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REVIEW 3 major objections 5 minor 69 references

Rhombohedral graphene's correlated phases are four electron crystals—Wigner, nodal, self-doped, and hole-lattice—selected by the Mexican-hat band shape.

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2026-08-04 01:07 UTC pith:TBZKW7CB

load-bearing objection Likely-robust model-level crystal phase diagram for Mexican-hat bands, but the experimental mapping to rhombohedral graphene rests on a single-band, zero-Berry-curvature model that the authors themselves flag as incomplete. the 3 major comments →

arxiv 2608.00167 v1 pith:TBZKW7CB submitted 2026-07-31 cond-mat.str-el

Rhombohedral Graphene: A Tale of Many Crystals

Ahmed Abouelkomsan , Filippo Gaggioli , Daniele Guerci , Liang Fu This is my paper
classification cond-mat.str-el
keywords rhombohedral grapheneWigner crystalself-doped crystalanticrystalMexican-hat dispersionneural-network variational Monte CarloHartree-Fockstrong correlations
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.

Rhombohedral graphene develops a conduction band that can be flattened into a Mexican-hat shape by a displacement field, and this paper argues that the shape of that band, not topology or flavor ordering, controls which strongly correlated phase appears at a given density. The authors introduce a simple ratio, r_eff, of Coulomb energy to average kinetic energy, computed from the noninteracting band, and show that it marks out the same region of the density-displacement-field plane where experiments see insulating and superconducting states. Using neural-network variational Monte Carlo and Hartree-Fock calculations, they find four distinct electron crystals: the ordinary Wigner crystal, a nodal Wigner crystal with ring-shaped wavefunctions, a self-doped 'phantom' crystal with fewer than one electron per unit cell, and an 'anticrystal' that is a triangular lattice of holes in an electron liquid. They identify the experimentally observed metallic crystal near n around 0.4 x 10^12 cm^-2 as the self-doped crystal and the resistive state near n around 0.6-0.8 x 10^12 cm^-2, flanked by chiral superconductivity, as the anticrystal. If correct, the experimental phase diagram is explained by a single-band Coulomb model, and r_eff becomes a predictive guide for finding correlated phases.

Core claim

The central claim is that the rich experimental phase diagram of rhombohedral graphene, including metallic and insulating electron crystals and their proximity to superconductivity, can be reproduced by a minimal model of electrons in a single spin- and valley-polarized band with a Mexican-hat dispersion, interacting through the Coulomb potential. In the annular Fermi sea that this dispersion produces, the authors find four crystalline ground states at increasing density: the ordinary triangular Wigner crystal; the nodal Wigner crystal, whose localized orbitals are ring-shaped with a node at radius about 2.4048/k0; the self-doped 'phantom' crystal, a metallic crystal with filling nu around 0

What carries the argument

The central object is the dimensionless ratio r_eff = E_int/E_kin, the Coulomb energy at the mean interparticle spacing divided by the average occupied kinetic energy; it is derived from the noninteracting band dispersion and locates strongly correlated regions without any interacting calculation. The argument is carried by the Mexican-hat dispersion epsilon(k) = -alpha|k|^2 + gamma|k|^4, which at displacement fields above a critical value produces an annular Fermi sea whose inner hole-like and outer electron-like pockets have very different masses; this two-length-scale fermiology generates the unconventional crystals. The interacting calculations combine a self-attention neural-network var

Load-bearing premise

The load-bearing premise is that real rhombohedral graphene, with its spin, valley, and Berry-curvature structure, can be reduced to a single spin- and valley-polarized band with a fitted Mexican-hat dispersion and no topological effects.

What would settle it

STM imaging of the resistive state at n around 0.7 x 10^12 cm^-2 and D around 60 meV that shows no triangular hole lattice, or a lattice with a period incompatible with nu around 2, would falsify the anticrystal identification; alternatively, a Hartree-Fock calculation that includes Berry curvature and all flavor sectors finding no phantom or anticrystal states at the claimed parameters would falsify the phase diagram.

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

If this is right

  • The r_eff map predicts that strong correlation in four-, five-, and six-layer rhombohedral graphene appears in the same density-displacement-field regions where experiments observe insulating and superconducting states, so it can guide where to look for new correlated phases.
  • The resistive state near n around 0.6-0.8 x 10^12 cm^-2 that lies between two chiral superconducting domes is identified as the metallic anticrystal, placing crystalline hole order at the doorstep of superconductivity.
  • The metallic crystal reported experimentally at n around 0.4 x 10^12 cm^-2 with roughly 15% hole doping is identified as the self-doped phantom crystal, not a conventional Wigner crystal.
  • The nodal Wigner crystal's real-space density has a ring of minima around each maximum, with first node radius 2.4048/k0, a signature that scanning tunneling microscopy could test.
  • Because the phantom and anticrystal phases arise from competition between interparticle spacing and the Mexican-hat momentum scale k0, they should be tunable continuously with displacement field.

