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REVIEW 4 major objections 4 minor 70 references

Twisted hBN between two TMD layers can produce a chiral p-wave exciton condensate that is both a Chern-number-one anomalous Hall insulator and a counterflow superfluid, at one hole per moiré cell, with no magnetic field.

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2026-08-04 17:12 UTC pith:WLQFDHRI

load-bearing objection A credible mean-field proposal for a zero-field chiral p-wave excitonic insulator in TMD/hBN/TMD stacks; the physics is borrowed from MoTe2/WSe2 theory, the geometry is new, and the main risk is the uncontrolled single-band truncation. the 4 major comments →

arxiv 2509.11041 v1 pith:WLQFDHRI submitted 2025-09-14 cond-mat.mes-hall

Topological excitonic insulators in electron bilayers modulated by twisted hBN

Yongxin Zeng , Allan H. MacDonald , Nemin Wei This is my paper
classification cond-mat.mes-hall
keywords exciton condensatequantum anomalous Hall effecttwisted hBNtransition metal dichalcogenide bilayersmoiré potentialchiral p-wave pairingcounterflow superfluiditymean-field phase diagram
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper predicts that two TMD monolayers separated by a twisted hBN spacer can form an equilibrium exciton condensate at zero magnetic field: at one hole per moiré unit cell, interlayer Coulomb interactions spontaneously establish phase coherence between the electrically isolated layers. The central result is a mean-field phase diagram showing that near zero displacement field and intermediate moiré modulation strength, the ground state is a chiral p-wave exciton condensate—a quantum anomalous Hall insulator with Chern number |C|=1 that also carries counterflow superfluidity. This matters because spontaneous interlayer coherence at zero field has been predicted but never confirmed in semiconductor bilayers, and the proposed geometry avoids the usual requirement of lattice alignment: the twisted hBN's ferroelectric moiré imprints opposite potentials on the two layers automatically. If the prediction holds, the p-wave state would be a concrete realization of a simultaneous topological insulator and exciton superfluid in a two-dimensional bilayer.

Core claim

On the paper's own terms, the discovery is that a bilayer of two TMD monolayers separated by a twisted hBN spacer, with interlayer tunneling suppressed and opposite triangular-lattice moiré modulations imposed on the two layers, spontaneously develops interlayer coherence at total filling ν=1. In the mean-field ground state, the layer pseudospin winds in momentum space in a chiral pattern: vortices at the κ and κ′ points carry opposite layer polarizations, producing a skyrmion texture with Chern number |C|=1 and therefore a quantum anomalous Hall effect without any Landau levels or magnetic field. The same state is an exciton condensate with counterflow superfluidity, since electrons and hol

What carries the argument

The load-bearing object is the momentum-space layer pseudospin texture n_k = (sin θ_k cos φ_k, sin θ_k sin φ_k, cos θ_k) defined by the interlayer-coherent mean-field state. The paper shows that when pairing occurs at momentum shift Q = ±κ, the maximum of one band aligns with the minimum of the other, and the resulting mean-field Hamiltonian reduces to nearest-neighbor interlayer coherence amplitudes on a honeycomb lattice—formally the graphene tight-binding model with complex hopping. The winding of the pseudospin texture then determines the pairing symmetry: p_x ± i p_y vortices at κ and κ′ with opposite layer polarizations give a skyrmion texture and Chern number |C|=1 (the p-EI), while v

Load-bearing premise

The prediction depends on the mean-field approximation being reliable in the region where the p-wave state appears; the authors themselves note that quantum fluctuations are likely to enlarge the metallic region and push the p-EI/nematic boundary to stronger interactions, so if fluctuations wipe out the p-wave state near zero displacement field, the central claim fails.

What would settle it

Measure Hall resistance and counterflow conductance in a device tuned to one hole per moiré cell, V_D=0, and V_m around 7 meV; if there is no quantized Hall plateau and no superfluid counterflow response, the predicted p-wave exciton condensate is absent. Equivalently, an unbiased numerical solution of the projected two-band model at those parameters that finds a metal or a topologically trivial insulator would falsify the mean-field prediction.

