Pith. sign in

REVIEW 4 major objections 6 minor 51 references

MoireStudio claims a single Python package can build accurate tight-binding and continuum models, with full relaxation, for arbitrary combinations of two-dimensional materials at a fraction of direct DFT cost.

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 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review

2026-08-03 02:41 UTC pith:FNJVIRCP

load-bearing objection MoireStudio is a plausible integrated twistronics package, but the 'universal' claim outruns the evidence and the code isn't visible. the 4 major comments →

arxiv 2602.09739 v1 pith:FNJVIRCP submitted 2026-02-10 cond-mat.mtrl-sci

MoireStudio: A Universal Twisted Electronic Structure Calculation Package

Junxi Yu , Yichen Liu , Cheng-Cheng Liu This is my paper
classification cond-mat.mtrl-sci
keywords twistronicsmoiré superlatticestight-binding modelcontinuum k·p modellattice relaxationcommensurate angle searchtwo-dimensional materialselectronic structure software
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 paper describes MoireStudio, a Python software package built to eliminate the technical barriers in twistronics calculations. It claims that from one JSON input file and a few Wannier-function outputs for untwisted bilayer stackings, a user can search commensurate angles, generate relaxed moiré structures, and construct both tight-binding and continuum Hamiltonians for any two-dimensional material pair, including rectangular lattices and heterostructures. The central bet is that the local-stacking approximation plus an analytical relaxation field is accurate enough to serve as a universal workflow. If true, researchers could study materials like black phosphorus, CrPS4, and graphene/hBN without hand-written scripts, and small-angle systems with tens of thousands of atoms become tractable on a small cluster. The paper demonstrates this on twisted bilayer graphene, MoS2, CrPS4, and twisted MoTe2, including topological phase diagrams with relaxation.

Core claim

The authors propose that the entire twisted electronic structure pipeline—commensurate angle search, moiré geometry generation, relaxation, and Hamiltonian construction—can be automated under the local-stacking approximation. Interlayer hopping between orbitals is assumed to depend only on their separation, and the coupling function is fitted from a handful of untwisted bilayer Wannier Hamiltonians, then mapped onto the twisted geometry. Relaxation is handled analytically rather than by large supercell force calculations: in-plane displacements enter as phase factors and out-of-plane displacements as coupling-strength corrections. On that basis the authors report band structures for twisted

What carries the argument

The workhorse is the two-center interlayer coupling function (Eq. 4): a Gaussian-exponential form t_ij(r) = h0 exp[-l(rz-d0)^2/r0^2] exp[-(rx^2+ry^2)/r0^2], whose parameters (h0, d0, r0) are fitted from a few non-equivalent untwisted stacking Hamiltonians. Together with the assumption that the intralayer Hamiltonian is unchanged by twist, this function converts any moiré geometry into a tight-binding Hamiltonian. The second load-bearing component is the analytical relaxation displacement field (Eq. 7): Fourier sums over moiré reciprocal vectors for in-plane and out-of-plane displacements, each with a single elastic constant (κ∥, κ⊥). These displacements are folded into both the tight-binding

Load-bearing premise

The package's claim to work for arbitrary 2D materials rests on the assumption that a twisted bilayer behaves as a patchwork of rigid untwisted local stackings, with interlayer hopping between any two orbitals depending only on the distance between their atoms and captured by a single fitted exponential form; if that fails for an untested material, the generated Hamiltonians will be wrong even though the software runs without complaint.

What would settle it

Take a twisted bilayer of a material with strongly directional interlayer bonding—say, a layered oxide or a compound with significant p-d hybridization—build the MoireStudio tight-binding Hamiltonian from only a few untwisted stackings, and compare the resulting band structure to a direct DFT calculation of the same commensurate supercell at a moderate twist angle near 20°. If the two disagree by more than the tolerance achieved in the paper's own TBG and MoS2 examples, the universal-accuracy claim is falsified.

