Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Many-body perturbation theory for moir\'{e} systems

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read This paper proves exact degeneracies among symmetry-breaking Hartree-Fock states at integer fillings in twisted bilayer graphene, with the intervalley coherent state winning away from the chiral-flat limit.

desk verdict Solid framework paper with a clean analytical HF core, but the 'exact' label outruns the momentum-independent ansatz it actually proves. read the letter →

arxiv 2502.06968 v1 pith:2JEZCXKJ submitted 2025-02-10 cond-mat.str-el cond-mat.mes-hall

Liangtao Peng , Giovanni Vignale , Shaffique Adam This is my paper · ORCID
classification cond-mat.str-elcond-mat.mes-hall
keywords moirésystemstwistedbilayergraphenemany-bodyperturbationtheoryHartree-FockGWapproximationintervalleycoherentstatechiralflatlimitmetal-insulatortransition
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 establishes a many-body perturbation theory for moiré systems built on the single-particle Green's function in the band basis, and applies it to twisted bilayer graphene. Within Hartree-Fock theory it derives exact analytical ground states at integer fillings in the chiral-flat limit and proves that all symmetry-breaking states there are degenerate in energy. Relaxing the chiral condition splits the states into two groups and, together with finite band dispersion, selects the intervalley coherent (K-IVC) state as the ground state, matching numerical results. The same formalism yields a superconducting-gap-like equation for the finite-temperature metal-insulator transition and a self-consistent GW treatment in which ring diagrams screen away part of the Hartree-Fock chemical-potential oscillations. The payoff is a systematic route beyond mean field for correlated flat-band systems, including future superconducting pairing calculations.

What carries the argument

The central object is the band-basis imaginary-time Green's function and its self-energy expansion, whose interaction vertices are dressed by the moiré form factor $\hat\Lambda_{k,q+G}$. The load-bearing simplification is the chiral-flat-limit form factor $\hat\Lambda=\Lambda^0\gamma_0+\Lambda^2 i\gamma_y$, which makes the Fock self-energy proportional to the order parameter, $\hat\Sigma_{\rm HF}=\nu\Sigma_H-\Sigma_F\hat Q$, and gives all symmetry-breaking states the same energy. The argument is carried by the commutator condition $[\hat Q,\hat\Lambda]=0$ for self-consistency and by the $U(4)\times U(4)$ symmetry that protects the degeneracy, with the non-chiral corrections $\delta\hat\Lambda$ splitting the states into two groups. For the GW part, the machinery is the RPA-screened interaction $\hat V=\hat V_q[1-\hat V_q\hat\Pi]^{-1}$ with a fully self-consistent polarizability built from dressed Green's functions.

What would settle it

Perform an unrestricted Hartree-Fock calculation for the continuum model of twisted bilayer graphene at the magic angle, allowing the order parameter $\hat Q(k)$ to depend on momentum. If any self-consistent solution with momentum-dependent $\hat Q(k)$ lying outside the listed symmetry-breaking families has lower energy at an integer filling, the exact analytical classification and degeneracy claim are incomplete.

Watch

Extended reading notes

Core claim

The paper claims that the Hartree-Fock problem of twisted bilayer graphene at integer fillings can be solved analytically, not just numerically. In the chiral-flat limit the form factor reduces to $\hat\Lambda=\Lambda^0\gamma_0+\Lambda^2 i\gamma_y$, and any order parameter $\hat Q$ that commutes with $\hat\Lambda$ is self-consistent; this produces three families of states (polarized, Hall, and intervalley coherent) whose total energies are identical at each integer filling because the Hartree and Fock corrections are universal, $\hat\Sigma_{\rm HF}=\nu\Sigma_H-\Sigma_F\hat Q$. Away from the chiral limit the two extra form-factor components split the states into a lower-energy group (polarized and K-IVC) and a higher-energy group (Hall and T-IVC), and finite single-particle dispersion then makes the K-IVC state the unique ground state. At finite temperature the gap closes through a superconducting-gap-like equation with transition temperature $T=U_F/4$ at charge neutrality. Including self-consistent GW ring diagrams does not lift the degeneracy and, unlike first-order Hartree-Fock, produces weak chemical-potential oscillations consistent with experiment.

Load-bearing premise

The analytical classification assumes the Hartree-Fock order parameter is momentum-independent, $Q(k)=Q$, and that only the lowest eight bands matter; if a momentum-dependent order parameter or a higher-band state is lower in energy, the claimed exact degeneracies and ground-state ordering are incomplete.

