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Exact models of chiral flat-band superconductors

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A local attraction between opposite-parity orbitals makes flat-band superconductivity exactly solvable.

desk verdict A genuinely new exact-solvability construction for flat-band superconductivity, with a real gap in the treatment of the singular pairing form factor that needs to be fixed before the general claim is accepted. read the letter →

arxiv 2508.21127 v2 pith:YQK3Z7N3 submitted 2025-08-28 cond-mat.str-el cond-mat.supr-con

Zhaoyu Han , Jonah Herzog-Arbeitman , Qiang Gao , Eslam Khalaf This is my paper
classification cond-mat.str-elcond-mat.supr-con
keywords flat-bandsuperconductivityexactlysolvablemodelsinversionsymmetryquantumgeometrytopologicalparticle-holedualitysuperfluidstiffnessrhombohedralgraphene
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

This paper claims that a single flavor-polarized flat band with inversion symmetry can host superconductivity that is exactly solvable: if two local orbitals have opposite inversion parities, a local attractive interaction between them, after projection onto the flat band, has zero-energy ground states of the form (η†)^N|vac>, with η† built from the band's Bloch wavefunctions. This matters because superconductivity in rhombohedral graphene and twisted MoTe2 appears in flavor-polarized, nearly flat bands, where standard weak-coupling BCS reasoning does not apply. The construction yields exact statements about the superconducting state: its stiffness is proportional to the attraction strength, it inherits the parent band's topology, and each model has two nearly degenerate pairing modes related by particle-hole duality that may have different Chern numbers. These features reproduce, at a qualitative level, the two distinct superconducting domes observed in rhombohedral graphene.

What carries the argument

The load-bearing object is the ratio-symmetric pairing form factor f(k)=u_A*(k)/u_B(-k), which inversion symmetry makes antisymmetric, f(k)=-f(-k). Around this form factor the paper constructs the local two-orbital attraction and its counterterm so that the projected Hamiltonian is a sum of positive semi-definite operators S_R† S_R with S_R=c†_RA c_RB; the commutation relation [S_R, η†]=0 in the flat-band subspace is what makes (η†)^N|vac> exact zero-energy ground states. The second key object is the particle-hole/time-reversal duality (C and T) that maps the A↔B exchanged Hamiltonian to the original one, relating fillings ν↔1-ν and giving two nearly degenerate SC modes. For explicit models

What would settle it

Compute the expectation value ⟨(η)^N H (η†)^N⟩ on a finite lattice using the Weierstrass compactified model at a filling where u_B(-k) has a zero; if the energy density does not vanish as system size grows, or if the BCS state norm diverges, the exact-solvability claim fails for that regularization.

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Extended reading notes

Core claim

The central claim is that for any pair of orbitals A,B with opposite inversion parities (p_A p_B = -1) in an isolated flat band, the projected Hamiltonian H_AB = -V Σ_R n_RA n_RB plus single-particle counterterms is exactly solvable. It is positive semi-definite, and the pairing operator η† = ∫ dk [u_A*(k)/u_B(-k)] γ†_k γ†_-k commutes with each projected local term in the sense that lets the Hamiltonian act on (η†)^N|vac> and then annihilate the vacuum, so these states have zero energy and are ground states. The ground states form a BCS family |z> with arbitrary phase, breaking U(1); in two dimensions their Majorana Chern number is C_maj = C_BdG + C_FB + 2C_<, and for two-orbital models C_Bd

Load-bearing premise

The pairing form factor f(k)=u_A*(k)/u_B(-k) can be singular where u_B(-k)=0; the argument that these singularities cause no divergence in physical observables must hold, for example through the lattice compactification, for the exact ground states to be well defined in the thermodynamic limit.

Editorial extensions

If this is right

  • Any inversion-symmetric isolated flat band with a local attraction between opposite-parity orbitals has exact zero-energy superconducting ground states at all even fillings, independent of band topology.
  • When the parent flat band is topological, the superconducting state is topological too (C_maj = C_FB in the two-orbital lower-band case), implying chiral Majorana edge modes.
  • Each model contains two nearly degenerate SC modes related by particle-hole duality, with D_s(ν)=D_s(1-ν), so the two modes appear as two SC domes tunable by charge-transfer gap and density.
  • The superfluid stiffness is bounded above and below by the interaction scale V times a quantum-geometric factor; in the dilute limit the upper bound coincides with the exact two-particle bound-state mass.
  • The exact BCS ground states have off-diagonal long-range order and arbitrary phase, so the model exhibits genuine U(1) symmetry breaking despite being strongly coupled.

