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REVIEW 3 major objections 6 minor 73 references

G(Wa)NN claims to be the first deep-learning model that generates electronic Hamiltonians directly in an orthogonal Wannier basis, turning a crystal structure into a sparse spin-orbit coupled tight-binding matrix.

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2026-08-01 15:29 UTC pith:RFLIBAQO

load-bearing objection A large-scale, credible ML effort for orthogonal Wannier Hamiltonians with a real engineering contribution, but the 'first' and '10K+ atom' claims outrun the evidence, and the missing graph cutoff is a genuine unresolved issue. the 3 major comments →

arxiv 2607.18403 v1 pith:RFLIBAQO submitted 2026-07-20 cond-mat.mtrl-sci

Towards a universal model for spin-orbit coupled Wannier Hamiltonians

Alexander C. Tyner This is my paper
classification cond-mat.mtrl-sci PACS 71.15.-m71.70.Ej72.10.-d
keywords Wannier Hamiltoniandeep learningspin-orbit couplingequivariant graph neural networkorthogonal basiskernel polynomial methodtight-bindingtransport simulation
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 introduces G(Wa)NN, a graph neural network trained on over 111,000 Wannier Hamiltonians (150M+ hop matrices) spanning 69 elements. Its central claim: given only a crystal structure, it produces a spin-orbit coupled electronic Hamiltonian in an orthogonal Wannier basis, with the sparsity needed for linear-scaling transport calculations. A companion package, Tailwater, handles subspace projection, KPM spectral functions, and local fine-tuning. If the claim holds, it closes a gap between first-principles accuracy and device-scale simulations. Validation targets include band gaps (0.08 eV MAE on a 2,000-material set), density of states, and a topological invariant Z2=1 with a surface Dirac cone for Bi2Se3.

Core claim

G(Wa)NN is an E(3)-equivariant message-passing network that predicts each Hamiltonian block—on-site and inter-site, including periodic images—as a complex matrix in a fixed spinful Wannier basis. Because the basis is orthogonal, the predicted Hamiltonian is strictly sparse. The training objective combines a real-space matrix-element loss with a momentum-space loss that weights states near the band edges, enforces correct level spacings, and protects symmetry-required band crossings. The paper demonstrates that the predicted Hamiltonians reproduce DFT band structures and density of states, identify the topological invariant of a topological insulator, and can be fine-tuned on a small custom d

What carries the argument

The central object is the spinful Wannier Hamiltonian block, reconstructed by inverting Clebsch–Gordan decomposition: the network emits covariant coefficients that are combined with precomputed coupling tensors into orbital-resolved blocks, and spin is attached through spin-scalar and spin-vector channels. The orthogonal, localized basis is what keeps matrices sparse and enables strict linear-scaling spectral methods. A second load-bearing piece is the momentum-space loss, which anchors eigenvalues to band edges, preserves level spacings, and protects crossings.

Load-bearing premise

The model assumes that all important Hamiltonian matrix elements are determined by atoms within some finite graph neighborhood, but the paper never states the cutoff radius or shows that predictions converge as that radius grows.

What would settle it

Take a material known to have significant long-range hoppings, generate its Hamiltonian with the model using a series of increasing graph cutoffs, and compare the predicted band structure and transport coefficients against a converged DFT-Wannier reference; if spectra or small energy splittings shift beyond tolerance when the cutoff is enlarged, the locality assumption fails.

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

If this is right

  • If the model generalizes as claimed, electronic-structure and transport simulations on systems with 10,000+ atoms become routine, with spectral observables computed in strictly linear time.
  • Band-gap screening on material databases could be done at model speed with accuracy close to dedicated ML gap predictors (0.08 eV MAE).
  • The sparse orthogonal Hamiltonian format drops directly into existing tight-binding and KPM pipelines, so topological invariants, surface spectra, and spin-Hall conductivities can be computed without an intervening DFT step.
  • Local fine-tuning on user datasets—including with reduced orbital bases—lets the base model be adapted to different functionals or out-of-distribution moiré systems at negligible cost.

