pith. machine review for the scientific record. sign in

arxiv: 2605.05292 · v1 · submitted 2026-05-06 · ❄️ cond-mat.str-el

Recognition: unknown

Microsopic Theory of Spin Polarons in Chern Ferromagnets

Eslam Khalaf, Qiang Gao

Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords spin polaronsChern ferromagnetsquantum geometryexact eigenstatesvariational wavefunctionsmoiré materialsskyrmionsflat bands
0
0 comments X

The pith

Spin polarons binding any number of spin flips are exact eigenstates in ideal Chern-1 bands with contact interactions.

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

The paper constructs closed-form wavefunctions for spin polaron states in SU(2) Chern ferromagnets that bind an arbitrary number of spin flips to a charge-e excitation. In the ideal limit of a flat Chern-1 band with normal-ordered contact interactions, these states are proven to be exact eigenstates with energy matching single-hole states. This provides a microscopic basis for understanding doped Chern ferromagnets in moiré materials. The wavefunctions allow efficient computation for large systems and many flips. When away from ideal conditions, a variational approach with a size parameter shows how quantum geometry affects stability.

Core claim

In an ideal Chern-1 band with a normal-ordered contact interaction, these polarons are exact eigenstates of the Hamiltonian with the same energy as single-hole excitations. The momentum-space wavefunctions admit two equivalent representations: a ratio of Jastrow factors of Weierstrass functions of relative momenta or an antisymmetrized geminal product of particle-hole wavefunctions. The latter enables efficient evaluation of overlaps and expectation values for large system sizes and many spin flips. Benchmarking in the lowest Landau level shows the single-spin-flip ansatz achieves high overlap with exact diagonalization while multi-spin-flip energies interpolate toward the skyrmion regime.

What carries the argument

The spin polaron wavefunction in momentum space, represented equivalently as a ratio of Jastrow factors of Weierstrass functions or an antisymmetrized geminal product of particle-hole wavefunctions.

If this is right

  • These exact eigenstates enable direct computation of properties of doped Chern ferromagnets without approximation in the ideal limit.
  • The variational family with a size parameter and geometry-informed dressing can be used to study realistic moiré systems.
  • Binding energies decrease slowly and the bound state size increases as quantum geometry becomes more concentrated when dispersion is suppressed.
  • Multi-spin-flip energies connect smoothly to the large-texture skyrmion regime.
  • The wavefunctions lay groundwork for variational descriptions of multi-polaron excitations and phases in topological flat bands.

Where Pith is reading between the lines

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

  • Similar exact or variational polaron constructions might extend to other flat-band systems with nontrivial topology beyond Chern insulators.
  • Materials engineering to minimize interaction-generated dispersion while preserving quantum geometry could stabilize these bound states for potential applications in topological quantum matter.
  • Numerical tests in specific moiré platforms could check whether the predicted binding persists and influences magnetic or transport properties at finite doping.

Load-bearing premise

The single-particle dispersion generated by interactions can be suppressed or tuned independently of the quantum geometry of the band.

What would settle it

Exact diagonalization on a finite-size Chern band with non-uniform geometry and unsuppressed interaction dispersion, where the variational polaron energy lies above the single-hole plus unbound spin-flip continuum.

Figures

Figures reproduced from arXiv: 2605.05292 by Eslam Khalaf, Qiang Gao.

Figure 1
Figure 1. Figure 1: FIG. 1. Lowest Landau level (LLL) benchmark. (a,b) For a single-spin-flip ( view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Variational binding energy for polarons with different spin view at source ↗
read the original abstract