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

If this is right

  • At total filling ν=1, the TMD/hBN/TMD stack is predicted to be an insulator with spontaneous interlayer coherence over a wide range of displacement fields and modulation strengths, with no single-particle tunneling and no applied magnetic field.
  • Near V_D=0 and V_m around 6–8 meV (with d=4 nm, ε=6, a_m=8 nm), the ground state is the p-EI: a quantum anomalous Hall insulator with Chern number |C|=1 and counterflow superfluidity.
  • Because the twisted hBN spacer imprints the moiré potential remotely, no lateral alignment between TMD layers or between TMD and hBN is required—removing the main experimental obstacle to zero-field exciton condensation in semiconductor bilayers.
  • The p-EI should show a quantized Hall drag effect, analogous to bilayer quantum Hall exciton condensates, providing a clear experimental signature that would also support the excitonic scenario proposed for MoTe2/WSe2 heterobilayers.
  • The qualitative structure of the phase diagram—layer-polarized, s-wave, nematic, p-wave, and metallic regions—persists across changes in layer asymmetry, dielectric constant, interlayer distance, and moiré length, with the p-EI favored at weaker interactions and near V_D=0.

Where Pith is reading between the lines

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

  • If the p-wave condensate is confirmed, the same geometry is a natural platform to search for fractional Chern states at other fillings, since the spontaneous coherence generates topologically nontrivial bands without band-structure engineering.
  • The honeycomb-lattice analogy suggests that the p-EI is equivalent to a spontaneously generated complex nearest-neighbor hopping; a concrete extension would be to measure chiral edge conduction directly and test whether the edge current direction is set by the sign of V_D or by sample-specific disorder.
  • The authors' mean-field caveat points to a direct theoretical test: run an unbiased many-body simulation of the same single-band model at the predicted p-EI parameters (e.g., V_m≈7 meV, V_D=0, d=4 nm, ε=6) to see whether the Chern-number-one interlayer coherent state survives beyond the mean-field approximation.
  • Because the hBN spacer guarantees independent electrical contacts, counterflow measurements in this geometry could cleanly separate spontaneous interlayer coherence from single-particle tunneling artifacts, making it a sharper test of exciton condensation than tunnel-coupled heterobilayers.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes a moiré device in which two TMD monolayers are separated by a twisted hBN multilayer that suppresses interlayer tunneling and imprints opposite triangular-lattice moiré potentials on the two layers. Using a continuum model plus dual-gate-screened Coulomb interactions and self-consistent Hartree-Fock in a single miniband per layer, the authors compute the ν=1 phase diagram as a function of modulation strength V_m and displacement field V_D. They identify layer-polarized, s-wave excitonic insulator, nematic excitonic insulator, metallic, and chiral p-wave excitonic insulator (p-EI) phases. The p-EI, found near V_D=0 and intermediate V_m, is characterized by a skyrmionic layer-pseudospin texture, Chern number |C|=1, and counterflow superfluidity. Phase diagrams for different α, d, ε, and a_m are collected in the Supplemental Material.

Significance. If the prediction is robust, the work would be a substantial advance: a zero-field equilibrium exciton condensate with spontaneous interlayer coherence, no single-particle tunneling, and coexisting quantum anomalous Hall and counterflow superfluid responses. The paper gives a concrete experimental signature (quantized Hall drag) and the mean-field calculations are self-consistent and internally consistent. The authors also honestly acknowledge that Hartree-Fock overestimates symmetry-broken states. However, the central p-EI claim rests on two approximations that are not fully controlled—the single-miniband projection at weak modulation and the neglect of spin/valley fluctuations—and the paper's abstract/conclusion are considerably stronger than these caveats.