Reading between the lines

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

  • The construction should extend to systems with approximate or emergent inversion symmetry, such as the SO(2)-symmetric low-energy models of rhombohedral graphene or twisted MoTe2, since only the nesting relation p_α u_α(k)=u_α(-k)e^{iξ(k)} is needed; the authors hint at this but do not carry it out.
  • The predicted Chern numbers for L-layer rhombohedral graphene (C_maj=1 and C_maj=2L-3 for the two domes) could be tested by thermal Hall or edge-transport measurements if the two domes are indeed the particle-hole conjugate pair.
  • The exact states could serve as benchmarks for numerical methods on strongly coupled flat-band superconductors, for example by comparing the bootstrap lower bound with the variational upper bound at finite sizes.
  • A finite-size scaling study near the singular points of the form factor, where u_B(-k)=0, would sharpen the thermodynamic-limit argument; the paper's lattice compactification makes this a concrete numerical test.
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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 paper introduces a class of exactly solvable models for superconducting ground states in a single isolated flat band with inversion symmetry. For two orbitals A,B of opposite parity, a local attractive density-density interaction plus a carefully chosen single-particle counterterm projects onto a positive semi-definite quartic Hamiltonian in the flat-band subspace. The central claim is that the pairing operator η† = ∫(u_A*/u_B(-k)) γ†_k γ†_{-k} commutes with the projected inter-orbital hopping terms in the required sense, so that (η†)^N|vac> are zero-energy ground states. The authors construct a BCS coherent state from these number-sector ground states, analyze the Majorana Chern number, derive an analytical variational upper bound for the superfluid stiffness, and supplement it with a numerical many-body-bootstrap lower bound. They illustrate the construction with a two-band continuum model of Chern number -M, a lattice compactification of that model, and a multi-orbital effective model related to rhombohedral graphene.

Significance. If the central construction is fully valid, this is a significant contribution to strong-coupling flat-band superconductivity: it gives a rare family of exact superconducting ground states whose pairing structure is controlled by the flat-band quantum geometry, with topological order inherited from the parent band and with two particle-hole conjugate pairing modes that may have different Majorana Chern numbers. The paper contains several concrete strengths: the positive-semidefinite plus commutation proof is explicit; the two-particle sector is solved exactly; the variational stiffness bound is derived from a gauge-response calculation, and the bootstrap lower bound provides a genuine numerical check. The construction is model-building rather than circular: the solvability is engineered by design, and no parameters are fitted. However, the validity of the main theorem is currently conditioned on a regularization of the singular pairing form factor f(k)=u_A*/u_B(-k), which the manuscript promises but does not supply. That issue is load-bearing for the exact-ground-state claim in the continuum formulation and for the general statement 'for any pair of orbitals.'

major comments (3)
  1. [Main text, Eq. (5) and Eq. (12); SM Sec. II.B and Eq. (115)] The pairing kernel f(k)=u_A*(k)/u_B(-k) is singular when u_B(-k)=0. In the primary continuum example, Eq. (12) with M odd has f(k) ~ k^{-M} for A=1,B=2, so for M=1 the norm of η†|vac> is ∫ d^2k |f|^2, which diverges logarithmically in both the infrared and the ultraviolet. Thus (η†)^N|vac> is not a normalizable vector in the thermodynamic-limit Fock space without an additional prescription. The main text explicitly promises to show that this does not cause divergences in physical observables, but the provided main text and SM excerpt do not contain such a proof. The commutator argument in SM Sec. II.B holds as a distributional identity, but it does not establish that η† is a well-defined operator. SM Eq. (115) also contains the denominator ∫ |u_A|^2/|u_B|^2, which diverges for the wavefunction of Eq. (12), so the assertion that physical quantities are automatically finite cannot be taken
  2. [Main text, Eq. (5) and Eq. (12); SM Sec. III.B] The claim 'for any pair of orbitals A,B with opposite parities' is broader than what is established. The single-band construction requires F(k)=u_A*(k)/u_B(-k) to define a legitimate many-body operator with the antisymmetry F(k)=-F(-k). When u_B has zeros, the commutator identity [η†, P S_R P]=0 is only formal unless the singularities are regulated. In the two-band model Eq. (12), the two dual choices A=1,B=2 and A=2,B=1 give form factors k^{-M} and k^M respectively; the latter is regular while the former is not. The particle-hole dual pair are therefore not on equal footing in the continuum model. The authors should either extend the regularization proof to cover the singular dual or explicitly restrict the exact-solvability theorem to models where f(k) is a bounded, or at least locally integrable, function with controlled singularities, with all physical observables shown to be finite.
  3. [SM Sec. IV.E and main text Eq. (11)] The variational stiffness upper bound is derived using the parity structure of ∂ ln(u_A* u_B) and δ ln f, and the statement that δf=0 is optimal at A=0. This derivation assumes that all momentum-space integrals are absolutely convergent and that integration by parts has no boundary terms. When f has poles, the intermediate expressions (e.g., the two-particle mass in SM Eq. (115)) are not manifestly finite, so the bound D_s,var in the main text has the same regularization problem. The numerical plots in Fig. 2 use the compactified lattice model, but the analytical formula Eq. (11) is presented as a general result. The manuscript should state the domain of validity of Eq. (11) and supply the corresponding regularization for the continuum case.
minor comments (5)
  1. [Abstract] The phrase 'flavorless orbitals' is unclear; presumably 'flavor-less' or 'same-flavor' is intended. Please clarify.
  2. [SM Eq. (68)] The notation V/V in the counterterm expression is confusing. Use a distinct symbol for the number of unit cells or the BZ volume, e.g., N_c or Ω, to avoid division of identical letters.
  3. [Main text Eq. (6) and Eq. (8)] The parameter z controls the filling through ν(Re z) = ⟨1⟩_z, but the domain of Re z (e.g., whether z→±∞ corresponds to the dilute limits) is stated only in words. It would help to define the limiting fillings explicitly.
  4. [Fig. 2 and Fig. 3] The statement that the maximum stiffness is achieved when the 'hot spots' of quantum geometry are 'marginally filled' is empirical. Please define 'marginal filling' precisely and state the numerical criterion used in the figure.
  5. [Appendix C] The Schur-complement truncation to the A orbitals is acknowledged to be uncontrolled. The Chern-number predictions for rhombohedral graphene should be stated as properties of the effective model, not as quantitative predictions for the original multi-orbital Hamiltonian.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: exact solvability is a transparently constructed model property, and all derived quantities are computed from the stated Hamiltonian.