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

Share X Bluesky LinkedIn Reddit HN

If this is right

  • A single input file takes a researcher from monolayer/bilayer Wannier data to a twisted band structure, removing the need for custom scripts and reducing the technical floor for entering twistronics.
  • The package extends automated moiré modeling to rectangular lattices, low-symmetry materials, and heterostructures—cases that the usual hexagonal-trilayer tools cannot handle.
  • Sparse, parallel solvers make small-angle systems with more than 10,000 atoms per cell tractable: the paper reports a magic-angle graphene band structure in about ten minutes on 64 cores.
  • Analytical relaxation is included for both tight-binding and continuum models, letting users see how band flatness and Chern numbers shift when in-plane and out-of-plane atomic displacements are turned on.
  • Outputs in standard exchange formats put the generated Hamiltonians into the hands of many-body and transport codes, opening a path from geometry to correlated-phase predictions.

Where Pith is reading between the lines

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

  • The 'universal' label is only as broad as the two-center, distance-only interlayer coupling; materials with strongly directional interlayer hybridization or significant screening could require more than the fitted Gaussian form, and the paper does not validate such a case.
  • The same local-stacking parameterization feeds both the tight-binding and continuum workflows, so MoireStudio could in principle support high-throughput screening: any 2D material with Wannier outputs could get a twisted band structure without new methodology.
  • The Fourier expansion of relaxation phases in the continuum model points to a systematic route for including higher-order moiré harmonic couplings; comparing TB and continuum results at large twist angles would test whether that hierarchy converges quickly.
  • Because relaxation is computed analytically rather than by force-field relaxation, the package might also serve as a fast pre-screener for which twisted systems are worth expensive DFT relaxation studies.

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 / 6 minor

Summary. MoireStudio is a Python package that aims to provide a universal workflow for twisted (moiré) bilayer electronic-structure calculations. The package has three modules: a geometry module for commensurate-angle search and moiré structure generation, a tight-binding (TB) module that builds twisted Hamiltonians from monolayer/bilayer Wannier data using a fitted interlayer coupling function and optional relaxation, and a continuum k·p module that constructs plane-wave Hamiltonians with user-supplied or fitted parameters. Relaxation is incorporated through an analytical displacement field, with separate in-plane and out-of-plane contributions. The paper presents example calculations for twisted bilayer graphene, twisted MoS2, twisted CrPS4, black phosphorus, graphene/hBN, and twisted MoTe2, showing band structures, Chern numbers, and relaxation effects that agree with prior DFT and experimental results.

Significance. If its claims hold, MoireStudio would be a genuinely useful integrated tool: it consolidates several steps that are usually scattered across scripts, provides a JSON-based reproducible input format, interfaces with Wannier90 and VASP, and demonstrates a parallel sparse solver capable of handling magic-angle-size moiré cells. The four example families (hexagonal, rectangular, low-symmetry, heterostructure) illustrate a plausible design range, and the reproduced TBG flat bands and tMoTe2 Chern phase diagram are encouraging signs. However, the central 'applicable to arbitrary 2D materials' claim rests on a strong two-center isotropic interlayer coupling approximation and a relaxation model with user-supplied parameters; neither is validated for a material where angular, orbital-direction-dependent hybridization matters. The paper is a software announcement, so the absence of a public code repository, test suite, and benchmark scripts is a significant gap for the claimed contribution.