We develop a microscopic theory of charged excitations in an SU(2) Chern ferromagnet and obtain closed-form wavefunctions for a hierarchy of charge-$e$ spin polaron states binding an arbitrary number of spin flips. In an ideal Chern-$1$ band with a normal-ordered contact interaction, we show that these polarons are exact eigenstates of the Hamiltonian with the same energy as single-hole excitations. Away from this ideal limit, we promote these states to a variational family by introducing a single size parameter and a geometry-informed single-particle dressing. Our momentum-space wavefunctions admit two equivalent representations: a ratio of Jastrow factors of Weierstrass functions of relative momenta or an antisymmetrized geminal product of particle-hole wavefunctions. The latter enables efficient evaluation of overlaps and expectation values for large system sizes and many spin flips. Benchmarking in the lowest Landau level, the single-spin-flip ansatz achieves $\gtrsim 99\%$ overlap with exact diagonalization and accurately captures binding energies, while the multi-spin-flip energies interpolate smoothly toward the large-texture (skyrmion) regime. For Chern bands with tunable quantum geometry, we find that interaction-generated single particle dispersion quickly destabilizes the spin polarons once quantum geometry becomes sufficiently non-uniform. When such dispersion is suppressed, however, the bound states persist deeper into the non-uniform regime, with the binding energy slowly decreasing and the bound state becoming larger as the quantum geometry becomes more concentrated. Our results provide a microscopic foundation for analyzing doped Chern ferromagnets in moir\'e platforms and lay the groundwork for variational wavefunctions of multi-polaron excitations and phases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a microscopic theory of charged excitations in SU(2) Chern ferromagnets, deriving closed-form wavefunctions for a hierarchy of charge-e spin polaron states that bind an arbitrary number of spin flips. In an ideal Chern-1 band with normal-ordered contact interaction, these polarons are shown to be exact eigenstates of the Hamiltonian with the same energy as single-hole excitations. The wavefunctions admit equivalent representations as ratios of Jastrow factors of Weierstrass functions of relative momenta or as antisymmetrized geminal products of particle-hole wavefunctions. Away from the ideal limit, the states are promoted to a variational family via a single size parameter and geometry-informed single-particle dressing. Benchmarks in the lowest Landau level demonstrate ≥99% overlap with exact diagonalization for the single-spin-flip case, with multi-flip energies interpolating to the skyrmion regime. For tunable quantum geometry, the work finds that interaction-generated dispersion destabilizes the polarons unless suppressed, in which case bound states persist with slowly decreasing binding energy and increasing size.

Significance. If the central results hold, the paper supplies a valuable microscopic foundation for analyzing doped Chern ferromagnets in moiré platforms and provides groundwork for variational studies of multi-polaron excitations and phases. The analytical demonstration that the polaron wavefunctions are exact eigenstates in the ideal limit (with energy matching single-hole excitations) is a clear strength, as is the efficient geminal-product representation that enables large-system calculations. The high-fidelity LLL benchmarks against exact diagonalization and the smooth interpolation to the skyrmion limit add substantial credibility to the variational ansatz. These elements distinguish the work from purely numerical or phenomenological approaches.

minor comments (2)
  1. [Section on variational extension] The transition from the ideal-limit exact eigenstate proof to the variational treatment with a single size parameter would benefit from an explicit statement of how the geometry-informed dressing is constructed (e.g., which single-particle orbitals are modified and by what criterion).
  2. [Tunable geometry results] In the discussion of non-uniform Chern bands, the quantitative criterion for 'sufficiently non-uniform' quantum geometry (where dispersion destabilizes the polarons) could be stated more precisely, perhaps with a plot or table of binding energy versus a geometry parameter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of our results on spin polarons in Chern ferromagnets. We are pleased that the referee recognizes the value of the closed-form wavefunctions, the exact eigenstate property in the ideal limit, the efficient geminal representation, and the benchmarks against exact diagonalization and the skyrmion regime. We appreciate the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs explicit closed-form wavefunctions for spin polarons as ratios of Jastrow factors of Weierstrass functions or antisymmetrized geminal products. In the ideal Chern-1 limit with normal-ordered contact interaction, it analytically shows these states are annihilated by the projected interaction Hamiltonian, yielding exactly the single-hole eigenvalue. This follows from the holomorphic structure enabled by uniform quantum geometry, not from any fit, self-definition, or unverified self-citation. The single size parameter is introduced explicitly only for the variational family away from the ideal limit. LLL benchmarks provide independent numerical validation. The derivation chain is self-contained with no load-bearing step that reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the ideal Chern-1 band with normal-ordered contact interaction for exactness and on the ability to suppress interaction-generated dispersion for stability in non-uniform geometry; one free variational parameter is introduced for the size.

free parameters (1)
  • size parameter
    Single adjustable parameter introduced to promote the exact states to a variational family away from the ideal limit.
axioms (2)
  • domain assumption SU(2) symmetry of the Chern ferromagnet
    Used to define the spin-polaron states binding spin flips.
  • domain assumption Normal-ordered contact interaction in ideal Chern-1 band
    Required for the states to be exact eigenstates with the same energy as single holes.

pith-pipeline@v0.9.0 · 5599 in / 1423 out tokens · 73224 ms · 2026-05-08T15:44:46.480429+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 6 canonical work pages

  1. [1]

    spin polaron

    and so is any state with any number of holes [37]. On the other hand, it is non-trivial to construct zero modes that involve electron operatorsc † r,↑ which generally do not com- mute with ˆM(r)[38] . For charge+eand one flipped spin, the unique (up to gauge choices) nontrivial operator satisfying[ ˆO, ˆM(r)] = 0is [34] Ψ(1) ξ E = ˆSξ cξ,↓ |↓⟩, ˆSξ = Z d2...

