Pith. sign in

REVIEW 2 major objections 4 minor 99 references

Relaxation-driven flat bands and topology in moir\'e transition metal dichalcogenide heterobilayers

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read Lattice relaxation alone opens topological bands in moiré WSe2/WS2.

desk verdict Compelling framework, but the strain tensor is likely built from the wrong displacement field—recompute before trusting the topology. read the letter →

arxiv 2608.08917 v1 pith:PWERYOVK submitted 2026-08-09 cond-mat.mes-hall cond-mat.str-el

Mitchell Luskin , Max Geier , Liang Fu , Ziyan Zhu This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-el
keywords moiréheterobilayerslatticerelaxationpseudomagneticfieldChernnumbersflatbandstransitionmetaldichalcogenidesquantumanomalousHalleffectfractionalinsulators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Standard continuum models of moiré transition metal dichalcogenide heterobilayers produce topologically trivial bands, and this paper argues the reason is that they leave out an intrinsic effect: atomic relaxation creates strain that acts on electrons like a magnetic field. The paper builds a continuum model that splits relaxation into three channels—a relaxed moiré potential with higher Fourier harmonics, a scalar deformation potential, and a pseudomagnetic vector potential—and shows in WSe2/WS2 that the pseudomagnetic channel alone opens a topological gap between the third and fourth valence bands, with Chern numbers +1 and −1 over a broad range of twist angle and lattice mismatch. The scalar and potential corrections narrow bandwidths and enlarge gaps, and the enhanced charge gap survives neural-network variational Monte Carlo calculations with Coulomb interactions. If these results hold, moiré heterobilayers are a new class of topological materials whose topology comes from relaxation itself, not from external strain or an intrinsic rigid-limit Chern band.

What carries the argument

The load-bearing object is the relaxed continuum Hamiltonian $$\hat H = \hat H_{\mathrm{mono}} + V_{m,\mathrm{relax}}(\mathbf r) + \hat H_\epsilon,$$ where the strain contribution is written in pseudo-gauge form as $\Phi(\mathbf r) + \{\hat k_\alpha, A_\alpha(\mathbf r)\}$. Relaxation enters through a displacement field obtained by minimizing elastic energy plus a DFT-parameterized generalized stacking fault energy; the strain tensor then produces a scalar deformation potential $\Phi = C\epsilon_0$ and a vector potential $\mathbf A = -\tfrac12 D(\epsilon_1, \epsilon_2)$ whose curl is the pseudomagnetic field $\mathbf B = \nabla\times\mathbf A$. The $\mathbf A$ term is what opens the topological gap at the quadratic band touching, while the relaxed moiré potential and scalar potential are topology-neutral but flatten bands and enlarge gaps.

What would settle it

A direct test is to measure the Hall conductance of a 3R-stacked WSe2/WS2 device near $\theta = 1^\circ$ and $\delta = 0.02$ as a function of filling up to $\nu = 6$; the model predicts a quantized plateau $\sigma_{xy} = e^2/h$ when the chemical potential lies in the gap between the third and fourth valence bands. A null result across the predicted topological region would indicate the strain field, the stacking-fault landscape, or the band calculation is overestimating the gap.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that lattice relaxation in moiré TMD heterobilayers is not a minor correction but the mechanism that makes the bands topological. The pseudomagnetic field generated by the relaxation-induced strain—not the relaxed moiré potential and not the scalar deformation potential—opens a gap between the third and fourth valence bands with Chern numbers $C_3 = +1$ and $C_4 = -1$ over a connected region of the twist-angle and lattice-mismatch plane, including the natural mismatch of WSe2/WS2 near $\delta \approx 0.04$. The top two valence bands remain trivial, which matches the experimentally observed trivial Mott insulator at filling $\nu = 1$, while the third band becomes a Chern band whose Berry curvature and quantum metric are smoothed by relaxation toward the ideal limit. The topological gap survives many-body interactions in neural-network variational Monte Carlo calculations at filling $\nu = 2$, and the net pseudomagnetic flux per moiré cell vanishes, so the topology arises from redistributed Berry curvature rather than net Landau-level quantization.

Load-bearing premise

The prediction rests on the strain field computed from a continuum elasticity model whose generalized stacking fault energy and elastic constants come from DFT; if that stacking-energy landscape, especially the AB/BA asymmetry, is inaccurate, the pseudomagnetic field and the topological gap it opens would change.