major comments (4)
  1. [§2.2, Eq. (4)] The load-bearing assumption for the 'universal' claim is that interlayer coupling t_ij(r) depends only on distance, with no dependence on the bond direction relative to orbital character. This is a strong form of the two-center approximation that omits the Slater-Koster angular factors needed for p_x/p_y and d orbitals. All electronic-structure validations in §4.2–§4.3 involve states dominated by p_z or chalcogen p_z orbitals; twisted black phosphorus is used only for geometry (Fig. 2). A concrete test would be to compare MoireStudio TB bands with direct DFT for a twisted bilayer of, e.g., black phosphorus or a d-electron system. Without such a test, the 'arbitrary 2D materials' claim in Section 1 is unsubstantiated.
  2. [§2.2, Eq. (4)] Eq. (4) has a formal error as printed. The text defines l as 'the sign of r_z − d_ij^0'. Then for r_z < d_ij^0, l is negative and the factor exp[−l(r_z−d_ij^0)^2/(r_ij^0)^2] becomes exp[+|...|], which grows without bound as the distance decreases. This is likely a typo (the intended form probably has an absolute value or a constant sign), but because this is the core interlayer coupling function used to construct every TB Hamiltonian, the equation must be corrected and the fitting procedure in §3.5 checked against the intended functional form.
  3. [§4.3, Figs. 8 and 9] The relaxation-enabled results are presented as a core feature, but the displacement field in Eq. (7) depends on user-supplied κ∥ and κ⊥ and the example uses order_num=1. The manuscript does not state what κ values were used in Figs. 8–9 or how a user determines them from DFT or experiment. Because the paper claims 'precise incorporation of full relaxation effects,' this is load-bearing: without a fitting protocol or validated defaults, the relaxation module's predictions are conditional on ad hoc input.
  4. [§3.2, §5] For a computational package paper, no public repository URL, no test suite, and no benchmark input/output files are given; the only availability statement is 'pip install MoireStudio'. Consequently, the timing claim in §4.2 ('approximately ten minutes' on 64 cores) and the Wannier input examples cannot be reproduced. I recommend the authors include the release repository, test examples, and a machine-readable specification of the JSON schema before publication.
minor comments (6)
  1. [§2.2, Eq. (4)] The symbol l is used both as an orbital index (i, j) and as the sign function; this invites confusion. Rename the sign variable, e.g., s_ij.
  2. [§2.4, Eq. (7)] The relaxation form is shown only for hexagonal lattices. The paper claims support for rectangular and arbitrary lattices, but no generalization of Eq. (7) is given for non-hexagonal cases. A sentence clarifying the range of validity would help.
  3. [Fig. 2 caption] The caption says '(a)-(d) Example ... (e)-(h) The minimum commensurable structure', but the text says 'as shown in Figs. 2(a)-(d)' for the generated structures. Please clarify which panels show angle distributions and which show structures.
  4. [Introduction] Typo: 'have given risen to' should be 'have given rise to'.
  5. [§3.3] 'V ASP format' should be 'VASP format'.
  6. [§2.3] The definition of GGG_j and ggg_j is terse; in particular, the numbering of reciprocal lattice points 'in order of increasing distance counterclockwise' should be stated more precisely, since the continuum model depends on this convention.

Circularity Check

0 steps flagged

No significant circularity: core equations are explicit model assumptions benchmarked against external DFT/experiment; no prediction reduces to its fitting input.

full rationale

The paper's derivation chain is a software implementation of established or explicitly approximate models, not a chain in which an output is identical to an input by construction. (i) The commensurability condition (Sec. 2.1, Eqs. 1-3) is a standard crystallographic rationality condition. (ii) The twisted TB Hamiltonian (Sec. 2.2, Eq. 4) uses the local-stacking approximation and the two-center approximation, with interlayer coupling parameters fitted to untwisted AA/AB bilayer Hamiltonians. The twisted bands are then genuine predictions for a different structure, and the package is benchmarked against external DFT for twisted CrPS4 (Fig. 5) and against experiment/DFT for magic-angle TBG (Fig. 6). The cited source [34] for Eq. (4) is the authors' own prior work, but it is not the only support: the underlying two-center approximation is attributed to the external ref. [36], and the fitted form is externally falsified in the examples. (iii) The continuum model (Secs. 2.3, 4.3) follows the standard Bistritzer-MacDonald/Wu et al. construction [20,21,22]; parameters such as inter_amps and intra_t_amps are supplied or taken from prior literature, and the resulting Chern numbers are compared with earlier DFT/experimental results. (iv) The relaxation module (Sec. 2.4, Eq. 7) uses the authors' analytical relaxation framework [35], but the kappa coefficients are user inputs or DFT-derivable parameters, not quantities secretly determined by the predicted band structures or Chern numbers. The relaxed structures and topologies are checked against previous DFT/MD and experiments. The main limitation is external validity of the 'universal' claim for arbitrary 2D materials: the distance-only two-center ansatz may fail when orbital-direction-dependent hybridization matters. That is a correctness or scope risk, not a circularity. No equation in the paper equals its own fitting input, and no load-bearing result depends solely on a self-citation chain.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim depends on fitted interlayer coupling parameters, user-supplied relaxation coefficients, and several modeling ansatze inherited from the authors' prior work. No new physical entities are introduced.

