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REVIEW 4 major objections 5 minor 60 references

The low-energy moiré bands of twisted semiconductors are governed by the valley character of the parent band edge together with stacking symmetry.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Parent valley character plus stacking symmetry, not chemistry, organizes bandwidth scaling and topological bands across more than 1,000 twisted semiconductor moiré structures.

T0 review reviewed 2026-08-01 challenge →

load-bearing objection A genuinely useful materials database with a plausible valley-based taxonomy, but the central scaling law rests on assertion rather than fits, and the pipeline transferability is the weak link. the 4 major comments →

arxiv 2607.25172 v1 pith:BW3ITJGS submitted 2026-07-28 cond-mat.mtrl-sci cond-mat.mes-hall

A Catalogue of Topological Moir\'{e} Bands in Twisted Semiconductors

Jiaheng Li , Yan Zhang , Jiaxuan Liu , Caiyuan Ye , Tiannian Zhu , Zhong Fang , Hongming Weng , Quansheng Wu This is my paper
classification cond-mat.mtrl-sci cond-mat.mes-hall
keywords twisted bilayer semiconductorsmoiré bandsvalley charactertopological minibandsChern numberZ2 indexhigh-throughput computationJanus bilayers
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper tries to establish that the wide variety of moiré minibands in twisted semiconductors follows a simple hierarchy: the momentum-space location of the parent monolayer's band edge (the valley) plus the stacking symmetry of the twisted bilayer determines whether the bands are flat, how their width scales with twist angle, and whether they carry nontrivial topology. To back this up, the authors built a high-throughput pipeline that relaxes twisted structures and computes moiré electronic structure for 43 experimentally realized monolayers and 91 bilayer prototypes, yielding over 1,000 angle-resolved band structures. The payoff is a predictive taxonomy: Γ-valley systems show nearly quadratic bandwidth scaling with twist angle, K-valley systems can host isolated valley Chern minibands depending on stacking, and M-valley systems are idiosyncratic with kagome-like physics. If correct, this gives materials designers a rule of thumb for choosing parent semiconductors that will produce flat topological moiré bands.

Core claim

The central discovery is that the low-energy moiré electronic structure of twisted semiconductors is organized primarily by the valley character of the parent band edge together with stacking symmetry. In Γ-valley systems, miniband width follows a nearly quadratic twist-angle scaling W(θ) ∝ θ², reflecting a folding-dominated kinetic-energy scale. In K-valley systems, parallel (AA) stacking permits spin-aligned interlayer tunneling that lifts valley degeneracy and yields isolated minibands with nonzero valley Chern numbers |C_K| = 1, whereas antiparallel (AB) stacking preserves inversion symmetry and suppresses such isolation. M-valley systems are material-specific, governed by non-symmorphic

What carries the argument

The central object is the valley character of the parent monolayer band edge — the high-symmetry momentum (Γ, K, M, or off-symmetry Q) where the valence or conduction band extremum sits — combined with the stacking symmetry (parallel AA vs antiparallel AB) of the twisted bilayer. The argument is carried by a high-throughput computational pipeline: structural relaxation via generalized stacking-fault energy surfaces and machine-learned force fields, followed by construction of first-principles tight-binding Hamiltonians and moiré band calculations using a truncated atomic plane-wave method, which yields bandwidths, Chern numbers, Z₂ indices, and quantum geometry without fitting.

Load-bearing premise

The entire classification assumes that the machine-learning-relaxed structures and the truncated atomic plane-wave tight-binding Hamiltonians built from them faithfully represent the true low-energy moiré physics of all 43 materials; if the relaxation or the tight-binding approximation misassigns a valley or hybridization in any compound, its category in the hierarchy would be an artifact of the pipeline.

