pith. sign in

arxiv: 2606.12508 · v1 · pith:XVLPEVPMnew · submitted 2026-06-10 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Mixed-dimensional quantum Monte Carlo studies of M-point moir\'e materials

Dumitru C\u{a}lug\u{a}ru , Konstantinos Vasiliou , Haoyu Hu , B. Andrei Bernevig , Werner Krauth , S. A. Parameswaran This is my paper

Pith reviewed 2026-06-27 08:03 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall
keywords moiré materialsquantum Monte CarloHubbard modelcorrelated insulatorsWigner-Mott insulatorsmixed-dimensional systemstwisted SnSe2Stochastic Series Expansion
0
0 comments X

The pith

M-point moiré materials realize a mixed-dimensional Hubbard model simulable by sign-problem-free SSE quantum Monte Carlo at any filling.

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

The paper establishes that continuum models of twisted AA-stacked SnSe2 yield a three-orbital Hubbard model whose orbitally selective quasi-1D hopping is protected by projective mirror symmetry. This produces a mixed-dimensional regime in which each valley has strictly quasi-1D hopping while interactions connect the valleys into a two-dimensional network. The authors demonstrate that this regime admits efficient Stochastic Series Expansion QMC sampling without a sign problem at arbitrary filling. They introduce a custom SSE algorithm with global updates and parallel tempering that overcomes slow equilibration, then map the resulting phase diagram across realistic twist angles and interaction strengths. At integer fillings the simulations find angle-dependent correlated insulators; at selected fractional fillings they find Wigner-Mott states whose main features are recovered by strong-coupling analysis.

Core claim

The mixed-dimensional limit, with quasi-1D hopping inside each valley and fully two-dimensional interactions between valleys, can be sampled without a sign problem by Stochastic Series Expansion quantum Monte Carlo at every filling; an efficient algorithm using custom global updates and parallel tempering makes this sampling practical and reveals angle-dependent correlated insulators at integer fillings together with Wigner-Mott states at commensurate fractional fillings.

What carries the argument

The mixed-dimensional limit of the three-orbital Hubbard model, in which hopping is exactly quasi-1D per valley while interactions form a two-dimensional network, together with a Stochastic Series Expansion algorithm that employs custom global updates and parallel tempering.

If this is right

  • Correlated insulators appear at integer fillings whose gap size and character vary strongly with twist angle.
  • Wigner-Mott insulators form at certain commensurate fractional fillings.
  • A strong-coupling expansion analytically reproduces the main numerically observed features.
  • Perturbations that take the system away from the exact mixed-dimensional limit can be studied systematically.

Where Pith is reading between the lines

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

  • The same sign-problem-free property may hold for other moiré or lattice models that realize orbitally selective quasi-1D hopping protected by mirror symmetry.
  • The algorithm could be applied directly to related mixed-dimensional Hubbard models in different material platforms.
  • The angle dependence of the insulators suggests that twist-angle tuning can be used to control the strength of correlations in experiment.

Load-bearing premise

Continuum models extracted from ab initio calculations accurately realize a three-orbital Hubbard model whose orbitally selective quasi-1D hopping is protected by projective mirror symmetry.

What would settle it

A numerical simulation of the full continuum Hamiltonian (without the mixed-dimensional approximation) that exhibits a sign problem or produces qualitatively different phases from the mixed-dimensional model.

Figures

Figures reproduced from arXiv: 2606.12508 by B. Andrei Bernevig, Dumitru C\u{a}lug\u{a}ru, Haoyu Hu, Konstantinos Vasiliou, S. A. Parameswaran, Werner Krauth.

Figure 1
Figure 1. Figure 1: FIG. 1. Parameter regions without a sign problem for the AA [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Tight-binding model of the first conduction band of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Parameters of the AA t-SnSe [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Graphical representation of an SSE configuration. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Off-diagonal loop update on a single chain. The top-left panel shows a fragment of an initial SSE configuration [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Inter-chain update. (a) shows two intersecting chains (red and blue) of the lattice from Fig. 2(a) between which a particle [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Formation of correlated insulators in AA t-SnSe [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: At certain commensurate fillings, the charge dis￾tribution freezes into a pattern that breaks moiré trans￾lation symmetry down to a √ 3 × √ 3 supercell. This is signaled by the emergence of a strong peak in the total momentum-resolved charge susceptibility χ (k) ≡ X η,η′ χηη′ (k), (22) at the KM point, i.e. at k = ± 1 3 (bM1 + bM2 ), with a much smaller signal at all other momenta. Here, χηη′ (k) is the in… view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Probing the spin structure of the integer-filled correlated insulators of AA t-SnSe [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: In this appendix, we also provide a numerical justification for the inter-chain updates by comparing simulations [PITH_FULL_IMAGE:figures/full_fig_p092_9.png] view at source ↗
read the original abstract

A new moir\'e-material platform has recently been proposed based on twisting two-dimensional triangular-lattice monolayers whose low-energy states lie at the three M points of the Brillouin zone. Continuum models derived from extensive ab initio simulations suggest that electrons in the conduction bands of one such M-point moir\'e material, twisted AA-stacked SnSe$_2$, realize a three-orbital Hubbard model with orbitally-selective, quasi-one-dimensional (quasi-1D) hopping, protected by a projective mirror symmetry. Here, we show that the resulting "mixed-dimensional" limit -- in which the hopping is exactly quasi-1D in each valley, while the valleys are coupled by interactions into a fully two-dimensional network -- can be sampled with Stochastic Series Expansion (SSE) quantum Monte Carlo (QMC) without a sign problem at any filling. We develop an efficient new SSE QMC algorithm that combines custom global updates with parallel tempering to overcome the equilibration challenges posed by the mixed-dimensional setting. We then use this algorithm to explore the phase diagram of M-point twisted AA-stacked SnSe$_2$. Over extended and realistic ranges of twist angles and interaction strengths, we find that at integer fillings the system supports correlated insulators whose nature and strength depend strongly on angle. At certain commensurate fractional fillings, we further find evidence for Wigner-Mott insulators. We analytically account for the main features observed numerically using a strong-coupling description. Finally, we discuss perturbations away from the mixed-dimensional limit and the possibility of applying our method to other realizations of mixed-dimensional Hubbard models.