Editorial extensions

If this is right

  • At integer filling $\nu = 6$, the model predicts a quantum anomalous Hall state with quantized Hall conductance $\sigma_{xy} = e^2/h$ in 3R-stacked WSe2/WS2 across a broad, connected region of twist angle and lattice mismatch.
  • Partial filling of the third valence band is a candidate for fractional Chern insulator states, because relaxation reduces the bandwidth and drives the trace-condition violation and Berry-curvature fluctuation toward the ideal Chern limit.
  • The top two valence bands remain topologically trivial, consistent with the observed trivial Mott insulator at $\nu = 1$; the relaxation-enhanced charge gap at $\nu = 2$ is a quantitative many-body prediction accessible to transport and compressibility measurements.
  • Previous continuum heterobilayer models found trivial bands because they omitted the strain-induced gauge fields; the paper implies that including them is necessary, not optional, for describing topology in these systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the central claim holds, many existing heterobilayer samples may already contain unexplored topological states at fillings near $\nu = 6$, since the pseudomagnetic field is strongest in the same small-angle regime where flat bands are flattest.
  • The decomposition into three relaxation channels suggests a tuning knob: substrate coupling, pressure, or twist angle could move a sample across the topological phase boundary, giving an in-situ switch between trivial and Chern regimes.
  • A testable extension would be to probe the predicted alternating-sign pseudomagnetic field pattern at domain-wall intersections, looking for position-dependent spectral shifts despite the vanishing net flux per moiré cell.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The manuscript develops a continuum model for moiré TMD heterobilayers that includes lattice relaxation through three channels: a relaxed moiré potential with higher Fourier harmonics, a scalar deformation (pseudoelectric) potential, and a pseudomagnetic vector potential. Applying the model to WSe2/WS2, the authors find that the pseudomagnetic field alone opens a topological gap between the third and fourth valence bands with Chern numbers C3=+1 and C4=-1 over a broad range of twist angle and lattice mismatch, while the other two channels flatten bands and enhance gaps. The paper also computes quantum-geometric diagnostics, predicts favorable conditions for fractional Chern insulators, and reports neural-network variational Monte Carlo charge gaps at filling ν=2 for the rigid and relaxed models.

Significance. If the central claim holds, the paper would resolve a notable discrepancy: prior continuum models of TMD heterobilayers found trivial bands, whereas this work argues that intrinsic lattice relaxation alone generates topology. The derivation from a tight-binding starting point is careful and the decomposition into three relaxation channels is a useful conceptual framework. The authors provide Chern-number phase diagrams over dense grids, DFT-parametrized inputs for both the moiré potential and the generalized stacking fault energy, and public code and data availability statements for the many-body calculations. The end-to-end pipeline from DFT energetics to continuum bands to many-body observables is a genuine strength. However, one load-bearing modeling choice in the strain Hamiltonian appears inconsistent with the derivation, and the many-body test addresses the ν=2 charge gap rather than the topological gap Δ34, so the current manuscript overstates some of its support.

major comments (2)
  1. [SM S1.2 and SM S3.2; main-text Eq. (5)] The strain Hamiltonian is derived from the intralayer deformation of the layer hosting the low-energy holes. In SM Eq. (S10), the hopping correction is proportional to u_ij = ∂_i u_j with u the displacement of that layer, so the pseudogauge field in Eq. (5) should be constructed from the WSe2 layer strain ε^(2)_ij = (∂_i u2_j + ∂_j u2_i)/2. However, SM S3.2 states: 'The strain tensor εij(r) used in the continuum electronic model is computed from the relative displacement Δu = u1 − u2 via spectral differentiation.' Since the moiré potential correctly uses Δu but the intralayer strain correction does not, this appears to be an internal inconsistency. In the twist-dominated limit the two layers relax in an approximately opposite manner, so ε(Δu) can overestimate the WSe2 strain by a factor of order 2, and the spatial pattern is altered whenever the acoustic component is nonzero. Because Fig. 2(b) attributes the C3=+1/C4=-1 gap entirely to this pseudomagnetic field, the central topological claim and the phase diagrams in Fig. 3 depend directly on this choice. The authors should recompute the strain Hamiltonian using the active-layer displacement field (or clearly justify why the relative-displacement strain is the correct object for the intralayer hopping correction).
  2. [Fig. 1(e)-(f) and abstract] The many-body charge gap in Fig. 1(e) is computed at filling ν=2, which probes the top two valence bands. Those bands are topologically trivial throughout the parameter range studied, so this calculation tests the relaxation enhancement of a trivial gap, not the survival of the topological gap Δ34 or the Chern numbers C3=±1. The abstract's statement that the bandgap enhancements 'survive many-body interactions using neural-network variational Monte Carlo calculations' is therefore supported only for the top two bands. The authors should either soften the claim to specify the charged gap at ν=2 or provide a many-body calculation that directly addresses the topological gap between the third and fourth valence bands.
minor comments (4)
  1. [Main text, Fig. 3 caption] The main text says gray regions indicate bands with adjacent bands within a 3 meV tolerance, while the Fig. 3 caption says Δ34 < 0.3 meV and SM S3 states a 0.8 meV threshold; these tolerances should be harmonized and defined in one place.
  2. [SM S2.4] The plane-wave basis is truncated at 163 moiré reciprocal lattice vectors; a convergence test in the number of basis functions for the Chern numbers and the gap Δ34 would strengthen the phase diagram claims, particularly near the gap-closing boundaries.
  3. [References] Reference [54] contains the placeholder identifier 'arXiv:2509.XXXXX' and should be completed before publication; reference [11] should also be checked for completeness.
  4. [Fig. 2 label] The label 'Alle,ects' in the fourth panel of Fig. 2 contains a typo; it should read 'All effects'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: Chern numbers are genuine outputs of an independent DFT-to-continuum pipeline.

