Pith. sign in

REVIEW 3 major objections 5 minor 80 references

Gradient-based optimization of non-Abelian fractional quantum states in patterned superlattices

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Gradient-optimized patterned superlattices in bilayer graphene create Moore-Read non-Abelian fractional Chern insulator states at half filling, as shown by exact diagonalization.

desk verdict A solid new inverse-design workflow for Moore-Read FCIs in patterned superlattices, carried by ED on just two small clusters; send it to referees and use the reports to demand a third cluster, released data, and per-point cluster-size details. read the letter →

arxiv 2608.09245 v1 pith:7AJQALJ5 submitted 2026-08-10 cond-mat.mes-hall cond-mat.str-el

Yifei Guan , Lichen Yu , Zizhuang Liu , Xin Lu , Jianpeng Liu This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.str-el
keywords non-AbelianfractionalCherninsulatorMoore-Readstatepatterneddielectricsuperlatticebilayergraphenegradient-basedinversedesignquantumgeometryparticleentanglementspectrumtopologicalflatband
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

The paper aims to show that non-Abelian fractional topological order can be engineered from scratch by optimizing the physical dimensions of a patterned dielectric superlattice. The authors define a loss function that measures how closely a chosen flat band's quantum geometry resembles the first excited Landau level, and they minimize it by gradient descent over the etched-hole radius and depth in bilayer-graphene devices. At $\nu=1/2$ filling of the optimized band, exact diagonalization produces the signatures of the Moore-Read state: sixfold quasi-degenerate ground-state manifolds on 28-site tori, twofold manifolds on 26-site tori, and particle entanglement spectra with low-lying level counts matching Moore-Read quasihole counting. These signatures appear for triangular, honeycomb, and kagome superlattice patterns over large regions of the parameter space, so the workflow is a concrete experimental route to non-Abelian anyons without a magnetic field.

What carries the argument

The engine of the design is the loss function $L = |(1/2\pi)\int_{\mathrm{BZ}} d^2 k \, \mathrm{tr}\,g(k) - 3| + \alpha \sigma(\Omega)$ with $\alpha=0.5$, where $g$ is the quantum metric and $\sigma(\Omega)$ the dimensionless standard deviation of the Berry curvature of the highest valence band. It encodes the target that a flat Chern band have the quantum geometry of the first excited Landau level: uniform Berry curvature, Chern number $\pm 1$, and $\int \mathrm{tr}\,g = 3$. The gradient of $L$ with respect to the hole radius and depth is evaluated by finite differences, each evaluation requiring a finite-element electrostatic simulation of the etched substrate followed by a continuum $k\cdot p$ band-structure calculation of Bernal bilayer graphene. The many-body identification of the Moore-Read phase uses the $(2,4)$-admissible generalized Pauli principle to predict torus ground-state degeneracies and the particle entanglement spectrum to compare low-lying level counts with quasihole counting.

What would settle it

Run exact diagonalization at half filling on the same 28-site and 26-site torus geometries for a parameter point with low loss but a substantially larger bandwidth, or with a different screening length; if the sixfold/twofold ground-state degeneracy and the 18,571/2,522 low-lying particle-entanglement levels no longer match Moore-Read counting, the loss function alone is not a sufficient criterion.

Watch

Extended reading notes

Core claim

At half filling, the highest valence band of bilayer graphene coupled to a structurally optimized patterned dielectric superlattice realizes a Moore-Read-type fractional Chern insulator. Exact diagonalization gives a sixfold quasi-degenerate ground-state manifold on the 28-site torus, split between three momentum sectors with two states each, and the expected twofold manifold on the 26-site torus with an unpaired Majorana sector; particle entanglement spectra show 18,571 low-lying levels below the entanglement gap for the $N_A=4$ cut and 2,522 for the $N_A=3$ cut, both matching Moore-Read quasihole counting. The paper shows that these states appear only after optimizing the hole radius and depth, and that across the $(L_s,\delta V)$ maps, low loss values correlate strongly with Moore-Read order, while unoptimized or high-loss structures do not form such states.

Load-bearing premise

The load-bearing premise is that a single-particle band whose quantum geometry mimics the first Landau level is enough to produce the Moore-Read many-body state at half filling; the paper itself notes that the loss is a strong descriptor rather than a sufficient criterion, because band dispersion and other ingredients also matter.