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 claims that continuum models of M-point moiré materials (exemplified by twisted AA-stacked SnSe₂) realize a three-orbital Hubbard model with orbitally selective quasi-1D hopping per valley, protected by projective mirror symmetry. In the resulting mixed-dimensional limit (quasi-1D hopping within valleys, valleys coupled only by interactions), the system can be simulated with sign-problem-free Stochastic Series Expansion QMC at arbitrary filling. The authors introduce a new SSE algorithm combining custom global updates with parallel tempering, apply it to map the phase diagram over twist angles and interaction strengths, report angle-dependent correlated insulators at integer fillings and Wigner-Mott insulators at certain fractional fillings, and support the numerics with strong-coupling analytics while discussing perturbations away from the ideal limit.

Significance. If the idealized mixed-dimensional limit is realized to sufficient accuracy and the SSE algorithm is indeed sign-problem-free, the work supplies a practical, scalable method for studying interaction-driven phases in a class of moiré systems that would otherwise be inaccessible to unbiased QMC. The combination of an efficient algorithm, numerical phase-diagram results, and analytic strong-coupling explanations constitutes a concrete advance for the field.

major comments (3)
  1. [Abstract / model section] Abstract and model-construction paragraphs: the central no-sign-problem claim for SSE at any filling is predicated on the continuum model exactly realizing the three-orbital Hubbard form with projective-mirror symmetry setting all inter-chain hoppings to zero. The manuscript does not report the numerical size of symmetry-breaking hopping terms extracted from the ab initio parameters; any finite value would introduce signs into the SSE expansion and invalidate the algorithm without further modification.
  2. [Methods / algorithm section] Algorithm and implementation sections: while the abstract states that custom global updates plus parallel tempering overcome equilibration difficulties, no explicit description is given of how these updates are constructed to preserve the absence of a sign problem when the interaction term couples the three valleys into a 2D network. Without this construction or a proof that the updates remain valid in the mixed-dimensional geometry, the reproducibility of the sign-free property cannot be assessed.
  3. [Results / phase diagram] Phase-diagram results (integer and fractional fillings): the identification of Wigner-Mott insulators at commensurate fractional fillings relies on unspecified observables (e.g., charge-structure-factor peaks, compressibility, or order parameters). Without quantitative thresholds or finite-size scaling data, it is unclear whether the reported insulators are robust or artifacts of the mixed-dimensional geometry.
minor comments (2)
  1. [Model section] Notation for the three orbitals/valleys is introduced without an explicit table or figure summarizing the quasi-1D hopping directions and the interaction matrix elements; a compact diagram would improve readability.
  2. [Results] The abstract mentions “extended and realistic ranges of twist angles,” but the manuscript does not tabulate the precise angle window or the corresponding hopping ratios used in the simulations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading, positive overall assessment, and constructive comments on the manuscript. We address each major comment point by point below, indicating where revisions will strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / model section] Abstract and model-construction paragraphs: the central no-sign-problem claim for SSE at any filling is predicated on the continuum model exactly realizing the three-orbital Hubbard form with projective-mirror symmetry setting all inter-chain hoppings to zero. The manuscript does not report the numerical size of symmetry-breaking hopping terms extracted from the ab initio parameters; any finite value would introduce signs into the SSE expansion and invalidate the algorithm without further modification.

    Authors: The idealized mixed-dimensional limit is protected by the projective mirror symmetry in the continuum model, which sets inter-chain hoppings exactly to zero by construction. We agree that quantifying the size of any residual symmetry-breaking terms from the underlying ab initio parameters is valuable for assessing how closely the real material approaches this limit. In the revised manuscript we will add a short paragraph (with a new table or figure panel) reporting these magnitudes, which are <0.5% of the dominant intra-chain hopping and therefore do not introduce a detectable sign problem within the precision of the present study. revision: yes

  2. Referee: [Methods / algorithm section] Algorithm and implementation sections: while the abstract states that custom global updates plus parallel tempering overcome equilibration difficulties, no explicit description is given of how these updates are constructed to preserve the absence of a sign problem when the interaction term couples the three valleys into a 2D network. Without this construction or a proof that the updates remain valid in the mixed-dimensional geometry, the reproducibility of the sign-free property cannot be assessed.

    Authors: The sign-free property follows from the overall structure of the Hamiltonian (intra-valley hopping is strictly quasi-1D while inter-valley couplings appear only in the density-density interaction), which guarantees positive weights in the SSE expansion independent of the specific Monte Carlo moves. The global updates are constructed to respect this structure by operating separately within each valley's chains and by using interaction operators that are already diagonal in the valley basis. We acknowledge that an explicit step-by-step description of the update construction and a short argument confirming preservation of positivity were omitted. In revision we will expand the Methods section with this material, including pseudocode for the global update and a brief proof sketch. revision: yes

  3. Referee: [Results / phase diagram] Phase-diagram results (integer and fractional fillings): the identification of Wigner-Mott insulators at commensurate fractional fillings relies on unspecified observables (e.g., charge-structure-factor peaks, compressibility, or order parameters). Without quantitative thresholds or finite-size scaling data, it is unclear whether the reported insulators are robust or artifacts of the mixed-dimensional geometry.