  2. [2]

    Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y . Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxi- ras, R. C. Ashoori, and P. Jarillo-Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature556, 80 (2018)

    2018

  3. [3]

    Y . Cao, V . Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y . Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxi- ras, R. C. Ashoori, and P. Jarillo-Herrero, Unconventional su- perconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018)

    2018

  4. [4]

    Yankowitz, S

    M. Yankowitz, S. Chen, H. Polshyn, Y . Zhang, K. Watanabe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuning su- perconductivity in twisted bilayer graphene, Science363, 1059 (2019)

    2019

  5. [5]

    X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir, I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang, A. Bachtold, A. H. MacDonald, and D. K. Efetov, Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene, Nature574, 653 (2019)

    2019

  6. [6]

    A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney, K. Watan- abe, T. Taniguchi, M. A. Kastner, and D. Goldhaber-Gordon, Emergent ferromagnetism near three-quarters filling in twisted bilayer graphene, Science365, 605 (2019)

    2019

  7. [7]

    Serlin, C

    M. Serlin, C. L. Tschirhart, H. Polshyn, Y . Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Intrin- sic quantized anomalous hall effect in a moir ´e heterostructure, Science367, 900 (2020)

    2020

  8. [8]

    Y . Tang, L. Li, T. Li, Y . Xu, S. Liu, K. Barmak, K. Watan- abe, T. Taniguchi, A. H. MacDonald, J. Shan, and K. F. Mak, Simulation of hubbard model physics in wse2/ws2 moir´e super- lattices, Nature579, 353 (2020)

    2020

  9. [9]

    E. C. Regan, D. Wang, C. Jin, M. I. Bakti Utama, B. Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta, M. Blei, J. D. Carlstr ¨om, K. Watanabe, T. Taniguchi, S. Tongay, M. F. Crommie, A. Zettl, and F. Wang, Mott and generalized wigner crystal states in wse2/ws2 moir´e superlattices, Nature579, 359 (2020)

    2020

  10. [10]

    X. Liu, Z. Hao, E. Khalaf, J. Y . Lee, Y . Ronen, H. Yoo, D. Haei Najafabadi, K. Watanabe, T. Taniguchi, A. Vishwanath, and P. Kim, Tunable spin-polarized correlated states in twisted double bilayer graphene, Nature583, 221 (2020)

    2020

  11. [11]

    Y . Cao, D. Rodan-Legrain, O. Rubies-Bigord `a, J. M. Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Electric Field Tunable Correlated States and Magnetic Phase Transi- tions in Twisted Bilayer-Bilayer Graphene, arXiv e-prints , arXiv:1903.08596 (2019)

    work page arXiv 1903

  12. [12]

    M. He, Y . Li, J. Cai, Y . Liu, K. Watanabe, T. Taniguchi, X. Xu, and M. Yankowitz, Symmetry breaking in twisted double bi- layer graphene, Nature Physics17, 26 (2021)

    2021

  13. [13]

    S. Chen, M. He, Y .-H. Zhang, V . Hsieh, Z. Fei, K. Watan- abe, T. Taniguchi, D. H. Cobden, X. Xu, C. R. Dean, and M. Yankowitz, Electrically tunable correlated and topological states in twisted monolayer–bilayer graphene, Nature Physics 17, 374 (2021)

    2021

  14. [14]

    F. Xu, Z. Sun, T. Jia, C. Liu, C. Xu, C. Li, Y . Gu, K. Watan- abe, T. Taniguchi, B. Tong, J. Jia, Z. Shi, S. Jiang, Y . Zhang, X. Liu, and T. Li, Observation of Integer and Fractional Quan- tum Anomalous Hall Effects in Twisted Bilayer MoTe 2, Phys. Rev. X13, 031037 (2023)

    2023

  15. [15]

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

    2023

  16. [16]

    J. Cai, E. Anderson, C. Wang, X. Zhang, X. Liu, W. Holtzmann, Y . Zhang, F. Fan, T. Taniguchi, K. Watanabe, Y . Ran, T. Cao, L. Fu, D. Xiao, W. Yao, and X. Xu, Signatures of fractional quantum anomalous Hall states in twisted MoTe2, Nature622, 63 (2023)