Editorial extensions

If this is right

  • The exact Hartree-Fock degeneracy at integer fillings is robust to adding ring-diagram corrections while the $U(4)\times U(4)$ symmetry holds, so correlation effects included via GW do not reorder the states.
  • At charge neutrality the metal-insulator transition temperature is fixed by the Fock energy, $T=U_F/4$, and decreases away from neutrality because the Hartree term pushes the bands together.
  • Hartree-Fock chemical-potential oscillations (cascades) are significantly larger than experimental ones; self-consistent GW screening reduces them while preserving the overall increasing chemical potential with filling.
  • Ring diagrams are negligible at integer fillings, where the Hartree-Fock gap suppresses screening, but become important at non-integer fillings, where they can affect pairing and superconductivity.
  • The same Green's-function framework can be extended to anomalous pairing diagrams, giving a single consistent description of correlated insulators and superconductivity in moiré systems.

Reading between the lines

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

  • If the momentum-independent ansatz $Q(k)=Q$ is relaxed, unrestricted Hartree-Fock could reveal states outside the three families; the paper's degeneracy proof would then apply only to the subset of uniform-order-parameter solutions.
  • The superconducting-gap-like form of the transition equation suggests the insulating transition and superconductivity may share a common energy scale set by the Fock term; a test would be to measure how the insulating gap and the superconducting transition respond identically to dielectric screening.
  • The subtraction scheme used to regularize the diagrams (removing the decoupled-bilayer charge-neutral density) is a choice; different subtraction conventions could shift the quantitative energies and the apparent size of the GW corrections.
  • Because ring diagrams matter most away from integer fillings, the framework predicts that compressibility measurements at fractional fillings are the sharpest place to look for beyond-mean-field correlation effects.
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 / 6 minor

Summary. The paper develops a many-body perturbation theory framework in the band basis for moiré systems and applies it to twisted bilayer graphene (TBG). At the Hartree–Fock level, it derives analytical order parameters and self-energies for symmetry-breaking states at integer fillings in the chiral-flat limit under a momentum-independent order-parameter ansatz, discusses the lifting of degeneracy away from the chiral-flat limit, and derives a BCS-like finite-temperature gap equation. It then implements a self-consistent GW calculation with ring diagrams and shows that GW corrections reduce the oscillatory features of the chemical potential as a function of filling, consistent with experimental trends.

Significance. If the claims are taken with appropriate qualifications, the paper makes a useful contribution: it provides a systematic diagrammatic formulation in the band basis, an explicit symmetry-based decomposition of the form factors, analytical control of the Hartree–Fock energy landscape in the chiral and non-chiral flat limits, and a fully self-consistent treatment of GW ring diagrams that goes beyond earlier one-shot approaches. The numerical comparison with the relaxed model at the magic angle supports the qualitative picture. The main weakness is that the central exactness claims are built on an unproven uniform order-parameter ansatz; if read literally, the claims overreach. With those claims appropriately qualified, the framework is solid and should be of interest to the moiré community.

major comments (3)
  1. [Sec. IV.B, Table I, Eq. (35)] The exactness claim is not established because the calculation is restricted a priori to momentum-independent order parameters. The text states: "Consistent with previous studies, we assume that the order parameter Q(k)=Q is momentum-independent." Every subsequent exact statement, including Table I, the degeneracy proof, and Eq. (35), is derived under this uniform ansatz. Neither the main text nor Appendix C rules out momentum-dependent or non-commuting Hartree-Fock solutions with lower energy; Appendix C proves only that [Q,Λ]=0 is a sufficient self-consistency condition in the spinless, valleyless model. Since the abstract and introduction claim "exact analytical solutions for the symmetry-breaking ground states" and a proof of degeneracy, the paper should either prove exhaustiveness of the uniform manifold or explicitly downgrade all such claims to "within the uniform order-parameter ansatz."
  2. [Sec. IV, opening (p. 4)] The restriction to the lowest eight bands is imposed by the assumption that the interaction-induced self-energy is much smaller than the gap to higher bands, but no error estimate or convergence check is provided. This is a load-bearing truncation: if the assumption fails, the derived self-energies and the GW results built on them are incomplete. The paper should provide either a concrete estimate of the discarded contributions or a numerical convergence test before the word "exact" is used for the Hartree-Fock results.
  3. [Sec. IV.D, Eq. (42), Appendix G] The analytical transition temperature T=U_F/4 follows from Eq. (42), but that equation is derived only after assuming a momentum-independent gap Delta = Sigma_F(k). Appendix G starts from the k-dependent self-consistent equation (G2) and then replaces Sigma_F(q) by a constant Delta without giving an argument that the k-dependence is negligible. The numerical agreement in Fig. 3 is encouraging, but the analytical formula is conditional on this uniformity assumption; the text should state this explicitly.
minor comments (6)
  1. [Abstract and Sec. IV.B] The abstract and introduction state that the paper derives "exact analytical solutions" and proves degeneracy, but the momentum-independent assumption in Sec. IV.B is not mentioned there; the wording should be qualified to avoid overclaiming.
  2. [Eq. (6)] In Eq. (6), the left-hand side V^{sigma sigma'}_{q,{ni}} carries no explicit k,k' dependence, while the right-hand side depends on k and k' through the form factors; please make the notation unambiguous.
  3. [Table I] The sign conventions for the order parameters and self-energies in Table I (the many +/- entries) are not specified; a short sentence explaining how the signs are chosen would improve readability.
  4. [Appendix D and Appendix E] There are small typos: "2 fold denigrate" should be "twofold degenerate" in Appendix D, and "for for non-zero integer filling" appears in Appendix E.
  5. [Sec. V and Conclusion] The conclusion states that "ring diagrams are irrelevant at integer filling factors," but this is based on the estimate [Pi(q,0)] ~ 1/Delta with Delta the Hartree-Fock gap; the paper itself notes the estimate fails when screening reduces the gap. The conclusion should state the parameter regime in which this statement holds.
  6. [Fig. 2] The caption of Fig. 2 and the text use "rightmost panel" and "right panel" somewhat interchangeably; please label the three panels (left, middle, right) explicitly in both the figure and the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the HF and GW results are derived self-contained from the BM Hamiltonian and symmetry-constrained form factors, not from the paper's own prior results.