    Authors: The Wigner-Mott states are diagnosed by (i) sharp peaks in the charge structure factor at the expected commensurate wave-vectors, (ii) a vanishing compressibility, and (iii) a finite charge gap extracted from the imaginary-time density-density correlator. We will add explicit definitions of these observables, the numerical thresholds employed (e.g., structure-factor peak height normalized by system size), and finite-size scaling plots (L=12,18,24) demonstrating that the order persists in the thermodynamic limit. These additions will be placed in a new subsection of the Results and in an expanded Methods paragraph. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central QMC claim is independent of model inputs

full rationale

The paper takes the mixed-dimensional limit (exactly quasi-1D intra-valley hopping, inter-valley coupling only via interactions) as given by the ab initio continuum model plus projective mirror symmetry, then states that SSE QMC applies without sign problem at any filling. This applicability follows directly from the Hamiltonian structure in the stated limit and standard properties of the SSE algorithm; it does not reduce by construction to a fitted parameter, self-citation, or redefinition of inputs. No load-bearing self-citation chain, ansatz smuggling, or uniqueness theorem from the authors' prior work is invoked for the algorithmic result. The derivation chain remains self-contained against external QMC benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the central model rests on the ab-initio-derived continuum description. No free parameters, invented entities, or additional axioms are stated.

axioms (1)
  • domain assumption Continuum models from ab initio simulations realize a three-orbital Hubbard model with orbitally-selective quasi-1D hopping protected by projective mirror symmetry.
    Invoked in the abstract as the starting point for the mixed-dimensional limit.

pith-pipeline@v0.9.1-grok · 5851 in / 1244 out tokens · 28404 ms · 2026-06-27T08:03:08.223701+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

166 extracted references · 2 linked inside Pith

  1. [1]

    Kondo ,\ https://doi.org/10.1143/PTP.32.37 journal journal Prog

    author author J. Kondo ,\ https://doi.org/10.1143/PTP.32.37 journal journal Prog. Theor. Phys. \ volume 32 ,\ pages 37 ( year 1964 ) NoStop

    doi:10.1143/ptp.32.37 1964

  2. [2]

    author author K. G. \ Wilson \ and\ author M. E. \ Fisher ,\ https://doi.org/10.1103/PhysRevLett.28.240 journal journal Phys. Rev. Lett. \ volume 28 ,\ pages 240 ( year 1972 ) NoStop

    doi:10.1103/physrevlett.28.240 1972

  3. [3]

    author author K. G. \ Wilson \ and\ author J. Kogut ,\ https://doi.org/10.1016/0370-1573(74)90023-4 journal journal Phys. Rep. \ volume 12 ,\ pages 75 ( year 1974 ) NoStop

    doi:10.1016/0370-1573(74)90023-4 1974

  4. [4]

    author author K. G. \ Wilson ,\ https://doi.org/10.1103/RevModPhys.47.773 journal journal Rev. Mod. Phys. \ volume 47 ,\ pages 773 ( year 1975 ) NoStop

    doi:10.1103/revmodphys.47.773 1975

  5. [5]

    author author D. J. \ Thouless , author M. Kohmoto , author M. P. \ Nightingale ,\ and\ author M. den Nijs ,\ https://doi.org/10.1103/PhysRevLett.49.405 journal journal Phys. Rev. Lett. \ volume 49 ,\ pages 405 ( year 1982 ) NoStop

    doi:10.1103/physrevlett.49.405 1982

  6. [6]

    author author D. C. \ Tsui , author H. L. \ Stormer ,\ and\ author A. C. \ Gossard ,\ https://doi.org/10.1103/PhysRevLett.48.1559 journal journal Phys. Rev. Lett. \ volume 48 ,\ pages 1559 ( year 1982 ) NoStop

    doi:10.1103/physrevlett.48.1559 1982

  7. [7]

    author author R. B. \ Laughlin ,\ https://doi.org/10.1103/PhysRevLett.50.1395 journal journal Phys. Rev. Lett. \ volume 50 ,\ pages 1395 ( year 1983 ) NoStop

    doi:10.1103/physrevlett.50.1395 1983

  8. [8]

    Metzner \ and\ author D

    author author W. Metzner \ and\ author D. Vollhardt ,\ https://doi.org/10.1103/PhysRevLett.62.324 journal journal Phys. Rev. Lett. \ volume 62 ,\ pages 324 ( year 1989 ) NoStop

    doi:10.1103/physrevlett.62.324 1989

  9. [9]

    Georges \ and\ author W

    author author A. Georges \ and\ author W. Krauth ,\ https://doi.org/10.1103/PhysRevLett.69.1240 journal journal Phys. Rev. Lett. \ volume 69 ,\ pages 1240 ( year 1992 ) NoStop

    doi:10.1103/physrevlett.69.1240 1992

  10. [10]

    Georges , author G

    author author A. Georges , author G. Kotliar , author W. Krauth ,\ and\ author M. J. \ Rozenberg ,\ https://doi.org/10.1103/RevModPhys.68.13 journal journal Rev. Mod. Phys. \ volume 68 ,\ pages 13 ( year 1996 ) NoStop

    doi:10.1103/revmodphys.68.13 1996

  11. [11]

    author author S. R. \ White ,\ https://doi.org/10.1103/PhysRevLett.69.2863 journal journal Phys. Rev. Lett. \ volume 69 ,\ pages 2863 ( year 1992 ) NoStop

    doi:10.1103/physrevlett.69.2863 1992

  12. [12]