Where Pith is reading between the lines

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

  • Inference: If the single-band, zero-Berry-curvature assumption holds, r_eff could be used as a cheap screening tool for correlated phases in other multilayer and moire graphene systems, where band flattening is also controlled by electric fields.
  • Inference: The proximity of the anticrystal to the superconducting domes hints that hole-crystal fluctuations, rather than the underlying band alone, may mediate pairing; this is a testable scenario if superconductivity disappears when the anticrystal is suppressed.
  • Inference: The analytical Bessel-function form for the nodal Wigner crystal suggests that the orbital shape could be extracted directly from STM images, providing a quantitative check that goes beyond energy comparisons.
  • Inference: If topology were included and the experimental resistive state turned out to be an anomalous Hall crystal instead, the model-level phase diagram could survive but the identification of experimental states would need revision.

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 / 5 minor

Summary. The paper proposes a dimensionless indicator r_eff = E_int/E_kin, computed from the noninteracting band dispersion, to locate strongly correlated regions of rhombohedral multilayer graphene in the density–displacement-field plane. It then studies the interacting ground states using two complementary methods: self-attention neural-network variational Monte Carlo (NN-VMC) on a simplified Mexican-hat dispersion and Hartree–Fock (HF) calculations using the full tetralayer graphene band structure. The central claim is that, in the annular-Fermi-sea regime, the model hosts four crystalline states—ordinary Wigner crystal, nodal Wigner crystal, self-doped 'phantom' crystal with filling ν≈0.9, and 'anticrystal' with ν≈2 described as a triangular hole lattice—and that these states correspond to experimentally observed metallic and resistive states near the superconducting domes. The paper also derives an analytic approximation for the nodal Wigner crystal and discusses experimental signatures via STM.

Significance. If the model-level phase diagram is correct, the paper makes a useful contribution by identifying a rich set of interaction-driven crystals in a single-band continuum model with Mexican-hat dispersion, and by showing that two very different numerical approaches (NN-VMC and restricted/unrestricted HF) find consistent phases. The analytic treatment of the nodal Wigner crystal in SM Appendix D is a nice, falsifiable result, and the paper is transparent about its model limitations. However, the broader significance claimed in the abstract and discussion depends on identifying the experimental resistive and metallic states with the model's 'phantom crystal' and 'anticrystal.' That experimental mapping is the weakest link: the model explicitly excludes flavor order and band topology, which are known to be important in the same experimental regime. The r_eff indicator is also presented with a calibration that appears partly post hoc. The work is therefore of high interest but needs revision before the experimental claims can be regarded as established.

major comments (3)
  1. [Discussion; SM Appendix A.1] The sentence 'Our work identifies this resistive state... as an anticrystal' is a load-bearing experimental claim, but it is made within a model that explicitly 'set[s] aside band topological effects' and assumes a single spin- and valley-polarized band. The same (n,D) region is known from refs. 50–59 to host anomalous Hall crystals, Chern insulators, and fractional Chern insulators, none of which are representable in the model. Because r_eff and the variational ground states carry no flavor or Berry-curvature information, the mapping to the experimental resistive state is not falsifiable as presented. To support the identification, the authors should either provide a concrete calculation or argument that flavor/quantum-geometry corrections are subleading in this density range, or state a falsifiable distinguishing prediction (e.g., triangular hole lattice with no anomalous Hall response
  2. [Eq. (1); Fig. 1(c,d)] The r_eff≥20 contour in Fig. 1(c,d) is presented as matching the experimental insulating/superconducting region, but the threshold value 20 and the choice ε=5 are not derived from the theory; the threshold is imported from 2DEG phenomenology and the dielectric constant is an adjustable screening parameter. Since r_eff scales with 1/ε, a different plausible ε can substantially move the contour. As presented, the match appears calibrated to the experimental features the indicator is then used to explain. The authors should show the stability of the qualitative r_eff contour over a range of physically plausible ε and threshold values, or demonstrate a predictive use outside the region used to set the threshold.
  3. [Fig. 3; SM Fig. S5; SM Appendix C] The paper states that Hartree–Fock calculations using the full band structure 'reproduce all the unconventional crystalline states identified with our NN-VMC simulations.' However, the NN-VMC phase diagram is for N=25 particles with the simplified Mexican-hat dispersion and ε=5, while the HF phase diagram uses the full tetralayer dispersion and ε=12, and the HF fillings are obtained under a triangular-unit-cell restriction (with unrestricted HF only for particular system sizes). Differences in ε alone change the Coulomb energy scale by about a factor of 2.4, so the two phase diagrams are not directly comparable. A quantitative cross-check at the same model and same ε—for example, HF with the Mexican-hat dispersion at ε=5—is needed to substantiate the claimed agreement and to identify which phase boundaries are robust across both methods.
minor comments (5)
  1. [Abstract and Fig. 1 caption] Typos: 'self-doped Wiger crystal' should be 'Wigner', and 'rhombohederal' should be 'rhombohedral'.
  2. [Eq. (1) and SM A.2] The definition of r_eff depends on measuring the kinetic energy relative to the band minimum (so that E_kin≥0). This should be stated in the main text near Eq. (1), not only in the SM.
  3. [Fig. 3 caption] 'topology chances' should be 'topology changes'.
  4. [SM Appendix B] There is a typo in 'weakly interacting reime' (should be 'regime'). Also, the caption of SM Fig. S5 should specify the color/phase labeling more explicitly.
  5. [Main text, NN-VMC section] For reproducibility, it would help to state explicitly that the NN-VMC supercell was fixed to a triangular shape with equal aspect ratio and N=25, and to comment on possible finite-size/supercell-shape bias on the observed triangular crystals.