Where Pith is reading between the lines

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

  • The paper never states the graph cutoff radius that defines neighboring blocks for Hamiltonian prediction; until a locality-convergence test is published, the claim of universal transferability to systems with long-range hybridization is premature.
  • The fine-tuning result suggests the pre-trained model could serve as a cheap preconditioner or starting point for hybrid-functional or many-body corrections, though only one example is shown.
  • The same sparse orthogonal Hamiltonian could feed NEGF and Chebyshev transport codes beyond KPM, potentially extending the acceleration to device-level current calculations.

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

3 major / 6 minor

Summary. The paper introduces G(Wa)NN, an E(3)-equivariant graph neural network that predicts spin-orbit coupled electronic Hamiltonians in an orthogonal Wannier basis directly from crystal structures. The model is trained on more than 111,000 Wannier90-generated Hamiltonians spanning 69 elements, drawn from MP-Traj, C2DB, GNOME, and Alexandria. The authors report a Hamiltonian matrix-element MAE of 10 meV, a normalized DOS MAE of 0.04 on a 2000-material GNOME validation set, a band-gap MAE of 0.08 eV, and successful reproduction of the Z2=1 topological phase and surface Dirac cone of Bi2Se3. They also present a local finetuning workflow for WSe2 bilayers and introduce Tailwater, a Python package for subspace projection and kernel-polynomial-method (KPM) spectral computations. The central claim is that the orthogonal, sparse Wannier representation enables linear-scaling inference and transport simulations at scales of 10K+ atoms, which would be unprecedented for machine-learned electronic-structure models.

Significance. If the central claims are substantiated, this is a significant contribution: direct prediction in an orthogonal, localized Wannier basis avoids the overlap-matrix problem of LCAO-based Hamiltonian models and enables the use of linear-scaling KPM methods for spectral and transport observables. The dataset size and diversity, the focus on gauge-invariant observables rather than raw matrix elements, the topological benchmark on Bi2Se3, and the open-source Tailwater package are clear strengths. The finetuning demonstration on a 78-atom twisted WSe2 system is a useful practical result. However, the paper's headline claims about universality, strict linear scaling, and 10K+ atom capability currently outrun the evidence presented; the missing locality/convergence analysis and the unclear train/test separation are load-bearing issues that must be addressed before the results can be fully credited.

major comments (3)
  1. [§II and §IV] The training set is stated to be built from structures of MP-Traj, C2DB, GNOME, and Alexandria (§II), while the validation set used for all headline benchmarks is described as '2000 materials from the GNOME dataset' (§IV, Figs. 4 and 5). The Fig. 5 caption asserts these materials were 'not seen in training,' but the paper never reports the train/validation split protocol, duplicate removal, or any filtering that would prevent GNOME structures from appearing in both training and validation. If GNOME-derived structures are in the training pool, the reported DOS MAE, band-gap MAE, and R2 values are leakage-inflated. Please state the exact split, the number of GNOME structures in the training set, and the procedure used to ensure no overlap or near-duplicate leakage.
  2. [§III and §V] The model predicts inter-site Hamiltonian blocks H_ij(R) 'per directed edge between atom i and a periodic image of atom j' (§III), but the graph construction—specifically the neighbor cutoff radius or edge criterion—is never reported. The Wannier spread check of Sec. II (<1.5|v_max|) does not control the decay of hopping amplitudes, so it does not justify omission of longer-range blocks. Without a stated cutoff and a convergence study in terms of physical observables (DOS, band gap, small energy splittings, transport coefficients), the sparse Hamiltonian produced for large systems may contain systematic errors that the averaged DOS/band-gap metrics are insensitive to. Please report the cutoff and provide convergence tests demonstrating that the truncated Hamiltonian faithfully reproduces target spectra and topology.
  3. [Abstract, §V.C] The abstract claims 'optimized inference and linear-scaling methods for orthogonal Hamiltonians unlock transport simulations at massive scales (10K+ atoms),' and §V.C states a primary goal of '1K-10K+ atoms.' However, no wall-clock scaling plot is shown for either inference or KPM spectral evaluation, and no calculation at 10K atoms is demonstrated. The runtime comparison in Fig. 7(e-f) is a small set of generation times, not a scaling analysis. 'Strict linear scaling' is asserted but not established. Please provide asymptotic scaling measurements for the full pipeline (graph construction, message passing, Hamiltonian assembly, and KPM observables) and an explicit demonstration at or near 10K atoms, or soften the claims accordingly.
minor comments (6)
  1. [Abstract and §I] The phrase 'first deep-learning model capable of generating the electronic Hamiltonian of solid-state systems in an orthogonal Wannier basis' is stronger than the text supports, given that Ref. [30] already generates Wannier Hamiltonians for a bilayer system. Recommend qualifying with 'at scale' or 'as a foundational model.'
  2. [§II] The sentence 'we ensure the total number of states is equal to the number of Wannier functions plus ten additional states' should say 'the number of Kohn-Sham bands' rather than 'states,' which is confusing in a spinor context.
  3. [§III.C] The composite loss weights λΔ and λ× are said to be 'set so that the spacing and crossing terms refine rather than overwhelm the eigenvalue error,' but their numerical values are not reported. Please provide the values or the tuning procedure.
  4. [§IV] The 10 meV Hamiltonian matrix-element MAE is mentioned without defining the gauge alignment procedure used before computing the error. Since the model is trained with a gauge-dependent target, this metric is meaningful only with a stated gauge convention; please clarify.
  5. [§V.C] The statement 'This sparse matrix can be evaluated directly to extract eigenvalues' should mention that sparse eigensolvers are used, and that their scaling depends on condition number and spectrum. The manuscript's linear-scaling claim should not conflate matrix sparsity with eigensolver scaling.
  6. [§V.B] The code snippet includes USER and PSWRD arguments, implying the API requires credentials. Please state more clearly the current access model (open weights vs. hosted API only) and whether the trained G(Wa)NN weights will be released.