major comments (4)
  1. [Model / SM Sec. I.A] The single-band projection is uncontrolled in the parameter region of the p-EI. The paper keeps only the topmost miniband per layer (SM Eq. S1–S2) assuming interactions do not cause significant band mixing. In Fig. 3 the p-EI appears for V_m ≈ 5–8 meV, a_m = 8 nm, ε = 6. The separation to the next miniband is O(V_m/3) ≈ 2–3 meV, while e²/(ε a_m) ≈ 30 meV. Since the interaction exceeds this gap by an order of magnitude, the truncation is not justified at these parameters. A two-band or full continuum HF calculation is needed to establish that the p-EI is not a projection artifact.
  2. [Discussion, third paragraph] The authors correctly state that Hartree-Fock overestimates symmetry-broken states and that quantum fluctuations will enlarge the metallic region and move the p-EI/NEI boundary to larger V_m. However, the abstract and introduction present the p-EI as a definitive prediction, despite this acknowledged caveat. The parameters highlighted in Fig. 4 (V_m=7 meV, V_D=0) lie precisely in the most fragile regime. Please either provide a beyond-mean-field estimate (e.g., RPA or QMC) for the p-EI region, or temper the central claim accordingly.
  3. [Mean-field theory, paragraph on pairing momentum Q] The choice of pairing momentum Q=κ is load-bearing for the p-EI state, but the paper only states that 'the ground state energy is minimized when Q=±κ' without showing the energy comparison. Please provide the energy versus Q scan or a symmetry argument that rules out other momenta (e.g., Q=0). Without this evidence, the phase diagram is restricted to an assumed pairing channel.
  4. [Model, spin neglect] The spin (valley) degree of freedom is neglected with the argument that spin-order energy scales are smaller than layer-order scales (footnote [49]). At ν=1 in a spinful system, the spinless p-EI state requires full spin polarization. The cited Ref. [56] supports kinetic ferromagnetism in a tunneling-coupled MoTe2/WSe2 model; its applicability to the hBN-separated bilayer is not demonstrated. A calculation, or at least a more detailed symmetry argument, is needed to show that spin-unpolarized competing states do not preempt the QAH phase.
minor comments (4)
  1. [Introduction, first paragraph] Typo: 'transiton' should be 'transition'. In the Model section, 'Hamiltonain' should be 'Hamiltonian'.
  2. [Fig. 2(b)] The band-structure plot would benefit from labeling the topmost miniband and the gap to the next band. The y-axis range (0–80 meV) obscures the relevant miniband gap, which is central to the projection issue.
  3. [SM Sec. I.B] The statement that a 15×15 k-grid and reciprocal-lattice cutoff at 4|g_1| 'ensures convergence' is not supported by a convergence test. Provide at least one check of the ground-state energy or phase boundary versus k-grid size.
  4. [Fig. 4] The pseudospin texture plots are dense; please define the normalization of n_z and clarify the color scale. A zoomed inset near one vortex would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the p-EI phase emerges from self-consistent Hartree-Fock with scanned parameters, not from fitting or self-citation.

full rationale

The derivation chain is self-contained: the authors start from a continuum model with stated physical parameters (m = 0.5 m_e, psi = pi/2 from twisted-hBN symmetry, alpha, d, a_m, epsilon), project each layer to its topmost miniband, and solve the Hartree-Fock equations self-consistently. The p-EI state is not an input; it emerges as a self-consistent solution in a region of the (V_m, V_D) phase diagram. The pairing momentum Q is not assumed to be kappa: the paper states 'We find that the ground state energy is minimized when Q = +/- kappa', i.e., it is selected by energy minimization. The topological Chern number is computed from the Berry curvature of the resulting self-consistent state, not imposed. Model inputs are physical or symmetry-motivated rather than fitted to reproduce the p-EI phase, and the phase diagram is compared with independent lattice-model work (Ref. [56]). The only same-author citation that plays a modeling role is Ref. [29] for the spinless approximation, but footnote [49] supplies an energy-scale argument (t^2/U vs t^2/V) and the paper explicitly says restoring spin would introduce competing spin-independent exciton order without qualitative changes; thus this self-citation is not load-bearing. The single-band projection is an approximation that may be numerically uncontrolled, but it is an openly stated assumption, not an input-output equivalence; concerns about its validity are correctness risks, not circularity. No fitted parameter is renamed as a prediction, and no uniqueness theorem is imported from the authors' prior work. Therefore the central derivation is not circular.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

The central claim rests on five modeling assumptions: single-band projection, spinless treatment, zero interlayer tunneling, first-harmonic remote moire potential, and HF accuracy. None of these are fitted to the target p-EI state; they are standard or explicitly flagged approximations. The free parameters are physical and geometric inputs scanned over a range, not fitted values, which keeps the circularity burden low. No new particles, forces, or conserved quantities are introduced.