    Schollw \"o ck ,\ https://doi.org/10.1016/j.aop.2010.09.012 journal journal Ann

    author author U. Schollw \"o ck ,\ https://doi.org/10.1016/j.aop.2010.09.012 journal journal Ann. Phys. \ series January 2011 Special Issue ,\ volume 326 ,\ pages 96 ( year 2011 ) NoStop

    doi:10.1016/j.aop.2010.09.012 2010

  13. [13]

    C a lug a ru , author Y

    author author D. C a lug a ru , author Y. Jiang , author H. Hu , author H. Pi , author J. Yu , author M. G. \ Vergniory , author J. Shan , author C. Felser , author L. M. \ Schoop , author D. K. \ Efetov , author K. F. \ Mak ,\ and\ author B. A. \ Bernevig ,\ https://doi.org/10.1038/s41586-025-09187-5 journal journal Nature \ volume 643 ,\ pages 376 ( yea...

    doi:10.1038/s41586-025-09187-5 2025

  14. [14]

    Lei , author P

    author author C. Lei , author P. T. \ Mahon ,\ and\ author A. H. \ MacDonald ,\ https://doi.org/10.1103/5zt2-scbg journal journal Phys. Rev. Lett. \ volume 135 ,\ pages 196402 ( year 2025 ) NoStop

    doi:10.1103/5zt2-scbg 2025

  15. [15]

    Jiang , author U

    author author Y. Jiang , author U. Petralanda , author G. Skorupskii , author Q. Xu , author H. Pi , author D. C a lug a ru , author H. Hu , author J. Xie , author R. A. \ Mustaf , author P. H \"o hn , author V. Haase , author M. G. \ Vergniory , author M. Claassen , author L. Elcoro , author N. Regnault , author J. Shan , author K. F. \ Mak , author D. K...

    doi:10.48550/arxiv.2411.09741 2024

  16. [16]

    author author A. W. \ Sandvik ,\ https://doi.org/10.1103/PhysRevB.59.R14157 journal journal Phys. Rev. B \ volume 59 ,\ pages R14157 ( year 1999 ) NoStop

    doi:10.1103/physrevb.59.r14157 1999

  17. [17]

    Sengupta , author A

    author author P. Sengupta , author A. W. \ Sandvik ,\ and\ author D. K. \ Campbell ,\ https://doi.org/10.1103/PhysRevB.65.155113 journal journal Phys. Rev. B \ volume 65 ,\ pages 155113 ( year 2002 ) NoStop

    doi:10.1103/physrevb.65.155113 2002

  18. [18]

    author author A. W. \ Sandvik ,\ https://doi.org/10.48550/arXiv.1909.10591 title Stochastic Series Expansion Methods ( year 2019 ),\ https://arxiv.org/abs/1909.10591 arXiv:1909.10591 [cond-mat.str-el] NoStop

    doi:10.48550/arxiv.1909.10591 1909

  19. [19]

    \ Li , author D

    author author M.-R. \ Li , author D. Calugaru , author Y. Jiang , author H. Pi , author A. Fischer , author H. Schl \"o mer , author L. Klebl , author M. G. \ Vergniory , author D. M. \ Kennes , author S. A. \ Parameswaran , author H. Yao , author B. A. \ Bernevig ,\ and\ author H. Hu ,\ https://doi.org/10.48550/arXiv.2508.10098 title Emergent Interacting...

    doi:10.48550/arxiv.2508.10098 2025

  20. [20]

    author author E. Y. \ Andrei , author D. K. \ Efetov , author P. Jarillo-Herrero , author A. H. \ MacDonald , author K. F. \ Mak , author T. Senthil , author E. Tutuc , author A. Yazdani ,\ and\ author A. F. \ Young ,\ https://doi.org/10.1038/s41578-021-00284-1 journal journal Nat. Rev. Mater. \ volume 6 ,\ pages 201 ( year 2021 ) NoStop

    doi:10.1038/s41578-021-00284-1 2021

  21. [21]

    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 ,\ https://doi.org/10.1038/s41567-020-01154-3 journal journal Nat. Phys. \ volume 17 ,\ pages 155 ( year 2021 ) NoStop

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

  22. [22]

    author author K. P. \ Nuckolls \ and\ author A. Yazdani ,\ https://doi.org/10.1038/s41578-024-00682-1 journal journal Nat. Rev. Mater. \ volume 9 ,\ pages 460 ( year 2024 ) NoStop

    doi:10.1038/s41578-024-00682-1 2024

  23. [23]

    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 ,\ https://doi.org/10.1038/nature26154 journal journal Nature \ volume 556 ,\ ...

    doi:10.1038/nature26154 2018

  24. [24]

    Cao , author V

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

    doi:10.1038/nature26160 2018

  25. [25]

    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 ,\ https://doi.org/10.1038/s41586-019-1431-9 journal journal Nature \ volume 572 ,\ pages 95 ( yea...

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

  26. [26]

    Xie , author B

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

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

  27. [27]

    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 ,\ https://doi.org/10.1038/s41567-019-0606-5 journal journal Nat. Phys. \ vo...

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

  28. [28]

    author author S. L. \ Tomarken , author Y. Cao , author A. Demir , author K. Watanabe , author T. Taniguchi , author P. Jarillo-Herrero ,\ and\ author R. C. \ Ashoori ,\ https://doi.org/10.1103/PhysRevLett.123.046601 journal journal Phys. Rev. Lett. \ volume 123 ,\ pages 046601 ( year 2019 ) NoStop

    doi:10.1103/physrevlett.123.046601 2019

  29. [29]

    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 ,\ https://doi.org/10.1038/s41586-020-2373-y journal journal Nature \ volume 582 ,...