Circularity Check

0 steps flagged

No significant circularity: supervised ML benchmark with held-out external validation; self-citations are motivational only.

full rationale

G(Wa)NN is an empirical machine-learning surrogate rather than a derivation: the structure-to-Hamiltonian map is learned from Wannier90 outputs and assessed on a held-out 2000-material GNOME validation set with external baselines (CGCNN, ALIGNN, coGN) and a Bi2Se3 Z2Pack/topological-surface check. The objective (Eqs. (10)-(12)) is a standard supervised loss on target matrix elements and target Bloch eigenvalues, so no 'prediction' is constructed from fitted outputs. Eq. (7) is an exact Clebsch-Gordan basis expansion for a (2ℓα+1)×(2ℓβ+1) block, not a fit disguised as a result. DOS and band-gap benchmarks compare predicted vs. target Wannier/DFT spectra, and the parity plot explicitly states the validation set was not seen in training. Self-citations (Refs. [12], [13], [63]) support background motivation—interconnect search and moiré materials—and are corroborated by external references; none is the load-bearing justification for model accuracy. The undisclosed graph cutoff for inter-site blocks is a robustness/convergence gap, but no equation or fitted value makes the reported observables equal to training targets by construction. I therefore find no circular step.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The framework introduces a model and software package, not new physical entities. The central claims rest on standard DFT/Wannier90 methodology plus several hand-chosen modeling decisions (fixed orbital basis, frozen window, graph locality, heuristic data filtering) that are not fully validated.

