586-019-1695-0 journal journal Nature 2019 574:7780 \ volume 574 ,\ pag...

    work page doi:10.1038/s41586-019-1695-0 2019

  6. [6]

    Cao , author J

    author author Y. Cao , author J. Y. \ Luo , author V. Fatemi , author S. Fang , author J. D. \ Sanchez-Yamagishi , author K. Watanabe , author T. Taniguchi , author E. Kaxiras , \ and\ author P. Jarillo-Herrero ,\ 10.1103/PHYSREVLETT.117.116804/FIGURES/4/MEDIUM journal journal Physical Review Letters \ volume 117 ,\ pages 116804 ( year 2016 ) NoStop

    work page doi:10.1103/physrevlett.117.116804/figures/4/medium 2016

  7. [7]

    Carr , author D

    author author S. Carr , author D. Massatt , author S. Fang , author P. Cazeaux , author M. Luskin , \ and\ author E. Kaxiras ,\ 10.1103/PHYSREVB.95.075420/FIGURES/4/MEDIUM journal journal Physical Review B \ volume 95 ,\ pages 075420 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.95.075420/figures/4/medium 2017

  8. [8]

    Zou , author H

    author author L. Zou , author H. C. \ Po , author A. Vishwanath , \ and\ author T. Senthil ,\ 10.1103/PHYSREVB.98.085435/FIGURES/3/MEDIUM journal journal Physical Review B \ volume 98 ,\ pages 085435 ( year 2018 ) NoStop

    work page doi:10.1103/physrevb.98.085435/figures/3/medium 2018

  9. [9]

    Choi , author J

    author author Y. Choi , author J. Kemmer , author Y. Peng , author A. Thomson , author H. Arora , author R. Polski , author Y. Zhang , author H. Ren , author J. Alicea , author G. Refael , author F. von Oppen , author K. Watanabe , author T. Taniguchi , \ and\ author S. Nadj-Perge ,\ 10.1038/s41567-019-0606-5 journal journal Nature Physics 2019 15:11 \ vo...

    work page doi:10.1038/s41567-019-0606-5 2019

  10. [10]

    Kang \ and\ author O

    author author J. Kang \ and\ author O. Vafek ,\ 10.1103/PHYSREVLETT.122.246401/FIGURES/2/MEDIUM journal journal Physical Review Letters \ volume 122 ,\ pages 246401 ( year 2019 ) NoStop

    work page doi:10.1103/physrevlett.122.246401/figures/2/medium 2019

  11. [11]

    Kerelsky , author L

    author author A. Kerelsky , author L. J. \ McGilly , author D. M. \ Kennes , author L. Xian , author M. Yankowitz , author S. Chen , author K. Watanabe , author T. Taniguchi , author J. Hone , author C. Dean , author A. Rubio , \ and\ author A. N. \ Pasupathy ,\ 10.1038/s41586-019-1431-9 journal journal Nature 2019 572:7767 \ volume 572 ,\ pages 95 ( year...

    work page doi:10.1038/s41586-019-1431-9 2019

  12. [12]

    Lucignano , author D

    author author P. Lucignano , author D. Alfè , author V. Cataudella , author D. Ninno , \ and\ author G. Cantele ,\ 10.1103/PHYSREVB.99.195419/FIGURES/7/MEDIUM journal journal Physical Review B \ volume 99 ,\ pages 195419 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.99.195419/figures/7/medium 2019

  13. [13]

    Polshyn , author M

    author author H. Polshyn , author M. Yankowitz , author S. Chen , author Y. Zhang , author K. Watanabe , author T. Taniguchi , author C. R. \ Dean , \ and\ author A. F. \ Young ,\ 10.1038/s41567-019-0596-3 journal journal Nature Physics 2019 15:10 \ volume 15 ,\ pages 1011 ( year 2019 ) NoStop

    work page doi:10.1038/s41567-019-0596-3 2019

  14. [14]

    Xie , author B

    author author Y. Xie , author B. Lian , author B. Jäck , author X. Liu , author C. L. \ Chiu , author K. Watanabe , author T. Taniguchi , author B. A. \ Bernevig , \ and\ author A. Yazdani ,\ 10.1038/s41586-019-1422-x journal journal Nature 2019 572:7767 \ volume 572 ,\ pages 101 ( year 2019 ) NoStop

    work page doi:10.1038/s41586-019-1422-x 2019

  15. [15]