full rationale

The derivation chain is self-contained and non-circular. The continuum Hamiltonian (Eq. 1) is assembled from a monolayer effective-mass term, a DFT-fitted moiré potential, and a strain Hamiltonian whose scalar and pseudogauge couplings C and D are obtained from the tight-binding/Schrieffer-Wolff projection of Fang et al.'s strained monolayer Hamiltonian (SM S2.1). The strain field entering Eq. (5) is computed independently from a continuum relaxation model (SM S3) whose GSFE and elastic constants are DFT-derived. The Chern numbers C3=+1 and C4=-1, the phase diagram, and the quantum-geometry diagnostics are all computed from this Hamiltonian with no parameter adjusted to reproduce the topological gap; the decomposition in Fig. 2 is a controlled on/off study of each relaxation channel, not a fit. Self-citations (Carr et al. for the configuration-space relaxation framework; Geier et al./PeriodicWave for the NNVMC solver; Fu's universality theorems) are methodological and none is invoked as a uniqueness theorem to forbid alternatives or force the topological choice. The limitation statements in SM S4 (finite 3x3 supercell omitting the M point, restriction to theta >= 2 deg due to magnetism) are honest checks and do not create circularity. The skeptic's concern that SM S3.2 computes the electronic strain from Δu = u1 - u2 rather than the WSe2-layer displacement u2 is a modeling-consistency/correctness issue: Eq. (S10) does specify the intralayer strain of the active layer, and using the relative displacement could alter the magnitude or pattern of the pseudomagnetic field, but this would weaken or shift a prediction; it does not make the prediction equal to its input. Hence no circular step is present; the relevant external risk is the accuracy of the DFT and tight-binding parametrizations, not logical circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim inherits its quantitative content from DFT-calculated moiré potentials, GSFE, elastic constants, and a literature tight-binding parameterization. These are external inputs, not outputs of the model, so the claim's credibility is tied to their accuracy. There are no invented particles or forces; the pseudomagnetic field is a derived effective field from strain.

free parameters (4)
  • Moiré potential Fourier coefficients Vs, ϕs (shells 1-3) = SCAN-rVV10: V1=3.659 meV, φ1=50.89°; V2=-0.279 meV; V3=0.603 meV (Table S1)
    Fitted to DFT valence band edge over a 9x9 stacking grid; used to define rigid and relaxed moiré potentials.
  • GSFE Fourier coefficients c1..c5 = c4 = -3.68 meV (3R), +0.70 meV (2H); other coefficients not fully listed
    Fitted to DFT GSFE landscape; controls the relaxation displacement field and hence the strain that produces the pseudomagnetic field.
  • Elastic constants K1,G1,K2,G2 = K1=53.37, G1=33.23, K2=46.87, G2=31.21 eV/Ų
    Derived from DFT stretching/shearing of monolayers; inputs to the continuum elasticity relaxation model.
  • Effective mass m* and strain couplings C, D = m*=0.55 me; C=-2.25 eV; D=8.32 eV·Å
    Taken from the Fang et al. tight-binding parametrization; define the kinetic term and the pseudogauge field amplitude.
assumptions (4)
  • domain assumption The moiré potential at each point depends only on the local stacking vector (local stacking approximation).
    Used throughout to map DFT stacking energies onto the moiré real-space potential (SM S2.2.1).
  • domain assumption The continuum elasticity model with linearized strain and GSFE energy captures the equilibrium relaxation displacement field.
    SM S3: energy minimization over displacement fields u1,u2; ignores atomic-scale reconstruction beyond the continuum approximation.
  • domain assumption The low-energy two-band expansion and Schrieffer-Wolff projection onto the valence band are valid because f1=1.65 eV is much larger than the moiré bandwidth.
    SM S1.4 and S2.1; permits the single-band Hamiltonian with effective mass and strain couplings.
  • domain assumption DFT (SCAN-rVV10) computed stacking energies and GSFE are quantitatively accurate for WSe2/WS2.
    The entire parameter set rests on DFT; if the DFT errors are large, the phase boundaries shift.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Relaxation-driven flat bands and topology in moir\'e transition metal dichalcogenide heterobilayers." pith.science (2026). https://pith.science/paper/PWERYOVK

@misc{pith2026260808917,
  author       = {Pith},
  title        = {Pith review of: Relaxation-driven flat bands and topology in moir\'e transition metal dichalcogenide heterobilayers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PWERYOVK}},
  note         = {Machine review of arXiv:2608.08917}
}
abstract