Circularity Check

0 steps flagged

No significant circularity: phase diagram comes from energy minimization with cross-checks; r_eff is an explicitly defined diagnostic, and experimental identifications are interpretive rather than fitted outputs.

full rationale

The paper's central phase diagram is not circular. r_eff (Eq. 1) is an explicit ratio E_int/E_kin; using it to locate regions of strong correlation is a definitional diagnostic, not a prediction derived from itself. The actual crystalline phases (WC, nodal WC, phantom, anticrystal) are obtained by minimizing the many-body energy: NN-VMC on the Mexican-hat dispersion and HF on the full tetralayer dispersion, with the HF calculation explicitly stated to 'reproduce all the unconventional crystalline states identified with our NN-VMC simulations' (main text). Although the NN-VMC uses alpha,gamma fitted to the full band (SM A.1), this is a controlled reduction, and the HF phase diagram uses the full band structure including trigonal warping, so the phases do not reduce to the fitted coefficients. The phantom crystal is independently consistent with refs [46,47], which are not by the present authors. The experimental mapping ('we identify this resistive state ... as an anticrystal'; 'metallic crystals ... self-doped Wigner crystal') is interpretive: it is not used to fit any parameter, and the phase identification comes from the computed nu and band structure. Minor self-citations ([18] for r_eff, [43] for the self-attention architecture) are not load-bearing because r_eff is explicitly defined in Eq. 1 and the neural-network ansatz is a generic numerical method. The acknowledged omission of Berry curvature/flavor order (SM A.1; Discussion) is a correctness risk for the experimental identification, not a circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 3 invented entities

The paper's experimental-level claims rest on a two-parameter fitted dispersion, chosen screening constants (ε=5 vs ε=12), a calibrated r_eff threshold, and a single-flavor model with topology set aside. The new phases themselves carry falsifiable real-space handles (hence independent_evidence=True), but their identification with specific experimental states is an interpretive layer built on top of the energy minimization, not a direct consequence of it.