full rationale

The paper constructs a family of Hamiltonians and proves exact results for them, which is the normal content of an exactly solvable model rather than circular inference. The central step is the definition of H_AB in Eq. (2) with an explicit single-particle counterterm chosen so that the projected interaction is positive semidefinite and, using the inversion-symmetry nesting relation of Eq. (1), commutes with the pairing operator of Eq. (5). The ground state (η†)^N|vac> is then a theorem, not a fitted input. No parameter is extracted from data and later repackaged as a prediction. The superfluid stiffness upper bound (Eq. 11) and the bootstrap lower bound are derived consequences; they do not enter the definition of the model. The self-citations to the authors' prior QGN construction [67], the many-body bootstrap method [74], and the lattice compactification [92] are methodological or contextual and do not assume the present result. The singular form-factor regularization issue is a mathematical-completeness concern about the operator kernel in unbounded continuum models, which the paper explicitly addresses through the lattice compactification of Eq. (13); this is a rigor gap, not a circularity, because the claimed prediction is not equivalent to an input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The exact solvability theorem rests on standard many-body arguments (PSD plus commutation) and the domain assumptions listed above. No new particles or mediators are introduced, and no experimental numbers are fitted. The model-specific counterterm is the main engineered input.

free parameters (3)
  • Interaction strength V
    Overall scale of the exactly solvable Hamiltonian. The central theorem holds for any V>0, so V is an input, not a fitted parameter.
  • Momentum scale k0 in the compactified toy model = scanned values in Fig. 2
    Tunable parameter controlling the concentration of quantum geometry in the lattice compactification. Chosen by hand for the numerical stiffness study, not fitted to experimental data.
  • Chern integer M and charge transfer gap delta
    Model parameters of the two-band example in Eq. 12. They set topology and the energy offset between orbitals but are not fitted to experimental data.
assumptions (6)
  • domain assumption The flat band is isolated and the interaction and bandwidth satisfy Delta >> V >> W, so projection onto the flat-band subspace is valid to leading order.
    SM Eq. (5) and the projection setup. Without this separation of scales the model is not exact.
  • domain assumption An inversion-symmetric local orbital basis exists with orbital parities p_alpha, and the flat-band wavefunction obeys p_alpha u_alpha(k)=u_alpha(-k) e^{i xi(k)} with xi(k) odd.
    Eq. (1). This compatibility condition makes the pairing operator commute with the projected interaction.
  • ad hoc to paper There exist two orbitals A,B of opposite parity with a local attractive interaction -V sum n_A n_B plus the specific counterterm H_AB,2.
    Eq. (2). The counterterm is engineered so that the projected Hamiltonian is positive semi-definite and the pairing operator is exact. This is the fine-tuned part of the construction.
  • domain assumption The vacuum state has remote bands occupied or empty as specified, and the many-body ground state is (eta^dagger)^N |vac>.
    SM Sec. II. Required for the zero-energy PSD argument.
  • domain assumption The quantum many-body bootstrap method of Ref. [74] provides valid lower bounds on superfluid stiffness, and finite-size results extrapolate smoothly to the thermodynamic limit.
    Fig. 3 and SM Sec. IV. The extrapolation is numerical, not a theorem.
  • ad hoc to paper For the rhombohedral graphene application, the Schur complement and truncation to A orbitals in Appendix C give a reliable effective model.
    App. C. The approximation is uncontrolled except in the small-|k|, small u_d/t1 limit.