free parameters (7)
  • effective mass m = 0.5 m_e
    Chosen as a representative TMD hole mass; sets band dispersion and phase-boundary locations but not the qualitative phase structure.
  • moire potential shape psi = pi/2
    Set by hBN ferroelectric domain symmetry; controls which potential extrema holes occupy and is essential for the honeycomb layer-sublattice locking that underlies p-EI.
  • modulation strength V_m = swept 4-10 meV in Fig. 3; p-EI near 6-7 meV
    Main control axis of the phase diagram; p-EI exists only in a window of intermediate V_m. It is a tunable experimental knob, not fitted to the target state.
  • layer asymmetry alpha = 1 in main diagram; 0.5 in Fig. S1(a)
    Ratio of bottom/top moire potential strength. alpha=1 is the symmetric case with the broadest p-EI region; other values give qualitatively similar phase diagrams.
  • interlayer distance d = 4 nm (3-5 nm in SM)
    Controls interlayer Coulomb coupling strength. The p-EI region shifts with d but persists in the SM scans.
  • moire length a_m = 8 nm (6-10 nm in SM)
    Sets miniband width; 8 nm is a representative value, with other values producing qualitatively similar phase diagrams.
  • dielectric constant epsilon and gate distance d_g = epsilon=6, d_g=50 nm
    Environmental screening inputs. The paper states d_g does not affect main results, and epsilon variations are shown in the SM.
axioms (5)
  • domain assumption Only the topmost miniband of each layer is retained; interactions do not cause significant band mixing
    Stated in the Mean-field theory section before projection. If band mixing is strong near phase boundaries, the HF phase diagram and p-EI topology could change.
  • domain assumption Spin and valley degrees of freedom are neglected
    The authors argue the spin-order energy scale is much smaller than the layer-order scale (Ref. [49]) and cite kinetic ferromagnetism in Ref. [56]; the nu=1 QAH state is obtained within this spinless model.
  • domain assumption Twisted hBN fully suppresses interlayer tunneling
    The model contains no interlayer single-particle hopping; the claim of spontaneous U(1) breaking and counterflow superfluidity relies on this isolation.
  • domain assumption First-harmonic triangular moire potential with psi=pi/2 and opposite signs in the two layers captures remote hBN imprinting
    Used in Eqs. (1)-(3). Supported by hBN domain symmetry and remote-imprint experiments, but the actual potential strength and functional form are not measured in this paper.
  • domain assumption Hartree-Fock mean-field ground state is a faithful guide to the phase diagram
    All phase boundaries come from self-consistent HF. The authors concede in the Discussion that HF overestimates symmetry-broken states, so the p-EI region may shrink with fluctuations.

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read the original abstract

Equilibrium interlayer exciton condensation is common in bilayer quantum Hall systems and is characterized by spontaneous phase coherence between isolated layers. It has been predicted that similar physics can occur in the absence of a magnetic field in some two-dimensional semiconductor bilayers. In this work we consider the case of two transition metal dichalcogenide (TMD) monolayers separated by a twisted hexagonal boron nitride (hBN) bilayer or multilayer. The hBN layers suppress tunneling between the TMD layers so that phase coherence is spontaneous when it is present. When twisted, the hBN layers also form a ferroelectric moir\'e pattern that applies opposite triangular-lattice modulation potentials to the two TMD layers. We show via mean-field theory that at total hole filling per moir\'e unit cell $\nu=1$, this geometry can favor a chiral p-wave exciton condensate state in which the quantum anomalous Hall effect coexists with counter-flow superfluidity. We present a mean-field phase diagram for TMD hole bilayers modulated by twisted hBN, discuss the conditions needed for the realization of the p-wave condensate state, and propose experiments that could confirm its presence.