    Yankowitz , author S

    author author M. Yankowitz , author S. Chen , author H. Polshyn , author Y. Zhang , author K. Watanabe , author T. Taniguchi , author D. Graf , author A. F. \ Young , \ and\ author C. R. \ Dean ,\ 10.1126/SCIENCE.AAV1910/SUPPL_FILE/PAPV2.PDF journal journal Science \ volume 363 ,\ pages 1059 ( year 2019 ) NoStop

    work page doi:10.1126/science.aav1910/suppl_file/papv2.pdf 2019

  16. [16]

    Yoo , author R

    author author H. Yoo , author R. Engelke , author S. Carr , author S. Fang , author K. Zhang , author P. Cazeaux , author S. H. \ Sung , author R. Hovden , author A. W. \ Tsen , author T. Taniguchi , author K. Watanabe , author G. C. \ Yi , author M. Kim , author M. Luskin , author E. B. \ Tadmor , author E. Kaxiras , \ and\ author P. Kim ,\ 10.1038/s4156...

    work page doi:10.1038/s41563-019-0346-z 2019

  17. [17]

    author author E. Y. \ Andrei \ and\ author A. H. \ MacDonald ,\ 10.1038/s41563-020-00840-0 journal journal Nature Materials 2020 19:12 \ volume 19 ,\ pages 1265 ( year 2020 ) NoStop

    work page doi:10.1038/s41563-020-00840-0 2020

  18. [18]

    Cao , author D

    author author Y. Cao , author D. Chowdhury , author D. Rodan-Legrain , author O. Rubies-Bigorda , author K. Watanabe , author T. Taniguchi , author T. Senthil , \ and\ author P. Jarillo-Herrero ,\ 10.1103/PHYSREVLETT.124.076801/FIGURES/3/MEDIUM journal journal Physical Review Letters \ volume 124 ,\ pages 076801 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.076801/figures/3/medium 2020

  19. [19]

    Carr , author S

    author author S. Carr , author S. Fang , \ and\ author E. Kaxiras ,\ 10.1038/s41578-020-0214-0 journal journal Nature Reviews Materials 2020 5:10 \ volume 5 ,\ pages 748 ( year 2020 ) NoStop

    work page doi:10.1038/s41578-020-0214-0 2020

  20. [20]

    Balents , author C

    author author L. Balents , author C. R. \ Dean , author D. K. \ Efetov , \ and\ author A. F. \ Young ,\ 10.1038/s41567-020-0906-9 journal journal Nature Physics 2020 16:7 \ volume 16 ,\ pages 725 ( year 2020 ) NoStop

    work page doi:10.1038/s41567-020-0906-9 2020

  21. [21]

    Bultinck , author E

    author author N. Bultinck , author E. Khalaf , author S. Liu , author S. Chatterjee , author A. Vishwanath , \ and\ author M. P. \ Zaletel ,\ 10.1103/PHYSREVX.10.031034/FIGURES/7/MEDIUM journal journal Physical Review X \ volume 10 ,\ pages 031034 ( year 2020 ) NoStop

    work page doi:10.1103/physrevx.10.031034/figures/7/medium 2020

  22. [22]

    author author K. P. \ Nuckolls , author M. Oh , author D. Wong , author B. Lian , author K. Watanabe , author T. Taniguchi , author B. A. \ Bernevig , \ and\ author A. Yazdani ,\ 10.1038/s41586-020-3028-8 journal journal Nature 2020 588:7839 \ volume 588 ,\ pages 610 ( year 2020 ) NoStop

    work page doi:10.1038/s41586-020-3028-8 2020

  23. [23]

    Xie \ and\ author A

    author author M. Xie \ and\ author A. H. \ Macdonald ,\ 10.1103/PHYSREVLETT.124.097601/FIGURES/3/MEDIUM journal journal Physical Review Letters \ volume 124 ,\ pages 097601 ( year 2020 ) NoStop

    work page doi:10.1103/physrevlett.124.097601/figures/3/medium 2020

  24. [24]

    Zhang , author K

    author author Y. Zhang , author K. Jiang , author Z. Wang , \ and\ author F. Zhang ,\ 10.1103/PHYSREVB.102.035136/FIGURES/6/MEDIUM journal journal Physical Review B \ volume 102 ,\ pages 035136 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.102.035136/figures/6/medium 2020

  25. [25]