Editorial extensions

If this is right

  • Patterned dielectric superlattices become a realistic zero-field platform for non-Abelian fractional Chern insulator states, with the relevant device parameters (hole radius and depth) set by nanofabrication rather than by twist-angle alignment.
  • The gradient-based inverse-design loop is not limited to bilayer graphene; the paper argues it extends to other van der Waals multilayers, so the same workflow could target other fractional phases.
  • The reported many-body gaps of order 0.3–0.4 meV with ground-state spreads as small as 0.002 meV give concrete energy scales for future experimental probes.
  • Fabrication must follow the optimized structural parameters: exact diagonalization of the unoptimized initial structures does not show Moore-Read degeneracy or entanglement signatures.
  • The strong correlation between small loss values and the appearance of Moore-Read order suggests the single-particle loss can be used as a fast screening tool over candidate devices before expensive many-body calculations.

Reading between the lines

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

  • An untested corollary of the paper's own caveat—that the loss is a strong descriptor rather than a sufficient criterion—is that scanning low-loss points with systematically varied bandwidth would map how far the first-LL geometry condition can be relaxed before Moore-Read order is lost.
  • The sign rule found between the triangular pattern (positive $\delta V$) and honeycomb/kagome patterns (negative $\delta V$), traced to the structure factor, gives a simple design rule that could be applied to other superlattice patterns without full optimization.
  • The same optimization loop could be run with a different Landau-level or model-wavefunction target, generating device designs for Laughlin-type or other topological orders and testing whether the inverse-design pipeline generalizes.
  • A transport-level signature, such as thermal Hall conductance quantization or interference measurements, is not computed in the paper but would be the natural experimental check of the non-Abelian order.
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

3 major / 5 minor

Summary. The manuscript proposes a gradient-based inverse-design workflow for patterned dielectric superlattices on bilayer graphene, optimizing the etched-hole radius and depth so that the highest valence band acquires a quantum geometry close to that of the first excited Landau level. At half filling of this band, exact diagonalization on tilted tori reports a sixfold quasi-degenerate ground-state manifold for N_s=28, N_e=14 and a twofold manifold for N_s=26, N_e=13, with particle entanglement spectra showing 18,571 and 2,522 low-lying levels below the entanglement gap, respectively. These signatures are interpreted as Moore-Read-type non-Abelian fractional Chern insulator states. The workflow is applied to triangular, honeycomb, and kagome superlattices, and many-body phase diagrams in the (L_s, δV) plane are presented. The paper explicitly acknowledges that the loss function is a strong descriptor rather than a sufficient criterion for Moore-Read order.

Significance. If the finite-size identification holds, this is a valuable contribution: it provides a realistic, experimentally accessible platform for non-Abelian FCIs without twist-angle fine-tuning, and it demonstrates a device-level inverse-design loop that couples FEM electrostatics, a realistic bilayer-graphene continuum model, and exact diagonalization. The strengths include the explicit optimization of experimentally relevant parameters, the positive comparison between optimized and unoptimized structures, and the even-odd electron-number dependence of the ground-state degeneracy, which is a sharp Moore-Read signature. The principal weakness is that the central many-body claim rests on only two small tilted-torus clusters, while the extended phase diagrams rely on single-cluster-size diagnostics at most parameter points, so the breadth of the conclusions currently exceeds the finite-size evidence.