free parameters (6)
  • Graph edge cutoff radius = not stated
    Model constructs graph edges between atoms and periodic images; no cutoff is specified, yet all Hamiltonian blocks are generated from these edges.
  • Loss weights λΔ, λ× = not stated
    Eq. (12) uses λΔ and λ× set so that spacing/crossing terms "refine rather than overwhelm"; exact values are omitted.
  • Architecture hyperparameters (layers, channels, ℓmax) = not stated
    The e3nn network is described generically; no counts of layers/channels or maximum angular momentum are given, so reproduction requires guessing.
  • Per-element orbital basis assignment = s for Z≤2; sp for 2<Z≤18; spd for Z>18
    Tab. I fixes a uniform basis for each element; this is a hand-chosen modeling assumption that may omit semicore or f states.
  • Frozen window termination = 1 eV above Fermi
    Wannier90 disentanglement window ends at EF+1 eV; choice affects target fidelity in the conduction region and the reported noise there.
  • Wannier spread threshold = 1.5 × |vmax|
    Data-quality filter; Wannier Hamiltonians with spread above this are rejected.
axioms (6)
  • domain assumption Wannier90 disentanglement plus maximal localization yields an accurate orthogonal Wannier Hamiltonian for the chosen frozen window.
    Assumed throughout Section II and used to generate all training targets; only spot-checked against DFT bands (MAE<0.05 eV).
  • domain assumption States within 1 eV above the Fermi energy capture all physically relevant low-energy physics.
    Used to set the frozen window and to justify subspace projection for transport.
  • ad hoc to paper A fixed s/sp/spd orbital basis per element is complete enough to represent the low-energy electronic structure of all compounds in the dataset.
    Tab. I assigns the same orbitals to every compound of an element, which may fail for elements needing f or semicore states (excluded lanthanides and Z>86 partly mitigate this).
  • ad hoc to paper Hamiltonian matrix elements depend only on a finite local graph neighborhood of atoms.
    The GNN predicts blocks from edges in a graph; no cutoff radius is stated, and no analysis shows that hoppings beyond the neighborhood are negligible.
  • domain assumption PBE-DFT with fully relativistic norm-conserving pseudopotentials, 60 Ry cutoff, and 6×6×6 k-grid provides converged reference Hamiltonians.
    Used to generate all training labels; the k-grid convergence is only explicitly justified for the smallest unit cell.
  • ad hoc to paper MP-Traj out-of-equilibrium frames selected with min bond-length difference >0.05 Å and max 5 frames/material are representative of finite-temperature and reconstructed structures.
    Selection rule is heuristic; no analysis of coverage or bias is provided.

pith-pipeline@v1.3.0-alltime-deepseek · 17598 in / 14882 out tokens · 121792 ms · 2026-08-01T15:29:52.609393+00:00 · methodology

0 comments
read the original abstract

While machine learning interatomic potentials (MLiPs) have matured to revolutionize material science, deep learning models for electronic structure are just beginning to emerge and restricted, almost exclusively, to non-orthogonal basis Hamiltonians. We introduce G(Wa)NN, the first deep-learning model capable of generating the electronic Hamiltonian of solid-state systems in an orthogonal Wannier basis. G(Wa)NN is trained on an unprecedented, diverse dataset of more than 111K Wannier Hamiltonians (150M+ hopping matrices) spanning 69 elements. The combination of optimized inference and linear-scaling methods for orthogonal Hamiltonians unlock transport simulations at massive scales (10K+ atoms). Crucially, the framework supports local finetuning, allowing users to adapt the base model to custom Wannier Hamiltonian datasets. To seamlessly translate these predictions into physical observables, we introduce Tailwater, a Python package providing an API interface to G(Wa)NN alongside a high performance post-processing library. Tailwater enables automated projection of the predicted Hamiltonian into an arbitrary low-energy subspace-directly mirroring familiar Wannier90 workflows-and includes a suite of Kernel Polynomial Method (KPM) functions that exploit the orthogonal basis to achieve strict linear scaling for spectral observables. The Tailwater ecosystem, with the G(Wa)NN model at its core, aims to help bridge the gap between deep learning and macro-scale quantum transport simulations.

discussion (0)

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Reference graph

Works this paper leans on

73 extracted references · 1 canonical work pages

  1. [1]

    The Wannier Bridge: from Ab Initio to Mesoscale3

    High-Accuracy/Small scale2. The Wannier Bridge: from Ab Initio to Mesoscale3. Large/Device ScaleAb initio Methods Wavefunction Methods (e.g. CCSD(T))Plane-Wave DFT w/ Hybrid Functionals (e.g. HSE06, PBE0)Plane-Wave DFT (e.g. PBE, LDA)State-of-the-art accuracy at high cost (< 10nm) Construct compact Hamiltonian w/ ab initio accuracy Plane-Wave DFT Hamilton...

  2. [2]

    / outputs

    Momentum-space loss The momentum-space termLk acts on the per-keigen- valuesϵp n andϵt n of the predicted and target Bloch Hamil- tonians (n the band index, thekdependence implicit) and combines three complementary terms. a. Band-edge–anchored eigenvalue error.A weighted eigenvalue MAE (Leig) that assigns full weight to states near the valence- and conduc...

    2000

  3. [3]

    stack and shift

    Example: Topological Insulator Bi 2Se3 To demonstrate the utility and accuracy of the G(Wa)NN-Tailwater ecosystem, Fig.(6) illustrates an example workflow using the rhombohedral (R¯3m) topo- logical insulatorBi2Se3—a prototypical material widely studied in community packages like WannierTools [51]. In Bi2Se3, the non-trivial topological phase (Z2 = [1; 0,...