What would settle it

Measure the bandwidth versus twist angle for three Γ-valley compounds classified as quadratic using a method that does not assume the truncated atomic plane-wave tight-binding model (e.g., full DFT on small commensurate cells); if the actual exponent deviates significantly from 2, the scaling rule fails. For K-valley AA compounds predicted to host isolated |C_K| = 1 minibands, an independent probe such as ARPES or exact diagonalization that finds no isolated Chern band would falsify the stacking-hybridization mechanism.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the valley-and-stacking hierarchy holds, Γ-valley twisted semiconductors are the most predictable class: their miniband widths shrink roughly as θ², and certain stacking configurations harbour Z₂ = 1 insulating minibands over wide twist-angle windows.
  • K-valley homobilayers with parallel (AA) stacking become the natural platform for isolated valley Chern minibands, since AA stacking allows spin-aligned interlayer tunneling; AB-stacked K-valley bilayers are expected to remain topologically trivial in the topmost minibands.
  • M-valley systems will not fit a single-scaling description; their moiré bands are dominated by non-symmorphic symmetries and can realize kagome or quasi-one-dimensional physics, requiring a different classification toolbox.
  • In Janus bilayers, choosing which surface faces the partner layer (e.g., Te–Te vs Te–Cl vs Cl–Cl in BiTeCl) can flip the parent valley character and thereby switch the resulting moiré topology, providing an experimental tuning knob.
  • Quantum-geometry screening of the database identifies candidates for fractionalized phases — AA-stacked MoTe₂ among Chern systems and AB-stacked BiTeI, MgBr₂, CaI₂ among Z₂ systems — whose flat bands are more Landau-level-like.

Where Pith is reading between the lines

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

  • A testable extension is to use the valley-and-stacking rule as a screening criterion: given a candidate monolayer, reading off its band-edge valley from a standard DFT band structure would predict whether flat topological minibands are likely at small twist angles, without running a full moiré calculation.
  • The quadratic W(θ) ∝ θ² scaling for Γ-valley systems suggests a simple folding picture in which the moiré potential plays a minor role; if so, even non-twisted periodic modulations that fold the Γ point would produce similarly flat bands, a connection the paper does not explore.
  • The surface-termination knob in Janus bilayers points toward a general design strategy: asymmetric termination can deliberately move band extrema off high-symmetry points, turning a systematic Γ-valley material into a complex Q-valley one, and vice versa, which may be used to toggle between flat-band and dispersive regimes in the same compound.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper presents a high-throughput computational framework that combines screening of experimentally known monolayers, GSFE/MLFF-based structural relaxation, and truncated atomic plane-wave (TAPW) tight-binding simulations to construct a database of moiré band structures for 43 twisted semiconductors and 91 bilayer prototypes, yielding over 1,000 angle-resolved band structures. The central claim is that the low-energy moiré electronic structure is organized primarily by the parent monolayer's valley character together with stacking symmetry: Γ-valley systems show a nearly quadratic twist-angle scaling W(θ)∝θ² attributed to folding-dominated kinetics; K-valley systems exhibit stacking-controlled isolated valley Chern minibands; M-valley systems are material-specific and display non-symmorphic kagome-like or quasi-one-dimensional behavior. The paper also reports topology (Chern, Z₂) and quantum geometry for selected minibands, and demonstrates that surface termination in Janus BiTeCl can switch the relevant valley character. The work is framed as a predictive taxonomy and is accompanied by a public web database.

Significance. If the reported hierarchy is robust, this is a useful organizing principle for a rapidly growing field, and the database itself is a valuable community resource. The paper makes explicit, falsifiable predictions: for example, AA-stacked K-valley systems should host isolated valley Chern minibands, certain Γ-valley AB stackings should host robust Z₂ phases, and Janus termination can switch valley character. The authors should be credited for constructing a large, publicly accessible dataset with band structures and topological data, and for grounding the claim in a systematic enumeration of stacking configurations. However, the significance is conditional on the transferability of the TAPW/MLFF pipeline to the full 43-material set; the current evidence is strong mainly for MoTe₂/WSe₂. The quantitative scaling law and the class sizes are also not yet established from the data as presented, which limits the degree to which the taxonomy can be accepted as predictive.