Moir\'e transition metal dichalcogenide (TMD) heterobilayers are commonly modeled by a continuum theory that yields topologically trivial bands, in contrast to their homobilayer counterparts which host topological bands and fractional Chern insulators (FCI). We show this conclusion is an artifact of neglecting the pseudomagnetic field generated by lattice relaxation, an effect intrinsic to every moir\'e material. We develop a continuum model that resolves relaxation into three channels: a modified moir\'e potential with higher Fourier harmonics, a pseudoelectric (scalar deformation) potential, and a pseudomagnetic (vector) potential. Using WSe$_2$/WS$_2$ as a prototype, we find that the pseudomagnetic field alone gaps the third and fourth valence bands with Chern numbers $\pm 1$ over a broad range of twist angle and lattice mismatch, while the moir\'e potential correction and pseudoelectric potential narrow the bandwidth and enhance the bandgaps, which survive many-body interactions using neural-network variational Monte Carlo calculations. Relaxation also smoothens the Berry curvature and quantum metric relative to the rigid model, moving the band closer to the ideal Chern limit, beneficial for the quantum anomalous Hall effect, FCI states, and flat-band superconductivity when filled to higher bands. Our work establishes a new framework that connects first-principles calculations, through the continuum model, to many-body observables. Using this framework, we show moir\'e heterobilayers as a new class of topological materials whose topology is driven entirely by intrinsic lattice relaxation.

Figures

Figures reproduced from arXiv: 2608.08917 by the authors.

Figure 1
Figure 1. FIG. 1: Overview of the relaxed continuum model. (a) Moiré patterns formed by pure twisting ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Decomposition of relaxation effects on the moiré band [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Relaxation-induced band structure changes and topology [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Ideal band condition of the third valence band across the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

99 extracted references · 39 canonical work pages

  1. [1]

    S. Dai, Y. Xiang, and D. J. Srolovitz, Twisted bilayer graphene: Moiréwith a twist, Nano Letters16, 5923 (2016)

    2016

  2. [2]

    N. N. T. Nam and M. Koshino, Lattice relaxation and en- ergy band modulation in twisted bilayer graphene, Phys. Rev. B96, 075311 (2017)

    2017

  3. [3]

    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

  4. [4]

    M. H. Naik and M. Jain, Ultraflatbands and shear soli- tons in moiré patterns of twisted bilayer transition metal dichalcogenides, Phys. Rev. Lett.121, 266401 (2018)

    2018

  5. [5]

    H. Yoo, R. Engelke, S. Carr, S. Fang, K. Zhang, P. Cazeaux, S. H. Sung, R. Hovden, A. W. Tsen, T. Taniguchi, K. Watanabe, G.-C. Yi, M. Kim, M. Luskin, E. B. Tadmor, E. Kaxiras, and P. Kim, Atomicandelectronicreconstructionatthevanderwaals interface in twisted bilayer graphene, Nature Materials 18, 448 (2019)

    2019

  6. [6]

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

    2019

  7. [7]

    V. V. Enaldiev, V. Zólyomi, C. Yelgel, S. J. Magorrian, and V. I. Fal’ko, Stacking domains and dislocation net- works in marginally twisted bilayers of transition metal dichalcogenides, Phys. Rev. Lett.124, 206101 (2020)

    2020

  8. [8]

    Leconte, S

    N. Leconte, S. Javvaji, J. An, A. Samudrala, and J. Jung, Relaxation effects in twisted bilayer graphene: A multi- scale approach, Phys. Rev. B106, 115410 (2022)

    2022

Show all 99 references
  1. [9]

    H. Li, S. Li, M. H. Naik, J. Xie, X. Li, J. Wang, E. Re- gan, D. Wang, W. Zhao, S. Zhao, S. Kahn, K. Yu- migeta, M. Blei, T. Taniguchi, K. Watanabe, S. Ton- gay, A. Zettl, S. G. Louie, F. Wang, and M. F. Crom- mie, Imaging moiréflat bands in three-dimensional recon- structed w...

    2021

  2. [10]

    Van Winkle, I

    M. Van Winkle, I. M. Craig, S. Carr, M. Dandu, K. C. Bustillo, J. Ciston, C. Ophus, T. Taniguchi, K. Watan- abe, A. Raja, S. M. Griffin, and D. K. Bediako, Rota- tional and dilational reconstruction in transition metal dichalcogenide moirébilayers, Nature Communications 14, 29...

    2023

  3. [11]

    D. R. Klein, U. Zondiner, A. Keren, J. Birkbeck, A. In- bar, J. Xiao, Y. Zamir, M. Sidorova, M. M. Al Ezzi, L. Peng, K. Watanabe, T. Taniguchi, S. Adam, and S. Ilani, Imaging the sub-moirépotential using an atomic single electron transistor, Nature650, 875 (2026)

    2026

  4. [12]

    Weston, Y

    A. Weston, Y. Zou, V. Enaldiev, A. Summerfield, N. Clark, V. Zólyomi, A. Graham, C. Rost, O. Kazakova, V. V. Korolkov, T. Sherber, M. Sherber, V. I. Fal’ko, A. K. Geim, and S. J. Haigh, Atomic reconstruction in twisted bilayers of transition metal dichalcogenides, Nat. Nanotec...