major comments (3)
  1. [Non-Abelian FCI states (Fig. 3 and Fig. S4)] The central existence claim rests on ED at only two tilted tori: N_s=28, N_e=14 and N_s=26, N_e=13. The sixfold/twofold ground-state manifolds and the PES counts (18,571 and 2,522) are finite-size signatures; without a third, larger cluster (e.g., N_s=32 or N_s=38) at the three representative points, accidental small-system coincidences cannot be excluded. Please add at least one additional system size and report the evolution of the many-body gap and PES counting across system sizes.
  2. [Many-body phase diagrams (Fig. 4)] The phase diagrams in Fig. 4 label every (L_s, δV) point as Moore-Read or non-Moore-Read, but the caption does not state which cluster size (N_s=26 or 28) was used at each point, and the diagnostics are finite-size by construction. The claim of a robust Moore-Read region over an extended parameter area is therefore not supported for points checked at only one cluster size. Please specify the N_s used per point and verify representative boundary and non-MR points at a second cluster size.
  3. [Discussion and Eq. (2)] The paper correctly states that the loss function L is 'a strong descriptor rather than a sufficient criterion.' Given that the optimization workflow's promise is to design non-Abelian states, the average-loss separation (1.2 versus 4.3 for triangular, and similar for the others) is only suggestive. The exceptions visible in Fig. 4 show that small L does not always yield Moore-Read order and larger L does not always preclude it. To make the predictive claim load-bearing, provide a quantitative assessment (e.g., a threshold with false-positive and false-negative rates, or an out-of-sample test), or explicitly limit the claim to the ED-verified structures rather than to a general optimization guarantee.
minor comments (5)
  1. [Fig. 4 caption] Please state the cluster size N_s used for each parameter point, and clarify that white regions were not calculated because the HVB Chern number is not ±1.
  2. [PES counting paragraph] The counts 18,571 and 2,522 are quoted without a derivation or citation for the expected Moore-Read quasihole counting on these tori; adding the counting formula would strengthen the identification.
  3. [Eq. (2)] The dimensionless standard deviation σ(Ω) of the Berry curvature should be explicitly normalized (e.g., by the mean |Ω|), so that the meaning of α=0.5 and the reported value σ(Ω)=0.8 are unambiguous.
  4. [Supplemental Material S3.1] The inverse screening length κ=(120 nm)^{-1} and the single-valley-single-spin projection are significant modeling choices; please add one sentence on how varying κ or including the other valley/spin flavor would affect the ED results.
  5. [Data availability] A data-availability statement would be helpful: the FEM and ED workflows are complex, and no code or data are currently released, which makes independent verification difficult.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Moore-Read assignment rests on exact-diagonalization and entanglement-spectrum checks that are not inputs to the single-particle loss function.

full rationale

The derivation chain is: (1) FEM electrostatics plus a k·p continuum model produce single-particle bands; (2) the loss function in Eq. (2) is defined from first-LL quantum geometry, namely |(1/2π)∫trg − 3| + ασ(Ω), with α = 0.5 fixed a priori; (3) gradient descent over the hole radius and depth minimizes this loss; (4) exact diagonalization at half filling on Ns = 28 and Ns = 26 tilted tori yields sixfold and twofold quasi-degenerate ground-state manifolds together with PES low-lying counts of 18,571 and 2,522 levels; (5) these counting results are compared with independent Moore-Read quasihole and generalized-Pauli-principle benchmarks. The central existence claim is carried by step (4), which is not an input to step (2): the loss function never sees ground-state degeneracies, entanglement gaps, or PES counts, and the many-body Hamiltonian in Eq. (3) is solved rather than tuned. The paper itself explicitly limits the single-particle proxy in the 'Many-body phase diagrams' section: the loss is 'a strong descriptor rather than a sufficient criterion, because other ingredients, such as band dispersion, also influence the many-body ground states.' That caveat supports non-circularity because the ED calculation is treated as the arbiter rather than as a restatement of the loss. The self-citations in the modeling lineage, such as Refs. 68–69 for the first-LL metric condition, supply a standard and independently checkable geometric target and do not force the many-body result. The main limitation is finite-size evidence at only two cluster sizes, which is a robustness concern, not a circularity concern.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a chain of modeling choices: a continuum model for bilayer graphene, an FEM electrostatic model for the patterned substrate, a single-band single-valley projection of interactions, finite-size exact diagonalization, and an ad hoc loss function. No new physical entities are introduced. The optimized hole radius and depth, the loss weight alpha, and the screening length kappa are the main tunable inputs.

