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REVIEW 3 major objections 4 minor 57 references

Quantum Theory of Exciton Magnetic Moment: Interaction and Topological Effects

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper derives the full first-order orbital magnetic moment of an exciton and shows that two previously missing quantum contributions explain why the p-exciton valley g-factor in biased bilayer graphene is an order of magnitude smaller

desk verdict A genuinely new formalism for exciton orbital magnetic moments, with real predictive gains in BBG, but the unquantified kernel assumption and a ~5σ s-exciton miss mean 'excellent agreement' is too strong. read the letter →

arxiv 2509.07284 v3 pith:TDADSMFA submitted 2025-09-08 cond-mat.mes-hall cond-mat.mtrl-sci

Gurjyot Sethi , Jiawei Ruan , Fang Zhang , Weichen Tang , Chen Hu , Mit Naik , Steven G. Louie This is my paper
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords excitonmagneticmomentvalleyg-factororbitalmagnetismBerryphasecorrectionenvelopewindingcenter-of-massquantumgeometrybiasedbilayergrapheneGW-BSE
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 develops a quantum theory of the exciton orbital magnetic moment within the GW-BSE many-body approach, treating the position operator carefully enough to avoid the divergences that plague earlier work. It shows that the exciton magnetic moment is not just a weighted difference of electron and hole band moments: three physically distinct pieces contribute, including two that are entirely new—one from the phase winding of the exciton's internal envelope function and one from the quantum geometry of the exciton's center-of-mass motion. Applied to biased bilayer graphene, the full formula gives valley g-factors of 20.32 for the s-exciton and 1.79 for the p-exciton, closely matching measured values of 19.8 and 1.4. The p-exciton's small g-factor arises from a near-cancellation: its envelope winding and center-of-mass pseudospin contributions oppose the single-particle moment difference. If correct, the work establishes that electron-hole interactions and band-topology effects are essential, not optional, for understanding the magnetic response of excitons.

What carries the argument

The central object is Eq. 6, the five-term formula for the exciton orbital magnetic moment. Two regularization steps carry the argument: replacing the position operator with a periodic analog r̂_q = −i∇_q e^{iq·r̂} (equivalent to a periodic magnetic field, then q→0), and building a localized exciton wave packet in the center-of-mass coordinate before delocalizing back to an exciton Bloch state. The five terms are regrouped into three labeled pieces: single-particle (Berry-corrected electron and hole moments), relative-motion (envelope-function phase winding in k-space), and center-of-mass (Q-space envelope winding plus exciton band Berry connection). These pieces do distinct physical work: t

What would settle it

A concrete numerical test: compute the first-order field-induced correction to the electron-hole interaction kernel for biased bilayer graphene within the same GW-BSE framework and compare its contribution to Eq. 6. If it changes the predicted g_p by more than the experimental uncertainty of ±0.8, the five-term expression is incomplete. Experimentally, measuring the exciton valley splitting over a wider magnetic-field range and checking strict linearity would also detect a first-order kernel renormalization.

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Extended reading notes

Core claim

The central result is Eq. 6, a gauge-invariant expression for the exciton orbital magnetic moment μ^X_{nQ} containing five terms that regroup into three contributions: a Berry-phase-corrected single-particle electron-hole moment difference, a relative-motion term from the k-space envelope function's phase winding, and center-of-mass terms from the Q-space envelope winding and the exciton band Berry connection. The derivation regularizes the position operator by first introducing a periodic magnetic field and then taking the long-wavelength limit, and it regularizes the exciton's extended center-of-mass Bloch state by constructing a localized exciton wave packet before delocalizing. In biased

Load-bearing premise

The main load-bearing assumption is that the magnetic field does not change the electron-hole interaction kernel (K̃_eh ≈ K_eh near Eq. 3), so the first-order energy shift comes only from single-particle and envelope wavefunction terms; if the kernel itself shifts at first order, those extra terms would enter on equal footing.

Editorial extensions

If this is right

  • The full formula reproduces measured valley g-factors in biased bilayer graphene: g_s = 20.32 vs 19.8 ± 0.1 and g_p = 1.79 vs 1.4 ± 0.8, where simpler approximations give 15.06 for both or 18.45/13.03.
  • The p-exciton's small g-factor is explained as a cancellation: envelope winding contributes −5.95 μ_B and center-of-mass motion −4.78 μ_B, nearly canceling the 9.84 μ_B single-particle contribution.
  • The single-particle piece itself contains a Berry-phase correction to the density of states that is substantial (+2.23 for s, +3.32 for p) and has been neglected in earlier treatments.
  • The center-of-mass contribution implies that exciton band quantum geometry, including the exciton Berry connection and pseudospin texture, directly influences the magnetic response.
  • The formalism is general to any dimensionality and magnetic field orientation, and its second-order extension is expected to give a full theory of exciton diamagnetic response.