    2020

  5. [13]

    Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan, and K. F. Mak, Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices, Nature579, 353 (2020)

    2020

  6. [14]

    E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu- migeta, M. Blei, J. D. Carlström, K. Watanabe, T. Taniguchi, S. Tongay, M. Crommie, A. Zettl, and F. Wang, Mott and generalized wigner crystal states in wse2/ws2 moirésuperlatti...

    2020

  7. [15]

    Huang, T

    X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian, T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, S.- F. Shi, and Y.-T. Cui, Correlated insulating states at fractional fillings of the ws2/wse2 moirélattice, Nature Physics17, 715 (2021)

    2021

  8. [16]

    C. Jin, E. C. Regan, A. Yan, M. Iqbal Bakti Utama, D. Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi, K. Watanabe, T. Taniguchi, S. Tongay, A. Zettl, and F. Wang, Observation of moiréexcitons in wse2/ws2 het- erostructure superlattices, Nature567, 76 (2019)

    2019

  9. [17]

    K. Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim, A. Rai, D. A. Sanchez, J. Quan, A. Singh, J. Emb- ley, A. Zepeda, M. Campbell, T. Autry, T. Taniguchi, K. Watanabe, N. Lu, S. K. Banerjee, K. L. Silverman, S. Kim, E. Tutuc, L. Yang, A. H. MacDonald, and X. Li, Evidence for moi...

    2019

  10. [18]

    F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hub- bard model physics in transition metal dichalcogenide moiré bands, Phys. Rev. Lett.121, 026402 (2018)

    2018

  11. [19]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological insulators in twisted transition metaldichalcogenidehomobilayers,Phys.Rev.Lett.122, 086402 (2019)

    2019

  12. [20]

    Angeli and A

    M. Angeli and A. H. MacDonald,γvalley transition metal dichalcogenide moiré bands, Proceedings of the National Academy of Sciences118, e2021826118 (2021), https://www.pnas.org/doi/pdf/10.1073/pnas.2021826118

    2021 doi

  13. [21]

    Devakul, V

    T. Devakul, V. Crépel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature Communications12, 6730 (2021)

    2021

  14. [22]

    Zhang, C

    X.-W. Zhang, C. Wang, X. Liu, Y. Fan, T. Cao, and D. Xiao, Polarization-driven band topology evolution in twisted mote2 and wse2, Nature Communications15, 4223 (2024)

    2024

  15. [23]

    N. Mao, C. Xu, J. Li, T. Bao, P. Liu, Y. Xu, C. Felser, L. Fu, and Y. Zhang, Transfer learning relaxation, elec- tronic structure and continuum model for twisted bilayer mote2, Communications Physics7, 262 (2024)

    2024

  16. [24]

    Zhang, K

    X.-W. Zhang, K. Yang, C. Wang, X. Liu, T. Cao, and D. Xiao, Twist-angle transferable continuum model and second flat chern band in twisted mote2 and wse2, npj Quantum Materials10, 110 (2025)

    2025

  17. [25]

    Angeli, G

    M. Angeli, G. R. Schleder, and E. Kaxiras, Twistronics of janus transition metal dichalcogenide bilayers, Phys. Rev. B106, 235159 (2022)

    2022

  18. [26]

    Xie, C.-P

    Y.-M. Xie, C.-P. Zhang, J.-X. Hu, K. F. Mak, and K. T. Law, Valley-polarized quantum anomalous hall state in moirémote 2/wse2 heterobilayers, Phys. Rev. Lett.128, 026402 (2022)

    2022

  19. [27]

    Z. Bi, N. F. Q. Yuan, and L. Fu, Designing flat bands by strain, Phys. Rev. B100, 035448 (2019)

    2019

  20. [28]

    A. P. Reddy, F. Alsallom, Y. Zhang, T. Devakul, and 7 L. Fu, Fractional quantum anomalous hall states in twisted bilayermote 2 andwse 2, Phys. Rev. B108, 085117 (2023)

    2023

  21. [29]

    Crépeland L.Fu,Anomalous hallmetalandfractional chern insulator in twisted transition metal dichalco- genides, Phys

    V. Crépeland L.Fu,Anomalous hallmetalandfractional chern insulator in twisted transition metal dichalco- genides, Phys. Rev. B107, L201109 (2023)

    2023

  22. [30]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, Y. He, X. Xu, Y. Ran, T.Cao,andD.Xiao,Fractionalcherninsulatorintwisted bilayermote 2, Phys. Rev. Lett.132, 036501 (2024)

    2024

  23. [31]

    B. A. Foutty, C. R. Kometter, T. Devakul, A. P. Reddy, K. Watanabe, T. Taniguchi, L. Fu, and B. E. Feldman, Mapping twist-tuned multiband topology in bilayer wse 2>, Science384, 343 (2024), https://www.science.org/doi/pdf/10.1126/science.adi4728

    2024 doi

  24. [32]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional chern in- sulator in moirémote2, Nature622, 69 (2023)