free parameters (4)
  • alpha (loss weight) = 0.5
    Hand-chosen relative weight for Berry-curvature fluctuations in Eq. (2); affects which structures are selected and therefore which many-body states appear.
  • inverse screening length kappa = (120 nm)^-1
    Chosen to model finite-thickness screening in the Coulomb potential, Eq. (S11); changes interaction strength and the stability of the FCI phase.
  • optimized hole radius r_hole = e.g., 13.9 nm for triangular Ls=60nm, dV=2V; varies with pattern and parameter point
    The device design variable optimized by gradient descent at each (Ls, dV); the final value determines the band geometry and the many-body outcome.
  • optimized hole depth h_hole = e.g., 20.1 nm for the same triangular case; varies
    Second device design variable optimized by gradient descent; enters the electrostatic potential and hence the band properties.
assumptions (5)
  • domain assumption The Slater-Koster-derived k.p continuum model (SM S1) accurately describes the low-energy electronic structure of Bernal bilayer graphene under a superlattice potential.
    The central single-particle input; errors in this model would propagate directly into the band geometry and the exact-diagonalization results.
  • domain assumption The FEM electrostatic simulation captures the device electrostatics, and retaining Fourier components through seventh order reproduces the FEM potential to better than 0.01%.
    The optimized r_hole and h_hole values and the resulting potentials depend on this model; no experimental validation is provided.
  • domain assumption Projecting Coulomb interactions onto a single spin-valley flavor and the highest valence band is sufficient at half filling (Eq. 3 and SM S3.1); other bands and intervalley scattering are negligible.
    If interband or intervalley coupling matters, the projected model could miss competing states or alter the topological order.
  • domain assumption The Moore-Read ground-state degeneracy and PES counting at Ns=26 and Ns=28 are representative of the thermodynamic limit.
    Only two small clusters are used; no finite-size scaling is presented.
  • ad hoc to paper The loss function L (Eq. 2), targeting first-LL quantum geometry, is a sufficient single-particle proxy for Moore-Read order.
    The authors explicitly state in the Discussion that the loss is a strong descriptor rather than a sufficient criterion, so the workflow's generality rests on this empirical correlation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gradient-based optimization of non-Abelian fractional quantum states in patterned superlattices." pith.science (2026). https://pith.science/paper/7AJQALJ5

@misc{pith2026260809245,
  author       = {Pith},
  title        = {Pith review of: Gradient-based optimization of non-Abelian fractional quantum states in patterned superlattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7AJQALJ5}},
  note         = {Machine review of arXiv:2608.09245}
}
read the original abstract

The realization of fractional Chern insulator (FCI) states in moir\'e heterostructures has attracted intense interest in the study of correlated states emerging from topological flat bands. So far, most experimentally realized FCI states may be interpreted as lattice analogues of fractional quantum Hall (FQH) states hosting Abelian anyonic excitations. Realizing non-Abelian FCI states is an important challenge in the field. Patterned dielectric superlattices provide a versatile platform for engineering topological flat bands. Such systems offer substantial structural flexibility and tunability, because their lattice patterns, periods, and other structural parameters can all be designed and fabricated. Here we propose to realize non-Abelian FCI states in patterned dielectric superlattices coupled to bilayer graphene. Specifically, we provide a realistic workflow based on a gradient-descent algorithm to design non-Abelian fractional states in bilayer graphene superlattices. The experimentally relevant structural parameters of the superlattices are gradient-optimized to favor a flat Chern band with quantum-geometric properties reminiscent of those of the first excited Landau level. Exact diagonalization calculations at 1/2 filling of the optimized flat Chern band naturally yield non-Abelian FCI states. We apply this workflow to triangular, honeycomb, and kagome patterned superlattices and find robust non-Abelian FCI states over a large region of the parameter space spanned by the superlattice constant and vertical potential drop. Our work thus establishes an experimentally feasible framework for exploring non-Abelian FCIs in realistic patterned-superlattice devices. It also demonstrates the potential of device-level inverse design to engineer correlated topological matter beyond the Abelian paradigm.

Figures

Figures reproduced from arXiv: 2608.09245 by the authors.