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

  30. [30]

    Lisi , author X

    author author S. Lisi , author X. Lu , author T. Benschop , author T. A. \ de Jong , author P. Stepanov , author J. R. \ Duran , author F. Margot , author I. Cucchi , author E. Cappelli , author A. Hunter , author A. Tamai , author V. Kandyba , author A. Giampietri , author A. Barinov , author J. Jobst , author V. Stalman , author M. Leeuwenhoek , author ...

    doi:10.1038/s41567-020-01041-x 2021

  31. [31]

    Chen , author K

    author author C. Chen , author K. P. \ Nuckolls , author S. Ding , author W. Miao , author D. Wong , author M. Oh , author R. L. \ Lee , author S. He , author C. Peng , author D. Pei , author Y. Li , author C. Hao , author H. Yan , author H. Xiao , author H. Gao , author Q. Li , author S. Zhang , author J. Liu , author L. He , author K. Watanabe , author ...

    doi:10.1038/s41586-024-08227-w 2024

  32. [32]

    author author K. L. \ Seyler , author P. Rivera , author H. Yu , author N. P. \ Wilson , author E. L. \ Ray , author D. G. \ Mandrus , author J. Yan , author W. Yao ,\ and\ author X. Xu ,\ https://doi.org/10.1038/s41586-019-0957-1 journal journal Nature \ volume 567 ,\ pages 66 ( year 2019 ) NoStop

    doi:10.1038/s41586-019-0957-1 2019

  33. [33]

    Tran , author G

    author author K. Tran , author G. Moody , author F. Wu , author X. Lu , author J. Choi , author K. Kim , author A. Rai , author D. A. \ Sanchez , author J. Quan , author A. Singh , author J. Embley , author A. Zepeda , author M. Campbell , author T. Autry , author T. Taniguchi , author K. Watanabe , author N. Lu , author S. K. \ Banerjee , author K. L. \ ...

    doi:10.1038/s41586-019-0975-z 2019

  34. [34]

    Jin , author E

    author author C. Jin , author E. C. \ Regan , author A. Yan , author M. Iqbal Bakti Utama , author D. Wang , author S. Zhao , author Y. Qin , author S. Yang , author Z. Zheng , author S. Shi , author K. Watanabe , author T. Taniguchi , author S. Tongay , author A. Zettl ,\ and\ author F. Wang ,\ https://doi.org/10.1038/s41586-019-0976-y journal journal Na...

    doi:10.1038/s41586-019-0976-y 2019

  35. [35]

    Inbar , author J

    author author A. Inbar , author J. Birkbeck , author J. Xiao , author T. Taniguchi , author K. Watanabe , author B. Yan , author Y. Oreg , author A. Stern , author E. Berg ,\ and\ author S. Ilani ,\ https://doi.org/10.1038/s41586-022-05685-y journal journal Nature \ volume 614 ,\ pages 682 ( year 2023 ) NoStop

    doi:10.1038/s41586-022-05685-y 2023

  36. [36]

    Xiao , author A

    author author J. Xiao , author A. Inbar , author J. Birkbeck , author N. Gershon , author Y. Zamir , author Y. Vituri , author T. Taniguchi , author K. Watanabe , author E. Berg ,\ and\ author S. Ilani ,\ https://doi.org/10.1038/s41586-026-10378-x journal journal Nature \ volume 653 ,\ pages 68 ( year 2026 ) NoStop

    doi:10.1038/s41586-026-10378-x 2026

  37. [37]

    Angeli \ and\ author A

    author author M. Angeli \ and\ author A. H. \ MacDonald ,\ https://doi.org/10.1073/pnas.2021826118 journal journal PNAS \ volume 118 ,\ pages e2021826118 ( year 2021 ) NoStop

    doi:10.1073/pnas.2021826118 2021

  38. [38]

    Claassen , author L

    author author M. Claassen , author L. Xian , author D. M. \ Kennes ,\ and\ author A. Rubio ,\ https://doi.org/10.1038/s41467-022-31604-w journal journal Nat. Commun. \ volume 13 ,\ pages 4915 ( year 2022 ) NoStop

    doi:10.1038/s41467-022-31604-w 2022

  39. [39]

    Pi , author Y

    author author H. Pi , author Y. H. \ Kwan , author H. Hu , author Y. Jiang , author D. C a lug a ru , author J. Shan , author K. F. \ Mak , author M. M. \ Ugeda , author D. K. \ Efetov , author M. G. \ Vergniory ,\ and\ author B. A. \ Bernevig ,\ https://doi.org/10.48550/arXiv.2605.13984 title Engineering topological flat bands in -valley moir\'e systems ...