    2023

  25. [33]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous hall effect, Nature622, 74 (2023)

    2023

  26. [34]

    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, and T. Li, Observation of integer and fractional quantum anomalous hall effects in twisted bilayermote 2, Phys. Rev. X13, 031037 (2023)

    2023

  27. [35]

    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. Yao, and X. Xu, Signatures of fractional quantum anomalous hall states in twisted mote2, Nature622, 63 (2023)

    2023

  28. [36]

    T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. De- vakul, K. Watanabe, T. Taniguchi, L. Fu, J. Shan, and K. F. Mak, Quantum anomalous hall effect from inter- twined moirébands, Nature600, 641 (2021)

    2021

  29. [37]

    (2026), see Supplemental Material for the full deriva- tion of the relaxed continuum model, computational de- tails, DFT parametrization, emergent gauge field maps, band structures at multiple twist angles, bandwidth and bandgap phase diagrams, superfluid stiffness phase dia- ...

    2026

  30. [38]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quan- tum many-body problem with artificial neu- ral networks, Science355, 602 (2017), https://www.science.org/doi/pdf/10.1126/science.aag2302

    2017 doi

  31. [39]

    Schütt, P.-J

    K. Schütt, P.-J. Kindermans, H. E. Sauceda Felix, S. Chmiela, A. Tkatchenko, and K.-R. Müller, Schnet: A continuous-filter convolutional neural network for mod- eling quantum interactions, inAdvances in Neural Infor- mation Processing Systems, Vol. 30, edited by I. Guyon, U.V....

    2017

  32. [40]

    Luo and B

    D. Luo and B. K. Clark, Backflow transformations via neural networks for quantum many-body wave functions, Phys. Rev. Lett.122, 226401 (2019)

    2019

  33. [42]

    Hermann, Z

    J. Hermann, Z. Schätzle, and F. Noé, Deep-neural- network solution of the electronic schrödinger equation, Nature Chemistry12, 891 (2020)

    2020

  34. [43]

    K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nature Communications11, 2368 (2020)

    2020

  35. [44]

    von Glehn, J

    I. von Glehn, J. S. Spencer, and D. Pfau, A self-attention ansatz for ab-initio quantum chemistry, inThe Eleventh International Conference on Learning Representations (2023)

    2023

  36. [45]

    L. L. Viteritti, R. Rende, and F. Becca, Transformer vari- ational wave functions for frustrated quantum spin sys- tems, Phys. Rev. Lett.130, 236401 (2023)

    2023

  37. [46]

    Gao and S

    N. Gao and S. Günnemann, Generalizing neural wave functions, inProceedings of the 40th International Con- ference on Machine Learning, Proceedings of Machine Learning Research, Vol. 202, edited by A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, and J. Scarlett (PMLR...

    2023

  38. [47]

    Cassella, H

    G. Cassella, H. Sutterud, S. Azadi, N. D. Drummond, D. Pfau, J. S. Spencer, and W. M. C. Foulkes, Discov- ering quantum phase transitions with fermionic neural networks, Phys. Rev. Lett.130, 036401 (2023)

    2023

  39. [48]

    Wilson, S

    M. Wilson, S. Moroni, M. Holzmann, N. Gao, F. Wu- darski, T. Vegge, and A. Bhowmik, Neural network ansatz for periodic wave functions and the homogeneous electron gas, Phys. Rev. B107, 235139 (2023)

    2023

  40. [49]

    Smith, Y

    C. Smith, Y. Chen, R. Levy, Y. Yang, M. A. Morales, and S. Zhang, Unified variational approach description of ground-state phases of the two-dimensional electron gas, Phys. Rev. Lett.133, 266504 (2024)

    2024

  41. [50]

    R. Li, H. Ye, D. Jiang, X. Wen, C. Wang, Z. Li, X. Li, D. He, J. Chen, W. Ren, and L. Wang, A computational framework for neural network-based variational monte carlowithforwardlaplacian,NatureMachineIntelligence 6, 209 (2024)

    2024

  42. [51]

    Pescia, J

    G. Pescia, J. Nys, J. Kim, A. Lovato, and G. Carleo, Message-passing neural quantum states for the homoge- neous electron gas, Phys. Rev. B110, 035108 (2024)

    2024

  43. [52]

    J. Kim, G. Pescia, B. Fore, J. Nys, G. Carleo, S. Gan- dolfi, M. Hjorth-Jensen, and A. Lovato, Neural-network quantum states for ultra-cold fermi gases, Communica- tions Physics7, 148 (2024)

    2024

  44. [53]

    Geier, K

    M. Geier, K. Nazaryan, T. Zaklama, and L. Fu, Self- attention neural network for solving correlated electron problems in solids, Phys. Rev. B112, 045119 (2025)

    2025

  45. [54]

    Avdoshkin, M

    A. Avdoshkin, M. Geier, and L. Fu, A neural solver for spinful fermi systems (2025), arXiv:2509.XXXXX [cond- mat.supr-con]

    2025

  46. [55]