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Cite this review

Pith. "Pith review of Exact models of chiral flat-band superconductors." pith.science (2026). https://pith.science/paper/YQK3Z7N3

@misc{pith2026250821127,
  author       = {Pith},
  title        = {Pith review of: Exact models of chiral flat-band superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YQK3Z7N3}},
  note         = {Machine review of arXiv:2508.21127}
}
read the original abstract

Recent experiments have reported the surprising observation of superconductivity in flavor polarized, nearly flat bands (FBs) of rhombohedral graphene. Motivated by these findings, we introduce a class of models for single-flavor FBs with inversion symmetry, where we show a local attractive interaction between orbitals with opposite parities leads to an exact superconducting ground state. We argue that this model can be relevant to realistic multi-flavor systems including short-range repulsion, since the main effect of such repulsion is to induce flavor polarization leaving possibly attractive residual interaction between different flavorless orbitals. The nature of the pairing is determined by the interplay between the FB quantum geometry and the interaction, and is often topological when the parent FB is so. Interestingly, each such model has two nearly degenerate pairing modes, whose energetic competition can be tuned by a change in the charge transfer gap between the two orbitals or electron density. These modes have the same angular momentum but different pairing amplitude structure and possibly different topology. We show that the superfluid stiffness is proportional to the attractive interaction scale using a combination of analytical variational upper bounds and numerical bootstrap lower bounds. We find empirically that the maximum superfluid stiffness is achieved when the hot spots of quantum geometry in the Brillouin zone are marginally filled.

Figures

Figures reproduced from arXiv: 2508.21127 by the authors.

Figure 1
Figure 1. FIG. 1. An illustration of the particle-hole duality. Two exac [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Variational upper bound on superfluid stiffness, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The lower bound on superfluid stiffness obtained by [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Reference graph

Works this paper leans on

134 extracted references · 45 canonical work pages · cited by 9 Pith papers

  1. [1]

    T. Han, Z. Lu, Z. Hadjri, L. Shi, Z. Wu, W. Xu, Y. Yao, A. A. Cotten, O. S. Sedeh, H. Weldeyesus, et al., Nature , 1 (2025)

    2025

  2. [2]

    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. Li, arXiv preprint arXiv:2507.22026 (2025)

    arXiv 2025

  3. [3]

    F. Xu, Z. Sun, J. Li, C. Zheng, C. Xu, J. Gao, T. Jia, K. Watanabe, T. Taniguchi, B. Tong, et al., arXiv preprint arXiv:2504.06972 (2025)

    arXiv 2025

  4. [4]

    B. Lian, Z. Wang, and B. A. Bernevig, Physical Review Letters 122, 257002 (2019)

    2019

  5. [5]

    You and A

    Y. You and A. Vishwanath, npj Quantum Materials 4, 16 (2019)

    2019

  6. [6]

    J. Yu, M. Xie, F. Wu, and S. D. Sarma, Physical Review B 107, L201106 (2023)

    2023

  7. [7]

    Christos, S

    M. Christos, S. Sachdev, and M. S. Scheurer, Nature Communications 14, 7134 (2023)

    2023

  8. [8]

    Putzer and M

    B. Putzer and M. S. Scheurer, Physical Review B 111, 144513 (2025)

    2025

Show all 134 references
  1. [9]

    Z. Chen, D. Chowdhury, and L. Fu, Physical Review B 111, L060501 (2025)

    2025

  2. [10]

    Schrade and L

    C. Schrade and L. Fu, Physical Review B 110, 035143 (2024)

    2024

  3. [11]

    Jiang, T

    G. Jiang, T. Heikkil¨ a, and P. T¨ orm¨ a, arXiv preprint arXiv:2504.03617 (2025)

    2025

  4. [12]

    Origin of superconductivity in rhombohedral trilayer graphene: Quasiparticle pairing within the inter-valley coherent phase,

    C. W. Chau, S. A. Chen, and K. T. Law, “Origin of superconductivity in rhombohedral trilayer graphene: Quasiparticle pairing within the inter-valley coherent phase,” (2024), arXiv:2404.19237 [cond-mat.supr-con]

    2024 arXiv

  5. [13]

    Here, we use QG to refer to the full gauge-invariant in- formation encoded in wavefunction overlaps between ar- bitrary momenta rather than the long wavelength quan- tities such as the quantum metric

  6. [14]

    Geier, M

    M. Geier, M. Davydova, and L. Fu, arXiv preprint arXiv:2409.13829 (2024)

    2024 arXiv

  7. [15]

    Guerci, A

    D. Guerci, A. Abouelkomsan, and L. Fu, arXiv preprint arXiv:2506.10938 (2025)

    2025

  8. [16]

    Wang, Z.-Q

    Y.-Q. Wang, Z.-Q. Gao, and H. Yang, arXiv preprint arXiv:2410.05384 (2024)

    2024 arXiv

  9. [17]

    Christos, P

    M. Christos, P. M. Bonetti, and M. S. Scheurer, arXiv preprint arXiv:2503.15471 (2025)

    2025

  10. [18]

    A. S. Patri and M. Franz, arXiv preprint arXiv:2505.21616 (2025)

    2025

  11. [19]

    Shavit and J

    G. Shavit and J. Alicea, Phys. Rev. Lett. 134, 176001 (2025)

    2025

  12. [20]

    Yang and Y.-H

    H. Yang and Y.-H. Zhang, Phys. Rev. B 112, L020506 (2025)

    2025

  13. [21]

    May-Mann, T

    J. May-Mann, T. Helbig, and T. Devakul, arXiv preprint arXiv:2503.05697 (2025)

    2025 arXiv

  14. [22]

    Dong and P

    Z. Dong and P. A. Lee, arXiv preprint arXiv:2503.11079 (2025)

    2025

  15. [23]