  4. [4]

    Gall, The search for the most conductive metal for narrow interconnect lines, Journal of Applied Physics127, 050901 (2020)

    D. Gall, The search for the most conductive metal for narrow interconnect lines, Journal of Applied Physics127, 050901 (2020)

    2020

  5. [5]

    C.-T. Chen, U. Bajpai, N. A. Lanzillo, C.-H. Hsu, H. Lin, and G. Liang, Topological semimetals for scaled back-end- of-line interconnect beyond Cu, in2020 IEEE Interna- tional Electron Devices Meeting (IEDM)(IEEE, 2020) pp. 32–4

    2020

  6. [6]

    Y. S. Zou, C. L. Gan, M.-H. Chung, and H. Takiar, A review of interconnect materials used in emerging mem- ory device packaging: first-and second-level interconnect materials, Journal of Materials Science: Materials in Elec- tronics32, 27133 (2021)

    2021

  7. [7]

    D. Gall, J. J. Cha, Z. Chen, H.-J. Han, C. Hinkle, J. A. Robinson, R. Sundararaman, and R. Torsi, Materials for interconnects, MRS Bulletin , 1 (2021)

    2021

  8. [8]

    J. H. Moon, E. Jeong, S. Kim, T. Kim, E. Oh, K. Lee, H. Han, and Y. K. Kim, Materials quest for advanced interconnect metallization in integrated circuits, Adv. Sci. 10, 2207321 (2023)

    2023

  9. [9]

    Batatia, P

    I. Batatia, P. Benner, Y. Chiang, A. M. Elena, D. P. Kovács, J. Riebesell, X. R. Advincula, M. Asta, M. Avay- lon, W. J. Baldwin,et al., A foundation model for atomistic materials chemistry, J. Chem. Phys.163, https://doi.org/10.1063/5.0297006 (2025)

    doi:10.1063/5.0297006 2025

  10. [10]

    Batatia, D

    I. Batatia, D. P. Kovacs, G. Simm, C. Ortner, and G. Csanyi, Mace: Higher order equivariant message pass- ing neural networks for fast and accurate force fields, inAdvances in Neural Information Processing Systems, Vol. 35, edited by S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh (Curran Associates, Inc.,

  11. [11]

    B. Deng, P. Zhong, K. Jun, J. Riebesell, K. Han, C. J. Bartel, and G. Ceder, Chgnet as a pretrained universal neural network potential for charge-informed atomistic modelling, Nature Machine Intelligence , 1–11 (2023). 13

    2023

  12. [12]

    Y.-L. Liao, B. Wood, A. Das, and T. Smidt, Equiformerv2: Improved equivariant transformer for scaling to higher- degree representations, inInternational Conference on Learning Representations, Vol. 2024, edited by B. Kim, Y. Yue, S. Chaudhuri, K. Fragkiadaki, M. Khan, and Y. Sun (2024) pp. 39282–39309

    2024

  13. [13]

    Chen and S

    C. Chen and S. P. Ong, A universal graph deep learning interatomic potential for the periodic table, Nat. Comput. Sci.2, 718 (2022)

    2022

  14. [14]

    Barros-Luque, M

    L. Barros-Luque, M. Shuaibi, X. Fu, B. M. Wood, M. Dzamba, M. Gao, A. Rizvi, M. Uyttendaele, C. L. Zit- nick, and Z. W. Ulissi, The open materials 2024 (omat24) inorganic materials dataset and models, Nat. Comput. Sci. , 1 (2026)

    2024

  15. [15]

    M. T. Kiani, A. C. Tyner, A. Jog, C.-T. Chen, J. M. Rondinelli, and J. J. Cha, Searching for materials for next-generation on-chip interconnects, Newton1, 10.1016/j.newton.2025.100133 (2025)

    arXiv 2025

  16. [16]

    A. C. Tyner, W. Rogers, P.-H. Shih, Y.-H. Tu, G. Liang, H. Lin, C.-T. Chen, and J. M. Rondinelli, Accelerated discovery of topological conductors for nanoscale inter- connects, Adv. Sci.13, e20535 (2026)