    Foster, Z

    A. Foster, Z. Schätzle, P. B. Szabó, L. Cheng, J. Köh- ler, G. Cassella, N. Gao, J. Li, F. Noé, and J. Her- mann, An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking (2025), arXiv:2506.19960 [physics.chem-ph]

    2025 arXiv

  47. [56]

    S. Fang, S. Carr, M. A. Cazalilla, and E. Kaxiras, Elec- tronicstructuretheoryofstrainedtwo-dimensionalmate- rials with hexagonal symmetry, Phys. Rev. B98, 075106 (2018)

    2018

  48. [57]

    Y. D. Chong, X.-G. Wen, and M. Soljačić, Effective the- ory of quadratic degeneracies, Phys. Rev. B77, 235125 (2008)

    2008

  49. [58]

    K. Sun, H. Yao, E. Fradkin, and S. A. Kivelson, Topo- logical insulators and nematic phases from spontaneous symmetry breaking in 2d fermi systems with a quadratic band crossing, Phys. Rev. Lett.103, 046811 (2009). 8

    2009

  50. [59]

    Morales-Durán, N

    N. Morales-Durán, N. C. Hu, P. Potasz, and A. H. Mac- Donald, Nonlocal interactions in moiré hubbard systems, Phys. Rev. Lett.128, 217202 (2022)

    2022

  51. [60]

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

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

    2014

  52. [61]

    T. S. Jackson, G. Möller, and R. Roy, Geometric stability of topological lattice phases, Nature Communications6, 8629 (2015)

    2015

  53. [62]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact landau level description of geometry and interaction in a flatband, Phys. Rev. Lett.127, 246403 (2021)

    2021

  54. [63]

    P. J. Ledwith, A. Vishwanath, and D. E. Parker, Vor- texability: A unifying criterion for ideal fractional chern insulators, Phys. Rev. B108, 205144 (2023)

    2023

  55. [64]

    Morales-Durán, J

    N. Morales-Durán, J. Wang, G. R. Schleder, M. Angeli, Z. Zhu, E. Kaxiras, C. Repellin, and J. Cano, Pressure- enhanced fractional chern insulators along a magic line in moiré transition metal dichalcogenides, Phys. Rev. Res. 5, L032022 (2023)

    2023

  56. [65]

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Anomalous hall crys- tals in rhombohedral multilayer graphene. i. interaction- driven chern bands and fractional quantum hall states at zero magnetic field, Phys. Rev. Lett.133, 206503 (2024)

    2024

  57. [66]

    C. Xu, J. Li, Y. Xu, Z. Bi, and Y. Zhang, Maximally localized wannier functions, interaction models, and fractional quantum anomalous hall effect in twisted bilayer mote<sub>2</sub>, Proceedings of the Na- tional Academy of Sciences121, e2316749121 (2024), https://www.pnas.org...

    2024 doi

  58. [67]

    P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vish- wanath, Fractional chern insulator states in twisted bi- layer graphene: An analytical approach, Phys. Rev. Res. 2, 023237 (2020)

    2020

  59. [68]

    Guinea, M

    F. Guinea, M. I. Katsnelson, and A. K. Geim, Energy gaps and a zero-field quantum hall effect in graphene by strain engineering, Nature Physics6, 30 (2010)

    2010

  60. [69]

    N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. H. C. Neto, and M. F. Crommie, Strain-induced pseudo–magnetic fields greater than 300 tesla in graphene nanobubbles, Science329, 544 (2010), https://www.science.org/doi/pdf/10.1126/science.1191700

    2010 doi

  61. [70]

    Low and F

    T. Low and F. Guinea, Strain-induced pseudomagnetic field for novel graphene electronics, Nano Letters10, 3551 (2010)

    2010

  62. [71]

    Vozmediano, M

    M. Vozmediano, M. Katsnelson, and F. Guinea, Gauge fields in graphene, Physics Reports496, 109 (2010)

    2010

  63. [72]

    Relaxation-driven band topology in moiré transition metal dichalcogenide heterobilayers

    M. Geier and K. Nazaryan, PeriodicWave (2025). Supplemental Material for “Relaxation-driven band topology in moiré transition metal dichalcogenide heterobilayers” CONTENTS S1. Continuum model derivation and computational details 2 S1.1. Tight-binding Hamiltonian and Fourier tr...