major comments (4)
  1. [Universal scaling and valley-controlled hierarchy] The central claim W(θ)∝θ² for Γ-valley systems is asserted as 'consistent with a folding-dominated regime' but no quantitative fit is provided. Fig. 2 shows heatmaps only; the text does not report power-law exponents, residuals, or a no-potential folding baseline for individual compounds. Because this scaling is a headline result, please provide log-log fits per material with quoted exponents and goodness-of-fit measures, and overlay a folding-only reference curve in Fig. 2 or a supplement figure. Without this, the reader cannot distinguish a genuine quasi-universal law from an approximate trend.
  2. [Computational workflow] The entire database relies on GSFE/MLFF relaxation followed by TAPW tight-binding Hamiltonians. The TAPW implementation is cited as Ref. [54], 'In preparation,' and the method appears to have been benchmarked mainly on MoTe₂ and WSe₂ via Refs. [22,44,52]. No per-material validation against direct DFT supercell calculations, independent continuum models, or experiment is shown for the broader set of halides and chalcogenides. A stratified validation is needed: at least one representative per valley class (Γ, K, M, Q) with a full-DFT or alternative moiré calculation at a small twist angle, checking parent band-edge ordering, moiré bandwidth, and topological invariants. If the pipeline misassigns valley character or misestimates interlayer hybridization in less-studied compounds, the reported valley hierarchy and the associated Chern/Z₂ assignments could be properties of the pipeline rather
  3. [Valley classification] The paper argues for a 'universal' valley-and-stacking hierarchy, but explicitly prioritizes high-symmetry valleys and sets aside off-symmetry Q-valley systems, describing their behavior only as 'intricate' and driven by 'sensitive interplay.' The manuscript does not state how many of the 43 monolayers or 91 prototypes fall into the Q-valley class, nor whether the remaining materials actually obey the taxonomy. Without this information, the claimed organizing principle is not testable for a potentially large fraction of the database. Please report the class sizes and provide a systematic account of Q-valley systems, or explicitly restrict the universality claim to high-symmetry-valley systems.
  4. [Twist-induced topology from valley structure] For M-valley systems the text states they 'host a sophisticated topological landscape governed by emergent non-symmorphic symmetries' and 'manifest unique kagome or quasi-one-dimensional physics,' citing Refs. [49,50], but no M-valley moiré band structure, symmetry analysis, or topological quantity from the database is shown in this work. Since the taxonomy includes M-valley systems as a distinct class, at least one representative M-valley example from the new database should be presented, or the text should clearly attribute this statement to prior literature rather than to the current high-throughput results.
minor comments (5)
  1. [Fig. 2 caption and surrounding text] Several strings such as '/uni0000001a/uni00000011/...' appear in the figure caption and in the text near Fig. 2. This appears to be an encoding artifact and must be fixed before publication.
  2. [References] Ref. [54] is cited as 'In preparation' for the TAPW implementation. This is not verifiable; please either provide the full technical details in the Supplementary Information or update the reference once available.
  3. [Methods / Definition of 'energetic isolation'] The Fig. 2 caption states that absence of symbols identifies minibands that 'lack energetic isolation' in K- and Γ-valleys, but the numerical criterion for energetic isolation is not defined in the main text. Please specify the band-isolation energy threshold and the twist-angle sampling grid used for the database.
  4. [Scaling wording] The text uses both 'quadratic scaling' and 'nearly quadratic' for W(θ). Please define the quantitative range of twist angles over which this scaling is claimed, and clarify how 'nearly' is assessed.
  5. [Data availability] The Data Availability statement refers to Supplementary Information [56], but the reference is a generic placeholder. Please point to the specific sections and file names of the database.

Circularity Check

0 steps flagged

No significant circularity: the valley-and-stacking taxonomy is an empirical correlation from over 1,000 computed band structures, not a fit renamed as a prediction; the main risks are pipeline transferability and an in-preparation method reference, neither of which makes the derivation circular.

full rationale

The paper's central claim is that low-energy moiré electronic structure is organized by parent valley character and stacking symmetry. This is presented as a database-driven empirical correlation: valley labels are read from parent monolayer DFT band edges, while moiré bandwidths, Chern numbers, and Z2 invariants are computed from TAPW tight-binding Hamiltonians on relaxed geometries. No fitted parameter is renamed as a prediction: the W(θ)∝θ² statement is reported as a scaling 'consistent with a folding-dominated kinetic-energy scale', and the K-valley Chern/Z2 statements are computed consequences of interlayer hybridization and stacking symmetry, with independent prior support (e.g., Ref. 25). The TAPW method is not solely a self-citation chain: Ref. 53 is an independent implementation, and Refs. 22 and 52 provide external or reproducible benchmarks for the small-angle behavior of MoTe2/WSe2. The same-group in-preparation reference [54] is a missing-support issue, not a circular step. The main genuine weakness is that the transferability of the GSFE/MLFF relaxation and TAPW pipeline to all 43 materials is not validated per material; however, that is a correctness/robustness risk, not a case where an output is equivalent to an input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim depends on one new conceptual entity, the valley-and-stacking hierarchy, but that is a classification, not a physical entity. The material-dependent predictions rest on DFT, MLFF, and TAPW approximations plus the chosen sampling and isolation criteria, which are the main free/assumed inputs.