Figure 1
Figure 1. FIG. 1. (a) The left panel shows the superlattice device: bilayer graphene is encapsulated by thin hBN layers. A periodically [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Single-particle phase diagrams of the gradient-optimized superlattice structures for (a) triangular, (b) honeycomb, and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The many-body spectrum and PES of the Moore [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Many-body phase diagrams of the HVB at 1/2 filling in the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 46 canonical work pages

  1. [1]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences108, 12233 (2011)

    2011

  2. [2]

    Tarnopolsky, A

    G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Phys. Rev. Lett.122, 106405 (2019)

    2019

  3. [3]

    Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All magic angles in twisted bilayer graphene are topological, Phys. Rev. Lett.123, 036401 (2019)

    2019

  4. [4]

    J. Ahn, S. Park, and B.-J. Yang, Failure of nielsen-ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle, Phys. Rev. X9, 021013 (2019)

    2019

  5. [5]

    J. Liu, J. Liu, and X. Dai, Pseudo landau level representation of twisted bilayer graphene: Band topology and implications on the correlated insulating phase, Phys. Rev. B99, 155415 (2019)

    work page 2019

  6. [6]

    Bultinck, S

    N. Bultinck, S. Chatterjee, and M. P. Zaletel, Mechanism for anomalous hall ferromagnetism in twisted bilayer graphene, Phys. Rev. Lett.124, 166601 (2020)

    work page 2020

  7. [7]

    Zhang, D

    Y.-H. Zhang, D. Mao, and T. Senthil, Twisted bilayer graphene aligned with hexagonal boron nitride: Anomalous hall effect and a lattice model, Phys. Rev. Research1, 033126 (2019)

    work page 2019

  8. [8]

    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

Show all 80 references
  1. [9]

    J. Liu, Z. Ma, J. Gao, and X. Dai, Quantum valley hall effect, orbital magnetism, and anomalous hall effect in twisted multilayer graphene systems, Phys. Rev. X9, 031021 (2019)

    2019

  2. [10]

    Wang and Z

    J. Wang and Z. Liu, Hierarchy of ideal flatbands in chiral twisted multilayer graphene models, Phys. Rev. Lett. 128, 176403 (2022)

    2022

  3. [11]

    P. J. Ledwith, A. Vishwanath, and E. Khalaf, Family of ideal chern flatbands with arbitrary chern number in chiral twisted graphene multilayers, Phys. Rev. Lett. 128, 176404 (2022)

    2022

  4. [12]

    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, Nat. Rev. Mater.6, 201 (2021)

    2021

  5. [13]

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

    2018

  6. [14]

    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 superlattices, Nature556, 80 (2018)

    2018

  7. [15]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, 8 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

  8. [16]

    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

  9. [17]

    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

  10. [18]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Knüppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, andJ.Shan,Thermodynamicevidenceoffractionalchern insulator in moiré mote2, Nature622, 69 (2023)

    2023

  11. [19]

    H. Park, W. Li, C. Hu, C. Beach, M. Gonçalves, J. F. Mendez-Valderrama, J. Herzog-Arbeitman, T. Taniguchi, K. Watanabe, D. Cobden, L. Fu, B. A. Bernevig, N. Regnault, J.-H. Chu, D. Xiao, and X. Xu, Observation of dissipationless fractional chern insulator, Nature Physics22, 389 (2026)

    2026

  12. [20]

    Z. Sun, F. Xu, J. Li, Y. Jiang, J. Gao, C. Xu, T. Jia, K. Cheng, J. Zhang, W. Tian, K. Watanabe, T. Taniguchi, J. Jia, S. Jiang, Y. Zhang, Y. Zhang, S. Lei, X. Liu, and T. Li, Twist-angle evolution from valley-polarized fractional topological phases to valley- degenerate super...

    2026

  13. [21]

    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

  14. [22]

    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, Even- and odd- denominator fractional quantum anomalous hall effect in graphene moire superlattices (2024), arXiv:2405.16944 [cond-mat.mes-hall]

    2024 arXiv

  15. [23]

    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 moiré superlattices, Nature 10.1038/s41586-024-08470-1 (2025)

    2025 doi

  16. [24]

    Z. Li, W. Wang, F. Wang, Z. Zhang, Q. Yang, K. Watanabe, T. Taniguchi, X. C. Xie, J. Wang, K. Liu, Z. Song, and X. Lu, Fractional high-chern insulator in twisted rhombohedral graphene, Nature655, 892 (2026)

    2026

  17. [25]

    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 moiré superlattice, Phys. Rev. X15, 031026 (2025)

    2025

  18. [26]

    Regnault and B

    N. Regnault and B. A. Bernevig, Fractional chern insulator, Phys. Rev. X1, 021014 (2011)

    2011

  19. [27]

    D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum hall effect in the absence of landau levels, Nature Communications2, 389 (2011)