free parameters (5)
  • α, γ (Mexican-hat dispersion coefficients) = Least-squares fits to tetralayer band structure as a function of D; representative NN-VMC values (α/EC, γ/EC) = (0.582,
    The simplified dispersion ε(k) = -αk² + γk⁴ determines k0 = √(α/2γ) and the entire NN-VMC and simplified-HF phase diagram. Fitted to the noninteracting band, not to the target phases, but the phase boundaries inherit any fit error.
  • Dielectric constant ε = ε = 5 for r_eff maps and NN-VMC; ε = 12 for the HF phase diagram
    Screening strength enters E_int directly (r_eff ∝ 1/ε) and shifts phase boundaries; the two methods use different values, and the experimental match of the r_eff≥20 contour is sensitive to this choice.
  • r_eff ≥ 20 strong-correlation threshold = 20
    Borrowed from the 2DEG r_s≥20 lore and stated to 'match well' the experimental insulating/superconducting region (Fig. 1c,d). This is a post-hoc calibration, not a derived criterion.
  • NN-VMC particle number N = 25
    All NN-VMC phase assignments use 25 particles in a triangular supercell with no finite-size extrapolation reported; crystallization energetics can be N-sensitive.
  • NN-VMC architecture hyperparameters = 4 layers, 4 heads, N_det = 2, dim 32, GELU, KFAC (SM Table I)
    Variational ansatz capacity choices affect whether competing phases are resolved; no ablation is shown.
axioms (6)
  • domain assumption Single-band model with spin- and valley-polarized electrons (SM App. A.1)
    The fourfold isospin degeneracy of rhombohedral graphene is collapsed to one flavor. Real experiments in this material are widely interpreted in terms of spin/valley ordering, so this is a strong simplification for the experimental mapping.
  • domain assumption Mexican-hat dispersion ε(k) = -αk² + γk⁴ captures the essential low-energy fermiology (Main text; SM A.1)
    Trigonal warping and higher-order band structure are absent in the NN-VMC and simplified-HF calculations; only the full-band HF (Fig. 3) includes them, and it is restricted to triangular cells.
  • domain assumption Berry curvature / band topology does not affect crystallization energetics (SM A.1; Discussion)
    Explicitly deferred to future work, yet the same material family shows topological and anomalous-Hall phases in nearby fillings; the paper's refs 50-59 document strong quantum-geometry effects.
  • domain assumption Uniform-dielectric long-range Coulomb interaction, no substrate or disorder (SM A.1, Eq. A4)
    Screening is treated through a single constant ε; the delicate balance between liquid and crystal phases could be altered by non-uniform screening from other flavors or the environment.
  • domain assumption The self-attention NN ansatz (with backflow-like orbital dependence) is expressive enough to capture all competing phases (Main text NN-VMC section; SM Table I, refs 42-43)
    No expressivity or convergence proof is given; phase discovery by VMC depends on the ansatz being able to represent the true ground state.
  • standard math Standard ground-state machinery: Hartree-Fock and variational Monte Carlo energy minimization (Main text; SM App. C)
    Both methods are variational upper bounds with well-known biases; the paper relies on their mutual agreement as evidence.
invented entities (3)
  • Anticrystal independent evidence
    purpose: Triangular lattice of holes in an electron liquid at ν≈2 (insulating), metallic when hole-doped to ν≈1.9; assigned to the experimental resistive state at n≈0.6-0.8×10^12 cm^-2 between the superconducting domes.
    Falsifiable handles: STM would show density minima on a triangular lattice with roughly two electrons per unit cell, and transport would show insulating behavior at ν=2 with a small hole Fermi surface away from it. Predicted hole doping ≈5%.
  • Phantom (self-doped) crystal independent evidence
    purpose: Crystal with more density peaks than electrons (ν≈0.9), metallic; assigned to the experimental metallic Wigner crystal at n≈0.4×10^12 cm^-2 with reported ~15% hole doping.
    Falsifiable handle: STM peak count should differ from electron count; concurrently reported in refs 46-47, which reduces novelty but strengthens independent support.
  • Nodal Wigner crystal independent evidence
    purpose: Wigner crystal whose single-particle orbital has a Bessel-function node ring, r_node ≈ 2.4048/k0, producing a hexagonal ring of density minima around each site.
    Falsifiable handle: STM density profile should show the characteristic ring of minima with radius set by k0, which the analytic derivation in App. D ties to the dispersion.

pith-pipeline@v1.3.0-alltime-deepseek · 17524 in / 18912 out tokens · 211696 ms · 2026-08-04T01:07:38.623344+00:00 · methodology

0 comments
read the original abstract

Experiments on rhombohedral graphene have uncovered an extraordinary wealth of correlated quantum phases - from chiral superconductors to electronic crystals - all within a single family of atomically thin materials. Here, we introduce a simple indicator, derived from the noninteracting band dispersion, that identifies strongly correlated regions in the phase diagram of rhombohedral graphene as a function of carrier density and displacement field. We develop a neural-network variational Monte Carlo method, combined with Hartree-Fock theory, to solve the interacting ground states. Our calculation reveals a variety of electron crystals with no classical analog. These include, at increasing density: Wigner crystal, self-doped Wiger crystal, as well as ''anticrystal'', a lattice of holes in an electron liquid. We discuss their experimental manifestations and possible connection to superconductivity.

Figures

Figures reproduced from arXiv: 2608.00167 by Ahmed Abouelkomsan, Daniele Guerci, Filippo Gaggioli, Liang Fu.

Figure 1
Figure 1. Figure 1: FIG. 1. Microscopic band structure (a) and illustration of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a)–(d) Ground-state charge density obtained from NN-VMC, illustrating four representative crystalline states. Blue [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Hartree Fock phase diagram for electrons with disper [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Hartree-Fock band structure of the self-doped Wigner [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

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Reference graph

Works this paper leans on

69 extracted references · 1 canonical work pages

  1. [1]

    T. Han, Z. Lu, G. Scuri, J. Sung, J. Wang, T. Han, K. Watanabe, T. Taniguchi, H. Park, and L. Ju, Cor- related insulator and chern insulators in pentalayer rhombohedral-stacked graphene, Nature Nanotechnology 19, 181 (2024)

    2024

  2. [2]

    Z. Lu, T. Han, Y. Yao, A. P. Reddy, J. Yang, J. Seo, K. Watanabe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anomalous Hall effect in multilayer graphene, Nature626, 759 (2024)

    2024

  3. [3]