free parameters (2)
  • Twist-angle sampling grid
    The set of commensurate angles per material is chosen for tractability; the W(θ)∝θ² claim and the breadth of topological windows depend on this grid, which is not specified in the main text.
  • Band-isolation energy criterion
    Assigning Chern/Z2 labels requires deciding which minibands are 'isolated'; the caption of Fig. 2 notes that symbols are omitted for bands lacking energetic isolation, but no quantitative threshold is given.
axioms (5)
  • domain assumption DFT band edges of parent monolayers determine the true valley character
    The taxonomy classifies every material by the Γ/K/M/Q valley of its parent VBM/CBM; a DFT error in band ordering would misclassify entire classes. Invoked throughout 'Valley classification'.
  • domain assumption GSFE+MLFF-relaxed geometries are close to true DFT-relaxed moiré geometries
    Pipeline relaxation step; accuracy is cited to Ref. [52] but not demonstrated for all 43 materials. Introduced in 'Computational Workflow'.
  • domain assumption TAPW tight-binding reproduces DFT-quality subbands for all screened materials
    TAPW is the engine for all >1,000 band structures; Refs. [53,44] validate on selected TMDs, while Ref. [54] is 'In preparation'. The transferability is assumed.
  • domain assumption Commensurate supercells at sampled angles represent experimental small-angle moiré
    Discrete commensurate angles are taken as representative of the continuous experimental twist angle; stated in 'Computational Workflow'.
  • standard math Standard topological invariants (Chern, Z2) reliably classify isolated moiré bands
    Used throughout 'Results'; standard band-topology methodology, not questioned within the paper.

reviewed 2026-08-01 · how reviews work

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Cite this review

Pith. "Pith review of A Catalogue of Topological Moir\'{e} Bands in Twisted Semiconductors." pith.science (2026). https://pith.science/paper/BW3ITJGS

@misc{pith2026260725172,
  author       = {Pith},
  title        = {Pith review of: A Catalogue of Topological Moir\'e Bands in Twisted Semiconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BW3ITJGS}},
  note         = {Machine review of arXiv:2607.25172}
}
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abstract

Twisted two-dimensional semiconductors provide a route to flat and topological moir\'e minibands, but systematic principles for organizing their material dependence have remained unclear. Here, we establish a high-throughput framework that integrates structural relaxation, first-principles electronic structure calculations, and moir\'e band topology. We apply this framework to 43 experimentally realized monolayers and 91 symmetry-inequivalent bilayer prototypes, yielding over 1,000 angle-resolved moir\'e electronic band structures. This database reveals that the low-energy moir\'e electronic structure is organized primarily by the valley character of the parent band edge together with stacking symmetry. In $\Gamma$-valley systems, the miniband width usually follows a nearly quadratic twist-angle scaling, consistent with a folding-dominated kinetic-energy scale. In $K$-valley systems, stacking-controlled interlayer hybridization governs whether parent Berry curvature is redistributed into isolated valley Chern minibands. By contrast, $M$-valley systems form a more material-specific class associated with anisotropic and symmetry-constrained band folding. The same valley-and-stacking hierarchy rationalizes the emergence or suppression of $\mathbb{Z}_2$ minibands, and surface termination in Janus bilayers provides a microscopic knob for changing the relevant valley character. These results establish a materials-level organizing principle for designing flat and topological moir\'e bands in twisted semiconductors.