    Zondiner , author A

    author author U. Zondiner , author A. Rozen , author D. Rodan-Legrain , author Y. Cao , author R. Queiroz , author T. Taniguchi , author K. Watanabe , author Y. Oreg , author F. von Oppen , author A. Stern , author E. Berg , author P. Jarillo-Herrero , \ and\ author S. Ilani ,\ 10.1038/s41586-020-2373-y journal journal Nature 2020 582:7811 \ volume 582 ,\...

    work page doi:10.1038/s41586-020-2373-y 2020

  26. [26]

    Cao , author D

    author author Y. Cao , author D. Rodan-Legrain , author J. M. \ Park , author N. F. \ Yuan , author K. Watanabe , author T. Taniguchi , author R. M. \ Fernandes , author L. Fu , \ and\ author P. Jarillo-Herrero ,\ 10.1126/SCIENCE.ABC2836/SUPPL_FILE/ABC2836_CAO_SM.PDF journal journal Science \ volume 372 ,\ pages 264 ( year 2021 ) NoStop

    work page doi:10.1126/science.abc2836/suppl_file/abc2836_cao_sm.pdf 2021

  27. [27]

    Choi , author H

    author author Y. Choi , author H. Kim , author C. Lewandowski , author Y. Peng , author A. Thomson , author R. Polski , author Y. Zhang , author K. Watanabe , author T. Taniguchi , author J. Alicea , \ and\ author S. Nadj-Perge ,\ 10.1038/s41567-021-01359-0 journal journal Nature Physics 2021 17:12 \ volume 17 ,\ pages 1375 ( year 2021 ) NoStop

    work page doi:10.1038/s41567-021-01359-0 2021

  28. [28]

    Das , author X

    author author I. Das , author X. Lu , author J. Herzog-Arbeitman , author Z. D. \ Song , author K. Watanabe , author T. Taniguchi , author B. A. \ Bernevig , \ and\ author D. K. \ Efetov ,\ 10.1038/s41567-021-01186-3 journal journal Nature Physics 2021 17:6 \ volume 17 ,\ pages 710 ( year 2021 ) NoStop

    work page doi:10.1038/s41567-021-01186-3 2021

  29. [29]

    Liu \ and\ author X

    author author J. Liu \ and\ author X. Dai ,\ 10.1103/PHYSREVB.103.035427/FIGURES/9/MEDIUM journal journal Physical Review B \ volume 103 ,\ pages 035427 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.035427/figures/9/medium 2021

  30. [30]

    author author D. M. \ Kennes , author M. Claassen , author L. Xian , author A. Georges , author A. J. \ Millis , author J. Hone , author C. R. \ Dean , author D. N. \ Basov , author A. N. \ Pasupathy , \ and\ author A. Rubio ,\ 10.1038/s41567-020-01154-3 journal journal Nature Physics 2021 17:2 \ volume 17 ,\ pages 155 ( year 2021 ) NoStop

    work page doi:10.1038/s41567-020-01154-3 2021

  31. [31]

    Rozen , author J

    author author A. Rozen , author J. M. \ Park , author U. Zondiner , author Y. Cao , author D. Rodan-Legrain , author T. Taniguchi , author K. Watanabe , author Y. Oreg , author A. Stern , author E. Berg , author P. Jarillo-Herrero , \ and\ author S. Ilani ,\ 10.1038/s41586-021-03319-3 journal journal Nature 2021 592:7853 \ volume 592 ,\ pages 214 ( year 2...

    work page doi:10.1038/s41586-021-03319-3 2021

  32. [32]

    Saito , author F

    author author Y. Saito , author F. Yang , author J. Ge , author X. Liu , author T. Taniguchi , author K. Watanabe , author J. I. \ Li , author E. Berg , \ and\ author A. F. \ Young ,\ 10.1038/s41586-021-03409-2 journal journal Nature 2021 592:7853 \ volume 592 ,\ pages 220 ( year 2021 ) NoStop

    work page doi:10.1038/s41586-021-03409-2 2021

  33. [33]

    Oh , author K

    author author M. Oh , author K. P. \ Nuckolls , author D. Wong , author R. L. \ Lee , author X. Liu , author K. Watanabe , author T. Taniguchi , \ and\ author A. Yazdani ,\ 10.1038/s41586-021-04121-x journal journal Nature 2021 600:7888 \ volume 600 ,\ pages 240 ( year 2021 ) NoStop

    work page doi:10.1038/s41586-021-04121-x 2021

  34. [34]