free parameters (3)
  • Interlayer coupling parameters h_ij^0, d_ij^0, r_ij^0 = fitted to DFT-Wannier Hamiltonians of several untwisted stacking configurations
    Eq. (4): the twisted TB Hamiltonian is built from these fitted parameters; accuracy of the package's TB bands depends on this fit.
  • Relaxation strength coefficients kappa_parallel and kappa_perp = user inputs in input.json, e.g. kappa_parallel=2e-3, kappa_perp=1e-3
    Eq. (7) and Section 3.1/4.1: relaxation displacement fields are proportional to 1/(theta^2 * kappa); the 'full relaxation' feature is controlled by these free coefficients rather than derived from first principles in this paper.
  • Continuum model couplings (inter_amps, inter_phas, intralayer amps/phases, effective mass) = example values: mass 0.62, inter_amps [-8.5, -8.5, -8.5]
    Section 3.6: the k.p workflow requires user-supplied or Wannier-fitted parameters; the demonstration tMoTe2 run uses hand-specified values.
axioms (5)
  • domain assumption Two-center approximation: interlayer coupling between orbitals i and j depends only on their distance vector.
    Section 2.2, Eq. (4). Not guaranteed for materials with strong covalent interlayer bonding or large orbital overlaps.
  • domain assumption Local stacking approximation: the twisted bilayer locally resembles an untwisted bilayer with an interlayer displacement.
    Section 2.2-2.3; underlies both TB and continuum Hamiltonian construction.
  • domain assumption Intralayer Hamiltonian is unchanged by twisting.
    Section 2.2; stated as generally negligible, but not tested for arbitrary materials.
  • ad hoc to paper Relaxation displacement field has the sinusoidal form of Eq. (7) with a limited number of harmonics (order_num=1 in the example).
    Section 2.4, Eq. (7); this comes from the authors' analytical framework [35] and is not independently derived or stress-tested in this paper.
  • standard math Commensurate superlattice requires the matrix L_t^{-1} R(theta) L_b to be rational; quasicrystals are excluded.
    Section 2.1, Eq. (3). Standard, but it limits the 'universal' claim to commensurate angles.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of MoireStudio: A Universal Twisted Electronic Structure Calculation Package." pith.science (2026). https://pith.science/paper/FNJVIRCP

@misc{pith2026260209739,
  author       = {Pith},
  title        = {Pith review of: MoireStudio: A Universal Twisted Electronic Structure Calculation Package},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FNJVIRCP}},
  note         = {Machine review of arXiv:2602.09739}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Twistronics is an emerging and captivating field in condensed matter physics and material science. However, accurately and efficiently calculating the electronic structures of twisted systems remains a significant challenge. To address this, we have developed MoireStudio, a universal Python-based computational package for twisted electronic structures. Its functionalities include commensurate structure search, structure generation, parameterization, and construction for tight-binding models and continuum models, and the precise incorporation of full relaxation effects. The package is applicable to arbitrary combinations of two-dimensional materials, including rectangular lattices and heterostructures. User-friendly and easy to use, MoireStudio supports parallel large-scale computations, provides visualization capabilities, and offers interfaces with third-party software. It is poised to become a convenient and powerful tool for researchers in twistronics fields.

Figures

Figures reproduced from arXiv: 2602.09739 by Cheng-Cheng Liu, Junxi Yu, Yichen Liu.

Figure 1
Figure 1. Figure 1: Workflow of MoireStudio. Starting from an input.json file, the system constructs and analyzes twisted material [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of MoireStudio structural module.(a)-(d) twisted bilayer graphene, twisted black phosphorene, twisted [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The influence of relaxation on the structure of tTMD. (a) Unrelaxed structure; (b) relaxed structure. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Band structure of (a) TBG and (b) twisted MoS [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Band structure of twisted bilayer CrPS4 at a twist angle of θ = 67.92◦ : (a) calculated by MoireStudio, (b) calculated by DFT (VASP)[42]. results[47, 36]. 4.3. Continuum Model Module In the continuum model section, we demonstrate the capabilities of MoireStudio us￾ing two highly studied twisted systems, TBG and tMoTe2, as examples. By setting "task": "band" in input.json, band structure calculations can be… view at source ↗
Figure 6
Figure 6. Figure 6: Band structure of magic-angle(1.05◦ ) TBG calculated using the parallel sparse matrix scheme in the TB module of MoireStudio with full relaxation. (a) (b) 1 -1 0 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Band structures calculated using the continuum model module of MoireStudio. (a) Band structure of TBG at [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effects of progressively incorporating relaxation on the low-energy bands of magic-angle TBG: (a) in-plane, (b) out-of-plane, and (c) combined relaxations. (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Change in the topology of the three highest valence bands of tMoTe [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗

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

51 extracted references · 7 canonical work pages

  1. [1]

    Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y . Luo, J. D. Sanchez-Y amagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, P . Jarillo-Herrero, Correlated in- sulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556 (7699) (2018) 80–84. doi:10.1038 /nature26154

    2018

  2. [2]

    Serlin, C

    M. Serlin, C. L. Tschirhart, H. Polshyn, Y . Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, A. F. Y oung, Intrinsic quantized anomalous Hall e ffect in a moiré heterostruc- ture, Science 367 (6480) (2020) 900–903. doi:10.1126 /science.aay5533

    2020

  3. [3]

    Y . Cao, V . Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, P . Jarillo-Herrero, Un- conventional superconductivity in magic-angle graphene superlattices, Nature 556 (7699) (2018) 43–50. doi:10.1038 /nature26160

    2018

  4. [4]

    Y ankowitz, S

    M. Y ankowitz, S. Chen, H. Polshyn, Y . Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Y oung, C. R. Dean, Tuning superconductivity in twisted bilayer graphene, Science 363 (6431) (2019) 1059–1064. doi:10.1126 /science.aav1910

    2019

  5. [5]

    X. Lu, P . Stepanov, W. Y ang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold, A. H. MacDonald, D. K. Efetov, Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature 574 (7780) (2019) 653–657. doi:10.1038 /s41586-019-1695-0

    2019

  6. [6]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y . Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y . Ran, T. Cao, L. Fu, D. Xiao, W. Y ao, X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature 622 (7981) (2023) 63–68. doi:10.1038 /s41586-023-06289-w

    2023

  7. [7]

    Y . Zeng, Z. Xia, K. Kang, J. Zhu, P . Knüppel, C. V aswani, K. Watanabe, T. Taniguchi, K. F. Mak, J. Shan, Thermodynamic evidence of fractional Chern insulator in moiré MoTe2, Nature 622 (7981) (2023) 69–73. doi:10.1038 /s41586-023-06452-3

    2023

  8. [8]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y . Gu, K. Watanabe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y . Zhang, X. Liu, T. Li, Observation of Integer and Fractional Quantum Anomalous Hall E ffects in Twisted Bilayer MoTe 2, Phys. Rev. X 13 (3) (2023) 031037. doi:10.1103/PhysRevX.13.031037. 17

    doi:10.1103/physrevx.13.031037 2023

  9. [9]

    Y . Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Barmak, J. Hone, A. J. Millis, A. Pasupathy, C. R. Dean, Superconductivity in 5.0◦ twisted bilayer WSe2, Nature 637 (8047) (2025) 839–845. doi:10.1038 /s41586-024- 08381-1

    2025

  10. [10]

    Z. Hao, A. M. Zimmerman, P . Ledwith, E. Khalaf, D. H. Najafabadi, K. Watan- abe, T. Taniguchi, A. Vishwanath, P . Kim, Electric field–tunable superconductivity in alternating-twist magic-angle trilayer graphene, Science 371 (6534) (2021) 1133–1138. doi:10.1126/science.abg0399

    work page doi:10.1126/science.abg0399 2021

  11. [11]

    K. Kang, B. Shen, Y . Qiu, Y . Zeng, Z. Xia, K. Watanabe, T. Taniguchi, J. Shan, K. F. Mak, Evidence of the fractional quantum spin Hall e ffect in moiré MoTe2, Nature 628 (8008) (2024) 522–526. doi:10.1038 /s41586-024-07214-5

    2024

  12. [12]

    Z. Lu, T. Han, Y . Y ao, A. P . Reddy, J. Y ang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, L. Ju, Fractional quantum anomalous Hall e ffect in multilayer graphene, Nature 626 (8000) (2024) 759–764. doi:10.1038 /s41586-023-07010-7

    2024

  13. [13]

    E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Y umigeta, M. Blei, J. D. Carlström, K. Watanabe, T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, F. Wang, Mott and generalized Wigner crystal states in WSe2 /WS2 moiré superlattices, Nature 579 (7799) (2020) 359–363. doi:10.1038 /s41586-020-2092-4

    2020

  14. [14]