Reading between the lines

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

  • If this cancellation mechanism is generic, other two-dimensional semiconductors with p- or d-like excitons and nontrivial band topology should show systematically smaller valley g-factors than s-excitons; measuring g-factors across an exciton Rydberg series would test this without new instrumentation.
  • The center-of-mass terms predict that exciton bands with nonzero Berry curvature, such as in moiré or flat-band systems, will carry magnetic-moment contributions even at zero center-of-mass momentum once valley mixing is present—this could be probed by magneto-optical spectroscopy of moiré excitons.
  • Because the derivation drops spin Zeeman terms for bright like-spin excitons, extending it with spin-orbit coupling may reveal additional valley-dependent spin corrections in monolayer transition metal dichalcogenides, where spin and valley are locked.
  • The unmodified-kernel assumption could be tested experimentally by checking whether exciton valley splittings stay linear in magnetic field over a wider field range; any nonlinearity beyond experimental error would signal either first-order kernel renormalization or the onset of second-order diamagnetic terms.
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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 / 4 minor

Summary. The paper develops a first-principles many-body theory of the orbital magnetic moment of excitons within the GW-BSE framework. The central result, Eq. (6), expresses the exciton magnetic moment as five terms, regrouped into three physical contributions: a Berry-phase-corrected single-particle electron-hole moment difference, an envelope-function winding term from the relative electron-hole motion, and a center-of-mass contribution from the exciton band Berry connection and Q-space pseudospin texture. The theory is applied to biased bilayer graphene, yielding valley g-factors of 20.32 for the s-exciton and 1.79 for the p-exciton, compared with experimental values of 19.8 ± 0.1 and 1.4 ± 0.8.

Significance. If Eq. (6) is correct, this is a substantial advance: it goes beyond heuristic envelope-weighted single-particle moment differences and includes quantum-geometric and interaction effects that are absent from prior treatments. The work is genuinely predictive: no fitted parameters enter the computed g-factors, and the explicit decomposition into single-particle, relative-motion, and COM contributions is physically transparent and falsifiable. The numerical identification of the p-exciton cancellation—where large positive and negative contributions nearly cancel—is an important and non-obvious result. The main limitations are verifiability and the unquantified treatment of the electron-hole interaction kernel in a magnetic field, both of which bear directly on the claimed quantitative agreement.

major comments (3)
  1. [Eq. (6) and surrounding text] The central five-term expression is derived in the supplementary text (sections 1–5), which is not available in this preprint. The main text only states the result and describes the regularization schematically. Because the order of limits (periodic-field long-wavelength limit and exciton wave-packet delocalization) is essential to Eq. (6), I cannot independently verify the central claim from the information provided. The authors should either include the full derivation in the paper or ensure the supplementary material is available to reviewers and readers.
  2. [Near Eq. (3)] The approximation K̃_eh ≈ K_eh is explicitly invoked but not quantified. A first-order-in-B change in the electron-hole interaction kernel contributes to the exciton energy shift at the same order as the five retained terms in Eq. (6). In biased bilayer graphene, the gap is only 161 meV and Coulomb interactions are strong, so a field-induced change in screening or exchange matrix elements is not obviously negligible. The numerical results make this concrete: the s-exciton g-factor is 20.32 versus 19.8 ± 0.1, a discrepancy of 0.52 μB (~5σ), and the p-exciton g-factor is a near cancellation of large terms (+6.52, +3.32, −5.95, −4.78 in Table 2). A kernel correction of order 1 μB would materially affect the prediction. The authors should provide a quantitative estimate or bound for ∂K_eh/∂B, or a symmetry/energetic argument for its irrelevance.
  3. [Table 1 and Eq. (13)] The claim of 'excellent agreement' with experiment is weakened by the s-exciton discrepancy. The computed value 20.32 differs from the reported 19.8 ± 0.1 by 0.52 μB, which is more than five experimental standard deviations. No theoretical uncertainty estimate (k-grid convergence, GW convergence, numerical evaluation of Berry connections, etc.) is provided. Without such an estimate, the statistical significance of the discrepancy cannot be assessed, and the quantitative-agreement claim is overstated. If the agreement is intended to be qualitative, that should be stated; if quantitative, the missing error budget should be supplied.
minor comments (4)
  1. [Section 3, text after Fig. 2b] Typo: 'no2 particle-hole asymmetry' should read 'no particle-hole asymmetry'.
  2. [Equations (4) and (6)] The notation is often hard to parse because of missing or ambiguous subscripts and superscripts (e.g., '𝛍𝑐𝑐′𝐤+𝐐 𝟐𝑐′' and similar terms). Please reformat using clear indices and parentheses so that each term in the five-term sum is unambiguous.
  3. [Table 1 footnote] The p-exciton experimental g-factor is not directly reported in ref. (20) but is inferred from the linewidth; this indirect extraction should be emphasized in the main text, not only in the table footnote, because it carries considerably larger systematic uncertainty than the quoted ±0.8 suggests.
  4. [Eq. (11) and Eq. (12)] The definitions of g-factor in these equations use the absolute value of a z-component difference but do not explicitly state that the field is along z and that only the orbital contribution is included; please add a sentence clarifying these conventions at first use.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the exciton magnetic moment formula is derived from a first-principles many-body perturbation calculation, and the computed g-factors are genuine predictions compared with experiment.