Figures

Figures reproduced from arXiv: 2509.11041 by Allan H. MacDonald, Nemin Wei, Yongxin Zeng.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the device structure. The [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Mean-field phase diagram in the space of modulation [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a-c) Berry curvature and (d-f) layer pseudospin distributions of (a,d) the p-EI state at [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

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Reference graph

Works this paper leans on

70 extracted references · 4 linked inside Pith

  1. [1]

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

    2019

  2. [2]

    Morales-Dur´ an, N

    N. Morales-Dur´ an, N. Wei, J. Shi, and A. H. MacDonald, Magic angles and fractional chern insulators in twisted homobilayer transition metal dichalcogenides, Phys. Rev. Lett.132, 096602 (2024)

    2024

  3. [3]

    J. Shi, N. Morales-Dur´ an, E. Khalaf, and A. H. MacDon- ald, Adiabatic approximation and aharonov-casher bands in twisted homobilayer transition metal dichalcogenides, Phys. Rev. B110, 035130 (2024)

    2024

  4. [4]

    Devakul, V

    T. Devakul, V. Cr´ epel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature communications12, 6730 (2021)

    2021

  5. [5]

    Zhang, N

    F. Zhang, N. Morales-Dur´ an, Y. Li, W. Yao, J.-J. Su, Y.- C. Lin, C. Dong, X. Liu, F.-X. R. Chen, H. Kim,et al., Experimental signature of layer skyrmions and implica- tions for band topology in twisted WSe2 bilayers, Nature Physics , 1 (2025)

    2025

  6. [6]

    Thompson, K

    E. Thompson, K. T. Chu, F. Mesple, X.-W. Zhang, C. Hu, Y. Zhao, H. Park, J. Cai, E. Anderson, K. Watan- abe,et al., Microscopic signatures of topology in twisted MoTe2, Nature Physics , 1 (2025)

    2025

  7. [7]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y. Zhang, F. Fan, T. Taniguchi, K. Watanabe,et al., Signatures of fractional quantum 6 anomalous Hall states in twisted MoTe2, Nature622, 63 (2023)

    2023

  8. [8]

    H. Park, J. Cai, E. Anderson, Y. Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu,et al., Observa- tion of fractionally quantized anomalous hall effect, Na- ture622, 74 (2023)

    2023

  9. [9]

    Y. Zeng, Z. Xia, K. Kang, J. Zhu, P. Kn¨ uppel, C. Vaswani, K. Watanabe, T. Taniguchi, K. F. Mak, and J. Shan, Thermodynamic evidence of fractional Chern insulator in moir´ e MoTe2, Nature622, 69 (2023)

    2023

  10. [10]

    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 In- teger and Fractional Quantum Anomalous Hall Effects in Twisted Bilayer MoTe2, Phys. Rev. X13, 031037 (2023)

    2023

  11. [11]

    We restrict our attention here to the case of equilibrium exciton condensates in which⟨ψ † t ψb⟩is time-independent. Related physics occurs in quasi-equilibrium [24] or non- equilibrium [66, 67] exciton condensates in which a bias voltage is applied between layers and⟨ψ † t ψb⟩has a time- dependent phase

  12. [12]

    Eisenstein and A

    J. Eisenstein and A. H. MacDonald, Bose–einstein con- densation of excitons in bilayer electron systems, Nature 432, 691 (2004)

    2004

  13. [13]

    Eisenstein, Exciton condensation in bilayer quantum hall systems, Annu

    J. Eisenstein, Exciton condensation in bilayer quantum hall systems, Annu. Rev. Condens. Matter Phys.5, 159 (2014)

    2014

  14. [14]

    I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Resonantly enhanced tunneling in a double layer quantum hall ferromagnet, Phys. Rev. Lett.84, 5808 (2000)

    2000

  15. [15]

    Kellogg, J

    M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Vanishing hall resistance at high magnetic field in a double-layer two-dimensional electron system, Phys. Rev. Lett.93, 036801 (2004)

    2004

  16. [16]

    Tutuc, M

    E. Tutuc, M. Shayegan, and D. A. Huse, Counterflow measurements in strongly correlated GaAs hole bilayers: Evidence for electron-hole pairing, Phys. Rev. Lett.93, 036802 (2004)

    2004

  17. [17]

    Kellogg, I

    M. Kellogg, I. B. Spielman, J. P. Eisenstein, L. N. Pfeif- fer, and K. W. West, Observation of quantized hall drag in a strongly correlated bilayer electron system, Phys. Rev. Lett.88, 126804 (2002)

    2002

  18. [18]

    J. Li, T. Taniguchi, K. Watanabe, J. Hone, and C. Dean, Excitonic superfluid phase in double bilayer graphene, Nature Physics13, 751 (2017)

    2017

  19. [19]