    J. Xie, Z. Huo, X. Lu, Z. Feng, Z. Zhang, W. Wang, Q. Y ang, K. Watanabe, T. Taniguchi, K. Liu, Z. Song, X. C. Xie, J. Liu, X. Lu, Tunable Fractional Chern Insulators in Rhombo- hedral Graphene Superlattices, arXiv:2405.16944 (2025). doi:10.48550 /arXiv.2405.16944

    Pith/arXiv arXiv doi:10.48550/arxiv.2405.16944 2025

  15. [15]

    M. H. Naik, M. Jain, Ultraflatbands and Shear Solitons in Moir \’e Patterns of Twisted Bilayer Transition Metal Dichalcogenides, Phys. Rev. Lett. 121 (26) (2018) 266401. doi:10.1103/PhysRevLett.121.266401

    work page doi:10.1103/physrevlett.121.266401 2018

  16. [16]

    Devakul, V

    T. Devakul, V . Crépel, Y . Zhang, L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nat Commun 12 (1) (2021) 6730. doi:10.1038 /s41467-021-27042-9

    2021

  17. [17]

    N. Mao, C. Xu, J. Li, T. Bao, P . Liu, Y . Xu, C. Felser, L. Fu, Y . Zhang, Transfer learning relaxation, electronic structure and continuum model for twisted bilayer MoTe2, Commun. Phys. 7 (1) (2024) 262. doi:10.1038 /s42005-024-01754-y

    2024

  18. [18]

    Suárez Morell, J

    E. Suárez Morell, J. D. Correa, P . V argas, M. Pacheco, Z. Barticevic, Flat bands in slightly twisted bilayer graphene: Tight-binding calculations, Phys. Rev. B 82 (12) (2010) 121407. doi:10.1103/PhysRevB.82.121407

    doi:10.1103/physrevb.82.121407 2010

  19. [19]

    Trambly De Laissardière, D

    G. Trambly De Laissardière, D. Mayou, L. Magaud, Localization of Dirac Electrons in Rotated Graphene Bilayers, Nano Lett. 10 (3) (2010) 804–808. doi:10.1021 /nl902948m

    2010

  20. [20]

    J. M. B. Lopes dos Santos, N. M. R. Peres, A. H. Castro Neto, Graphene Bi- layer with a Twist: Electronic Structure, Phys. Rev. Lett. 99 (25) (2007) 256802. doi:10.1103/PhysRevLett.99.256802

    doi:10.1103/physrevlett.99.256802 2007

  21. [21]

    Bistritzer, A

    R. Bistritzer, A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. U.S.A. 108 (30) (2011) 12233–12237. doi:10.1073 /pnas.1108174108. 18

    2011

  22. [22]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, A. H. MacDonald, Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers, Phys. Rev. Lett. 122 (8) (2019) 086402. doi:10.1103/PhysRevLett.122.086402

    doi:10.1103/physrevlett.122.086402 2019

  23. [23]

    J. Cao, M. Wang, S.-F. Qian, C.-C. Liu, Y . Y ao, Ab Initio four-band Wannier tight- binding model for generic twisted graphene systems, Phys. Rev. B 104 (8) (2021) L081403. doi:10.1103/PhysRevB.104.L081403

    work page doi:10.1103/physrevb.104.l081403 2021

  24. [24]

    Y . Li, Z. Zhan, X. Kuang, Y . Li, S. Y uan, TBPLaS: A tight-binding package for large-scale simulation, Computer Physics Communications 285 (2023) 108632. doi:10.1016/j.cpc.2022.108632

    arXiv 2023

  25. [25]

    Z. Ma, S. Li, Y .-W. Zheng, M.-M. Xiao, H. Jiang, J.-H. Gao, X. Xie, Topolog- ical flat bands in twisted trilayer graphene, Science Bulletin 66 (1) (2021) 18–22. doi:10.1016/j.scib.2020.10.004. URL https://linkinghub.elsevier.com/retrieve/pii/S2095927320306575

    work page doi:10.1016/j.scib.2020.10.004 2021

  26. [26]

    Y . Gao, Y . Zhang, D. Xiao, Tunable Layer Circular Photogalvanic Effect in Twisted Bilay- ers, Physical Review Letters 124 (7) (2020) 077401. doi:10.1103/PhysRevLett.124.077401. URL https://link.aps.org/doi/10.1103/PhysRevLett.124.077401

    doi:10.1103/physrevlett.124.077401 2020

  27. [27]