    Cr´ epel and L

    V. Cr´ epel and L. Fu, Science Advances 7, eabh2233 (2021) , https://www.science.org/doi/pdf/10.1126/sciadv.abh2233

    2021 doi

  16. [24]

    Guerci and L

    D. Guerci and L. Fu, arXiv preprint arXiv:2503.05863 (2025)

    2025 arXiv

  17. [25]

    Y. Chen, M. S. Scheurer, and C. Schrade, Phys. Rev. B 112, L060505 (2025)

    2025

  18. [26]

    Layer pseudospin superconductivity in twisted mote2,

    J.-X. Hu, A. Daido, Z.-T. Sun, Y.-M. Xie, and K. T. Law, “Layer pseudospin superconductivity in twisted mote2,” (2025), arXiv:2506.12767 [cond-mat.supr-con]

    2025 arXiv

  19. [27]

    F. D. M. Haldane, Phys. Rev. Lett. 51, 605 (1983)

    1983

  20. [28]

    S. A. Trugman and S. Kivelson, Phys. Rev. B 31, 5280 (1985)

    1985

  21. [29]

    R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)

    1983

  22. [30]

    Tarnopolsky, A

    G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Phys. Rev. Lett. 122, 106405 (2019)

    2019

  23. [31]

    P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vish- wanath, Phys. Rev. Research 2, 023237 (2020)

    2020

  24. [32]

    Ozawa and B

    T. Ozawa and B. Mera, , 1 (2021)

    2021

  25. [33]

    Mera and T

    B. Mera and T. Ozawa, Phys. Rev. B 104, 045104 (2021)

    2021

  26. [34]

    Mera and T

    B. Mera and T. Ozawa, Phys. Rev. B 104, 115160 (2021)

    2021

  27. [35]

    Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers,

    B. Mera, A. Zhang, and N. Goldman, “Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers,” arXiv:2106.00800

    arXiv

  28. [36]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Phys. Rev. Lett. 127, 246403 (2021)

    2021

  29. [37]

    P. J. Ledwith, A. Vishwanath, and E. Khalaf, Phys. Rev. Lett. 128, 176404 (2022)

    2022

  30. [38]

    P. J. Ledwith, A. Vishwanath, and D. E. Parker, Phys. Rev. B 108, 205144 (2023)

    2023

  31. [39]

    Regnault and B

    N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 (2011)

    2011

  32. [40]

    Neupert, L

    T. Neupert, L. Santos, C. Chamon, and C. Mudry, Phys. Rev. Lett. 106, 236804 (2011)

    2011

  33. [41]

    Sheng, Z.-C

    D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nat. Com- mun. 2, 389 (2011)

    2011

  34. [42]

    Qi, Phys

    X.-L. Qi, Phys. Rev. Lett. 107, 126803 (2011)

    2011

  35. [43]

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, Phys. Rev. B 85, 241308 (2012)

    2012

  36. [44]

    Y.-L. Wu, N. Regnault, and B. A. Bernevig, Phys. Rev. Lett. 110, 106802 (2013)

    2013

  37. [45]

    Recent Developments in Fractional Chern Insulators,

    Z. Liu and E. J. Bergholtz, “Recent Developments in Fractional Chern Insulators,” (2022), arXiv:2208.08449 [cond-mat, physics:math-ph, physics:quant-ph]

    2022 arXiv

  38. [46]

    S. A. Parameswaran, R. Roy, and S. L. Sondhi, C. R. Phys. 14, 816 (2013)

    2013

  39. [47]

    E. J. Bergholtz and Z. Liu, Int. J. Mod. Phys. B 27, 1330017 (2013) , https://doi.org/10.1142/S021797921330017X

    2013 doi

  40. [48]

    Tovmasyan, S

    M. Tovmasyan, S. Peotta, P. T¨ orm¨ a, and S. D. Huber, Phys. Rev. B 94, 245149 (2016)

    2016

  41. [49]

    Huhtinen, J

    K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevig, and P. T¨ orm¨ a, Phys. Rev. B 106, 014518 (2022)

    2022

  42. [50]

    Herzog-Arbeitman, A

    J. Herzog-Arbeitman, A. Chew, K.-E. Huhtinen, P. T¨ orm¨ a, and B. A. Bernevig, arXiv preprint arXiv:2209.00007 (2022)

    2022 arXiv

  43. [51]

    Peotta and P

    S. Peotta and P. T¨ orm¨ a, Nature Communications 6 (2015), 10.1038/ncomms9944

    2015 doi

  44. [52]

    Julku, S

    A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. T¨ orm¨ a,Phys. Rev. Lett. 117, 045303 (2016)

    2016

  45. [53]

    T¨ orm¨ a, S

    P. T¨ orm¨ a, S. Peotta, and B. A. Bernevig, Nature Re- views Physics 4, 528–542 (2022)

    2022

  46. [54]

    Peotta, K.-E

    S. Peotta, K.-E. Huhtinen, and P. T¨ orm¨ a, arXiv e-prints , arXiv:2308.08248 (2023) , arXiv:2308.08248 [cond-mat.quant-gas]

    2023

  47. [55]