    X. Liu, K. Watanabe, T. Taniguchi, B. I. Halperin, and P. Kim, Quantum hall drag of exciton condensate in graphene, Nature Physics13, 746 (2017)

    2017

  20. [20]

    R. Qi, Q. Li, Z. Zhang, J. Nie, B. Zou, Z. Cui, H. Kim, C. Sanborn, S. Chen, J. Xie,et al., Competition between excitonic insulators and quantum hall states in correlated electron–hole bilayers, Nature Materials , 1 (2025)

    2025

  21. [21]

    P. X. Nguyen, R. Chaturvedi, B. Zou, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Quantum oscillations in a dipolar excitonic insulator, Na- ture Materials , 1 (2025)

    2025

  22. [22]

    B. Zou, Y. Zeng, A. H. MacDonald, and A. Strashko, Electrical control of two-dimensional electron-hole fluids in the quantum hall regime, Phys. Rev. B109, 085416 (2024)

    2024

  23. [23]

    Zou and A

    B. Zou and A. H. MacDonald, Vortex lattice states of bilayer electron-hole fluids in quantizing magnetic fields, Phys. Rev. Lett.135, 086002 (2025)

    2025

  24. [24]

    Zeng and A

    Y. Zeng and A. H. MacDonald, Electrically controlled two-dimensional electron-hole fluids, Phys. Rev. B102, 085154 (2020)

    2020

  25. [25]

    L. Ma, P. X. Nguyen, Z. Wang, Y. Zeng, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Strongly correlated excitonic insulator in atomic double layers, Nature598, 585 (2021)

    2021

  26. [26]

    R. Qi, A. Y. Joe, Z. Zhang, Y. Zeng, T. Zheng, Q. Feng, J. Xie, E. Regan, Z. Lu, T. Taniguchi,et al., Thermody- namic behavior of correlated electron-hole fluids in van der waals heterostructures, Nature communications14, 8264 (2023)

    2023

  27. [27]

    R. Qi, A. Y. Joe, Z. Zhang, J. Xie, Q. Feng, Z. Lu, Z. Wang, T. Taniguchi, K. Watanabe, S. Tongay,et al., Perfect coulomb drag and exciton transport in an exci- tonic insulator, Science388, 278 (2025)

    2025

  28. [28]

    P. X. Nguyen, L. Ma, R. Chaturvedi, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Perfect coulomb drag in a dipolar excitonic insulator, Science388, 274 (2025)

    2025

  29. [29]

    Y. Zeng, N. Wei, and A. H. MacDonald, Layer pseudospin magnetism in a transition metal dichalcogenide double- moir´ e system, Phys. Rev. B106, 165105 (2022)

    2022

  30. [30]

    Zhang, D

    Y.-H. Zhang, D. N. Sheng, and A. Vishwanath, Su(4) chi- ral spin liquid, exciton supersolid, and electric detection in moir´ e bilayers, Phys. Rev. Lett.127, 247701 (2021)

    2021

  31. [31]

    Del Re and L

    L. Del Re and L. Classen, Field control of symmetry- broken and quantum disordered phases in frustrated moir´ e bilayers with population imbalance, Phys. Rev. Res.6, 023082 (2024)

    2024

  32. [32]

    Zhang, E

    Z. Zhang, E. C. Regan, D. Wang, W. Zhao, S. Wang, M. Sayyad, K. Yumigeta, K. Watanabe, T. Taniguchi, S. Tongay,et al., Correlated interlayer exciton insula- tor in heterostructures of monolayer WSe 2 and moir´ e WS2/WSe2, Nature physics18, 1214 (2022)

    2022

  33. [33]

    J. Gu, L. Ma, S. Liu, K. Watanabe, T. Taniguchi, J. C. Hone, J. Shan, and K. F. Mak, Dipolar excitonic insulator in a moir´ e lattice, Nature physics18, 395 (2022)

    2022

  34. [34]

    D. Chen, Z. Lian, X. Huang, Y. Su, M. Rashetnia, L. Ma, L. Yan, M. Blei, L. Xiang, T. Taniguchi,et al., Excitonic insulator in a heterojunction moir´ e superlattice, Nature Physics18, 1171 (2022)

    2022

  35. [35]