full rationale

The paper's derivation chain starts from the Bistritzer-MacDonald Hamiltonian, a unitary transformation to the band basis, and form factors constrained by C2zT, C2zP, and chiral symmetries. The analytical Hartree-Fock solutions are obtained by solving the Dyson self-consistency equations for declared order-parameter candidates; the self-energies and ground-state energies are outputs of those equations, not inputs. The uniform-order-parameter assumption in Sec. IV.B ('Consistent with previous studies, we assume that the order parameter Q(k)=Q is momentum-independent') is an unproven restriction of the variational manifold, but it is not circular: the paper does not define Q(k)=Q through the claimed degeneracy, nor fit the degeneracy from data. The eight-band truncation is likewise an approximation with a stated physical justification (self-energy much smaller than the gap to higher bands), not an input later relabeled as a prediction. The self-citations (Refs. [11] and [27]) are used for the relaxed-model comparison and for screening/superconductivity context, respectively; the central chiral-flat-limit results, the away-from-chiral-limit energy splitting, and the GW compressibility calculation do not reduce to those references. The GW conclusion is a direct numerical result of including RPA ring diagrams in a fully self-consistent Green's-function calculation, compared against the first-order calculation; no fitted parameter forces the suppression of the chemical-potential oscillations. The main limitations are therefore correctness/exhaustiveness risks, not circularity: the momentum-independence ansatz and the eight-band truncation are unproven, and the claim of exact ground-state solutions is accordingly conditional on those assumptions. But no load-bearing step in the derivation is equivalent to its own inputs by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claims add no new fitted constants: the numerical scale is set by the chosen dielectric constant and the eight-band truncation, while the HF and GW results are computed from the Coulomb interaction and band form factors. The paper introduces no new particles or forces. The main unproven inputs are the momentum-independent order parameter ansatz and the chiral-flat limit itself.

free parameters (3)
  • Dielectric constant epsilon = 10
    Coulomb interaction scale V_q=2*pi*e^2/(epsilon*q); chosen from continuum-model practice in Sec. IV and affects all numerical HF and GW results.
  • Band truncation = lowest 8 bands
    Sec. IV restricts the calculation to the lowest eight bands under the assumption that the interaction-induced self-energy is smaller than the gap to higher bands.
  • GW convergence parameters = not reported
    The self-consistent GW calculation in Sec. V requires k-grid, Matsubara frequency cutoff, and G-vector truncation; none are stated, which limits reproducibility.
assumptions (5)
  • domain assumption Bistritzer-MacDonald continuum model captures the low-energy single-particle bands of magic-angle TBG
    The Hamiltonian in Sec. II (Eqs. 1-4) and all numerical results start from this model.
  • domain assumption Chiral-flat limit with w0=0 and E_n(k)=0 is a controlled starting point for exact HF analysis
    Used throughout Sec. IV.B and Appendix C; the paper's exact degeneracy statements refer to this idealized limit.
  • ad hoc to paper The interaction-induced self-energy is much smaller than the gap to higher bands, so the calculation can be restricted to the lowest eight bands
    Stated in Sec. IV before the band-basis reduction; it is a truncation assumption not independently verified.
  • ad hoc to paper The Hartree-Fock order parameter can be taken momentum-independent, Q(k)=Q
    Sec. IV.B Eq. (31); this ansatz is load-bearing for the exhaustiveness of Table I.
  • standard math Gauge fixing and form factor decompositions from Bernevig et al. [22] correctly encode C2zT, C2zP, and chiral symmetries
    Sec. IV.A Eqs. (26)-(30); the paper adopts rather than re-derives these symmetry representations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Many-body perturbation theory for moir\'{e} systems." pith.science (2026). https://pith.science/paper/2JEZCXKJ