    J. Liu, Z. Fang, H. Weng, Q. Wu, DPmoire: A tool for constructing accurate ma- chine learning force fields in moiré systems, npj Comput Mater 11 (1) (2025) 248. doi:10.1038/s41524-025-01740-0

    work page doi:10.1038/s41524-025-01740-0 2025

  28. [28]

    H. Li, Z. Wang, N. Zou, M. Y e, R. Xu, X. Gong, W. Duan, Y . Xu, Deep-learning density functional theory Hamiltonian for e fficient ab initio electronic-structure calculation, Nat Comput Sci 2 (6) (2022) 367–377. doi:10.1038 /s43588-022-00265-6

    2022

  29. [29]

    T. Bao, R. Xu, H. Li, X. Gong, Z. Tang, J. Fu, W. Duan, Y . Xu, Deep-Learning Database of Density Functional Theory Hamiltonians for Twisted Materials, arXiv (2024). URL http://arxiv.org/abs/2404.06449

    Pith/arXiv arXiv 2024

  30. [30]

    B. Xie, J. Liu, Lattice distortions, moiré phonons, and relaxed electronic band struc- tures in magic-angle twisted bilayer graphene, Phys. Rev. B 108 (9) (2023) 094115. doi:10.1103/PhysRevB.108.094115

    work page doi:10.1103/physrevb.108.094115 2023

  31. [31]

    N. N. T. Nam, M. Koshino, Lattice relaxation and energy band modulation in twisted bilayer graphene, Phys. Rev. B 96 (7) (2017) 075311. doi:10.1103 /PhysRevB.96.075311

    2017

  32. [32]

    Tarnopolsky, A

    G. Tarnopolsky, A. J. Kruchkov, A. Vishwanath, Origin of Magic Angles in Twisted Bilayer Graphene, Phys. Rev. Lett. 122 (10) (2019) 106405. doi:10.1103/PhysRevLett.122.106405

    doi:10.1103/physrevlett.122.106405 2019

  33. [33]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, Y . He, X. Xu, Y . Ran, T. Cao, D. Xiao, Fractional Chern Insulator in Twisted Bilayer MoTe2, Phys. Rev. Lett. 132 (3) (2024) 036501. doi:10.1103/PhysRevLett.132.036501

    doi:10.1103/physrevlett.132.036501 2024

  34. [34]

    J. Y u, S. Qian, C.-C. Liu, General electronic structure calculation method for twisted sys- tems, Phys. Rev. B 111 (7) (2025) 075434. doi:10.1103 /PhysRevB.111.075434

    2025

  35. [35]

    J. Y u, B. Wang, C.-C. Liu, Relaxation and Its E ffects on Electronic Structure in Twisted Systems: An Analytical Perspective (2025). doi:10.48550 /arXiv.2509.13114. 19

    Pith/arXiv arXiv doi:10.48550/arxiv.2509.13114 2025

  36. [36]

    J. Jung, A. Raoux, Z. Qiao, A. H. MacDonald, Ab Initio theory of moiré superlat- tice bands in layered two-dimensional materials, Phys. Rev. B 89 (20) (2014) 205414. doi:10.1103/PhysRevB.89.205414

    work page doi:10.1103/physrevb.89.205414 2014

  37. [37]

    Zhang, C

    X.-W. Zhang, C. Wang, X. Liu, Y . Fan, T. Cao, D. Xiao, Polarization-driven band topology evolution in twisted MoTe2 and WSe2, Nat. Commun. 15 (1) (2024) 4223. doi:10.1038/s41467-024-48511-x

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

  38. [38]

    Marzari, D

    N. Marzari, D. V anderbilt, Maximally localized generalized wannier func- tions for composite energy bands, Phys. Rev. B 56 (20) (1997) 12847–12865. doi:10.1103/PhysRevB.56.12847

    doi:10.1103/physrevb.56.12847 1997

  39. [39]

    A. A. Mostofi, J. R. Y ates, Y .-S. Lee, I. Souza, D. V anderbilt, N. Marzari, Wannier90: A tool for obtaining maximally-localised wannier functions, Comput Phys Commun 178 (9) (2008) 685–699. doi:10.1016 /j.cpc.2007.11.016