Figures

Figures reproduced from arXiv: 2607.25172 by Caiyuan Ye, Hongming Weng, Jiaheng Li, Jiaxuan Liu, Quansheng Wu, Tiannian Zhu, Yan Zhang, Zhong Fang.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

Works this paper leans on

60 extracted references · 2 linked inside Pith

  1. [1]

    S. Carr, D. Massatt, S. Fang, P. Cazeaux, M. Luskin, and E. Kaxiras, Twistronics: Manipulating the electronic properties of two-dimensional layered structures through their twist angle, Phys. Rev. B95, 075420 (2017)

    2017

  2. [2]

    S. Carr, S. Fang, and E. Kaxiras, Electronic-structure methods for twisted moir´ e layers, Nat. Rev. Mater.5, 748 (2020)

    2020

  3. [3]

    E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moir´ e materials, Nat. Rev. Mater.6, 201 (2021)

    2021

  4. [4]

    D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. Basov, A. N. Pasupathy, and A. Rubio, Moir´ e heterostructures as a condensed- matter quantum simulator, Nat. Phys.17, 155 (2021)

    2021

  5. [5]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moir´ e bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. 108, 12233 (2011)

    2011

  6. [6]

    Regnault and B

    N. Regnault and B. A. Bernevig, Fractional Chern insu- lator, Phys. Rev. X1, 021014 (2011)

    2011

  7. [7]

    Vizner Stern, Y

    M. Vizner Stern, Y. Waschitz, W. Cao, I. Nevo, K. Watanabe, T. Taniguchi, E. Sela, M. Urbakh, O. Hod, and M. Ben Shalom, Interfacial ferroelectricity by van der Waals sliding, Science372, 1462 (2021)

    2021

  8. [8]

    C. Jin, E. C. Regan, A. Yan, M. Iqbal Bakti Utama, D. Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi, et al., Observation of moir´ e excitons in WSe2/WS2 het- erostructure superlattices, Nature567, 76 (2019)

    2019

  9. [9]

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

    2019

  10. [10]

    Y. Cao, V. Fatemi, S. p. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Uncon- ventional superconductivity in magic-angle graphene su- perlattices, Nature556, 43 (2018)

    2018

  11. [11]

    Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable correlated states and spin-polarized phases in twisted bilayer–bilayer graphene, Nature583, 215 (2020)

    2020

  12. [12]

    C. Wang, Y. Gao, H. Lv, X. Xu, and D. Xiao, Stacking domain wall magnons in twisted van der Waals magnets, Phys. Rev. Lett.125, 247201 (2020)

    2020

  13. [13]

    Song, Q.-C

    T. Song, Q.-C. Sun, E. Anderson, C. Wang, J. Qian, T. Taniguchi, K. Watanabe, M. A. McGuire, R. St¨ ohr, D. Xiao, et al., Direct visualization of magnetic domains and moir´ e magnetism in twisted 2D magnets, Science 374, 1140 (2021)

    2021

  14. [14]

    H. Xie, X. Luo, G. Ye, Z. Ye, H. Ge, S. H. Sung, E. Ren- nich, S. Yan, Y. Fu, S. Tian, et al., Twist engineer- ing of the two-dimensional magnetism in double bilayer chromium triiodide homostructures, Nat. Phys.18, 30 (2022)

    2022

  15. [15]

    B. Yang, Y. Li, H. Xiang, H. Lin, and B. Huang, Moir´ e magnetic exchange interactions in twisted magnets, Nat. Comput. Sci.3, 314 (2023)

    2023

  16. [16]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional Chern insulator in moir´ e MoTe2, Nature622, 69 (2023)

    2023

  17. [17]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe, et al., Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature622, 63 (2023)

    2023

  18. [18]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y. Gu, K. Watanabe, T. Taniguchi, B. Tong, et al., Observa- tion of integer and fractional quantum anomalous Hall effects in twisted bilayer MoTe2, Phys. Rev. X13, 031037 (2023)

    2023

  19. [19]

    Zhang, C

    X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted MoTe 2 and WSe 2, Nat. Commun.15, 4223 (2024)

    2024

  20. [20]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T. Cao, and D. Xiao, Fractional Chern insulator in twisted bilayer MoTe 2, Phys. Rev. Lett.132, 036501 (2024)

    2024

  21. [21]

    Redekop, C

    E. Redekop, C. Zhang, H. Park, J. Cai, E. Anderson, O. Sheekey, T. Arp, G. Babikyan, S. Salters, K. Watan- abe, et al., Direct magnetic imaging of fractional Chern insulators in twisted MoTe 2, Nature635, 584 (2024)

    2024

  22. [22]

    Y. Jia, J. Yu, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, and Q. Wu, Moir´ e fractional Chern insulators. I. First-principles calcula- tions and continuum models of twisted bilayer MoTe 2, Phys. Rev. B109, 205121 (2024)