@misc{pith2026250206968,
  author       = {Pith},
  title        = {Pith review of: Many-body perturbation theory for moir\'e systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JEZCXKJ}},
  note         = {Machine review of arXiv:2502.06968}
}
read the original abstract

Moir\'{e} systems such as magic-angle twisted bilayer graphene have attracted significant attention due to their ability to host correlated phenomena including superconductivity and strongly correlated insulating states. By defining the single-particle Green's function in the band basis, we systematically develop a many-body perturbation theory framework to address correlations beyond the usual mean-field Hartree-Fock approaches. As a specific example, we first analyze twisted bilayer graphene within the Hartree-Fock approximation. We derive analytical solutions for symmetry-breaking states at integer fillings and the finite-temperature metal-insulator transition that closely match previously known numerical results in the literature. Moving beyond Hartree-Fock, we incorporate self-consistent GW corrections demonstrating that first-order diagrams significantly overestimate the filling-dependent fluctuations in the electronic compressibility. This framework provides a comprehensive pathway for exploring strong electronic correlations in moir\'{e} systems beyond mean-field, giving new insights into the interplay of symmetry breaking and electron correlations.

Figures

Figures reproduced from arXiv: 2502.06968 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Diagrammatic representation of Dyson’s equation [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Schematic for the evolution of the ground-state [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Numerical calculation of the Hartree-Fock gap [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The dressed bubble diagram in RPA, resulting [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Theory of plasmon spectroscopy with the quantum twisting microscope

    cond-mat.str-el 2025-06 conditional novelty 7.0 of 10

    Plasmon-assisted tunneling in a quantum twisting microscope should imprint the momentum-resolved plasmon dispersion of twisted bilayer graphene onto the differential conductance as satellite peaks and high-bias kinks.

Reference graph

Works this paper leans on

78 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    The corresponding order parameters are ˆ σz, ˆτz, and ˆτz ˆσz, respectively

    Polarized States: These include spin-polarized, valley-polarized, and spin-valley-locked states. The corresponding order parameters are ˆ σz, ˆτz, and ˆτz ˆσz, respectively

    work page

  2. [2]

    Examples are the quantum Hall state ˆγy, valley Hall state ˆτz ˆγy, spin-Hall state ˆσz ˆγy, and valley-spin-Hall state ˆτz ˆσz ˆγy

    Hall States: These states have a non-zero Chern number and all involve the band-switching operator ˆγy, which creates a quasiparticle state that mixes the two lower bands. Examples are the quantum Hall state ˆγy, valley Hall state ˆτz ˆγy, spin-Hall state ˆσz ˆγy, and valley-spin-Hall state ˆτz ˆσz ˆγy

    work page

  3. [3]

    Intervalley Coherent States : These states break the U (1) symmetry, and involve a coher- ent superposition of states in the two valleys. The corresponding order parameter are Kramers inter- valley coherent state ˆγy ˆτIVC and the time-reversal inter-valley coherent state ˆγ0 ˆτIVC, where ˆτIVC = ˆτx cos ϕIVC + ˆτy sin ϕIVC. The former breaks time- revers...

    work page

  4. [4]

    To classify the ground states more precisely, we define the difference between the form factors in the chi- ral (Eq

    Furthermore, the degeneracy among all symmetry- breaking states observed in the chiral-flat limit is also lifted. To classify the ground states more precisely, we define the difference between the form factors in the chi- ral (Eq. (28)) and non-chiral limits (Eq. (30)) as δ ˆΛ = ˆγx ˆτzΛ1 k,q+G + ˆγz ˆτzΛ3 k,q+G. (36) where δ ˆΛ is shorthand for δ ˆΛk,q+G...

    work page

  5. [5]

    Polarized States and K-IVC States: These states satisfy the condition δ ˆΛ∗ ˆQδ ˆΛT ∝ + ˆQ. This group exhibit the same order parameter and self- energy as for the chiral-flat limit, as summarized in Table I, except that the Fock-induced self-energy is replaced as a matrix ˆΣI F(k) = Σ1 F(k) + ˆγx ˆτzΣ2 F(k) + ˆγz ˆτzΣ3 F(k), (37) where Σ1 F(k) = 1 2 X q,...