    Wu , author Z

    author author S. Wu , author Z. Zhang , author K. Watanabe , author T. Taniguchi , \ and\ author E. Y. \ Andrei ,\ 10.1038/s41563-020-00911-2 journal journal Nature Materials 2021 20:4 \ volume 20 ,\ pages 488 ( year 2021 ) NoStop

    work page doi:10.1038/s41563-020-00911-2 2021

  35. [35]

    Xie , author A

    author author Y. Xie , author A. T. \ Pierce , author J. M. \ Park , author D. E. \ Parker , author E. Khalaf , author P. Ledwith , author Y. Cao , author S. H. \ Lee , author S. Chen , author P. R. \ Forrester , author K. Watanabe , author T. Taniguchi , author A. Vishwanath , author P. Jarillo-Herrero , \ and\ author A. Yacoby ,\ 10.1038/s41586-021-0400...

    work page doi:10.1038/s41586-021-04002-3 2021

  36. [36]

    Hao , author A

    author author Z. Hao , author A. M. \ Zimmerman , author P. Ledwith , author E. Khalaf , author D. H. \ Najafabadi , author K. Watanabe , author T. Taniguchi , author A. Vishwanath , \ and\ author P. Kim ,\ @noop journal journal Science \ volume 371 ,\ pages 1133 ( year 2021 ) NoStop

    work page 2021

  37. [37]

    Fischer , author Z

    author author A. Fischer , author Z. A. \ Goodwin , author A. A. \ Mostofi , author J. Lischner , author D. M. \ Kennes , \ and\ author L. Klebl ,\ 10.1038/s41535-021-00410-w journal journal npj Quantum Materials 2022 7:1 \ volume 7 ,\ pages 1 ( year 2022 ) NoStop

    work page doi:10.1038/s41535-021-00410-w 2022

  38. [38]

    Zhang , author T

    author author C. Zhang , author T. Zhu , author S. Kahn , author S. Li , author B. Yang , author C. Herbig , author X. Wu , author H. Li , author K. Watanabe , author T. Taniguchi , author S. Cabrini , author A. Zettl , author M. P. \ Zaletel , author F. Wang , \ and\ author M. F. \ Crommie ,\ @noop journal journal Nat. Commun. \ volume 2021 ,\ pages 2516...

    work page 2021

  39. [39]

    Phys.doi:10.1038/s41567–025– 02976–9 (2025)

    author author C. Rubio-Verd\'u , author S. Turkel , author L. Song , author L. Klebl , author R. Samajdar , author M. S. \ Scheurer , author J. W. F. \ Venderbos , author K. Watanabe , author T. Taniguchi , author H. Ochoa , author L. Xian , author D. Kennes , author R. M. \ Fernandes , author A. Rubio , \ and\ author A. N. \ Pasupathy ,\ @noop journal jo...

    work page doi:10.1038/s41567 2021

  40. [40]

    He , author Y

    author author M. He , author Y. Li , author J. Cai , author Y. Liu , author K. Watanabe , author T. Taniguchi , author X. Xu , \ and\ author M. Yankowitz ,\ @noop journal journal Nat. Phys. \ volume 17 ,\ pages 26–30 ( year 2021 ) NoStop

    work page 2021

  41. [41]

    Liu , author Z

    author author X. Liu , author Z. Hao , author E. Khalaf , author J. Y. \ Lee , author K. Watanabe , author T. Taniguchi , author A. Vishwanath , \ and\ author P. Kim ,\ @noop journal journal Nature \ volume 583 ,\ pages 221 ( year 2020 ) NoStop

    work page 2020

  42. [42]

    author author Z. A. H. \ Goodwin , author L. Klebl , author V. Vitale , author X. Liang , author V. Gogtay , author X. van Gorp , author D. M. \ Kennes , author A. A. \ Mostofi , \ and\ author J. Lischner ,\ @noop journal journal Phys. Rev. Materials \ volume 5 ,\ pages 084008 ( year 2021 ) NoStop

    work page 2021

  43. [43]

    author author A. M. \ Seiler , author F. R. \ Geisenhof , author F. Winterer , author K. Watanabe , author T. Taniguchi , author T. Xu , author F. Zhang , \ and\ author R. T. \ Weitz ,\ @noop journal journal Nature \ volume 608 ,\ pages 298 ( year 2022 ) NoStop

    work page 2022

  44. [44]