    2023

  17. [17]

    H. Park, J. Cai, E. Anderson, Y . Zhang, J. Zhu, X. Liu, C. Wang, W. Holtzmann, C. Hu, Z. Liu, T. Taniguchi, K. Watanabe, J.-H. Chu, T. Cao, L. Fu, W. Yao, C.-Z. Chang, D. Cobden, D. Xiao, and X. Xu, Observation of fractionally quantized anomalous Hall effect, Nature622, 74 (2023)

    2023

  18. [18]

    H. Zhou, T. Xie, A. Ghazaryan, T. Holder, J. R. Ehrets, E. M. Spanton, T. Taniguchi, K. Watanabe, E. Berg, M. Serbyn,et al., Half-and quarter-metals in rhombohedral trilayer graphene, Na- ture598, 429 (2021)

    2021

  19. [19]

    Y . Choi, Y . Choi, M. Valentini, C. L. Patterson, L. F. Holleis, O. I. Sheekey, H. Stoyanov, X. Cheng, T. Taniguchi, K. Watan- abe, et al., Superconductivity and quantized anomalous hall ef- fect in rhombohedral graphene, Nature639, 342 (2025)

    2025

  20. [20]

    Z. Lu, T. Han, Y . Yao, A. P. Reddy, J. Yang, J. Seo, K. Watan- abe, T. Taniguchi, L. Fu, and L. Ju, Fractional quantum anoma- lous hall effect in multilayer graphene, Nature626, 759 (2024)

    2024

  21. [21]

    Zhang, D

    Y .-H. Zhang, D. Mao, Y . Cao, P. Jarillo-Herrero, and T. Senthil, Nearly flat chern bands in moir´e superlattices, Phys. Rev. B99, 075127 (2019)

    2019

  22. [22]

    Bultinck, E

    N. Bultinck, E. Khalaf, S. Liu, S. Chatterjee, A. Vishwanath, and M. P. Zaletel, Ground state and hidden symmetry of magic- angle graphene at even integer filling, Phys. Rev. X10, 031034 (2020)

    2020

  23. [23]

    Khalaf, S

    E. Khalaf, S. Chatterjee, N. Bultinck, M. P. Zaletel, and A. Vishwanath, Charged skyrmions and topological origin of superconductivity in magic-angle graphene, Sci. Adv.7, eabf5299 (2021)

    2021

  24. [24]

    Lian, Z.-D

    B. Lian, Z.-D. Song, N. Regnault, D. K. Efetov, A. Yazdani, and B. A. Bernevig, Twisted bilayer graphene. iv. exact insulator ground states and phase diagram, Phys. Rev. B103, 205414 (2021)

    2021

  25. [25]

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

    2021

  26. [26]

    S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Skyrmions and the crossover from the integer to fractional quantum hall effect at small zeeman energies, Phys. Rev. B47, 16419 (1993)

    1993

  27. [27]

    K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Spontaneous inter- layer coherence in double-layer quantum hall systems: Charged vortices and kosterlitz-thouless phase transitions, Phys. Rev. B 7 51, 5138 (1995)

    1995

  28. [28]

    H. A. Fertig, L. Brey, R. C ˆot´e, and A. H. MacDonald, Charged spin-texture excitations and the Hartree-Fock approximation in the quantum Hall effect, Physical Review B50, 11018 (1994)

    1994

  29. [29]

    Khalaf and A

    E. Khalaf and A. Vishwanath, Baby skyrmions in chern fer- romagnets and topological mechanism for spin-polaron forma- tion in twisted bilayer graphene, Nature Communications13, 10.1038/s41467-022-33673-3 (2022)

    work page doi:10.1038/s41467-022-33673-3 2022

  30. [30]

    Schindler, O

    F. Schindler, O. Vafek, and B. A. Bernevig, Trions in twisted bilayer graphene, Physical Review B105, 155135 (2022)

    2022

  31. [31]

    Y . H. Kwan, G. Wagner, N. Bultinck, S. H. Simon, and S. A. Parameswaran, Skyrmions in twisted bilayer graphene: Sta- bility, pairing, and crystallization, Phys. Rev. X12, 031020 (2022)

    2022

  32. [32]

    J. S. Hofmann, E. Khalaf, A. Vishwanath, E. Berg, and J. Y . Lee, Fermionic monte carlo study of a realistic model of twisted bilayer graphene, Phys. Rev. X12, 011061 (2022)

    2022

  33. [33]