    2026

  17. [17]

    M. T. Kiani and J. J. Cha, Shrinking interconnects beyond copper, Science390, 572 (2025)

    2025

  18. [18]

    H. Kim, S. Oh, S. An, J. Kim, T. Kim, S. Jeong, O. Kaya, T. Galvani, S. Roche, J. J. Cha,et al., Future interconnect materialsforhighlyintegratedsemiconductordevices,Nat. Rev. Elect. Eng. , 1 (2025)

    2025

  19. [19]

    Merchant, S

    A. Merchant, S. Batzner, S. S. Schoenholz, M. Aykol, G. Cheon, and E. D. Cubuk, Scaling deep learning for materials discovery, Nature624, 80 (2023)

    2023

  20. [20]

    Zhong, R

    Y. Zhong, R. Wang, X. Gong, and H. Xiang, A univer- sal spin–orbit-coupled hamiltonian model for accelerated quantum material discovery, Nat. Mach. Intell. , 1 (2026)

    2026

  21. [21]

    X. Gong, H. Li, N. Zou, R. Xu, W. Duan, and Y. Xu, General framework for e (3)-equivariant neural network representation of density functional theory hamiltonian, Nat. Commun.14, 2848 (2023)

    2023

  22. [22]

    C. Qian, V. Vitartas, J. R. Kermode, and R. J. Maurer, Equivariant electronic hamiltonian prediction with many-body message passing, npj Comput. Mater. https://doi.org/10.1038/s41524-026-02020-1 (2026)

    doi:10.1038/s41524-026-02020-1 2026

  23. [23]

    Weiße, G

    A. Weiße, G. Wellein, A. Alvermann, and H. Fehske, The kernel polynomial method, Rev. Mod. Phys.78, 275 (2006)

    2006

  24. [24]

    Pizzi, V

    G. Pizzi, V. Vitale, R. Arita, S. Blugel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibañez-Azpiroz, H. Lee, J.-M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Poncé, T. Pon- weiser, J. Qiao, F. Thole, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Sou...

    2020

  25. [25]

    Tiwari, B

    S. Tiwari, B. Cucco, V.-A. Ha, and F. Giustino, Meta- optimization of maximally-localized wannier functions, npj Comput. Mater. https://doi.org/10.1038/s41524-026- 02082-1 (2026)

    work page doi:10.1038/s41524-026- 2026

  26. [26]

    Giannozzi, O

    P. Giannozzi, O. Baseggio, P. Bonfà, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F. Ferrari Ruffino, A. Ferretti, N. Marzari, I. Timrov, A. Urru, and S. Baroni, Quantum espresso toward the exascale, J. Chem. Phys.152, 154105 (2020)

    2020

  27. [27]

    Giannozzi, O

    P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, M. Cococcioni, N. Colonna, I. Carnimeo, A. D. Corso, S. de Gironcoli, P. Delugas, R. A. D. Jr, A. Ferretti, A. Floris, G. Fratesi, G. Fugallo, R. Gebauer, U. Gerst- mann, F. Giustino, T. Gorni, J. Jia, M. Kawamura, H.-Y. Ko, A. Kokalj, E. Küçük...

    2017

  28. [28]

    Giannozzi, S

    P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococ- cioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. S...

    2009

  29. [29]

    Hafner, Ab-initio simulations of materials using vasp: Density-functional theory and beyond, J

    J. Hafner, Ab-initio simulations of materials using vasp: Density-functional theory and beyond, J. Comput. Chem. 29, 2044 (2008)

    2044

  30. [30]

    Ordejón, and D

    J.M.Soler, E.Artacho, J.D.Gale, A.García, J.Junquera, P. Ordejón, and D. Sánchez-Portal, The siesta method for ab initio order-n materials simulation, J. Phys.: Condens. Matt.14, 2745 (2002)

    2002

  31. [31]

    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 materials1, 011002 (2013)

    2013

  32. [32]

    J. E. Saal, S. Kirklin, M. Aykol, B. Meredig, and C. Wolverton, Materials design and discovery with high- throughput density functional theory: the open quantum materials database (oqmd), Jom65, 1501 (2013)

    2013

  33. [33]

    Y. Qi, W. Gong, and Q. Yan, Bridging deep learning force fields and electronic structures with a physics-informed approach, npj Comput. Mater.11, 177 (2025)