    2011

  20. [28]

    Neupert, L

    T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum hall states at zero magnetic field, Phys. Rev. Lett.106, 236804 (2011)

    2011

  21. [29]

    Tang, J.-W

    E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum hall states, Phys. Rev. Lett.106, 236802 (2011)

    2011

  22. [30]

    K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett.106, 236803 (2011)

    2011

  23. [31]

    Möller and N

    G. Möller and N. R. Cooper, Composite fermion theory for bosonic quantum hall states on lattices, Phys. Rev. Lett.103, 105303 (2009)

    2009

  24. [32]

    Qi, Generic wave-function description of fractional quantum anomalous hall states and fractional topological insulators, Phys

    X.-L. Qi, Generic wave-function description of fractional quantum anomalous hall states and fractional topological insulators, Phys. Rev. Lett.107, 126803 (2011)

    2011

  25. [33]

    Moore and N

    G. Moore and N. Read, Nonabelions in the fractional quantum hall effect, Nuclear Physics B360, 362 (1991)

    1991

  26. [34]

    Greiter, X.-G

    M. Greiter, X.-G. Wen, and F. Wilczek, Paired hall state at half filling, Phys. Rev. Lett.66, 3205 (1991)

    1991

  27. [35]

    Nayak and F

    C. Nayak and F. Wilczek, 2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum hall states, Nucl. Phys. B479, 529 (1996)

    1996

  28. [36]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-abelian anyons and topological quantum computation, Rev. Mod. Phys.80, 1083 (2008)

    2008

  29. [37]

    Kapit and E

    E. Kapit and E. Mueller, Exact parent hamiltonian for thequantumhallstatesinalattice,Phys.Rev.Lett.105, 215303 (2010)

    2010

  30. [38]

    Z. Liu, E. J. Bergholtz, and E. Kapit, Non-abelian fractional chern insulators from long-range interactions, Phys. Rev. B88, 205101 (2013)

    2013

  31. [39]

    D. Wang, Z. Liu, W.-M. Liu, J. Cao, and H. Fan, Fermionic non-abelian fractional chern insulators from dipolar interactions, Phys. Rev. B91, 125138 (2015)

    2015

  32. [40]

    He, W.-W

    A.-L. He, W.-W. Luo, H. Yao, and Y.-F. Wang, Non- abelian fractional chern insulator in disk geometry, Phys. Rev. B101, 165127 (2020)

    2020

  33. [41]

    A. P. Reddy, N. Paul, A. Abouelkomsan, and L. Fu, Non- abelian fractionalization in topological minibands, Phys. Rev. Lett.133, 166503 (2024)

    2024

  34. [42]

    Zhang and X.-Y

    L. Zhang and X.-Y. Song, Moore-read state in half- filled moiré chern band from three-body pseudopotential, Phys. Rev. B109, 245128 (2024)

    2024

  35. [43]

    C.-E. Ahn, W. Lee, K. Yananose, Y. Kim, and G. Y. Cho, Non-abelian fractional quantum anomalous hall states and first landau level physics of the second moiré band of twistedbilayermote 2,Phys.Rev.B110,L161109(2024)

    2024

  36. [44]

    C. Xu, N. Mao, T. Zeng, and Y. Zhang, Multiple chern bands in twistedmote 2 and possible non-abelian states, Phys. Rev. Lett.134, 066601 (2025)

    2025

  37. [45]

    Wang, X.-W

    C. Wang, X.-W. Zhang, X. Liu, J. Wang, T. Cao, and D. Xiao, Higher landau-level analogs and signatures of non-abelian states in twisted bilayermote2, Phys. Rev. Lett.134, 076503 (2025)

    2025

  38. [46]

    Chen, W.-W

    F. Chen, W.-W. Luo, W. Zhu, and D. N. Sheng, Robust non-abelian even-denominator fractional chern insulator in twisted bilayer mote2, Nature Communications16, 2115 (2025)

    2025

  39. [47]

    H. Liu, Z. Liu, and E. J. Bergholtz, Non-abelian fractional chern insulators and competing states in flat moiré bands, Phys. Rev. Lett.135, 106604 (2025)

    2025

  40. [48]