    Parker, P

    D. Parker, P. Ledwith, E. Khalaf, T. Soejima, J. Hauschild, Y . Xie, A. Pierce, M. P. Zaletel, A. Yacoby, and A. Vishwanath, Field-tuned and zero-field fractional chern insulators in magic angle graphene, arXiv preprint arXiv:2112.13837 (2021)

    work page arXiv 2021

  34. [34]

    D. E. Parker, T. Soejima, J. Hauschild, M. P. Zaletel, and N. Bultinck, Strain-induced quantum phase transitions in magic-angle graphene, Phys. Rev. Lett.127, 027601 (2021)

    2021

  35. [35]

    Supplemental material, see Supplemental Material for deriva- tions of the exact zero modes and their general-nwavefunc- tions, the mapping to generic ideal Chern bands, details of the variational ansatz and regulator, the diagrammatic evaluation scheme (including the partition formula for generaln), and ad- ditional results for moir´e band structures

  36. [36]

    This gives rise to a state whereS z is decreased by 1 butS total is unchanged

    Adding theq= 0Goldstone mode amounts to rotating the parent state. This gives rise to a state whereS z is decreased by 1 butS total is unchanged. However, for non-zero but smallq, this state has different total spinS total

  37. [37]

    For a density-density interaction, which differs from the normal-ordered interaction by a constant, a hole costs a con- stant positive energy

  38. [38]

    Since the holes in the same band cannot interact via contact interaction due to Pauli blocking

  39. [39]

    As discussed earlier, we are excluding states formed by a com- bination of a hole andngoldstone modes which satisfy this condition trivially

  40. [40]

    F. D. M. Haldane and E. H. Rezayi, Periodic laughlin-jastrow wave functions for the fractional quantized hall effect, Phys. Rev. B31, 2529 (1985)

    1985

  41. [41]

    A. J. Coleman, Structure of fermion density matrices. ii. anti- symmetrized geminal powers, J. Math. Phys.6, 1425 (1965)

    1965

  42. [42]

    Blaizot and G

    J.-P. Blaizot and G. Ripka, Quantum Theory of Finite Systems (MIT Press, Cambridge, MA, 1986)

    1986

  43. [43]

    P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Research2, 023237 (2020)

    2020

  44. [44]

    J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact lan- dau level description of geometry and interaction in a flatband, Phys. Rev. Lett.127, 246403 (2021)

    2021

  45. [45]

    P. J. Ledwith, A. Vishwanath, and D. E. Parker, V ortexability: A Unifying Criterion for Ideal Fractional Chern Insulators, arXiv e-prints , arXiv:2209.15023 (2022), arXiv:2209.15023 [cond- mat.str-el]

    work page arXiv 2022

  46. [46]

    Z. Yan, Q. Li, T. Soejima, and E. Khalaf, Anyon dispersion in aharonov-casher bands and implications for twisted MoTe2, arXiv preprint arXiv:2512.15863 (2025)

    work page arXiv 2025

  47. [47]

    Schleith, T

    M.-L. Schleith, T. Soejima, and E. Khalaf, Anyon dispersion from non-uniform magnetic field on the sphere, arXiv preprint arXiv:2506.11211 (2025)

    work page arXiv 2025

  48. [48]

    This will provide information about the polaron dispersion

    Strictly speaking, we should construct the variational wave- functions at fixed momentum and find the minimum energy as a function of momentum. This will provide information about the polaron dispersion. However, since the momentum wave- functions do not admit a simple geminal form, we take instead the polaron position as a variational parameter and minim...

  49. [49]

    Thus, the estimated extrapolated value is around54%

    This value is sensitive to the data extrapolations: we only in- clude data points that are linear in1/ √n, which we have to exclude data for smalln(polaron limit) and largen(finite size effect). Thus, the estimated extrapolated value is around54%

  50. [50]

    P. J. Ledwith, J. Dong, A. Vishwanath, and E. Khalaf, Nonlo- cal moments and mott semimetal in the chern bands of twisted bilayer graphene, Phys. Rev. X15, 021087 (2025)

    2025

  51. [51]

    Guinea and N

    F. Guinea and N. R. Walet, Continuum models for twisted bi- layer graphene: Effect of lattice deformation and hopping pa- rameters, Phys. Rev. B99, 205134 (2019)

    2019

  52. [52]

    B. A. Bernevig, B. Lian, A. Cowsik, F. Xie, N. Regnault, and Z.-D. Song, Twisted bilayer graphene. v. exact analytic many- body excitations in coulomb hamiltonians: Charge gap, gold- stone modes, and absence of cooper pairing, Phys. Rev. B103, 205415 (2021)