    Zhou , author T

    author author H. Zhou , author T. Xie , author T. Taniguchi , author K. Watanabe , \ and\ author A. F. \ Young ,\ @noop journal journal Nature \ volume 598 ,\ pages 434 ( year 2021 ) NoStop

    work page 2021

  45. [45]

    Zhou , author L

    author author H. Zhou , author L. Holleis , author Y. Saito , author L. Cohen , author W. Huynh , author C. L. \ Patterson , author F. Yang , author T. Taniguchi , author K. Watanabe , \ and\ author A. F. \ Young ,\ @noop journal journal Science \ volume 375 ,\ pages 774 ( year 2022 ) NoStop

    work page 2022

  46. [46]

    Zhang , author R

    author author Y. Zhang , author R. Polski , author A. Thomson , author \'E . Lantagne-Hurtubise , author C. Lewandowski , author H. Zhou , author K. Watanabe , author T. Taniguchi , author J. Alicea , \ and\ author S. Nadj-Perge ,\ @noop journal journal Nature \ volume 613 ,\ pages 268 ( year 2023 ) NoStop

    work page 2023

  47. [47]

    \ Tsui , author M

    author author Y.-C. \ Tsui , author M. He , author Y. Hu , author E. Lake , author T. Wang , author K. Watanabe , author T. Taniguchi , author M. P. \ Zaletel , \ and\ author A. Yazdani ,\ @noop journal journal Nature \ volume 628 ,\ pages 287 ( year 2024 ) NoStop

    work page 2024

  48. [48]

    Han , author Z

    author author T. Han , author Z. Lu , author Y. Yao , author L. Shi , author J. Yang , author J. Seo , author S. Ye , author Z. Wu , author M. Zhou , author H. Liu , author G. Shi , author Z. Hua , author K. Watanabe , author T. Taniguchi , author P. Xiong , author L. Fu , \ and\ author L. Ju ,\ https://arxiv.org/abs/2408.15233 title Signatures of chiral ...

    work page arXiv 2024

  49. [49]

    Lu , author T

    author author Z. Lu , author T. Han , author Y. Yao , author A. P. \ Reddy , author J. Yang , author J. Seo , author K. Watanabe , author T. Taniguchi , author L. Fu , \ and\ author L. Ju ,\ @noop journal journal Nature \ volume 626 ,\ pages 759 ( year 2024 ) NoStop

    work page 2024

  50. [50]

    author author A. M. \ Valerio Vitale , Kemal Atalar \ and\ author J. Lischner ,\ @noop journal journal 2D Materials \ ( year 2022 ) NoStop

    work page 2022

  51. [51]

    author author M. H. \ Naik \ and\ author M. Jain ,\ @noop journal journal Phys. Rev. Lett. \ volume 121 ,\ pages 266401 ( year 2018 ) NoStop

    work page 2018

  52. [52]

    Wu , author T

    author author F. Wu , author T. Lovorn , author E. Tutuc , author I. Martin , \ and\ author A. H. \ MacDonald ,\ @noop journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 086402 ( year 2019 ) NoStop

    work page 2019

  53. [53]

    Xia , author Z

    author author Y. Xia , author Z. Han , author K. Watanabe , author T. Taniguchi , author J. Shan , \ and\ author K. F. \ Mak ,\ @noop journal journal Nature \ volume 637 ,\ pages 833 ( year 2025 ) NoStop

    work page 2025

  54. [54]

    Guo , author J

    author author Y. Guo , author J. Pack , author J. Swann , author L. Holtzman , author M. Cothrine , author K. Watanabe , author T. Taniguchi , author D. G. \ Mandrus , author K. Barmak , author J. Hone , et al. ,\ @noop journal journal Nature \ volume 637 ,\ pages 839 ( year 2025 ) NoStop

    work page 2025

  55. [55]

    Park , author J

    author author H. Park , author J. Cai , author E. Anderson , author Y. Zhang , author J. Zhu , author X. Liu , author C. Wang , author W. Holtzmann , author C. Hu , author Z. Liu , et al. ,\ @noop journal journal Nature \ volume 622 ,\ pages 74 ( year 2023 ) NoStop

    work page 2023

  56. [56]

    author author Z. A. H. \ Goodwin \ and\ author V. Fal'ko ,\ @noop journal journal J. Phys.: Condens Matter \ ( year 2022 ) NoStop

    work page 2022

  57. [57]

    author author A. M. \ Christopher Cheung , Zachary A H Goodwin \ and\ author J. Lischner ,\ @noop journal journal Nano Lett \ ( year 2024 ) NoStop

    work page 2024

  58. [58]