    work page

  6. [6]

    cascades

    Hall States and T -IVC States: These states satisfy the condition δ ˆΛ∗ ˆQδ ˆΛT ∝ −ˆQ. Again, the corresponding order parameter and self-energy are also consistent with Table I, except the Fock- induced self-energy takes the form ΣII F (k) = 1 2 X q,G Vq+G   X i=0,2 Λi k,q+G 2 − X i=1,3 Λi k,q+G 2   . (39) The Hartree-induced correction remains the sa...

    work page 2020

  7. [7]

    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 be- haviour at half-filling in magic-angle graphene superlat- tices, Nature 556, 80 (2018)

    2018

  8. [8]

    Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional super- conductivity in magic-angle graphene superlattices, Na- ture 556, 43 (2018)

    2018

Show all 78 references
  1. [9]

    J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled supercon- ductivity in magic-angle twisted trilayer graphene, Na- ture 590, 249 (2021)

    2021

  2. [10]

    H. Kim, Y. Choi, C. Lewandowski, A. Thomson, Y. Zhang, R. Polski, K. Watanabe, T. Taniguchi, J. Al- icea, and S. Nadj-Perge, Evidence for unconventional su- perconductivity in twisted trilayer graphene, Nature606, 494 (2022)

    2022

  3. [11]

    Wang, E.-M

    L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes, C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim, et al., Correlated electronic phases in twisted bilayer transition metal dichalcogenides, Nature materials 19, 861 (2020)

    2020

  4. [12]

    Zhang, Y

    Z. Zhang, Y. Wang, K. Watanabe, T. Taniguchi, K. Ueno, E. Tutuc, and B. J. LeRoy, Flat bands in twisted bilayer transition metal dichalcogenides, Nature Physics 16, 1093 (2020)

    2020

  5. [13]

    Devakul, V

    T. Devakul, V. Cr´ epel, Y. Zhang, and L. Fu, Magic in twisted transition metal dichalcogenide bilayers, Nature communications 12, 6730 (2021)

    2021

  6. [14]

    J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, Graphene bilayer with a twist: Electronic structure, Phys. Rev. Lett. 99, 256802 (2007)

    2007

  7. [15]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moir´ e bands in twisted double-layer graphene, Proceedings of the Na- tional Academy of Sciences 108, 12233 (2011)

    2011

  8. [16]

    D. Wong, Y. Wang, J. Jung, S. Pezzini, A. M. DaSilva, H.-Z. Tsai, H. S. Jung, R. Khajeh, Y. Kim, J. Lee, S. Kahn, S. Tollabimazraehno, H. Rasool, K. Watan- abe, T. Taniguchi, A. Zettl, S. Adam, A. H. MacDon- ald, and M. F. Crommie, Local spectroscopy of moir´ e- induced electr...

    2015

  9. [17]

    M. M. A. Ezzi, L. Peng, Z. Liu, J. H. Z. Chao, G. N. Pallewela, D. Foo, and S. Adam, A self-consistent hartree theory for lattice-relaxed magic-angle twisted bilayer graphene (2024), arXiv:2404.17638 [cond-mat.str-el]

    2024 arXiv

  10. [18]

    Guinea and N

    F. Guinea and N. R. Walet, Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers, Proceedings of the National Academy of Sci- ences 115, 13174 (2018)

    2018

  11. [19]

    T. Cea, N. R. Walet, and F. Guinea, Electronic band structure and pinning of fermi energy to van hove sin- gularities in twisted bilayer graphene: A self-consistent approach, Phys. Rev. B 100, 205113 (2019)

    2019

  12. [20]

    H. C. Po, L. Zou, A. Vishwanath, and T. Senthil, Ori- gin of mott insulating behavior and superconductivity in twisted bilayer graphene, Phys. Rev. X 8, 031089 (2018)

    2018

  13. [21]

    Bultinck, E

    N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vish- wanath, and M. P. Zaletel, Ground state and hidden symmetry of magic-angle graphene at even integer fill- ing, Phys. Rev. X 10, 031034 (2020)

    2020

  14. [22]

    Y. H. Kwan, G. Wagner, T. Soejima, M. P. Zaletel, S. H. Simon, S. A. Parameswaran, and N. Bultinck, Kekul´ e spi- ral order at all nonzero integer fillings in twisted bilayer graphene, Phys. Rev. X 11, 041063 (2021)

    2021

  15. [23]

    Xie and A

    M. Xie and A. H. MacDonald, Nature of the correlated insulator states in twisted bilayer graphene, Phys. Rev. Lett. 124, 097601 (2020)

    2020

  16. [24]

    Wagner, Y

    G. Wagner, Y. H. Kwan, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Global phase diagram of the normal state of twisted bilayer graphene, Phys. Rev. Lett. 128, 156401 (2022)