    Jiang, P

    G. Jiang, P. T¨ orm¨ a, and Y. Barlas, Proceedings of the National Academy of Sciences 122, e2416726122 (2025) , 7 https://www.pnas.org/doi/pdf/10.1073/pnas.2416726122

    2025 doi

  48. [56]

    Jiang and Y

    G. Jiang and Y. Barlas, Phys. Rev. B 109, 214518 (2024)

    2024

  49. [57]

    S. A. Chen and K. T. Law, Phys. Rev. Lett. 132, 026002 (2024)

    2024

  50. [58]

    Mielke and H

    A. Mielke and H. Tasaki, Communications in Mathematical Physics 158, 341 (1993) , arXiv:cond- mat/9305026 [cond-mat]

    1993

  51. [59]

    Kang and O

    J. Kang and O. Vafek, Phys. Rev. Lett. 122, 246401 (2019)

    2019

  52. [60]

    K. Seo, V. N. Kotov, and B. Uchoa, Phys. Rev. Lett. 122, 246402 (2019)

    2019

  53. [61]

    Repellin, Z

    C. Repellin, Z. Dong, Y.-H. Zhang, and T. Senthil, Phys. Rev. Lett. 124, 187601

  54. [62]

    Bultinck, E

    N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vish- wanath, and M. P. Zaletel, Phys. Rev. X 10, 031034

  55. [63]

    TBG IV: Exact Insulator Ground States and Phase Diagram of Twisted Bilayer Graphene,

    B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yaz- dani, and B. A. Bernevig, “TBG IV: Exact Insulator Ground States and Phase Diagram of Twisted Bilayer Graphene,” arXiv:2009.13530

    2009 arXiv

  56. [64]

    B. A. Bernevig, B. Lian, A. Cowsik, F. Xie, N. Regnault, and Z.-D. Song, Phys. Rev. B 103, 205415 (2021)

    2021

  57. [65]

    P. J. Ledwith, E. Khalaf, and A. Vishwanath, Annals of Physics 435, 168646 (2021)

    2021

  58. [66]

    J. Kang, T. Oh, J. Lee, and B.-J. Yang, arXiv e-prints , arXiv:2402.07171 (2024) , arXiv:2402.07171 [cond-mat.str-el]

    2024 arXiv

  59. [67]

    Z. Han, J. Herzog-Arbeitman, B. A. Bernevig, and S. A. Kivelson, Phys. Rev. X 14, 041004 (2024)

    2024

  60. [68]

    Zhang, W

    J.-X. Zhang, W. O. Wang, L. Balents, and L. Savary, arXiv preprint arXiv:2504.03882 (2025)

    2025 arXiv

  61. [69]

    Sun, R.-P

    Z.-T. Sun, R.-P. Yu, S. A. Chen, J.-X. Hu, and K. Law, arXiv preprint arXiv:2408.00548 (2024)

    2024

  62. [70]

    Z. Han, S. A. Kivelson, and P. A. Volkov, Phys. Rev. Lett. 132, 226001 (2024)

    2024

  63. [71]

    We also note that in the FB limit the retardation effects of the bosons are not significant, so the interactions can be regarded instantaneous

  64. [72]

    Read and D

    N. Read and D. Green, Phys. Rev. B 61, 10267 (2000)

    2000

  65. [73]

    G. E. Volovik, The universe in a helium droplet, Vol. 117 (OUP Oxford, 2003)

    2003

  66. [74]

    Q. Gao, Z. Han, and E. Khalaf, arXiv preprint arXiv:2506.18969 (2025)

    2025 arXiv

  67. [75]

    Here we implicitly embed all the orbital positions to the inversion center

  68. [76]

    Alexandradinata, X

    A. Alexandradinata, X. Dai, and B. A. Bernevig, Phys. Rev. B 89, 155114 (2014)

    2014

  69. [77]

    C. Fang, M. J. Gilbert, and B. A. Bernevig, Phys. Rev. B 86, 115112 (2012)

    2012

  70. [78]

    The vacuum state here is defined as the lower remote bands being occupied but the FB and the higher remote bands being empty

  71. [79]

    Somma, H

    R. Somma, H. Barnum, G. Ortiz, and E. Knill, Phys. Rev. Lett. 97, 190501 (2006)

    2006

  72. [80]

    Nussinov and G

    Z. Nussinov and G. Ortiz, Phys. Rev. B 79, 214440 (2009)

    2009

  73. [81]

    P. W. Phillips, L. Yeo, and E. W. Huang, Nature Physics 16, 1175 (2020)

    2020

  74. [82]

    Iba˜ nez, J

    M. Iba˜ nez, J. Links, G. Sierra, and S.-Y. Zhao, Phys. Rev. B 79, 180501 (2009)

    2009

  75. [83]

    Lerma H., S

    S. Lerma H., S. M. A. Rombouts, J. Dukelsky, and G. Ortiz, Phys. Rev. B 84, 100503 (2011)

    2011

  76. [84]

    Dukelsky, C

    J. Dukelsky, C. Esebbag, and P. Schuck, Phys. Rev. Lett. 87, 066403 (2001)

    2001

  77. [85]