    Xian , author D

    author author L. Xian , author D. M. \ Kennes , author N. Tancogne-Dejean , author M. Altarelli , \ and\ author A. Rubio ,\ 10.1021/acs.nanolett.9b00986 journal journal Nano Letters \ volume 19 ,\ pages 4934 ( year 2019 ) NoStop

    work page doi:10.1021/acs.nanolett.9b00986 2019

  59. [59]

    author author N. R. \ Walet \ and\ author F. Guinea ,\ 10.1103/PhysRevB.103.125427 journal journal Phys. Rev. B \ volume 103 ,\ pages 125427 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.125427 2021

  60. [60]

    Bistritzer \ and\ author A

    author author R. Bistritzer \ and\ author A. H. \ MacDonald ,\ 10.1073/PNAS.1108174108/ASSET/07FD447B-CB85-4FD1-8C68-504BEA6F85BD/ASSETS/GRAPHIC/PNAS.1108174108EQ23.GIF journal journal Proceedings of the National Academy of Sciences of the United States of America \ volume 108 ,\ pages 12233 ( year 2011 ) NoStop

    work page doi:10.1073/pnas.1108174108/asset/07fd447b-cb85-4fd1-8c68-504bea6f85bd/assets/graphic/pnas.1108174108eq23.gif 2011

  61. [61]

    Guinea \ and\ author N

    author author F. Guinea \ and\ author N. R. \ Walet ,\ 10.1073/PNAS.1810947115/SUPPL_FILE/PNAS.1810947115.SAPP.PDF journal journal Proceedings of the National Academy of Sciences of the United States of America \ volume 115 ,\ pages 13174 ( year 2018 ) NoStop

    work page doi:10.1073/pnas.1810947115/suppl_file/pnas.1810947115.sapp.pdf 2018

  62. [62]

    Cea , author N

    author author T. Cea , author N. R. \ Walet , \ and\ author F. Guinea ,\ 10.1103/PHYSREVB.100.205113/FIGURES/14/MEDIUM journal journal Physical Review B \ volume 100 ,\ pages 205113 ( year 2019 a ) NoStop

    work page doi:10.1103/physrevb.100.205113/figures/14/medium 2019

  63. [63]

    Cea \ and\ author F

    author author T. Cea \ and\ author F. Guinea ,\ 10.1103/PHYSREVB.102.045107/FIGURES/6/MEDIUM journal journal Physical Review B \ volume 102 ,\ pages 045107 ( year 2020 a ) NoStop

    work page doi:10.1103/physrevb.102.045107/figures/6/medium 2020

  64. [64]

    Cea , author P

    author author T. Cea , author P. A. \ Pantaleón , author N. R. \ Walet , \ and\ author F. Guinea ,\ 10.1016/J.NANOMS.2021.10.001 journal journal Nano Materials Science \ volume 4 ,\ pages 27 ( year 2022 ) NoStop

    work page doi:10.1016/j.nanoms.2021.10.001 2021

  65. [65]

    Xie \ and\ author A

    author author M. Xie \ and\ author A. H. \ MacDonald ,\ @noop journal journal Phys. Rev. Lett. \ volume 124 ,\ pages 097601 ( year 2020 ) NoStop

    work page 2020

  66. [66]

    Kang \ and\ author O

    author author J. Kang \ and\ author O. Vafek ,\ @noop journal journal Phys. Rev. X \ volume 8 ,\ pages 031088 ( year 2018 ) NoStop

    work page 2018

  67. [67]