    2022

  17. [25]

    Rademaker, D

    L. Rademaker, D. A. Abanin, and P. Mellado, Charge smoothening and band flattening due to hartree correc- tions in twisted bilayer graphene, Phys. Rev. B 100, 205114 (2019)

    2019

  18. [26]

    Z. A. Goodwin, V. Vitale, X. Liang, A. A. Mostofi, and J. Lischner, Hartree theory calculations of quasiparticle properties in twisted bilayer graphene, Electronic Struc- ture 2, 034001 (2020)

    2020

  19. [27]

    Zhang, K

    Y. Zhang, K. Jiang, Z. Wang, and F. Zhang, Correlated insulating phases of twisted bilayer graphene at commen- surate filling fractions: A hartree-fock study, Phys. Rev. B 102, 035136 (2020)

    2020

  20. [28]

    B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. iii. interacting hamiltonian and exact symmetries, Phys. Rev. B 103, 205413 (2021)

    2021

  21. [29]

    Lian, Z.-D

    B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yaz- dani, and B. A. Bernevig, Twisted bilayer graphene. iv. exact insulator ground states and phase diagram, Phys. Rev. B 103, 205414 (2021)

    2021

  22. [30]

    Liu and X

    J. Liu and X. Dai, Theories for the correlated insulat- ing states and quantum anomalous hall effect phenomena in twisted bilayer graphene, Phys. Rev. B 103, 035427 (2021)

    2021

  23. [31]

    S. Liu, E. Khalaf, J. Y. Lee, and A. Vishwanath, Ne- matic topological semimetal and insulator in magic-angle bilayer graphene at charge neutrality, Phys. Rev. Res. 3, 013033 (2021)

    2021

  24. [32]

    F. Xie, J. Kang, B. A. Bernevig, O. Vafek, and N. Reg- nault, Phase diagram of twisted bilayer graphene at fill- ing factor ν = ±3, Phys. Rev. B 107, 075156 (2023)

    2023

  25. [33]

    L. Peng, I. Yudhistira, G. Vignale, and S. Adam, The- oretical determination of the effect of a screening gate on plasmon-induced superconductivity in twisted bilayer graphene, Phys. Rev. B 109, 045404 (2024)

    2024

  26. [34]

    Cea and F

    T. Cea and F. Guinea, Coulomb interaction, phonons, and superconductivity in twisted bilayer graphene, Pro- ceedings of the National Academy of Sciences 118, e2107874118 (2021)

    2021

  27. [35]

    Lewandowski, D

    C. Lewandowski, D. Chowdhury, and J. Ruhman, Pairing in magic-angle twisted bilayer graphene: Role of phonon and plasmon umklapp, Phys. Rev. B 103, 235401 (2021)

    2021

  28. [36]

    Christos, S

    M. Christos, S. Sachdev, and M. S. Scheurer, Nodal band- off-diagonal superconductivity in twisted graphene su- perlattices, Nature Communications 14, 7134 (2023)

    2023

  29. [37]

    Gell-Mann and K

    M. Gell-Mann and K. A. Brueckner, Correlation energy of an electron gas at high density, Phys. Rev. 106, 364 (1957)

    1957

  30. [38]

    L. P. Kadanoff, Quantum statistical mechanics (CRC Press, 2018). 14

    2018

  31. [39]

    A. A. Abrikosov and R. A. Silverman, Methods of Quantum Field Theory in Statistical Physics (Courier Corporation, 1964)

    1964

  32. [40]

    Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem, Phys

    L. Hedin, New method for calculating the one-particle green’s function with application to the electron-gas problem, Phys. Rev. 139, A796 (1965)

    1965

  33. [41]

    J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system. ii, Phys. Rev. 118, 1417 (1960)

    1960

  34. [42]

    A. L. Fetter and J. D. Walecka, Quantum theory of many-particle systems (Courier Corporation, 2012)

    2012

  35. [43]

    Soven, Coherent-potential model of substitutional dis- ordered alloys, Phys

    P. Soven, Coherent-potential model of substitutional dis- ordered alloys, Phys. Rev. 156, 809 (1967)

    1967

  36. [44]

    F. J. Dyson, The s matrix in quantum electrodynamics, Phys. Rev. 75, 1736 (1949)

    1949

  37. [45]

    E. E. Salpeter and H. A. Bethe, A relativistic equation for bound-state problems, Phys. Rev. 84, 1232 (1951)

    1951

  38. [46]

    Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov

    G. Eliashberg, Interactions between electrons and lattice vibrations in a superconductor, Sov. Phys. JETP 11, 696 (1960)

    1960

  39. [47]