    Z. Lu, T. Han, Y. Yao, Z. Hadjri, J. Yang, J. Seo, L. Shi, S. Ye, K. Watanabe, T. Taniguchi, and L. Ju, Extended quantum anomalous Hall states in graphene/hBN moir´ e superlattices, Nature637, 1090 (2025)

    2025

  4. [4]

    J. Xie, Z. Huo, X. Lu, Z. Feng, Z. Zhang, W. Wang, Q. Yang, K. Watanabe, T. Taniguchi, K. Liu, Z. Song, X. C. Xie, J. Liu, and X. Lu, Tunable fractional chern in- sulators in rhombohedral graphene superlattices (2025), arXiv:2405.16944 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2025

  5. [5]

    Y. Choi, Y. Choi, M. Valentini, C. L. Patterson, L. F. W. Holleis, O. I. Sheekey, H. Stoyanov, X. Cheng, T. Taniguchi, K. Watanabe, and A. F. Young, Supercon- ductivity and quantized anomalous hall effect in rhom- bohedral graphene, Nature639, 342–347 (2025)

    2025

  6. [6]

    S. H. Aronson, T. Han, Z. Lu, Y. Yao, J. P. Butler, K. Watanabe, T. Taniguchi, L. Ju, and R. C. Ashoori, Displacement field-controlled fractional chern insulators and charge density waves in a graphene/hbn moir´ e su- perlattice, Phys. Rev. X15, 031026 (2025)

    2025

  7. [7]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Sharifi Sedeh, H. Weldeyesus, et al., Signatures of chiral superconductivity in rhombohedral graphene, Nature643, 654 (2025)

    2025

  8. [8]

    Dutta, N

    S. Dutta, N. Auerbach, T. Han, Y. Zhou, G. Shavit, N.-S. Kander, Y. Myasoedov, M. E. Huber, K. Watan- abe, T. Taniguchi, L. Ju, and E. Zeldov, Reconfigurable chiral superconductivity (2026), arXiv:2605.13303 [cond- mat.mes-hall]

    Pith/arXiv arXiv 2026

  9. [9]

    R. Q. Nguyen, H.-T. Wu, E. Morissette, N. J. Zhang, P. Qin, K. Watanabe, T. Taniguchi, A. W. Hui, D. E. Feldman, and J. I. A. Li, A hierarchy of supercon- 6 ductivity and topological charge density wave states in rhombohedral graphene (2025), arXiv:2507.22026 [cond- mat.mes-hall]

    Pith/arXiv arXiv 2025

  10. [10]

    Qin, H.-T

    P. Qin, H.-T. Wu, R. Q. Nguyen, E. Morissette, N. J. Zhang, K. Watanabe, T. Taniguchi, and J. I. A. Li, Stripe order in the metallic and superconduct- ing phases of rhombohedral hexalayer graphene (2026), arXiv:2504.05129 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2026

  11. [11]

    Z. Hua, S. Ye, P. Pattanakanvijit, G. Shi, T. Han, E. Aitken, J. Yang, J. Seo, H. Liu, R. Hao, K. Xiao, J. Guo, V. T. Phong, K. Watanabe, T. Taniguchi, C. Huang, C. Lewandowski, L. Ju, P. Xiong, and Z. Lu, Multi-knob switchable chiral superconductivity quar- tet in rhombohedral graphene (2026), arXiv:2607.06520 [cond-mat.supr-con]

    Pith/arXiv arXiv 2026

  12. [12]

    R. Q. Nguyen, P. Qin, H.-T. Wu, S. Mishra, T. Wolf, J. Roll, E. Morissette, N. J. Zhang, S. Alkidim, K. Watanabe, T. Taniguchi, A. W. Hui, D. E. Feldman, A. MacDonald, and J. I. A. Li, Coexisting charge density wave and superconducting order in quantizing magnetic fields (2026), arXiv:2607.05039 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2026

  13. [13]

    T. Han, J. P. Butler, S. Ye, Z. Hua, S. Dutta, Z. Hadjri, Z. Wu, J. Yang, J. Seo, P. Pattanakanvijit, E. Aitken, K. Watanabe, T. Taniguchi, P. Xiong, E. Zeldov, Z. Lu, R. Ashoori, and L. Ju, Evidence of metallic wigner crys- tal in rhombohedral graphene (2026), arXiv:2604.00113 [cond-mat.mes-hall]

    arXiv 2026

  14. [14]

    Z. Zhou, K. Kr¨ otzsch, R. Ayache, Y. Li, S. Joy, K. Watanabe, T. Taniguchi, M. Heiblum, P. Roul- leau, and M. Banerjee, Competing orders driven by wigner crystal phase in rhombohedral graphene (2026), arXiv:2607.15014 [cond-mat.mes-hall]

    Pith/arXiv arXiv 2026

  15. [15]