    2025

  64. [73]

    S. Fang, S. Carr, M. A. Cazalilla, and E. Kaxiras, Electronic structure theory of strained two-dimensional materials with hexagonal symmetry, Phys. Rev. B98, 075106 (2018)

    2018

  65. [74]

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

    2018

  66. [75]

    Cazeauz, M

    P. Cazeauz, M. Luskin, and D. Massatt, Arch. Rat. Mech. Anal.235, 1289 (2019)

    2019

  67. [76]

    Cazeaux, D

    P. Cazeaux, D. Clark, R. Engelke, P. Kim, and M. Luskin, Relaxation and domain wall structure of bilayer moiré systems, Journal of Elasticity , 1 (2023)

    2023

  68. [77]

    F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard model physics in transition metal dichalcogenide moiré bands, Phys. Rev. Lett.121, 026402 (2018)

    2018

  69. [78]

    Fukui, Y

    T. Fukui, Y. Hatsugai, and H. Suzuki, Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn.74, 1674 (2005)

    2005

  70. [79]

    D. Pfau, J. S. Spencer, A. G. D. G. Matthews, and W. M. C. Foulkes, Ab initio solution of the many-electron Schr\"odinger equation with deep neural networks, Phys. Rev. Res.2, 033429 (2020)

    2020

  71. [80]

    von Glehn, J

    I. von Glehn, J. S. Spencer, and D. Pfau, A self-attention ansatz for ab-initio quantum chemistry, inThe Eleventh Inter- national Conference on Learning Representations(2023)

    2023

  72. [81]

    Geier, K

    M. Geier, K. Nazaryan, T. Zaklama, and L. Fu, Self-attention neural network for solving correlated electron problems in solids, Phys. Rev. B112, 045119 (2025)

    2025

  73. [82]

    Fu, A minimal and universal representation of fermionic wavefunctions (fermions = bosons + one) (2025), arXiv:2510.11431 [cond-mat.str-el]

    L. Fu, A minimal and universal representation of fermionic wavefunctions (fermions = bosons + one) (2025), arXiv:2510.11431 [cond-mat.str-el]

    2025

  74. [83]

    Fu, Fermi sets: Universal and interpretable neural architectures for fermions (2026), arXiv:2601.02508 [cond-mat.str-el]

    L. Fu, Fermi sets: Universal and interpretable neural architectures for fermions (2026), arXiv:2601.02508 [cond-mat.str-el]

    2026 arXiv

  75. [84]

    Avdoshkin, M

    A. Avdoshkin, M. Geier, and L. Fu, A neural solver for spinful fermi systems (2025), arXiv:2509.XXXXX [cond-mat.supr- con]

    2025

  76. [85]

    C. Yun, S. Bhojanapalli, A. S. Rawat, S. Reddi, and S. Kumar, Are transformers universal approximators of sequence-to- sequence functions?, inInternational Conference on Learning Representations(2020)

    2020

  77. [86]

    W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33 (2001)

    2001

  78. [87]

    Martens and R

    J. Martens and R. Grosse, Optimizing neural networks with kronecker-factored approximate curvature, inProceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML’15 (JMLR.org,

  79. [88]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nature Physics20, 1476 (2024)

    2024

  80. [89]

    Goldshlager, N

    G. Goldshlager, N. Abrahamsen, and L. Lin, A kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions, Journal of Computational Physics516, 113351 (2024)

    2024

  81. [90]

    Schmidt and S

    K. Schmidt and S. Fantoni, A quantum monte carlo method for nucleon systems, Physics Letters B446, 99 (1999)

    1999

  82. [91]

    Ambrosetti, P

    A. Ambrosetti, P. L. Silvestrelli, F. Toigo, L. Mitas, and F. Pederiva, Variational monte carlo for spin-orbit interacting systems, Phys. Rev. B85, 045115 (2012)

    2012

  83. [92]

    C. A. Melton, M. C. Bennett, and L. Mitas, Quantum Monte Carlo with variable spins, The Journal of Chemical Physics 144, 244113 (2016)

    2016

  84. [93]

    C. A. Melton, M. Zhu, S. Guo, A. Ambrosetti, F. Pederiva, and L. Mitas, Spin-orbit interactions in electronic structure quantum Monte Carlo methods, Phys. Rev. A93, 042502 (2016)

    2016

  85. [94]

    Adams, G

    C. Adams, G. Carleo, A. Lovato, and N. Rocco, Variational Monte Carlo calculations ofA≤4nuclei with an artificial neural-network correlator ansatz, Physical Review Letters127, 022502 (2021)

    2021

  86. [95]

    Gerard, M

    L. Gerard, M. Scherbela, H. Sutterud, M. Foulkes, and P. Grohs, Transferable Neural Wavefunctions for Solids (2024), arXiv:2405.07599 [physics.comp-ph]

    2024 arXiv

  87. [96]

    D. Luo, T. Zaklama, and L. Fu, Solving fractional electron states in twisted MoTe _2with deep neural network, arXiv preprint arXiv:2503.13585 (2025)

    2025 arXiv

  88. [97]

    Geier and K

    M. Geier and K. Nazaryan, PeriodicWave (2025)

    2025

  89. [98]

    Geier and L

    M. Geier and L. Fu, Predicting magnetism with first-principles ai (2026), arXiv:2602.09093 [cond-mat.str-el]

    2026

  90. [99]

    Peotta and P

    S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nat. Commun.6, 8944 (2015)

    2015

  91. [100]

    Törmä, S

    P. Törmä, S. Peotta, and B. A. Bernevig, Superconductivity, superfluidity and quantum geometry in twisted multilayer systems, Nat. Rev. Phys.4, 528 (2022)

    2022

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.