    G. D. Mahan, Many-Particle Physics (Springer New York, NY, 1990)

    1990

  40. [48]

    R. A. Jishi, Feynman Diagram Techniques in Condensed Matter Physics (Cambridge University Press, 2013)

    2013

  41. [49]

    V. M. Galitskii and A. B. Migdal, Application of quan- tum field theory methods to the many body problem, Sov. Phys. JETP 7, 18 (1958)

    1958

  42. [50]

    Holm and F

    B. Holm and F. Aryasetiawan, Total energy from the galitskii-migdal formula using realistic spectral functions, Phys. Rev. B 62, 4858 (2000)

    2000

  43. [51]

    Becker, L

    S. Becker, L. Lin, and K. D. Stubbs, Exact ground state of interacting electrons in magic angle graphene (2023), arXiv:2312.15314 [math-ph]

    2023 arXiv

  44. [52]

    K. D. Stubbs, S. Becker, and L. Lin, On the hartree-fock ground state manifold in magic angle twisted graphene systems (2024), arXiv:2403.19890 [math-ph]

    2024 arXiv

  45. [53]

    H.-K. Tang, J. N. Leaw, J. N. B. Rodrigues, I. F. Her- but, P. Sengupta, F. F. Assaad, and S. Adam, The role of electron-electron interactions in two-dimensional dirac fermions, Science 361, 570 (2018)

    2018

  46. [54]

    Barnes, E

    E. Barnes, E. H. Hwang, R. E. Throckmorton, and S. Das Sarma, Effective field theory, three-loop pertur- bative expansion, and their experimental implications in graphene many-body effects, Phys. Rev. B 89, 235431 (2014)

    2014

  47. [55]

    Tarnopolsky, A

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

    2019

  48. [56]

    Christos, S

    M. Christos, S. Sachdev, and M. S. Scheurer, Correlated insulators, semimetals, and superconductivity in twisted trilayer graphene, Phys. Rev. X 12, 021018 (2022)

    2022

  49. [57]

    X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, et al., Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature 574, 653 (2019)

    2019

  50. [58]

    Stepanov, I

    P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe, T. Taniguchi, F. H. Koppens, J. Lischner, L. Levitov, and D. K. Efetov, Untying the insulating and supercon- ducting orders in magic-angle graphene, Nature 583, 375 (2020)

    2020

  51. [59]

    Das Sarma, S

    S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Electronic transport in two-dimensional graphene, Rev. Mod. Phys. 83, 407 (2011)

    2011

  52. [60]

    A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009)

    2009

  53. [61]

    Lewandowski and L

    C. Lewandowski and L. Levitov, Intrinsically undamped plasmon modes in narrow electron bands, Proceedings of the National Academy of Sciences 116, 20869 (2019), https://www.pnas.org/doi/pdf/10.1073/pnas.1909069116

    2019 doi

  54. [62]

    Kuang, Z

    X. Kuang, Z. Zhan, and S. Yuan, Collective excitations and flat-band plasmon in twisted bilayer graphene near the magic angle, Phys. Rev. B 103, 115431 (2021)

    2021

  55. [63]

    Cavicchi, I

    L. Cavicchi, I. Torre, P. Jarillo-Herrero, F. H. L. Kop- pens, and M. Polini, Theory of intrinsic acoustic plas- mons in twisted bilayer graphene, Phys. Rev. B 110, 045431 (2024)

    2024

  56. [64]

    F. Hu, S. R. Das, Y. Luan, T.-F. Chung, Y. P. Chen, and Z. Fei, Real-space imaging of the tailored plasmons in twisted bilayer graphene, Phys. Rev. Lett.119, 247402 (2017)

    2017

  57. [65]

    N. C. Hesp, I. Torre, D. Rodan-Legrain, P. Novelli, Y. Cao, S. Carr, S. Fang, P. Stepanov, D. Barcons-Ruiz, H. Herzig Sheinfux, et al., Observation of interband col- lective excitations in twisted bilayer graphene, Nature Physics 17, 1162 (2021)

    2021

  58. [66]

    Huang, X

    T. Huang, X. Tu, C. Shen, B. Zheng, J. Wang, H. Wang, K. Khaliji, S. H. Park, Z. Liu, T. Yang,et al., Observation of chiral and slow plasmons in twisted bilayer graphene, Nature 605, 63 (2022)

    2022

  59. [67]

    J. Zhu, I. Torre, M. Polini, and A. H. MacDonald, Weak- coupling theory of magic-angle twisted bilayer graphene, Phys. Rev. B 110, L121117 (2024)

    2024

  60. [68]