    S. Niu, Y. Peng, and D. N. Sheng, Quantum phase diagram and non-abelian moore-read state in double twisted bilayer graphene, Phys. Rev. B112, 035415 (2025)

    2025

  41. [49]

    B. Li, Y. Ouyang, and F. Wu, Abelian and non-abelian fractionalized states in twistedmote 2: A generalized 9 landau-level theory, Phys. Rev. B113, 195129 (2026)

    2026

  42. [50]

    Forsythe, X

    C. Forsythe, X. Zhou, K. Watanabe, T. Taniguchi, A. Pasupathy, P. Moon, M. Koshino, P. Kim, and C. R. Dean, Band structure engineering of 2D materials using patterned dielectric superlattices, Nature Nanotechnology13, 566 (2018)

    2018

  43. [51]

    S.-C. Chen, R. Kraft, R. Danneau, K. Richter, and M.- H. Liu, Electrostatic superlattices on scaled graphene lattices, Communications Physics3, 71 (2020)

    2020

  44. [52]

    Y. Li, S. Dietrich, C. Forsythe, T. Taniguchi, K. Watanabe, P. Moon, and C. R. Dean, Anisotropic band flattening in graphene with one-dimensional superlattices, Nature Nanotechnology16, 525 (2021)

    2021

  45. [53]

    Barcons Ruiz, H

    D. Barcons Ruiz, H. Herzig Sheinfux, R. Hoffmann, I. Torre, H. Agarwal, R. K. Kumar, L. Vistoli, T. Taniguchi, K. Watanabe, A. Bachtold, and F. H. L. Koppens, Engineering high quality graphene superlattices via ion milled ultra-thin etching masks, Nature Communications13, 6926 (2022)

    2022

  46. [54]

    S. Wang, Z. Zhan, X. Fan, Y. Li, P. A. Pantaleón, C. Ye, Z. He, L. Wei, L. Li, F. Guinea, S. Yuan, and C. Zeng, Dispersion-selective band engineering in an artificial kagome superlattice, Phys. Rev. Lett.133, 066302 (2024)

    2024

  47. [55]

    J. Sun, S. A. Akbar Ghorashi, K. Watanabe, T. Taniguchi, F. Camino, J. Cano, and X. Du, Signature of correlated insulator in electric field controlled superlattice, Nano Letters24, 13600 (2024), pMID: 39432385, https://doi.org/10.1021/acs.nanolett.4c03238

    2024 doi

  48. [56]

    S. A. A. Ghorashi, A. Dunbrack, A. Abouelkomsan, J. Sun, X. Du, and J. Cano, Topological and stacked flat bands in bilayer graphene with a superlattice potential, Phys. Rev. Lett.130, 196201 (2023)

    2023

  49. [57]

    X. Lu, S. Zhang, Y. Wang, X. Gao, K. Yang, Z. Guo, Y. Gao, Y. Ye, Z. Han, and J. Liu, Synergistic correlated states and nontrivial topology in coupled graphene- insulator heterostructures, Nature Communications14, 5550 (2023)

    2023

  50. [58]

    S. A. A. Ghorashi and J. Cano, Multilayer graphene with a superlattice potential, Phys. Rev. B107, 195423 (2023)

    2023

  51. [59]

    Seleznev, J

    D. Seleznev, J. Cano, and D. Vanderbilt, Inducing topological flat bands in bilayer graphene with electric and magnetic superlattices, Phys. Rev. B110, 205115 (2024)

    2024

  52. [60]

    Z. Zhan, Y. Li, and P. A. Pantaleón, Designing band structures by patterned dielectric superlattices, Phys. Rev. B111, 045148 (2025)

    2025

  53. [61]

    T. Tan, A. P. Reddy, L. Fu, and T. Devakul, Designing topology and fractionalization in narrow gap semiconductor films via electrostatic engineering, Phys. Rev. Lett.133, 206601 (2024)

    2024

  54. [62]

    W. Miao, A. Rashidi, and X. Dai, Artificial moiré engineering for an ideal bhz model (2024), arXiv:2409.08540 [cond-mat.mes-hall]

    2024

  55. [63]

    Y. Shi, B. Xie, F. Ren, X. Cai, Z. Guo, Q. Li, X. Lu, N. Regnault, Z. Liu, and J. Liu, Fractional topological states in rhombohedral multilayer graphene modulated by kagome superlattice, Phys. Rev. Lett.135, 256603 (2025)