    Dukelsky, S

    J. Dukelsky, S. Pittel, and G. Sierra, Rev. Mod. Phys. 76, 643 (2004)

    2004

  78. [86]

    Ortiz, R

    G. Ortiz, R. Somma, J. Dukelsky, and S. Rombouts, Nuclear Physics B 707, 421 (2005)

    2005

  79. [87]

    Z. Wang, Y. Xu, H. Pu, and K. R. A. Hazzard, Phys. Rev. B 96, 115110 (2017)

    2017

  80. [88]

    D. J. Scalapino, S. R. White, and S. Zhang, Phys. Rev. B 47, 7995 (1993)

    1993

  81. [89]

    Mao and D

    D. Mao and D. Chowdhury, Proceedings of the Na- tional Academy of Sciences 120, e2217816120 (2023) , https://www.pnas.org/doi/pdf/10.1073/pnas.2217816120

    2023 doi

  82. [90]

    J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, and B. A. Bernevig, Nature Physics 20, 1262 (2024) , arXiv:2305.02340 [cond-mat.supr-con]

    2024 arXiv

  83. [91]

    Moore and N

    G. Moore and N. Read, Nuclear Physics B 360, 362 (1991)

    1991

  84. [92]

    P. J. Ledwith, J. Dong, A. Vishwanath, and E. Khalaf, Phys. Rev. X 15, 021087 (2025)

    2025

  85. [94]

    L. Chen, T. Mazaheri, A. Seidel, and X. Tang, Journal of Physics A: Mathematical and Theoretical 47, 152001 (2014)

    2014

  86. [95]

    B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006) , https://www.science.org/doi/pdf/10.1126/science.1133734

    2006 doi

  87. [96]

    T. Tan, A. P. Reddy, L. Fu, and T. Devakul, Phys. Rev. Lett. 133, 206601 (2024)

    2024

  88. [97]

    Soejima, J

    T. Soejima, J. Dong, A. Vishwanath, and D. E. Parker, arXiv preprint arXiv:2503.12704 (2025)

    2025

  89. [98]

    Song and B

    Z.-D. Song and B. A. Bernevig, Phys. Rev. Lett. 129, 047601 (2022)

    2022

  90. [99]

    Herzog-Arbeitman, J

    J. Herzog-Arbeitman, J. Yu, D. C˘ alug˘ aru, H. Hu, N. Regnault, C. Liu, O. Vafek, P. Coleman, A. Tsve- lik, Z.-d. Song, et al., arXiv preprint arXiv:2404.07253 (2024)

    2024 arXiv

  91. [100]

    Herzog-Arbeitman, Y

    J. Herzog-Arbeitman, Y. Wang, J. Liu, P. M. Tam, Z. Qi, Y. Jia, D. K. Efetov, O. Vafek, N. Regnault, H. Weng, Q. Wu, B. A. Bernevig, and J. Yu, Phys. Rev. B 109, 205122 (2024)

    2024

  92. [101]

    Z. Dong, A. S. Patri, and T. Senthil, Phys. Rev. B 110, 205130 (2024)

    2024

  93. [102]

    Tan and T

    T. Tan and T. Devakul, Phys. Rev. X 14, 041040 (2024)

    2024

  94. [103]

    Soejima, J

    T. Soejima, J. Dong, T. Wang, T. Wang, M. P. Zaletel, A. Vishwanath, and D. E. Parker, Phys. Rev. B 110, 205124 (2024)

    2024

  95. [104]

    B. A. Bernevig and Y. H. Kwan, arXiv e-prints , arXiv:2503.09692 (2025) , arXiv:2503.09692 [cond- mat.str-el]

    2025 arXiv

  96. [105]

    Mao and D

    D. Mao and D. Chowdhury, Phys. Rev. B 109, 024507 (2024)

    2024

  97. [106]

    Hazra, N

    T. Hazra, N. Verma, and M. Randeria, arXiv preprint arXiv:1811.12428 (2018)

    2018 arXiv

  98. [107]

    Barthel and R

    T. Barthel and R. H¨ ubener, Phys. Rev. Lett. 108, 200404 (2012)

    2012

  99. [108]

    Baumgratz and M

    T. Baumgratz and M. B. Plenio, New Journal of Physics 14, 023027 (2012) , arXiv:1106.5275 [quant-ph]

    2012 arXiv

  100. [109]

    Lin and M

    L. Lin and M. Lindsey, arXiv e-prints , arXiv:1910.00560 (2019), arXiv:1910.00560 [physics.comp-ph]

    1910 arXiv

  101. [110]

    D. A. Mazziotti, Phys. Rev. A 102, 052819 (2020)

    2020

  102. [111]

    A. Haim, R. Kueng, and G. Refael, arXiv e-prints , arXiv:2001.06510 (2020) , arXiv:2001.06510 [cond- mat.str-el]. 8

    2001 arXiv

  103. [112]

    D. A. Mazziotti, Phys. Rev. Lett. 130, 153001 (2023)

    2023

  104. [113]