    Koshino and E

    M. Koshino and E. McCann, Trigonal warping and berry’s phasenπin abc-stacked multilayer graphene, Phys. Rev. B80, 165409 (2009)

    2009

  16. [16]

    Zhang, B

    F. Zhang, B. Sahu, H. Min, and A. H. MacDonald, Band structure ofabc-stacked graphene trilayers, Phys. Rev. B 82, 035409 (2010)

    2010

  17. [17]

    See the Supplemental Material [URL] for details

  18. [18]

    Geier, M

    M. Geier, M. Davydova, and L. Fu, Chiral and topo- logical superconductivity in isospin polarized multilayer graphene (2024), arXiv:2409.13829 [cond-mat.supr-con]

    Pith/arXiv arXiv 2024

  19. [19]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. Sharifi Sedeh, H. Weldeyesus, J. Yang, J. Seo, S. Ye, M. Zhou, H. Liu, G. Shi, Z. Hua, K. Watan- abe, T. Taniguchi, P. Xiong, D. M. Zumb¨ uhl, L. Fu, and L. Ju, Signatures of chiral superconductivity in rhombo- hedral graphene, Nature643, 654–661 (2025)

    2025

  20. [20]

    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

    doi:10.1126/science.aag2302 2017

  21. [21]

    Carrasquilla and R

    J. Carrasquilla and R. G. Melko, Machine learning phases of matter, Nature Physics13, 431 (2017)

    2017

  22. [22]

    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

  23. [23]

    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

  24. [24]

    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

  25. [25]

    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

  26. [26]

    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

  27. [27]

    Valenti, Y

    A. Valenti, Y. Vituri, Y. Yang, D. E. Parker, T. Soe- jima, J. Dong, M. A. Morales, A. Vishwanath, E. Berg, and S. Zhang, Quantum geometry driven crystallization: A neural-network variational monte carlo study (2025), arXiv:2512.07947 [cond-mat.str-el]

    arXiv 2025

  28. [28]

    Gaggioli, P.-A

    F. Gaggioli, P.-A. Graham, and L. Fu, Electronic crystals and quasicrystals in semiconductor quantum wells: An AI-powered discovery (2025), 2512.10909 [cond-mat]

    arXiv 2025

  29. [29]

    Gaggioli, S

    F. Gaggioli, S. Azadi, and L. Fu, Accurate self-attention wavefunctions at large scale (2026), arXiv:2607.08616 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  30. [30]

    C.-T. Li, T. Ong, M. Geier, H. Lin, and L. Fu, Attention is all you need to solve chiral superconductivity (2025), arXiv:2509.03683 [cond-mat.supr-con]

    Pith/arXiv arXiv 2025

  31. [31]

    Y. Teng, D. D. Dai, and L. Fu, Solving the fractional quantum hall problem with self-attention neural network, Phys. Rev. B111, 205117 (2025)

    2025

  32. [32]

    Y. Qian, T. Zhao, J. Zhang, T. Xiang, X. Li, and J. Chen, Describing landau level mixing in fractional quantum hall states with deep learning, Phys. Rev. Lett.134, 176503 (2025)

    2025

  33. [33]

    Nazaryan, F

    K. Nazaryan, F. Gaggioli, Y. Teng, and L. Fu, Artificial intelligence for quantum matter: Finding a needle in a haystack (2025), arXiv:2507.13322 [cond-mat.str-el]

    arXiv 2025

  34. [34]

    A. P. Fadon, D. Pfau, J. S. Spencer, W. T. Lou, T. Ne- upert, and W. M. C. Foulkes, Extracting anyon statis- tics from neural network fractional quantum hall states (2025), arXiv:2512.15872 [cond-mat.str-el]

    arXiv 2025

  35. [35]

    Abouelkomsan and L

    A. Abouelkomsan and L. Fu, First-principles ai finds crystallization of fractional quantum hall liquids (2026), arXiv:2602.03927 [cond-mat.mes-hall]

    arXiv 2026

  36. [36]

    J. Zhu, Y. Huang, X. Hu, D. Xiao, and T. Cao, Crystallization in the fractional quantum hall regime with disorder-aware neural quantum states (2026), arXiv:2604.06316 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  37. [37]

    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

  38. [38]

    Foster, Z

    A. Foster, Z. Sch¨ atzle, P. B. Szab´ o, L. Cheng, J. K¨ ohler, G. Cassella, N. Gao, J. Li, F. No´ e, and J. Hermann, An ab initio foundation model of wavefunctions that accurately describes chemical bond breaking (2025), arXiv:2506.19960 [physics.chem-ph]

    Pith/arXiv arXiv 2025

  39. [39]