    2008

  40. [40]

    Souza, N

    I. Souza, N. Marzari, D. V anderbilt, Maximally localized wannier functions for entangled energy bands, Phys. Rev. B 65 (3) (2001) 035109. doi:10.1103 /PhysRevB.65.035109

    2001

  41. [41]

    T. Cole, S. Coh, D. V anderbilt, Python Tight Binding (PythTB) (Nov. 2025). doi:10.5281/zenodo.12721315. URL https://zenodo.org/records/12721315

    arXiv 2025

  42. [42]

    Kresse, J

    G. Kresse, J. Furthmüller, E fficient iterative schemes for ab initio total-energy cal- culations using a plane-wave basis set, Phys. Rev. B 54 (16) (1996) 11169–11186. doi:10.1103/PhysRevB.54.11169. URL https://link.aps.org/doi/10.1103/PhysRevB.54.11169

    doi:10.1103/physrevb.54.11169 1996

  43. [43]

    V . Wang, N. Xu, J.-C. Liu, G. Tang, W.-T. Geng, V ASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using V ASP code, Computer Physics Communications 267 (2021) 108033. doi:10.1016 /j.cpc.2021.108033

    arXiv 2021

  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, J. Y u, Universal Moiré-Model-Building Method without Fitting: Application to Twisted MoTe$_2$ and WSe$_2$ (2024). doi:10.48550 /arXiv.2411.08108

    Pith/arXiv arXiv doi:10.48550/arxiv.2411.08108 2024

  45. [45]

    Y . Jia, J. Y u, J. Liu, J. Herzog-Arbeitman, Z. Qi, H. Pi, N. Regnault, H. Weng, B. A. Bernevig, Q. Wu, Moiré fractional Chern insulators. I. First-principles calculations and continuum models of twisted bilayer MoTe2, Phys. Rev. B 109 (20) (2024) 205121. doi:10.1103/physrevb.109.205121

    doi:10.1103/physrevb.109.205121 2024

  46. [46]

    Y . Liu, J. Y u, C.-C. Liu, Twisted Magnetic V an der Waals Bilayers: An Ideal Platform for Altermagnetism, Phys. Rev. Lett. 133 (20) (2024) 206702. doi:10.1103/PhysRevLett.133.206702

    doi:10.1103/physrevlett.133.206702 2024

  47. [47]

    Koshino, N

    M. Koshino, N. F. Q. Y uan, T. Koretsune, M. Ochi, K. Kuroki, L. Fu, Maximally Localized Wannier Orbitals and the Extended Hubbard Model for Twisted Bilayer Graphene, Phys. Rev. X 8 (3) (2018) 031087. doi:10.1103 /PhysRevX.8.031087

    2018

  48. [48]

    Cantele, D

    G. Cantele, D. Alfè, F. Conte, V . Cataudella, D. Ninno, P . Lucignano, Structural relaxation and low-energy properties of twisted bilayer graphene, Phys. Rev. Research 2 (4) (2020) 043127. doi:10.1103 /PhysRevResearch.2.043127. 20

    2020

  49. [49]

    S. Lisi, X. Lu, T. Benschop, T. A. de Jong, P . Stepanov, J. R. Duran, F. Margot, I. Cucchi, E. Cappelli, A. Hunter, A. Tamai, V . Kandyba, A. Giampietri, A. Barinov, J. Jobst, V . Stal- man, M. Leeuwenhoek, K. Watanabe, T. Taniguchi, L. Rademaker, S. J. van der Molen, M. P . Allan, D. K. Efetov, F. Baumberger, Observation of flat bands in twisted bilayer ...

    2021

  50. [50]

    Lucignano, D

    P . Lucignano, D. Alfè, V . Cataudella, D. Ninno, G. Cantele, Crucial role of atomic corruga- tion on the flat bands and energy gaps of twisted bilayer graphene at the magic angle, Phys. Rev. B 99 (19) (2019) 195419. doi:10.1103 /PhysRevB.99.195419

    2019

  51. [51]

    N. P . Kazmierczak, M. V an Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. Brown, J. Cis- ton, T. Taniguchi, K. Watanabe, D. K. Bediako, Strain fields in twisted bilayer graphene, Nat. Mater. 20 (7) (2021) 956–963. doi:10.1038 /s41563-021-00973-w. 21

    2021

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.