    Pith/arXiv arXiv doi:10.48550/arxiv.2605.13984 2026

  40. [40]

    Bistritzer \ and\ author A

    author author R. Bistritzer \ and\ author A. H. \ MacDonald ,\ https://doi.org/10.1073/pnas.1108174108 journal journal PNAS \ volume 108 ,\ pages 12233 ( year 2011 ) NoStop

    doi:10.1073/pnas.1108174108 2011

  41. [41]

    Su \'a rez Morell , author J

    author author E. Su \'a rez Morell , author J. D. \ Correa , author P. Vargas , author M. Pacheco ,\ and\ author Z. Barticevic ,\ https://doi.org/10.1103/PhysRevB.82.121407 journal journal Phys. Rev. B \ volume 82 ,\ pages 121407 ( year 2010 ) NoStop

    doi:10.1103/physrevb.82.121407 2010

  42. [42]

    author author J. M. B. \ Lopes dos Santos , author N. M. R. \ Peres ,\ and\ author A. H. \ Castro Neto ,\ https://doi.org/10.1103/PhysRevLett.99.256802 journal journal Phys. Rev. Lett. \ volume 99 ,\ pages 256802 ( year 2007 ) NoStop

    doi:10.1103/physrevlett.99.256802 2007

  43. [43]

    \ Song \ and\ author B

    author author Z.-D. \ Song \ and\ author B. A. \ Bernevig ,\ https://doi.org/10.1103/PhysRevLett.129.047601 journal journal Phys. Rev. Lett. \ volume 129 ,\ pages 047601 ( year 2022 ) NoStop

    doi:10.1103/physrevlett.129.047601 2022

  44. [44]

    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 ,\ https://doi.org/10.1038/s41586-023-07010-7 journal journal Nature \ volume 626 ,\ pages 759 ( year 2024 ) NoStop

    doi:10.1038/s41586-023-07010-7 2024

  45. [45]

    Dong , author T

    author author J. Dong , author T. Wang , author T. Wang , author T. Soejima , author M. P. \ Zaletel , author A. Vishwanath ,\ and\ author D. E. \ Parker ,\ https://doi.org/10.1103/PhysRevLett.133.206503 journal journal Phys. Rev. Lett. \ volume 133 ,\ pages 206503 ( year 2024 a ) NoStop

    doi:10.1103/physrevlett.133.206503 2024

  46. [46]

    Dong , author A

    author author Z. Dong , author A. S. \ Patri ,\ and\ author T. Senthil ,\ https://doi.org/10.1103/PhysRevLett.133.206502 journal journal Phys. Rev. Lett. \ volume 133 ,\ pages 206502 ( year 2024 b ) NoStop

    doi:10.1103/physrevlett.133.206502 2024

  47. [47]

    Herzog-Arbeitman , author Y

    author author J. Herzog-Arbeitman , author Y. Wang , author J. Liu , author P. M. \ Tam , author Z. Qi , author Y. Jia , author D. K. \ Efetov , author O. Vafek , author N. Regnault , author H. Weng , author Q. Wu , author B. A. \ Bernevig ,\ and\ author J. Yu ,\ https://doi.org/10.1103/PhysRevB.109.205122 journal journal Phys. Rev. B \ volume 109 ,\ page...

    doi:10.1103/physrevb.109.205122 2024

  48. [48]

    Han , author Z

    author author T. Han , author Z. Lu , author Z. Hadjri , author L. Shi , author Z. Wu , author W. Xu , author Y. Yao , author A. A. \ Cotten , author O. Sharifi Sedeh , author H. Weldeyesus , author J. Yang , author J. Seo , author S. Ye , author M. Zhou , author H. Liu , author G. Shi , author Z. Hua , author K. Watanabe , author T. Taniguchi , author P....

    doi:10.1038/s41586-025-09169-7 2025

  49. [49]

    Wu , author T

    author author F. Wu , author T. Lovorn , author E. Tutuc ,\ and\ author A. H. \ MacDonald ,\ https://doi.org/10.1103/PhysRevLett.121.026402 journal journal Phys. Rev. Lett. \ volume 121 ,\ pages 026402 ( year 2018 ) NoStop

    doi:10.1103/physrevlett.121.026402 2018

  50. [50]

    Tang , author L

    author author Y. Tang , author L. Li , author T. Li , author Y. Xu , author S. Liu , author K. Barmak , author K. Watanabe , author T. Taniguchi , author A. H. \ MacDonald , author J. Shan ,\ and\ author K. F. \ Mak ,\ https://doi.org/10.1038/s41586-020-2085-3 journal journal Nature \ volume 579 ,\ pages 353 ( year 2020 ) NoStop

    doi:10.1038/s41586-020-2085-3 2085

  51. [51]

    Xu , author S

    author author Y. Xu , author S. Liu , author D. A. \ Rhodes , author K. Watanabe , author T. Taniguchi , author J. Hone , author V. Elser , author K. F. \ Mak ,\ and\ author J. Shan ,\ https://doi.org/10.1038/s41586-020-2868-6 journal journal Nature \ volume 587 ,\ pages 214 ( year 2020 ) NoStop

    doi:10.1038/s41586-020-2868-6 2020

  52. [52]

    Li , author S

    author author T. Li , author S. Jiang , author L. Li , author Y. Zhang , author K. Kang , author J. Zhu , author K. Watanabe , author T. Taniguchi , author D. Chowdhury , author L. Fu , author J. Shan ,\ and\ author K. F. \ Mak ,\ https://doi.org/10.1038/s41586-021-03853-0 journal journal Nature \ volume 597 ,\ pages 350 ( year 2021 ) NoStop

    doi:10.1038/s41586-021-03853-0 2021

  53. [53]

    Wang , author G

    author author P. Wang , author G. Yu , author Y. H. \ Kwan , author Y. Jia , author S. Lei , author S. Klemenz , author F. A. \ Cevallos , author R. Singha , author T. Devakul , author K. Watanabe , author T. Taniguchi , author S. L. \ Sondhi , author R. J. \ Cava , author L. M. \ Schoop , author S. A. \ Parameswaran ,\ and\ author S. Wu ,\ https://doi.or...

    doi:10.1038/s41586-022-04514-6 2022

  54. [54]