    J. Wang, J. Surace, I. Fr´ erot, B. Legat, M.-O. Renou, V. Magron, and A. Ac ´ ın, Phys. Rev. X 14, 031006 (2024)

    2024

  105. [114]

    Q. Gao, R. A. Lanzetta, P. Ledwith, J. Wang, and E. Khalaf, Physical Review X 15, 031034 (2025)

    2025

  106. [115]

    M. G. Scheer, arXiv preprint arXiv:2410.00810 (2024)

    2024 arXiv

  107. [116]

    Khoo and M

    Y. Khoo and M. Lindsey, Journal of Computational Physics 510, 113041 (2024)

    2024

  108. [117]

    Roy, Phys

    R. Roy, Phys. Rev. B 90, 165139 (2014)

    2014

  109. [118]

    F. Xie, Z. Song, B. Lian, and B. A. Bernevig, Phys. Rev. Lett. 124, 167002 (2020)

    2020

  110. [119]

    Herzog-Arbeitman, V

    J. Herzog-Arbeitman, V. Peri, F. Schindler, S. D. Hu- ber, and B. A. Bernevig, Phys. Rev. Lett. 128, 087002 (2022)

    2022

  111. [120]

    J. Yu, J. Herzog-Arbeitman, and B. A. Bernevig, arXiv e-prints , arXiv:2501.00100 (2024) , arXiv:2501.00100 [cond-mat.mes-hall]

    2024 arXiv

  112. [121]

    Kwon and B.-J

    S. Kwon and B.-J. Yang, Phys. Rev. B 109, L161111 (2024)

    2024

  113. [122]

    Y. H. Kwan, Z. Wang, G. Wagner, S. H. Simon, S. A. Parameswaran, and N. Bultinck, arXiv e-prints , arXiv:2406.15343 (2024) , arXiv:2406.15343 [cond- mat.str-el]

    2024

  114. [123]

    W. J. Jankowski, R.-J. Slager, and G. F. Lange, arXiv e-prints , arXiv:2501.16428 (2025) , arXiv:2501.16428 [cond-mat.mes-hall]

    2025

  115. [124]

    Prijon, S

    T. Prijon, S. D. Huber, and K.-E. Huhtinen, arXiv e-prints , arXiv:2507.16909 (2025) , arXiv:2507.16909 [cond-mat.supr-con]

    2025

  116. [125]

    Zeng and A

    Y. Zeng and A. J. Millis, arXiv e-prints , arXiv:2506.18081 (2025) , arXiv:2506.18081 [cond- mat.supr-con]. 9 End Matter Appendix A: A bound on Ds,var We can obtain an elementary upper bound for the Ds,var as follows. For convenience we define the FB pro- jection matrix Pαβ(k) ...

    2025

  117. [126]

    12 with M = 1

    This model can be viewed as a L- band generalization of Eq. 12 with M = 1. We start with a k · p minimal model of the band struc- ture of RLG around the K point. It is given by a 2 L×2L hopping matrix: t(k) =                     ud vF ¯k vFk u d t1 t1 2ud v...

  118. [127]

    The projected BCS states 13

  119. [128]

    flat band subspace

    The BCS states 14 C. The variational energy 15 D. Perturbative treatment 17 E. Simplifications for the simple constructions in inversion symme tric system 18 F. Simplification for quantum geometric nesting models 19 V. Upper bound on excitation spectrum 21 VI. Toy models 22 A. A...

  120. [129]

    The projected BCS states Here we discuss the properties of Antisymmetric Geminal Product ( AGP) states, or projected BCS states. For simplicity, we only consider the case where the cooper pair momentu m Q = 0: |N ⟩ = ( ˆη†) N |vac⟩/NN (121) ˆη† ≡ ∑ k;n,m≤Nflat Fnm(k)ˆγ† k,nˆγ†...

  121. [130]

    However, we are eventually interested in the thermodynamic limit N, N0 → ∞ with a fixed filling fraction ν = N/N0

    The BCS states The AGP states are better suited for the study of finite systems wit h numerical methods. However, we are eventually interested in the thermodynamic limit N, N0 → ∞ with a fixed filling fraction ν = N/N0. The AGP states are no longer convenient in this limit. Inste...

  122. [131]

    The projected, exactly solvable model is given by Eq. 75

  123. [132]

    Z. Han, J. Herzog-Arbeitman, B. A. Bernevig, and S. A. Kivels on, Phys. Rev. X 14, 041004 (2024)

    2024

  124. [133]

    Khamoshi, T

    A. Khamoshi, T. M. Henderson, and G. E. Scuseria, The Journal of Chemical Physics 151, 184103 (2019) , https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.5127850/15565529/184103 1 online.pdf

    2019 doi

  125. [134]

    Huhtinen, J

    K.-E. Huhtinen, J. Herzog-Arbeitman, A. Chew, B. A. Bernevi g, and P. T¨ orm¨ a,Phys. Rev. B 106, 014518 (2022)

    2022

  126. [135]

    F. D. M. Haldane, Journal of Mathematical Physics 59, 071901 (2018) , https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.5042618/15971731/071901 1 online.pdf

    2018 doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.