    Linteau, S

    D. Linteau, S. Moroni, G. Carleo, and M. Holzmann, Neural wave functions for high-pressure atomic hydrogen, Physical Review Research8, 10.1103/t72h-tkcx (2026)

    doi:10.1103/t72h-tkcx 2026

  40. [40]

    Abouelkomsan, M

    A. Abouelkomsan, M. Geier, and L. Fu, Topological or- der in neural wavefunctions, Physical Review B113, 10.1103/bzq1-123h (2026)

    work page doi:10.1103/bzq1-123h 2026

  41. [41]

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

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

    Pith/arXiv arXiv 2026

  42. [42]

    von Glehn, J

    I. von Glehn, J. S. Spencer, and D. Pfau, A self- attention ansatz for ab-initio quantum chemistry (2023), arXiv:2211.13672 [physics.chem-ph]. 7

    Pith/arXiv arXiv 2023

  43. [43]

    Geier, K

    M. Geier, K. Nazaryan, T. Zaklama, and L. Fu, Self- attention neural network for solving correlated elec- tron problems in solids, Physical Review B112, 045119 (2025)

    2025

  44. [44]

    Joy and B

    S. Joy and B. Skinner, Wigner crystallization in bernal bilayer graphene, arXiv preprint arXiv:2310.07751 (2023)

    Pith/arXiv arXiv 2023

  45. [45]

    S. Joy, L. Levitov, and B. Skinner, Chiral wigner crys- tal phases induced by berry curvature, Physical Review Letters135, 256502 (2025)

    2025

  46. [46]

    J. Dong, T. Soejima, D. E. Parker, and A. Vishwanath, Crystals caught doping: Metallic wigner crystals in rhombohedral graphene (2026), arXiv:2604.00114 [cond- mat.str-el]

    arXiv 2026

  47. [47]

    J. Feng, Z. Han, M. P. Zaletel, and Z. Dong, Self- doped crystal from preempted band-inversion transitions (2026), arXiv:2604.09820 [cond-mat.str-el]

    Pith/arXiv arXiv 2026

  48. [48]

    H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, Imag- ing two-dimensional generalized wigner crystals, Nature 597, 650 (2021)

    2021

  49. [49]

    Y.-C. Tsui, M. He, Y. Hu, E. Lake, T. Wang, K. Watan- abe, T. Taniguchi, M. P. Zaletel, and A. Yazdani, Direct observation of a magnetic-field-induced wigner crystal, Nature628, 287 (2024)

    2024

  50. [50]

    Sheng, A

    D. Sheng, A. P. Reddy, A. Abouelkomsan, E. J. Bergholtz, and L. Fu, Quantum anomalous hall crystal at fractional filling of moir´ e superlattices, Physical Review Letters133, 066601 (2024)

    2024

  51. [51]

    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, Physical Review Letters133, 10.1103/physrevlett.133.206503 (2024)

    doi:10.1103/physrevlett.133.206503 2024

  52. [52]

    Z. Dong, A. S. Patri, and T. Senthil, Theory of quan- tum anomalous hall phases in pentalayer rhombohedral graphene moir´ e structures, Physical Review Letters133, 206502 (2024)

    2024

  53. [53]

    B. Zhou, H. Yang, and Y.-H. Zhang, Fractional quan- tum anomalous hall effect in rhombohedral multilayer graphene in the moir´ eless limit, Physical Review Letters 133, 206504 (2024)

    2024

  54. [54]

    Soejima, J

    T. Soejima, J. Dong, A. Vishwanath, and D. E. Parker, lambda-jellium model for the anomalous hall crystal, arXiv preprint arXiv:2503.12704 (2025)

    arXiv 2025

  55. [55]

    Tan and T

    T. Tan and T. Devakul, Parent berry curvature and the ideal anomalous hall crystal, Physical Review X14, 041040 (2024)

    2024

  56. [56]

    Desrochers, M

    F. Desrochers, M. R. Hirsbrunner, J. Huxford, A. S. Pa- tri, T. Senthil, and Y. B. Kim, Elastic response and insta- bilities of anomalous hall crystals, Physical Review Let- ters136, 166503 (2026)

    2026

  57. [57]

    Y. Zeng, D. Guerci, V. Cr´ epel, A. J. Millis, and J. Cano, Sublattice structure and topology in spontaneously crys- tallized electronic states, Phys. Rev. Lett.132, 236601 (2024)

    2024

  58. [58]

    Soejima, J

    T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zale- tel, A. Vishwanath, and D. E. Parker, Anomalous hall crystals in rhombohedral multilayer graphene. ii. gen- eral mechanism and a minimal model, Physical Review B110, 205124 (2024)

    2024

  59. [59]

    Z. Dong, A. S. Patri, and T. Senthil, Stability of anoma- lous hall crystals in multilayer rhombohedral graphene, Physical Review B110, 205130 (2024)

    2024

  60. [60]

    Rhombohedral Graphene: A Tale of Many Crystals

    M. L. Kim and X.-G. Wen, Shape of wigner crystal and hole self-doping in a mexican-hat dispersion, to appear. 1 Supplemental materials for: “Rhombohedral Graphene: A Tale of Many Crystals” Ahmed Abouelkomsan1∗, Filippo Gaggioli 1∗, Daniele Guerci 1∗ and Liang Fu1 1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA-02139, USA ∗Thes...