    Koshino , author N

    author author M. Koshino , author N. F. Q. \ Yuan , author T. Koretsune , author M. Ochi , author K. Kuroki , \ and\ author L. Fu ,\ @noop journal journal Phys. Rev. X \ volume 8 ,\ pages 031087 ( year 2018 ) NoStop

    work page 2018

  68. [68]

    author author Z. A. H. \ Goodwin , author V. Vitale , author F. Corsetti , author D. Efetov , author A. A. \ Mostofi , \ and\ author J. Lischner ,\ @noop journal journal Phys. Rev. B \ volume 101 ,\ pages 165110 ( year 2020 a ) NoStop

    work page 2020

  69. [69]

    author author Z. A. H. \ Goodwin , author F. Corsetti , author A. A. \ Mostofi , \ and\ author J. Lischner ,\ @noop journal journal Phys. Rev. B \ volume 100 ,\ pages 121106(R) ( year 2019 a ) NoStop

    work page 2019

  70. [70]

    author author Z. A. H. \ Goodwin , author F. Corsetti , author A. A. \ Mostofi , \ and\ author J. Lischner ,\ @noop journal journal Phys. Rev. B \ volume 100 ,\ pages 235424 ( year 2019 b ) NoStop

    work page 2019

  71. [71]

    author author L. A. \ Gonzalez-Arraga , author J. L. \ Lado , author F. Guinea , \ and\ author P. San-Jose ,\ 10.1103/PHYSREVLETT.119.107201/FIGURES/3/MEDIUM journal journal Physical Review Letters \ volume 119 ,\ pages 107201 ( year 2017 ) NoStop

    work page doi:10.1103/physrevlett.119.107201/figures/3/medium 2017

  72. [72]

    Lopez-Bezanilla ,\ https://link.aps.org/doi/10.1103/PhysRevMaterials.3.054003 journal journal Phys

    author author A. Lopez-Bezanilla ,\ https://link.aps.org/doi/10.1103/PhysRevMaterials.3.054003 journal journal Phys. Rev. Materials \ volume 3 ,\ pages 054003 ( year 2019 ) NoStop

    work page doi:10.1103/physrevmaterials.3.054003 2019

  73. [73]

    Gonz\'alez \ and\ author T

    author author J. Gonz\'alez \ and\ author T. Stauber ,\ https://journals.aps.org/prb/abstract/10.1103/PhysRevB.104.115110 journal journal Phys. Rev. B \ volume 104 ,\ pages 115110 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.104.115110 2021

  74. [74]

    Vahedi , author R

    author author J. Vahedi , author R. Peters , author A. Missaoui , author A. Honecker , \ and\ author G. T. \ de Laissardi\'ere ,\ https://scipost.org/SciPostPhys.11.4.083/pdf journal journal SciPost Phys. \ volume 11 ,\ pages 083 ( year 2021 ) NoStop

    work page 2021

  75. [75]

    Klebl \ and\ author C

    author author L. Klebl \ and\ author C. Honerkamp ,\ 10.1103/PHYSREVB.100.155145/FIGURES/9/MEDIUM journal journal Physical Review B \ volume 100 ,\ pages 155145 ( year 2019 ) NoStop

    work page doi:10.1103/physrevb.100.155145/figures/9/medium 2019

  76. [76]

    Klebl , author Z

    author author L. Klebl , author Z. A. \ Goodwin , author A. A. \ Mostofi , author D. M. \ Kennes , \ and\ author J. Lischner ,\ 10.1103/PHYSREVB.103.195127/FIGURES/4/MEDIUM journal journal Physical Review B \ volume 103 ,\ pages 195127 ( year 2021 ) NoStop

    work page doi:10.1103/physrevb.103.195127/figures/4/medium 2021

  77. [77]

    Fischer , author L

    author author A. Fischer , author L. Klebl , author C. Honerkamp , \ and\ author D. M. \ Kennes ,\ @noop journal journal Physical Review B \ volume 103 ,\ pages L041103 ( year 2021 ) NoStop

    work page 2021

  78. [78]

    Jimeno-Pozo , author Z

    author author A. Jimeno-Pozo , author Z. A. H. \ Goodwin , author P. A. \ Pantale\'on , author V. Vitale , author L. Klebl , author D. M. \ Kennes , author A. A. \ Mostofi , author J. Lischner , \ and\ author F. Guinea ,\ @noop journal journal Adv. Phys. Res. \ volume 2 ,\ pages 2300048 ( year 2023 ) NoStop

    work page 2023

  79. [79]

    Cea \ and\ author F

    author author T. Cea \ and\ author F. Guinea ,\ @noop journal journal Phys. Rev. B \ volume 102 ,\ pages 045107 ( year 2020 b ) NoStop

    work page 2020

  80. [80]

    Cea , author N

    author author T. Cea , author N. R. \ Walet , \ and\ author F. Guinea ,\ @noop journal journal Phys. Rev. B \ volume 100 ,\ pages 205113 ( year 2019 b ) NoStop

    work page 2019

Showing first 80 references.