    2025

  34. [34]

    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

  35. [35]

    M. M. Ghahremanpour, P. J. Van Maaren, and D. Van Der Spoel, The alexandria library, a quantum-chemical database of molecular properties for force field develop- ment, Sci. data5, 180062 (2018)

    2018

  36. [36]

    J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett.78, 1396 (1997)

    1997

  37. [37]

    M. J. Van Setten, M. Giantomassi, E. Bousquet, M. J. Ver- straete, D. R. Hamann, X. Gonze, and G.-M. Rignanese, The pseudodojo: Training and grading a 85 element opti- mized norm-conserving pseudopotential table, Comput. Phys. Commun.226, 39 (2018)

    2018

  38. [38]

    H. J. Monkhorst and J. D. Pack, Special points for brillouin-zone integrations, Phys. Rev. B13, 5188 (1976). 14

    1976

  39. [39]

    Fang, T.-W

    Z. Fang, T.-W. Hsu, and Q. Yan, Dataset of tensorial optical and transport properties of materials from the wannier function method, Sci. Data12, 1092 (2025)

    2025

  40. [40]

    Geiger and T

    M. Geiger and T. Smidt, e3nn: Euclidean neural networks, arXiv:2207.09453 (2022)

    Pith/arXiv arXiv 2022

  41. [41]

    D. A. Varshalovich, A. N. Moskalev, and V. K. Kher- sonskii,Quantum theory of angular momentum(World Scientific, 1988)

    1988

  42. [42]

    Sbailò, Á

    L. Sbailò, Á. Fekete, L. M. Ghiringhelli, and M. Schef- fler, The nomad artificial-intelligence toolkit: turning materials-science data into knowledge and understanding, npj Comput. Mater.8, 250 (2022)

    2022

  43. [43]

    N. Lee, H. Noh, S. Kim, D. Hyun, G. S. Na, and C. Park, Density of states prediction of crystalline materials via prompt-guided multi-modal transformer, Advances in Neural Information Processing Systems36, 61678 (2023)

    2023

  44. [44]

    W. B. How, P. Febrer, S. Chong, A. Mazitov, F. Bigi, M. Kellner, S. Pozdnyakov, and M. Ceriotti, A universal machine learning model for the electronic density of states, Digit. Discov.5, 1635 (2026)

    2026

  45. [45]

    Xie and J

    T. Xie and J. C. Grossman, Crystal graph convolutional neural networks for an accurate and interpretable predic- tion of material properties, Phys. Rev. Lett.120, 145301 (2018)

    2018

  46. [46]

    Choudhary and B

    K. Choudhary and B. DeCost, Atomistic line graph neural network for improved materials property predictions, npj Computational Materials7, 185 (2021)

    2021

  47. [47]

    R. Ruff, P. Reiser, J. Stühmer, and P. Friederich, Con- nectivity optimized nested line graph networks for crystal structures, Digital Discovery3, 594 (2024)

    2024

  48. [48]

    A. Dunn, Q. Wang, A. Ganose, D. Dopp, and A. Jain, Benchmarkingmaterialspropertypredictionmethods: the matbench test set and automatminer reference algorithm, npj Comput. Mater.6, 138 (2020)

    2020

  49. [49]

    Informatics, Tailwater github repo, https://github.com/alextyner-tailwater/Tailwater (2026)

    T. Informatics, Tailwater github repo, https://github.com/alextyner-tailwater/Tailwater (2026)

    2026

  50. [50]

    Cholia, D

    S.P.Ong, W.D.Richards, A.Jain, G.Hautier, M.Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python materials genomics (pymatgen): A robust, open-source python library for materials analysis, Comput. Mater. Sci.68, 314 (2013)

    2013

  51. [51]

    Gresch, G

    D. Gresch, G. Autès, O. V. Yazyev, M. Troyer, D. Van- derbilt, B. A. Bernevig, and A. A. Soluyanov, Z2pack: Numerical implementation of hybrid wannier centers for identifying topological materials, Phys. Rev. B95, 075146 (2017)

    2017

  52. [52]

    C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Wain- tal, Kwant: a software package for quantum transport, New Journal of Physics16, 063065 (2014)

    2014

  53. [53]