  61. [61]

    Mexican hat

    A Minimal Model for Interacting Phases in Rhombohedral Graphene We investigate interaction-driven phases in rhombohedral graphene within a minimal model of spin- and valley- polarized electrons. We focus on the interplay between the band dispersion and strong electronic interactions, setting aside band topological effects to reduce the complexity of the p...

  62. [62]

    The last equation fixes the chemical potential, andd e = p 1/(πn) is the interparticle distance

    Interaction strength and correlation regime We introduce a simple guide to quantify the degree of correlation by comparing the average kinetic energy and the average Coulomb energy per particle: Ekin = 1 ℓ2n Z d2k (2π)2 f(k)ϵ(k), E int = ECℓ de , ℓ 2n= Z d2k (2π)2 f(k), f(k) = 1 e ϵ(k)−µ kB T + 1 ,(A7) wherekis dimensionless and the dispersion of the cond...

  63. [63]

    Non-interacting density-density susceptibility To characterize the intrinsic length scales of the annular Fermi gas with Mexican-hat dispersion, we compute the static density–density susceptibility: Π(q) = 1 ℓ2 Z d2k (2π)2 f(ϵ(k+q)−µ)−f(ϵ(k)−µ) ϵ(k)−ϵ(k+q) ,(A8) wherekis expressed in unit of 1/ℓand we have introduced the chemical potentialµfixing the dens...

    2048

  64. [64]

    The Hartree contribution is independent of the momentum distribution, EH [ˆn] =Ne(Ne −1) 2A V0,(C3) and is therefore removed as an irrelevant constant energy shift

    T ranslationally invariant liquids For a translationally invariant liquid,n k,k′ is diagonal, and the Hartree-Fock energy per particle reduces to: E[ˆn] = X k ϵ(k)nk + 1 2A k̸=pX kp nknp [V0 −V k−p],(C2) wheren k is thekspace occupation number. The Hartree contribution is independent of the momentum distribution, EH [ˆn] =Ne(Ne −1) 2A V0,(C3) and is there...

  65. [65]

    Moreover, we consider a triangular unit cell defined by primitive vectorsa 1 anda 2 6 FIG

    Restricted Hartree-F ock Restricted calculations are performed assuming the system develops a periodicity with lattice vectorsa 1,a 2 and unit cell areaA UC =a 1 ×a 2. Moreover, we consider a triangular unit cell defined by primitive vectorsa 1 anda 2 6 FIG. S6. Crystalline states found in NN-VMC calculations of the Hamiltonian (A3) with the dispersion re...

  66. [66]

    Unrestricted Hartree-F ock Additionally, to explore the full range of possible translational-symmetry-broken patterns, we perform unconstrained Hartree–Fock simulations. To minimize the Hartree-Fock energy, we compute the energy variation with respect to nk,k′: ∆E[{ˆn}] = X k,k′ δE[{ˆn}] δnk,k′ δnk,k′,(C10) where we have introduced: Fk,k′ = δE[{ˆn}] δnk,k...

  67. [67]

    The Hartree–Fock simulations presented below employ the dispersion relationϵ(k) of the lowest conduction band above charge neutrality in rhombohedral tetralayer graphene

    Summary of Hartree-F ock solutions In the following, we illustrate the energy landscape for different crystalline ground states obtained within restricted and unrestricted Hartree-Fock, and their respective band structure. The Hartree–Fock simulations presented below employ the dispersion relationϵ(k) of the lowest conduction band above charge neutrality ...

  68. [68]

    =γk 4 0 −(4γ/α)k 2 +γ(4γ/α 2)k4 + Ω2 γk 4 0 r2 =γk 4 0 −2(k2/k2

  69. [69]

    k2 −k 2 0 2 2k4 0 + Ω2 2γk 4 0 r2 # = 2γk 4 0 k2 0

    + Ω2 γk 4 0 r2 = 2γk 4 0 " k2 −k 2 0 2 2k4 0 + Ω2 2γk 4 0 r2 # = 2γk 4 0 k2 0 " k2 −k 2 0 2 2k2 0 + k2 0Ω2 2γk 4 0 r2 # ,(D2) 11 where in going from the first to the second line we added an irrelevant constant to complete the square. By redefining units of energy and introducing the inverse lenght squaredω 2 = Ω2/γk 4 0, we finally obtain the simple Hamil...