    2021

  53. [53]

    A. T. Pierce, Y . Xie, J. M. Park, E. Khalaf, S. H. Lee, Y . Cao, D. E. Parker, P. R. Forrester, S. Chen, K. Watanabe, T. Taniguchi, A. Vishwanath, P. Jarillo-Herrero, and A. Ya- coby, Unconventional sequence of correlated chern insulators in magic-angle twisted bilayer graphene, Nature Physics17, 1210 (2021)

    2021

  54. [54]

    J. Kang, B. A. Bernevig, and O. Vafek, Cascades between light and heavy fermions in the normal state of magic-angle twisted bilayer graphene, Phys. Rev. Lett.127, 266402 (2021)

    2021

  55. [55]

    J. Dong, T. Wang, T. Wang, T. Soejima, M. P. Zaletel, A. Vish- wanath, and D. E. Parker, Anomalous hall crystals in rhom- bohedral multilayer graphene. i. interaction-driven chern bands and fractional quantum hall states at zero magnetic field, Phys- ical Review Letters133, 206503 (2024)

    2024

  56. [56]

    Z. Dong, A. S. Patri, and T. Senthil, Theory of quantum anoma- lous hall phases in pentalayer rhombohedral graphene moir ´e structures, Physical Review Letters133, 206502 (2024)

    2024

  57. [57]

    B. Zhou, H. Yang, and Y .-H. Zhang, Fractional quantum anomalous hall effect in rhombohedral multilayer graphene in the moir ´eless limit, Physical Review Letters133, 206504 (2024)

    2024

  58. [58]

    Tan and T

    T. Tan and T. Devakul, Parent berry curvature and the ideal anomalous hall crystal, Physical Review X14, 041040 (2024)

    2024

  59. [59]

    Chatterjee, N

    S. Chatterjee, N. Bultinck, and M. P. Zaletel, Symmetry break- ing and skyrmionic transport in twisted bilayer graphene, Phys. Rev. B101, 165141 (2020). 1 Supplemental Material for ”Theory of microscopic spin polarons in Chern ferromagnets” CONTENTS Overview 1 S1. Review of Landau levels on the torus and Ideal bands 2 S1.1. Setup and conventions 2 S1.2. L...

    2020

  60. [60]

    = Z d2rϕ∗ ke(r)Rkh 1 ,kh 2 (r)(S66) Eq. S63 is fulfilled if and onlyR k,kh 1 ,kh 2 (r)vanishes upon antisymmetrizing relative to its three variables which, together with (S64), implies Rkh 1 ,kh 2 (r) =Aϕ kh 1 (r)fkh 2 (r)(S67) whereAis the antisymmetrization operator relative to the hole momentak h 1 andk h 2 . The requirement thatR k2,k3 is in the LLL p...

  61. [61]

    It is easy to extend this to a wavefunction satisfying the boundary condition by summing over reciprocal lattice shifts

    =Af kh 2 Z d2rϕ∗ ke(r)ϕkh 1 (r) =Af kh 2 δ(ke −k h 1)(S68) Here, we have assumedk h 1,2 are in the first BZ. It is easy to extend this to a wavefunction satisfying the boundary condition by summing over reciprocal lattice shifts. The wavefunction above represents a single hole attached to a Goldstone mode as can be verified by computing itsS 2 eigenvalue ...

  62. [62]

    Second, notice that the residues of the poles atu=ξandu=−ik 1 vanish upon antisymmetrization

    (S75) × A σ(k3 −iξ) Z d2ue − 1 2 u2+ i 2 (u¯k1−¯uk1)¯σ(k1 −iu)σ(k 2 −iu) σ(k2 −k 1 +i(u−ξ)) σ(k2 −k 1)σ(i(u−ξ)) =e i 2 (¯k3+¯k2−¯k1)ξe− 1 4 (k2 1+k2 2+k2 3) Z d2ue − 1 2 u2+ i 2 (u¯k1−¯uk1)|σ(k1 −iu)| 2Fu,ξ(k1,k 2,k 3) =e i 2 (¯k3+¯k2−¯k1)ξe 1 4 (k2 1−k2 2−k2 3) Z d2ue − 1 2 |u+ik1|2 |σ(k1 −iu)| 2Fu,ξ(k1,k 2,k 3)(S76) where Fu,ξ(k1,k 2,k 3) =A σ(k2 −k 1 +...