full rationale

The central derivation (Eq. 6) is self-contained. It starts from the GW-BSE exciton Hamiltonian, evaluates first-order energy shifts under a periodic magnetic field using a regularized position operator, and handles the COM divergence through a wave-packet construction. The five terms in Eq. 6 are algebraic consequences of this procedure, not inputs fitted to the target results. The GW-BSE band structure and exciton envelope functions are computed ab initio, and the resulting g-factors (20.32 for s, 1.79 for p) are compared with, not fitted to, experiment. The comparison is a genuine test, although the p-exciton experimental value is indirectly inferred from linewidth estimates in ref. 20. Self-citations (e.g., CWL optical selection rules, ref. 37, and exciton exchange physics, ref. 41) are used to interpret the signs and relative sizes of the contributions, but they are not load-bearing: the numerical calculation independently determines the envelope-function winding and exciton Berry connection. The paper's explicit assumption K̃_eh ≈ K_eh near Eq. 3 is a stated physical approximation rather than a circular reduction; it could affect numerical accuracy if a field-induced change in the e-h kernel is non-negligible, but it does not make the derived expression equivalent to its input by construction. Accordingly, no circularity step is identified and the score is low.

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

The central claim rests on the GW-BSE framework, the assumption that the e-h kernel is field-independent, and the new regularization procedure. No fitted parameters or new physical entities are introduced.

assumptions (4)
  • domain assumption GW-BSE many-body framework correctly describes exciton states and energies in BBG
    The entire calculation uses the ab initio GW and GW-BSE approximations (refs. 29,30) as the starting point; the central result is expressed within this framework.
  • domain assumption The electron-hole interaction kernel is unperturbed by the magnetic field (K̃_eh ≈ K_eh)
    Stated before Eq. 3; any first-order field-induced change in the kernel would modify the expression and is not included.
  • ad hoc to paper The periodic magnetic field and exciton wave-packet regularization, with the stated order of limits, yield the physical uniform-field magnetic moment
    The paper introduces this regularization procedure to handle divergences; the order of limits is stated as essential (Fig. 1a).
  • domain assumption DFT-LDA starting point plus GW quasiparticle corrections is accurate for BBG band structure
    The band gap and dispersions depend on the GW calculation; the Mexican-hat dispersion is key to the single-particle moment difference.

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Pith. "Pith review of Quantum Theory of Exciton Magnetic Moment: Interaction and Topological Effects." pith.science (2026). https://pith.science/paper/TDADSMFA

@misc{pith2026250907284,
  author       = {Pith},
  title        = {Pith review of: Quantum Theory of Exciton Magnetic Moment: Interaction and Topological Effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TDADSMFA}},
  note         = {Machine review of arXiv:2509.07284}
}
read the original abstract