    2024

  23. [23]

    Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Superconductivity in twisted bilayer WSe 2, Nature637, 833 (2025)

    2025

  24. [24]

    C. Xu, N. Mao, T. Zeng, and Y. Zhang, Multiple Chern bands in twisted MoTe2 and possible non-Abelian states, Phys. Rev. Lett.134, 066601 (2025)

    2025

  25. [25]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. Mac- Donald, Topological insulators in twisted transition metal dichalcogenide homobilayers, Phys. Rev. Lett.122, 086402 (2019)

    2019

  26. [26]

    Wang, E.-M

    L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim, et al., Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nat. Mater.19, 861 (2020)

    2020

  27. [27]

    Y. Xiao, J. Liu, and L. Fu, Moir´ e is more: access to new properties of two-dimensional layered materials, Matter 3, 1142 (2020)

    2020

  28. [28]

    K. F. Mak and J. Shan, Semiconductor moir´ e materials, Nat. Nanotech.17, 686 (2022)

    2022

  29. [29]

    K. Kang, B. Shen, Y. Qiu, Y. Zeng, Z. Xia, K. Watan- abe, T. Taniguchi, J. Shan, and K. F. Mak, Evidence of the fractional quantum spin Hall effect in moir´ e MoTe2, Nature628, 522 (2024)

    2024

  30. [30]

    Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Barmak, J. Hone, et al., Superconductivity in 5.0°twisted bilayer WSe2, Nature637, 839 (2025)

    2025

  31. [31]

    D. Zhai, H. Yu, and W. Yao, Twistronics and moir´ e su- perlattice physics in 2D transition metal dichalcogenides, 7 Rep. Prog. Phys.88, 084501 (2025)

    2025

  32. [32]

    A. K. Geim and I. V. Grigorieva, Van der Waals het- erostructures, Nature499, 419 (2013)

    2013

  33. [33]

    K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. Castro Neto, 2d materials and van der Waals het- erostructures, Science353, aac9439 (2016)

    2016

  34. [34]

    Kresse and J

    G. Kresse and J. Furthm¨ uller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B54, 11169 (1996)

    1996

  35. [35]

    A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, et al., Commentary: The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater.1, 011002 (2013)

    2013

  36. [36]

    Curtarolo, G

    S. Curtarolo, G. L. Hart, M. B. Nardelli, N. Mingo, S. Sanvito, and O. Levy, The high-throughput highway to computational materials design, Nat. Mater.12, 191 (2013)

    2013

  37. [37]

    S. R. Kalidindi and M. De Graef, Materials data science: current status and future outlook, Annu. Rev. Mater. Res.45, 171 (2015)

    2015

  38. [38]

    fourth paradigm

    A. Agrawal and A. Choudhary, Perspective: Materials informatics and big data: Realization of the “fourth paradigm” of science in materials science, APL Mater. 4(2016)

    2016

  39. [39]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature547, 298 (2017)

    2017

  40. [40]

    Haastrup, M

    S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P. S. Schmidt, N. F. Hinsche, M. N. Gjerding, D. Torelli, P. M. Larsen, A. C. Riis-Jensen, et al., The Computational 2D Materials Database: high-throughput modeling and dis- covery of atomically thin crystals, 2D Mater.5, 042002 (2018)

    2018

  41. [41]

    Mounet, M

    N. Mounet, M. Gibertini, P. Schwaller, D. Campi, A. Merkys, A. Marrazzo, T. Sohier, I. E. Castelli, A. Ce- pellotti, G. Pizzi, et al., Two-dimensional materials from high-throughput computational exfoliation of experimen- tally known compounds, Nat. Nanotech.13, 246 (2018)

    2018

  42. [42]

    Zhang, Y

    T. Zhang, Y. Jiang, Z. Song, H. Huang, Y. He, Z. Fang, H. Weng, and C. Fang, Catalogue of topological elec- tronic materials, Nature566, 475 (2019)

    2019

  43. [43]

    Vergniory, L

    M. Vergniory, L. Elcoro, C. Felser, N. Regnault, B. A. Bernevig, and Z. Wang, A complete catalogue of high- quality topological materials, Nature566, 480 (2019)

    2019

  44. [44]