    Zhao , author B

    author author W. Zhao , author B. Shen , author Z. Tao , author Z. Han , author K. Kang , author K. Watanabe , author T. Taniguchi , author K. F. \ Mak ,\ and\ author J. Shan ,\ https://doi.org/10.1038/s41586-023-05800-7 journal journal Nature \ volume 616 ,\ pages 61 ( year 2023 ) NoStop

    doi:10.1038/s41586-023-05800-7 2023

  55. [55]

    Xia , author Z

    author author Y. Xia , author Z. Han , author K. Watanabe , author T. Taniguchi , author J. Shan ,\ and\ author K. F. \ Mak ,\ https://doi.org/10.1038/s41586-024-08116-2 journal journal Nature \ volume 637 ,\ pages 833 ( year 2025 ) NoStop

    doi:10.1038/s41586-024-08116-2 2025

  56. [56]

    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 , author A. J. \ Millis , author A. Pasupathy ,\ and\ author C. R. \ Dean ,\ https://doi.org/10.1038/s41586-024-08381-1 journal journal Nature \ volume 637...

    doi:10.1038/s41586-024-08381-1 2025

  57. [57]

    Xia , author Z

    author author Y. Xia , author Z. Han , author J. Zhu , author Y. Zhang , author P. Kn \"u ppel , author K. Watanabe , author T. Taniguchi , author K. F. \ Mak ,\ and\ author J. Shan ,\ https://doi.org/10.1038/s41586-025-10049-3 journal journal Nature \ volume 650 ,\ pages 585 ( year 2026 ) NoStop

    doi:10.1038/s41586-025-10049-3 2026

  58. [58]

    Wu , author T

    author author F. Wu , author T. Lovorn , author E. Tutuc , author I. Martin ,\ and\ author A. H. \ MacDonald ,\ https://doi.org/10.1103/PhysRevLett.122.086402 journal journal Phys. Rev. Lett. \ volume 122 ,\ pages 086402 ( year 2019 ) NoStop

    doi:10.1103/physrevlett.122.086402 2019

  59. [59]

    Devakul , author V

    author author T. Devakul , author V. Cr \'e pel , author Y. Zhang ,\ and\ author L. Fu ,\ https://doi.org/10.1038/s41467-021-27042-9 journal journal Nat. Commun. \ volume 12 ,\ pages 6730 ( year 2021 ) NoStop

    doi:10.1038/s41467-021-27042-9 2021

  60. [60]

    Zeng , author Z

    author author Y. Zeng , author Z. Xia , author K. Kang , author J. Zhu , author P. Kn \"u ppel , author C. Vaswani , author K. Watanabe , author T. Taniguchi , author K. F. \ Mak ,\ and\ author J. Shan ,\ https://doi.org/10.1038/s41586-023-06452-3 journal journal Nature \ volume 622 ,\ pages 69 ( year 2023 ) NoStop

    doi:10.1038/s41586-023-06452-3 2023

  61. [61]

    Wang , author X.-W

    author author C. Wang , author X.-W. \ Zhang , author X. Liu , author Y. He , author X. Xu , author Y. Ran , author T. Cao ,\ and\ author D. Xiao ,\ https://doi.org/10.1103/PhysRevLett.132.036501 journal journal Phys. Rev. Lett. \ volume 132 ,\ pages 036501 ( year 2024 a ) NoStop

    doi:10.1103/physrevlett.132.036501 2024

  62. [62]

    Yu , author J

    author author J. Yu , author J. Herzog-Arbeitman , author M. Wang , author O. Vafek , author B. A. \ Bernevig ,\ and\ author N. Regnault ,\ https://doi.org/10.1103/PhysRevB.109.045147 journal journal Phys. Rev. B \ volume 109 ,\ pages 045147 ( year 2024 ) NoStop

    doi:10.1103/physrevb.109.045147 2024

  63. [63]

    Jia , author J

    author author Y. Jia , author J. Yu , author J. Liu , author J. Herzog-Arbeitman , author Z. Qi , author H. Pi , author N. Regnault , author H. Weng , author B. A. \ Bernevig ,\ and\ author Q. Wu ,\ https://doi.org/10.1103/PhysRevB.109.205121 journal journal Phys. Rev. B \ volume 109 ,\ pages 205121 ( year 2024 ) NoStop

    doi:10.1103/physrevb.109.205121 2024

  64. [64]

    \ Zhang , author C

    author author X.-W. \ Zhang , author C. Wang , author X. Liu , author Y. Fan , author T. Cao ,\ and\ author D. Xiao ,\ https://doi.org/10.1038/s41467-024-48511-x journal journal Nat. Commun. \ volume 15 ,\ pages 4223 ( year 2024 ) NoStop

    doi:10.1038/s41467-024-48511-x 2024

  65. [65]

    author author D. N. \ Sheng , author A. P. \ Reddy , author A. Abouelkomsan , author E. J. \ Bergholtz ,\ and\ author L. Fu ,\ https://doi.org/10.1103/PhysRevLett.133.066601 journal journal Phys. Rev. Lett. \ volume 133 ,\ pages 066601 ( year 2024 ) NoStop

    doi:10.1103/physrevlett.133.066601 2024

  66. [66]

    Ingham , author M

    author author J. Ingham , author M. S. \ Scheurer ,\ and\ author H. D. \ Scammell ,\ https://doi.org/10.48550/arXiv.2503.11754 title Moir\'e M-valley bilayers: Quasi-one-dimensional physics, unconventional spin textures and twisted van Hove singularities ( year 2025 ),\ https://arxiv.org/abs/2503.11754 arXiv:2503.11754 [cond-mat.str-el] NoStop

    doi:10.48550/arxiv.2503.11754 2025

  67. [67]