Combining magnetometry with optical spectroscopy has uncovered novel quantum phenomena and is emerging as a powerful probe of quantum materials. However, the theory of the magnetic response of excitons, correlated electron-hole pairs in insulators, remains incomplete due to insufficient treatment of electron-hole interactions and quantum geometric effects. In biased bilayer graphene, for instance, theoretical predictions of valley g-factors for p-excitons deviate from experiment by nearly an order of magnitude. Here, we develop a quantum theory of the exciton orbital magnetic moment, based on first-order perturbation theory within the GW plus Bethe-Salpeter Equation approach and a rigorous treatment of the position operator in the response of exciton states to a magnetic field. Our formalism reveals three distinct contributions that go beyond the heuristic approaches used in the literature: a Berry-phase-corrected single-particle electron and hole moment difference, a term from envelope-function winding linked to electron-hole relative motion, and a center-of-mass correction from exciton band quantum geometry, with the latter two being completely new effects not considered in previous studies. Our ab initio calculations yield results in excellent agreement with experiment, establishing the importance of interaction and quantum geometric effects in the magnetic response of excitons.

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Works this paper leans on

57 extracted references · 56 canonical work pages

  1. [1]

    J. A. Sobota, Y. He, Z.-X. Shen, Angle-resolved photoemission studies of quantum materials. Reviews of Modern Physics 93, (2021)

    work page 2021

  2. [2]

    S. W. Koch, M. Kira, G. Khitrova, H. M. Gibbs, Semiconductor excitons in new light. Nature Materials 5, 523–531 (2006)

    work page 2006

  3. [3]

    F. J. García de Abajo, Optical excitations in electron microscopy. Reviews of Modern Physics 82, 209–275 (2010)

    work page 2010

  4. [4]

    N. P. Wilson et al., Interlayer electronic coupling on demand in a 2D magnetic semiconductor. Nature Materials 20, 1657–1662 (2021)

    work page 2021

  5. [5]

    K. L. Seyler et al., Ligand-field helical luminescence in a 2D ferromagnetic insulator. Nature Physics 14, 277–281 (2017)

    work page 2017

  6. [6]

    M. Wu, Z. Li, T. Cao, S. G. Louie, Physical origin of giant excitonic and magneto-optical responses in two-dimensional ferromagnetic insulators. Nature Communications 10, (2019)

    work page 2019

  7. [7]

    J. R. Schaibley et al., Valleytronics in 2D materials. Nature Reviews Materials 1, (2016)

    work page 2016

  8. [8]

    Yin et al., Tunable and giant valley-selective Hall effect in gapped bilayer graphene

    J. Yin et al., Tunable and giant valley-selective Hall effect in gapped bilayer graphene. Science 375, 1398–1402 (2022)

    work page 2022

Show all 57 references
  1. [9]

    S. L. Bayliss et al., Optically addressable molecular spins for quantum information processing. Science 370, 1309–1312 (2020)

    2020

  2. [10]

    Liebich et al., Controlling Coulomb correlations and fine structure of quasi-one-dimensional excitons by magnetic order

    M. Liebich et al., Controlling Coulomb correlations and fine structure of quasi-one-dimensional excitons by magnetic order. Nature Materials 24, 384–390 (2025)

    2025

  3. [11]

    M. R. Wasielewski et al., Exploiting chemistry and molecular systems for quantum information science. Nature Reviews Chemistry 4, 490–504 (2020). 11

    2020

  4. [12]

    D. Y. Qiu, F. H. da Jornada, S. G. Louie, Optical Spectrum ofMoS2: Many-Body Effects and Diversity of Exciton States. Physical Review Letters 111, (2013)

    2013

  5. [13]

    K. F. Mak, D. Xiao, J. Shan, Light–valley interactions in 2D semiconductors. Nature Photonics 12, 451–460 (2018)

    2018

  6. [14]

    D. Xiao, W. Yao, Q. Niu, Valley-Contrasting Physics in Graphene: Magnetic Moment and Topological Transport. Physical Review Letters 99, (2007)

    2007

  7. [15]

    Schmidt et al., Magnetic-Field-Induced Rotation of Polarized Light Emission from MonolayerWS2

    R. Schmidt et al., Magnetic-Field-Induced Rotation of Polarized Light Emission from MonolayerWS2. Physical Review Letters 117, (2016)

    2016

  8. [16]

    MacNeill et al., Breaking of Valley Degeneracy by Magnetic Field in MonolayerMoSe2

    D. MacNeill et al., Breaking of Valley Degeneracy by Magnetic Field in MonolayerMoSe2. Physical Review Letters 114, (2015)

    2015

  9. [17]

    Aivazian et al., Magnetic control of valley pseudospin in monolayer WSe2

    G. Aivazian et al., Magnetic control of valley pseudospin in monolayer WSe2. Nature Physics 11, 148–152 (2015)

    2015

  10. [18]