    Y. Zeng, Z. Xia, R. Dery, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Exciton density waves in coulomb-coupled dual moir´ e lattices, Nature Materials 22, 175 (2023)

    2023

  36. [36]

    Yasuda, X

    K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Stacking-engineered ferroelectricity in bilayer boron nitride, Science372, 1458 (2021)

    2021

  37. [37]

    Vizner Stern, Y

    M. Vizner Stern, Y. Waschitz, W. Cao, I. Nevo, K. Watanabe, T. Taniguchi, E. Sela, M. Urbakh, O. Hod, and M. Ben Shalom, Interfacial ferroelectricity by van der waals sliding, Science372, 1462 (2021)

    2021

  38. [38]

    D. S. Kim, R. C. Dominguez, R. Mayorga-Luna, D. Ye, J. Embley, T. Tan, Y. Ni, Z. Liu, M. Ford, F. Y. Gao, et al., Electrostatic moir´ e potential from twisted hexago- nal boron nitride layers, Nature materials23, 65 (2024)

    2024

  39. [39]

    Zhang, J

    Z. Zhang, J. Xie, W. Zhao, R. Qi, C. Sanborn, S. Wang, S. Kahn, K. Watanabe, T. Taniguchi, A. Zettl,et al., En- gineering correlated insulators in bilayer graphene with a remote coulomb superlattice, Nature Materials23, 189 (2024)

    2024

  40. [40]

    M. He, J. Cai, H. Zheng, E. Seewald, T. Taniguchi, 7 K. Watanabe, J. Yan, M. Yankowitz, A. Pasupathy, W. Yao,et al., Dynamically tunable moir´ e exciton ry- dberg states in a monolayer semiconductor on twisted bilayer graphene, Nature Materials23, 224 (2024)

    2024

  41. [41]

    J. Gu, J. Zhu, P. Knuppel, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Remote imprinting of moir´ e lattices, Nature Materials23, 219 (2024)

    2024

  42. [42]

    X. Wang, C. Xu, S. Aronson, D. Bennett, N. Paul, P. J. Crowley, C. Collignon, K. Watanabe, T. Taniguchi, R. Ashoori,et al., Moir´ e band structure engineering us- ing a twisted boron nitride substrate, Nature Communi- cations16, 178 (2025)

    2025

  43. [43]

    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

  44. [44]

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

    2023

  45. [45]

    Y. Zeng, T. M. R. Wolf, C. Huang, N. Wei, S. A. A. Ghorashi, A. H. MacDonald, and J. Cano, Gate-tunable topological phases in superlattice modulated bilayer graphene, Phys. Rev. B109, 195406 (2024)

    2024

  46. [46]

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

    2024

  47. [47]

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

    2018

  48. [48]

    [68–70], for details of the mean-field calculations and phase dia- grams at different parameters

    See Supplemental Material, which includes Refs. [68–70], for details of the mean-field calculations and phase dia- grams at different parameters

  49. [49]

    As the two moir´ es are laterally shifted,Vdecreases whileUstays unchanged, enlarging the separation of energy scales

    For laterally aligned double-moir´ es described by a Hubbard-like model [29], the energy scale of spin order is∼t 2/Uwith hoppingtand intralayer onsite repulsion U, while the energy scale of layer order is∼t 2/Vwhere V≪Uis the interlayer onsite repulsion. As the two moir´ es are laterally shifted,Vdecreases whileUstays unchanged, enlarging the separation ...

  50. [50]

    F.-C. Wu, F. Xue, and A. H. MacDonald, Theory of two- dimensional spatially indirect equilibrium exciton con- densates, Phys. Rev. B92, 165121 (2015)

    2015

  51. [51]

    parity anomaly

    F. D. M. Haldane, Model for a quantum hall effect with- out landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett.61, 2015 (1988)

    2015

  52. [52]

    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

  53. [53]

    T. Li, S. Jiang, B. Shen, Y. Zhang, L. Li, Z. Tao, T. De- vakul, K. Watanabe, T. Taniguchi, L. Fu,et al., Quan- tum anomalous hall effect from intertwined moir´ e bands, Nature600, 641 (2021)

    2021

  54. [54]