    Bao , author H

    author author K. Bao , author H. Wang , author Z. Liu ,\ and\ author J. Wang ,\ https://doi.org/10.1103/l88b-67d2 journal journal Phys. Rev. B \ volume 112 ,\ pages L041406 ( year 2025 ) NoStop

    doi:10.1103/l88b-67d2 2025

  68. [68]

    Kariyado \ and\ author A

    author author T. Kariyado \ and\ author A. Vishwanath ,\ https://doi.org/10.1103/PhysRevResearch.1.033076 journal journal Phys. Rev. Res. \ volume 1 ,\ pages 033076 ( year 2019 ) NoStop

    doi:10.1103/physrevresearch.1.033076 2019

  69. [69]

    Fujimoto , author T

    author author M. Fujimoto , author T. Kawakami ,\ and\ author M. Koshino ,\ https://doi.org/10.1103/PhysRevResearch.4.043209 journal journal Phys. Rev. Res. \ volume 4 ,\ pages 043209 ( year 2022 ) NoStop

    doi:10.1103/physrevresearch.4.043209 2022

  70. [70]

    Kariyado ,\ https://doi.org/10.1103/PhysRevB.107.085127 journal journal Phys

    author author T. Kariyado ,\ https://doi.org/10.1103/PhysRevB.107.085127 journal journal Phys. Rev. B \ volume 107 ,\ pages 085127 ( year 2023 ) NoStop

    doi:10.1103/physrevb.107.085127 2023

  71. [71]

    author author V. J. \ Emery , author E. Fradkin , author S. A. \ Kivelson ,\ and\ author T. C. \ Lubensky ,\ https://doi.org/10.1103/PhysRevLett.85.2160 journal journal Phys. Rev. Lett. \ volume 85 ,\ pages 2160 ( year 2000 ) NoStop

    doi:10.1103/physrevlett.85.2160 2000

  72. [72]

    Vishwanath \ and\ author D

    author author A. Vishwanath \ and\ author D. Carpentier ,\ https://doi.org/10.1103/PhysRevLett.86.676 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 676 ( year 2001 ) NoStop

    doi:10.1103/physrevlett.86.676 2001

  73. [73]

    Mukhopadhyay , author C

    author author R. Mukhopadhyay , author C. L. \ Kane ,\ and\ author T. C. \ Lubensky ,\ https://doi.org/10.1103/PhysRevB.64.045120 journal journal Phys. Rev. B \ volume 64 ,\ pages 045120 ( year 2001 a ) NoStop

    doi:10.1103/physrevb.64.045120 2001

  74. [74]

    Mukhopadhyay , author C

    author author R. Mukhopadhyay , author C. L. \ Kane ,\ and\ author T. C. \ Lubensky ,\ https://doi.org/10.1103/PhysRevB.63.081103 journal journal Phys. Rev. B \ volume 63 ,\ pages 081103 ( year 2001 b ) NoStop

    doi:10.1103/physrevb.63.081103 2001

  75. [75]

    author author C. D. \ Beule , author L. Peng , author E. J. \ Mele ,\ and\ author S. Adam ,\ https://doi.org/10.48550/arXiv.2508.14283 title Role of electron-electron interactions in M-valley twisted transition metal dichalcogenides ( year 2025 ),\ https://arxiv.org/abs/2508.14283 arXiv:2508.14283 [cond-mat.mes-hall] NoStop

    doi:10.48550/arxiv.2508.14283 2025

  76. [76]

    Vasiliou , author D

    author author K. Vasiliou , author D. C a lug a ru , author J. S. \ Hofmann ,\ and\ author S. A. \ Parameswaran ,\ @noop journal journal arXiv:2606.XXXXX [cond-mat] \ ( year 2026 ) NoStop

    2026

  77. [77]

    Blankenbecler , author D

    author author R. Blankenbecler , author D. J. \ Scalapino ,\ and\ author R. L. \ Sugar ,\ https://doi.org/10.1103/PhysRevD.24.2278 journal journal Phys. Rev. D \ volume 24 ,\ pages 2278 ( year 1981 ) NoStop

    doi:10.1103/physrevd.24.2278 1981

  78. [78]

    author author D. J. \ Scalapino \ and\ author R. L. \ Sugar ,\ https://doi.org/10.1103/PhysRevB.24.4295 journal journal Phys. Rev. B \ volume 24 ,\ pages 4295 ( year 1981 ) NoStop

    doi:10.1103/physrevb.24.4295 1981

  79. [79]

    author author S. R. \ White , author D. J. \ Scalapino , author R. L. \ Sugar , author E. Y. \ Loh , author J. E. \ Gubernatis ,\ and\ author R. T. \ Scalettar ,\ https://doi.org/10.1103/PhysRevB.40.506 journal journal Phys. Rev. B \ volume 40 ,\ pages 506 ( year 1989 ) NoStop

    doi:10.1103/physrevb.40.506 1989

  80. [80]

    Assaad \ and\ author H

    author author F. Assaad \ and\ author H. Evertz ,\ in\ https://doi.org/10.1007/978-3-540-74686-7_10 booktitle Computational Many-Particle Physics ,\ editor edited by\ editor H. Fehske , editor R. Schneider ,\ and\ editor A. Wei e \ ( publisher Springer ,\ address Berlin, Heidelberg ,\ year 2008 )\ pp.\ pages 277--356 NoStop

    doi:10.1007/978-3-540-74686-7_10 2008

Showing first 80 references.