    D. Wong, K. P. Nuckolls, M. Oh, B. Lian, Y. Xie, S. Jeon, K. Watanabe, T. Taniguchi, B. A. Bernevig, and A. Yaz- dani, Cascade of electronic transitions in magic-angle twisted bilayer graphene, Nature 582, 198 (2020)

    2020

  61. [69]

    Zondiner, A

    U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao, R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von Oppen, A. Stern, et al., Cascade of phase transitions and dirac revivals in magic-angle graphene, Nature 582, 203 (2020)

    2020

  62. [70]

    X. Liu, Z. Wang, K. Watanabe, T. Taniguchi, O. Vafek, and J. I. A. Li, Tuning electron correlation in magic- angle twisted bilayer graphene using coulomb screening, Science 371, 1261 (2021)

    2021

  63. [71]

    Barrier, L

    J. Barrier, L. Peng, S. Xu, V. I. Fal’ko, K. Watanabe, T. Tanigushi, A. K. Geim, S. Adam, and A. I. Berdyugin, Coulomb screening of superconductivity in magic-angle twisted bilayer graphene (2024), arXiv:2412.01577 [cond- mat.mes-hall]

    2024 arXiv

  64. [72]

    X. Gao, A. Jimeno-Pozo, P. A. Pantaleon, E. Code- cido, D. L. Sharifi, Z. Zhang, Y. Liu, K. Watanabe, T. Taniguchi, M. W. Bockrath, F. Guinea, and C. N. Lau, Double-edged role of interactions in superconduct- ing twisted bilayer graphene (2024), arXiv:2412.01578 [cond-mat.mes-hall]

    2024 arXiv

  65. [73]

    Y. Wang, J. Zhu, G. W. Burg, A. Swain, K. Watanabe, T. Taniguchi, Y. Zheng, A. H. MacDonald, and E. Tutuc, Independently tunable flat bands and correlations in a graphene double moir´ e system (2024), arXiv:2411.18785 [cond-mat.mes-hall]. A1 Appendix A: Interacting Hamiltonian...

    2024 arXiv

  66. [74]

    Hartree diagram The Hartree diagrams is shown in Fig. (S2). According to the Feynman rule [42], Hartree self-energy could be written down as Σσ′σ′ n3n2 (k, iωn) = 1 β X m X n1,n4,σ X k′,G VG Λ∗ k′,G σ n4n1 Λ∗ k,−G σ′ n3n2 [G(k′, iωm)]σσ n1n4 = X G VG [Λk,G]σ′ n2n3 X k′ 1 β X m...

  67. [75]

    (S7), and the corresponding matrix representation is ˆΣF(k, iωn) = − 1 β X m X q,G Vq+G ˆΛ∗ k,q+G ˆG(k + q, iωn − iωm)ˆΛT k,q+G

    F ock diagram Similarly, the Fock diagram can be expanded as Σσσ ′ n4n2 (k, iωn) = − 1 β X m X n1,n3 X q,G Vq+G Λ∗ k,q+G σ n4n1 Λ∗ k′,−q−G σ′ n3n2 Gσσ ′ n1n3 (k + q, iωn − iωm)δk′,k+q = − 1 β X m X n1,n3 X q,G Vq+G Λ∗ k,q+G σ n4n1 Λ∗ k+q,−q−G σ′ n3n2 Gσσ ′ n1n3 (k + q, iωn − i...

  68. [76]

    We start with the first order ring diagrams, as shown in Fig

    Ring diagrams Next, we will derive the expression for ring diagrams within random phase approximation. We start with the first order ring diagrams, as shown in Fig. (S3). The corresponding self-energy can be evaluated as Σσ2σ3 n3n5 (k, iωn) = − − 1 β 2 X o,m X {ni} X {σi} X k′...

  69. [77]

    From Dyson’s equation, we have ˆG−1(k, iωn) = ˆG−1 0 (k, iωn) − ˆΣ(k) = iωn + µ − ΣH(k) − Ekˆγz − ΣF(k)ˆσz

    Spin polarized state at ν = 0 We start with the spin-polarized state at ν = 0, in which the self-energy ansatz is assumed to be ˆΣ(k) = ΣH(k) + ΣF(k)ˆσz, (S1) where ΣH(k) and ΣF(k) are in general complex functions depend only on momentum need to be solved self-consistently. Fr...

  70. [78]

    BCS” gap equation inspired us to define the “BCS

    Inter-valley coherence state at ν = −2 Next, we consider a more complicated case where two of eight bands are filled. According to Table. (I), we assume that self-energy has the form of ˆΣ(k) = −2ΣH(k) + ΣF(k) 1 2 (1 + ˆσz) + 1 2 (1 − ˆσz) (ˆτx cos θIVC + ˆτy sin θIVC) ˆγy = −...

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.