    Moldovan, M

    D. Moldovan, M. Anđelković, and F. Peeters, pybinding v0. 9.4: a python package for tight-binding calculations, Zenodo (2020)

    2020

  54. [54]

    Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and A. A. Soluyanov, Wanniertools: An open-source software pack- age for novel topological materials, Comput. Phys. Com- mun.224, 405 (2018)

    2018

  55. [55]

    S. S. Tsirkin, High performance wannier interpolation of berry curvature and related quantities with wannierberri code, npj Comput. Mater.7, 33 (2021)

    2021

  56. [56]

    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 moiré materials, Nature Reviews Materials6, 201 (2021)

    2021

  57. [57]

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

    2018

  58. [58]

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

    2018

  59. [59]

    Ghiotto, E.-M

    A. Ghiotto, E.-M. Shih, G. S. Pereira, D. A. Rhodes, B. Kim, J. Zang, A. J. Millis, K. Watanabe, T. Taniguchi, J.C.Hone,et al.,Quantumcriticalityintwistedtransition metal dichalcogenides, Nature597, 345 (2021)

    2021

  60. [60]

    K. F. Mak and J. Shan, Semiconductor moiré materials, Nature Nanotechnology17, 686 (2022)

    2022

  61. [61]

    S. F. Zhao, X. Cui, P. A. Volkov, H. Yoo, S. Lee, J. A. Gardener, A. J. Akey, R. Engelke, Y. Ronen, R. Zhong, et al., Time-reversal symmetry breaking superconductivity between twisted cuprate superconductors, Science382, 1422 (2023)

    2023

  62. [62]

    Pixley and P

    J. Pixley and P. A. Volkov, Twisted nodal superconduc- tors, arXiv preprint arXiv:2503.23683 (2025)

    Pith/arXiv arXiv 2025

  63. [63]

    D. Liu, M. Luskin, and S. Carr, Seeing moiré: Convolu- tional network learning applied to twistronics, Phys. Rev. Res.4, 043224 (2022)

    2022

  64. [64]

    T. Bao, R. Xu, H. Li, X. Gong, Z. Tang, J. Fu, W. Duan, and Y. Xu, Deep-learning database of density functional theory hamiltonians for twisted materials, arXiv preprint arXiv:2404.06449 (2024)

    Pith/arXiv arXiv 2024

  65. [65]

    H. Yang, R. Hu, H. Wu, X. He, Y. Zhou, Y. Xue, K. He, W. Hu, H. Chen, M. Gong,et al., Identification and struc- tural characterization of twisted atomically thin bilayer materials by deep learning, Nano Letters24, 2789 (2024)

    2024

  66. [66]

    Kaplan, A

    D. Kaplan, A. C. Tyner, E. Y. Andrei, and J. Pixley, Machine learning assisted high throughput prediction of moir\’e materials, arXiv preprint arXiv:2512.16892 (2025)

    arXiv 2025

  67. [67]

    Fang and E

    S. Fang and E. Kaxiras, Electronic structure theory of weakly interacting bilayers, Phys. Rev. B93, 235153 (2016)

    2016

  68. [68]

    S. Carr, S. Fang, Z. Zhu, and E. Kaxiras, Exact continuum model for low-energy electronic states of twisted bilayer graphene, Phys. Rev. Res.1, 013001 (2019)

    2019

  69. [69]

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

    2020

  70. [70]

    Zhong, H

    Y. Zhong, H. Yu, M. Su, X. Gong, and H. Xiang, Trans- ferable equivariant graph neural networks for the hamil- tonians of molecules and solids, npj Comput. Mater.9, 182 (2023)

    2023

  71. [71]

    Grimme, J

    S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametriza- tion of density functional dispersion correction (dft- d) for the 94 elements h-pu, J. Chem. Phys.132, https://doi.org/10.1063/1.3382344 (2010)

    doi:10.1063/1.3382344 2010

  72. [72]

    Yusufaly, D

    T. Yusufaly, D. Vanderbilt, and S. Coh, Tight-binding formalism in the context of the pythtb package, URL https://www. physics. rutgers. edu/pythtb/formalism. html (2013)

    2013

  73. [73]

    J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid func- tionals based on a screened coulomb potential, J. Chem. Phys.118, 8207 (2003)

    2003