    W. Zhao, K. Kang, Y. Zhang, P. Kn¨ uppel, Z. Tao, L. Li, C. L. Tschirhart, E. Redekop, K. Watanabe, T. Taniguchi,et al., Realization of the haldane chern in- sulator in a moir´ e lattice, Nature Physics20, 275 (2024)

    2024

  55. [55]

    Z. Tao, B. Shen, S. Jiang, T. Li, L. Li, L. Ma, W. Zhao, J. Hu, K. Pistunova, K. Watanabe, T. Taniguchi, T. F. Heinz, K. F. Mak, and J. Shan, Valley-Coherent Quan- tum Anomalous Hall State in AB-Stacked MoTe 2/WSe2 Bilayers, Phys. Rev. X14, 011004 (2024)

    2024

  56. [56]

    Dong and Y.-H

    Z. Dong and Y.-H. Zhang, Excitonic Chern insulator and kinetic ferromagnetism in a MoTe 2/WSe2 moir´ e bilayer, Phys. Rev. B107, L081101 (2023)

    2023

  57. [57]

    Xie, C.-P

    Y.-M. Xie, C.-P. Zhang, and K. T. Law, Topological px +ip y intervalley coherent state in moir´ e MoTe2/WSe2 heterobilayers, Phys. Rev. B110, 045115 (2024)

    2024

  58. [58]

    Y. Yang, M. A. Morales, and S. Zhang, Metal-insulator transition in a semiconductor heterobilayer model, Phys. Rev. Lett.132, 076503 (2024)

    2024

  59. [59]

    Y. Yang, M. A. Morales, and S. Zhang, Ferromagnetic semimetal and charge-density wave phases of interact- ing electrons in a honeycomb moir´ e potential, Phys. Rev. Lett.133, 266501 (2024)

    2024

  60. [60]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017)

    2017

  61. [61]

    X. Li, Y. Qian, W. Ren, Y. Xu, and J. Chen, Emergent wigner phases in moir´ e superlattice from deep learning, arXiv preprint arXiv:2406.11134 (2024)

    Pith/arXiv arXiv 2024

  62. [62]

    D. Luo, D. D. Dai, and L. Fu, Simulating moir´ e quantum matter with neural network, arXiv preprint arXiv:2406.17645 (2024)

    Pith/arXiv arXiv 2024

  63. [63]

    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

  64. [64]

    X. Li, Y. Chen, B. Li, H. Chen, F. Wu, J. Chen, and W. Ren, Deep learning sheds light on integer and fractional topological insulators, arXiv preprint arXiv:2503.11756 (2025)

    Pith/arXiv arXiv 2025

  65. [65]

    D. Luo, T. Zaklama, and L. Fu, Solving fractional elec- tron states in twisted MoTe 2 with deep neural network, arXiv preprint arXiv:2503.13585 (2025)

    Pith/arXiv arXiv 2025

  66. [66]

    Y. Zeng, V. Cr´ epel, and A. J. Millis, Keldysh field the- ory of dynamical exciton condensation transitions in nonequilibrium electron-hole bilayers, Phys. Rev. Lett. 132, 266001 (2024)

    2024

  67. [67]

    Z. Sun, Y. Murakami, F. Xuan, T. Kaneko, D. Goleˇ z, and A. J. Millis, Dynamical exciton condensates in bi- ased electron-hole bilayers, Phys. Rev. Lett.133, 217002 (2024)

    2024

  68. [68]

    Marzari and D

    N. Marzari and D. Vanderbilt, Maximally localized gen- eralized wannier functions for composite energy bands, Phys. Rev. B56, 12847 (1997)

    1997

  69. [69]

    Marzari, A

    N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized wannier functions: Theory and applications, Rev. Mod. Phys.84, 1419 (2012)

    2012

  70. [70]

    Topological excitonic insulators in electron bilayers modulated by twisted hBN

    J. Kang and O. Vafek, Non-abelian dirac node braiding and near-degeneracy of correlated phases at odd inte- ger filling in magic-angle twisted bilayer graphene, Phys. Rev. B102, 035161 (2020). 8 Supplemental Material for “Topological excitonic insulators in electron bilayers modulated by twisted hBN” I. DET AILS OF MEAN-FIELD CALCULA TIONS A. Single-band ...

    2020