    A. V. Stier et al., Magnetooptics of Exciton Rydberg States in a Monolayer Semiconductor. Physical Review Letters 120, (2018)

    2018

  11. [19]

    Srivastava et al., Valley Zeeman effect in elementary optical excitations of monolayer WSe2

    A. Srivastava et al., Valley Zeeman effect in elementary optical excitations of monolayer WSe2. Nature Physics 11, 141–147 (2015)

    2015

  12. [20]

    Ju et al., Tunable excitons in bilayer graphene

    L. Ju et al., Tunable excitons in bilayer graphene. Science 358, 907–910 (2017)

    2017

  13. [21]

    Woźniak, P

    T. Woźniak, P. E. Faria Junior, G. Seifert, A. Chaves, J. Kunstmann, Exciton g-factors of van der Waals heterostructures from first-principles calculations. Physical Review B 101, (2020)

    2020

  14. [22]

    F. Xuan, S. Y. Quek, Valley Zeeman effect and Landau levels in two-dimensional transition metal dichalcogenides. Physical Review Research 2, (2020)

    2020

  15. [23]

    Förste et al., Exciton g-factors in monolayer and bilayer WSe2 from experiment and theory

    J. Förste et al., Exciton g-factors in monolayer and bilayer WSe2 from experiment and theory. Nature Communications 11, (2020)

    2020

  16. [24]

    Deilmann, P

    T. Deilmann, P. Krüger, M. Rohlfing, Ab Initio Studies of Exciton g-factors: Monolayer Transition Metal Dichalcogenides in Magnetic Fields. Physical Review Letters 124, (2020)

    2020

  17. [25]

    E. I. Blount, Formalisms of Band Theory. Solid State Physics, 305–373 (1962)

    1962

  18. [26]

    Resta, Electrical polarization and orbital magnetization: the modern theories

    R. Resta, Electrical polarization and orbital magnetization: the modern theories. Journal of Physics: Condensed Matter 22, (2010)

    2010

  19. [27]

    J. Shi, G. Vignale, D. Xiao, Q. Niu, Quantum Theory of Orbital Magnetization and Its Generalization to Interacting Systems. Physical Review Letters 99, (2007)

    2007

  20. [28]

    P. E. Faria Junior, D. Hernangómez-Pérez, T. Amit, J. Fabian, S. Refaely-Abramson, Generalized many-body exciton g-factors: magnetic hybridization and non-monotonic Rydberg series in monolayer WSe2. 2025 (10.48550/arXiv.2505.18468)

    2025 doi arXiv

  21. [29]

    M. S. Hybertsen, S. G. Louie, Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Physical Review B 34, 5390–5413 (1986). 12

    1986

  22. [30]

    Rohlfing, S

    M. Rohlfing, S. G. Louie, Electron-hole excitations and optical spectra from first principles. Physical Review B 62, 4927–4944 (2000)

    2000

  23. [31]

    Mauri, S

    F. Mauri, S. G. Louie, Magnetic Susceptibility of Insulators from First Principles. Physical Review Letters 76, 4246–4249 (1996)

    1996

  24. [32]

    Vignale, Orbital paramagnetism of electrons in a two-dimensional lattice

    G. Vignale, Orbital paramagnetism of electrons in a two-dimensional lattice. Physical Review Letters 67, 358–361 (1991)

    1991

  25. [33]

    Chang, Q

    M.-C. Chang, Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical dynamics in magnetic Bloch bands. Physical Review B 53, 7010–7023 (1996)

    1996

  26. [34]

    W. Yao, Q. Niu, Berry Phase Effect on the Exciton Transport and on the Exciton Bose-Einstein Condensate. Physical Review Letters 101, (2008)

    2008

  27. [35]

    D. Xiao, J. Shi, Q. Niu, Berry Phase Correction to Electron Density of States in Solids. Physical Review Letters 95, (2005)

    2005

  28. [36]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, Q. Niu, Berry phase effects on electronic properties. Reviews of Modern Physics 82, 1959–2007 (2010)

    1959

  29. [37]

    T. Cao, M. Wu, S. G. Louie, Unifying Optical Selection Rules for Excitons in Two Dimensions: Band Topology and Winding Numbers. Physical Review Letters 120, (2018)

    2018

  30. [38]

    E. V. Castro et al., Biased Bilayer Graphene: Semiconductor with a Gap Tunable by the Electric Field Effect. Physical Review Letters 99, (2007)