    2025

  56. [64]

    A. Uri, S. Grover, Y. Cao, J. A. Crosse, K. Bagani, D. Rodan-Legrain, Y. Myasoedov, K. Watanabe, T. Taniguchi, P. Moon, M. Koshino, P. Jarillo-Herrero, and E. Zeldov, Mapping the twist-angle disorder and landau levels in magic-angle graphene, Nature581, 47 (2020)

    2020

  57. [65]

    N. P. Kazmierczak, M. Van Winkle, C. Ophus, K. C. Bustillo, S. Carr, H. G. Brown, J. Ciston, T. Taniguchi, K. Watanabe, and D. K. Bediako, Strain fields in twisted bilayer graphene, Nature materials20, 956 (2021)

    2021

  58. [66]

    R. H. Morf, Transition from quantum hall to compressible states in the second landau level: New light on theν= 5/2enigma, Phys. Rev. Lett.80, 1505 (1998)

    1998

  59. [67]

    Storni, R

    M. Storni, R. H. Morf, and S. Das Sarma, Fractional quantum hall state atν= 5 2 and the moore-read pfaffian, Phys. Rev. Lett.104, 076803 (2010)

    2010

  60. [68]

    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

  61. [69]

    Z. Liu, B. Mera, M. Fujimoto, T. Ozawa, and J. Wang, Theory of generalized landau levels and its implications for non-abelian states, Phys. Rev. X15, 031019 (2025)

    2025

  62. [70]

    P. J. Ledwith, E. Khalaf, and A. Vishwanath, Strong coupling theory of magic-angle graphene: A pedagogical introduction, Annals of Physics435, 168646 (2021)

    2021

  63. [71]

    B. A. Bernevig and N. Regnault, Emergent many- bodytranslationalsymmetriesofabelianandnon-abelian fractionally filled topological insulators, Phys. Rev. B85, 075128 (2012)

    2012

  64. [72]

    Li and F

    H. Li and F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification of topological order in non-abelian fractional quantum hall effect states, Physical Review Letters101, 010504 (2008)

    2008

  65. [73]

    Chandran, M

    A. Chandran, M. Hermanns, N. Regnault, and B. A. Bernevig, Bulk-edge correspondence in entanglement spectra, Phys. Rev. B84, 205136 (2011)

    2011

  66. [74]

    Sterdyniak, N

    A. Sterdyniak, N. Regnault, and G. Möller, Particle entanglement spectra for quantum hall states on lattices, Phys. Rev. B86, 165314 (2012)

    2012

  67. [75]

    A. G. Fonseca, E. Wang, S. Vaidya, P. J. Ledwith, A. Vishwanath, and M. Soljačić, Gradient-based search ofquantumphases: discoveringunconventionalfractional chern insulators (2026), arXiv:2509.10438 [cond-mat.str- el]

    2026

  68. [76]

    Moon and M

    P. Moon and M. Koshino, Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice, Phys. Rev. B90, 155406 (2014)

    2014

  69. [77]

    Moon and M

    P. Moon and M. Koshino, Optical absorption in twisted bilayer graphene, Physical Review B87, 205404 (2013)

    2013

  70. [78]

    J. Y. Lee, E. Khalaf, S. Liu, X. Liu, Z. Hao, P. Kim, and A. Vishwanath, Theory of correlated insulating behaviour and spin-triplet superconductivity in twisted double bilayer graphene, Nature Communications10, 5333 (2019)

    2019

  71. [79]

    Zhang, X

    S. Zhang, X. Dai, and J. Liu, Spin-polarized nematic order, quantum valley hall states, and field-tunable topological transitions in twisted multilayer graphene systems, Phys. Rev. Lett.128, 026403 (2022)

    2022

  72. [80]

    Gradient-based optimization of non-Abelian fractional quantum states in patterned superlattices

    C. Repellin, B. A. Bernevig, and N. Regnault,𭟋 2 fractional topological insulators in two dimensions, Phys. Rev. B90, 245401 (2014). 10 Supplemental Material for “Gradient-based optimization of non-Abelian fractional quantum states in patterned superlattices” S1. NON-INTERACTI...

    2014

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.