    Zhang, H

    Y. Zhang, H. Pi, J. Liu, W. Miao, Z. Qi, N. Regnault, H. Weng, X. Dai, B. A. Bernevig, Q. Wu, et al., Uni- versal moir´ e-model-building method without fitting: Ap- plication to twisted MoTe2 and WSe2, arXiv:2411.08108 (2024)

    Pith/arXiv arXiv 2024

  45. [45]

    Jiang, U

    Y. Jiang, U. Petralanda, G. Skorupskii, Q. Xu, H. Pi, D. C˘ alug˘ aru, H. Hu, J. Xie, R. A. Mustaf, P. H¨ ohn, et al., 2d theoretically twistable material database, arXiv preprint arXiv:2411.09741 (2024)

    Pith/arXiv arXiv 2024

  46. [46]

    Q. Xu, A. Fischer, N. Tancogne-Dejean, T. Zhang, E. V. n. Bostr¨ om, M. Claassen, D. M. Kennes, A. Ru- bio, and L. Xian, Engineering 2d square lattice Hubbard models in 90°twisted GeX/SnX (X = S, se) moir´ e super- lattices, Phys. Rev. X15, 041049 (2025)

    2025

  47. [47]

    S. Naik, M. H. Naik, I. Maity, and M. Jain, Twister: Construction and structural relaxation of commensu- rate moir´ e superlattices, Comput. Phys. Commun.271, 108184 (2022)

    2022

  48. [48]

    and E(3)-equivariant machine-learning force fields (MLFFs) [51], enabling efficient optimization of large su- percells with near first-principles accuracy [52]. Based on the relaxed geometries, electronic properties are eval- uated using density functional theory to construct rep- Monolayer structure Bilayer prototypes Truncatad atomic plane wave (b) Twis...

  49. [49]

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

    2018

  50. [50]

    C˘ alug˘ aru, Y

    D. C˘ alug˘ aru, Y. Jiang, H. Hu, H. Pi, J. Yu, M. G. Vergniory, J. Shan, C. Felser, L. M. Schoop, D. K. Efe- tov, et al., Moir´ e materials based on M-point twisting, Nature643, 376 (2025)

    2025

  51. [51]

    C. Lei, P. T. Mahon, and A. H. MacDonald, Moir´ e band theory for M-valley twisted transition metal dichalco- genides, Phys. Rev. Lett.135, 196402 (2025)

    2025

  52. [52]

    Musaelian, S

    A. Musaelian, S. Batzner, A. Johansson, L. Sun, C. J. Owen, M. Kornbluth, and B. Kozinsky, Learning local equivariant representations for large-scale atomistic dy- namics, Nat. Commun.14, 579 (2023)

    2023

  53. [53]

    J. Liu, Z. Fang, H. Weng, and Q. Wu, DPmoire: a tool for constructing accurate machine learning force fields in moir´ e systems, npj Comput. Mater.11, 248 (2025)

    2025

  54. [54]

    W. Miao, C. Li, X. Han, D. Pan, and X. Dai, Truncated atomic plane wave method for subband structure calcula- tions of moir´ e systems, Phys. Rev. B107, 125112 (2023)

    2023

  55. [55]

    Zhang, Q

    Y. Zhang, Q. Wu et al., In preparation (2026)

    2026

  56. [56]

    C. L. Kane and E. J. Mele,Z 2 topological order and the quantum spin Hall effect, Phys. Rev. Lett.95, 146802 (2005)

    2005

  57. [57]

    It also provides a comprehensive analysis of the electronic and topological properties of moir´ e superlattices across a wide range of experimentally feasible twist angles

    See Supplementary Information for details on the com- putational framework, as well as the electronic structures and energy landscapes of screened parent materials in monolayer, bilayer, and bulk forms. It also provides a comprehensive analysis of the electronic and topological properties of moir´ e superlattices across a wide range of experimentally feas...

  58. [58]

    F. D. M. Haldane, Geometrical description of the frac- tional quantum Hall effect, Phys. Rev. Lett.107, 116801 (2011)

    2011

  59. [59]

    Roy, Band geometry of fractional topological insula- tors, Phys

    R. Roy, Band geometry of fractional topological insula- tors, Phys. Rev. B90, 165139 (2014)

    2014

  60. [60]

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, C. R. Physique14, 816 (2013)

    2013

This paper was first reviewed by deepseek-v4-flash on August 1, 2026.