    2007

  31. [39]

    Zhang et al., Direct observation of a widely tunable bandgap in bilayer graphene

    Y. Zhang et al., Direct observation of a widely tunable bandgap in bilayer graphene. Nature 459, 820–823 (2009)

    2009

  32. [40]

    C.-H. Park, N. Marzari, Berry phase and pseudospin winding number in bilayer graphene. Physical Review B 84, (2011)

    2011

  33. [41]

    D. Y. Qiu, T. Cao, S. G. Louie, Nonanalyticity, Valley Quantum Phases, and Lightlike Exciton Dispersion in Monolayer Transition Metal Dichalcogenides: Theory and First-Principles Calculations. Physical Review Letters 115, (2015)

    2015

  34. [42]

    F. Wu, F. Qu, A. H. MacDonald, Exciton band structure of monolayerMoS2. Physical Review B 91, (2015)

    2015

  35. [43]

    L. Y. Liu et al., Direct Observation of Massless Excitons and Linear Exciton Dispersion. 2025 (10.48550/arXiv.2502.20454)

    2025 doi arXiv

  36. [44]

    D. Y. Qiu, G. Cohen, D. Novichkova, S. Refaely-Abramson, Signatures of Dimensionality and Symmetry in Exciton Band Structure: Consequences for Exciton Dynamics and Transport. Nano Letters 21, 7644–7650 (2021)

    2021

  37. [45]

    Yang et al., Spectroscopy signatures of electron correlations in a trilayer graphene/hBN moiré superlattice

    J. Yang et al., Spectroscopy signatures of electron correlations in a trilayer graphene/hBN moiré superlattice. Science 375, 1295–1299 (2022)

    2022

  38. [46]

    M. H. Naik et al., Intralayer charge-transfer moiré excitons in van der Waals superlattices. Nature 609, 52–57 (2022). 13

    2022

  39. [47]

    Sethi, Y

    G. Sethi, Y. Zhou, L. Zhu, L. Yang, F. Liu, Flat-Band-Enabled Triplet Excitonic Insulator in a Diatomic Kagome Lattice. Physical Review Letters 126, (2021)

    2021

  40. [48]

    Giannozzi et al., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials

    P. Giannozzi et al., QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Journal of Physics: Condensed Matter 21, (2009)

    2009

  41. [49]

    D. R. Hamann, Optimized norm-conserving Vanderbilt pseudopotentials. Physical Review B 88, (2013)

    2013

  42. [50]

    Sohier, M

    T. Sohier, M. Calandra, F. Mauri, Density functional perturbation theory for gated two-dimensional heterostructures: Theoretical developments and application to flexural phonons in graphene. Physical Review B 96, (2017)

    2017

  43. [51]

    Giovannetti, P

    G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, J. van den Brink, Substrate-induced band gap in graphene on hexagonal boron nitride:Ab initiodensity functional calculations. Physical Review B 76, (2007)

    2007

  44. [52]

    Deslippe et al., BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures

    J. Deslippe et al., BerkeleyGW: A massively parallel computer package for the calculation of the quasiparticle and optical properties of materials and nanostructures. Computer Physics Communications 183, 1269–1289 (2012)

    2012

  45. [53]

    F. H. da Jornada, D. Y. Qiu, S. G. Louie, Nonuniform sampling schemes of the Brillouin zone for many-electron perturbation-theory calculations in reduced dimensionality. Physical Review B 95, (2017)

    2017

  46. [54]

    Pizzi et al., Wannier90 as a community code: new features and applications

    G. Pizzi et al., Wannier90 as a community code: new features and applications. Journal of Physics: Condensed Matter 32, (2020)

    2020

  47. [55]

    Culcer, Y

    D. Culcer, Y. Yao, Q. Niu, Coherent wave-packet evolution in coupled bands. Physical Review B 72, (2005)

    2005

  48. [56]

    Y. H. Kwan, Y. Hu, S. H. Simon, S. A. Parameswaran, Exciton Band Topology in Spontaneous Quantum Anomalous Hall Insulators: Applications to Twisted Bilayer Graphene. Physical Review Letters 126, (2021)

    2021

  49. [57]

    P. Li, I. Appelbaum, Excitons without effective mass: Biased bilayer graphene. Physical Review B 99, (2019). Acknowledgements: We thank Xiaoxun Gong, Jack McArthur, and Shang Ren for the fruitful discussions. Funding: This work was primarily supported by the Center for Computa...

    2019

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