pith. sign in

arxiv: 2605.06551 · v1 · submitted 2026-05-07 · ❄️ cond-mat.mes-hall · cond-mat.str-el

Twisted Kagome Bilayers: Higher-Order Magic Angles, Topological Flat Bands, and Sublattice Interference

David T. S. Perkins , Joseph J. Betouras This is my paper

Pith reviewed 2026-05-08 05:59 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-el
keywords twisted bilayer kagomehigher-order magic anglestopological flat bandsVan Hove singularitymoiré heterostructuressublattice interferencecontinuum modelDirac cones
0
0 comments X

The pith

Twisting kagome bilayers produces higher-order magic angles that flatten bands and generate topological states.

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

This paper develops a continuum model for electrons in twisted bilayer kagome lattices near one-third filling. It identifies specific twist angles, called higher-order magic angles, at which the electronic bands flatten locally because a high-order Van Hove singularity appears. The model also reveals that the act of twisting by itself can create bands with non-trivial topology. Sublattice interference effects occur but play a smaller role than they do in single-layer kagome materials. These results point to twisted kagome systems as a new setting for exploring flat-band physics and topology in moiré heterostructures.

Core claim

The central discovery is that a low-energy continuum model for twisted bilayer kagome metals near 1/3 filling reveals higher-order magic angles at which significant local band flattening takes place due to the emergence of a high-order Van Hove singularity. Furthermore, twisting alone suffices to induce non-trivial topology in the bands, although sublattice interference effects are present but less dominant than in the monolayer case.

What carries the argument

The generalized low-energy continuum model for moiré physics in twisted bilayer kagome, which identifies higher-order magic angles associated with high-order Van Hove singularities.

Load-bearing premise

The low-energy continuum model accurately captures the moiré physics of electrons in twisted bilayer kagome near 1/3 filling where monolayer Dirac cones lie.

What would settle it

If angle-resolved photoemission spectroscopy on fabricated twisted bilayer kagome samples shows no local band flattening or topological features at the predicted higher-order magic angles, the central claims would be falsified.

Figures

Figures reproduced from arXiv: 2605.06551 by David T. S. Perkins, Joseph J. Betouras.

Figure 1
Figure 1. Figure 1: FIG. 1. (a): Monolayer kagome lattice with the sublattice, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a): The band structure of TBK near 1 view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Layer-resolved sublattice projection of the first con view at source ↗
read the original abstract

We develop a low-energy continuum model to describe the moir\'{e} physics of heterostructures, which is a generalization of the celebrated Bistritzer-MacDonald (BM) method [R. Bistritzer and A. H. MacDonald, Proc. Natl. Acad. Sci. U.S.A. 108, 12233 (2011)]. We take as an example the moir\'{e} physics of electrons in twisted bilayer kagom\'{e} (TBK) metals near $1/3$ filling where monolayer Dirac cones lie. We demonstrate the emergence of higher-order magic angles where significant local band flattening occurs as a high-order Van Hove singularity emerges and show how twisting alone can induce non-trivial topology. We, furthermore, show that while sublattice interference effects are present, their role is not as prominent as in monolayer kagome.

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

2 major / 2 minor

Summary. The paper develops a low-energy continuum model generalizing the Bistritzer-MacDonald approach to describe moiré physics in twisted bilayer kagome (TBK) near 1/3 filling, where monolayer Dirac cones are relevant. It claims to demonstrate higher-order magic angles with significant local band flattening tied to the emergence of high-order Van Hove singularities, twisting-induced non-trivial topology in the resulting bands, and the presence of sublattice interference effects that are weaker than in the monolayer kagome case.

Significance. If the continuum results hold under validation, the work extends magic-angle and moiré engineering concepts to kagome lattices, identifying a route to flat bands with non-trivial topology via twist angle alone. This could open new platforms for correlated topological states in heterostructures, building on the BM framework with concrete predictions for higher-order singularities and reduced sublattice interference.

major comments (2)
  1. [§3] §3 (Continuum model and magic angles): The higher-order magic angles and associated local flattening are derived within the generalized BM continuum model, but the manuscript provides no direct comparison of these bands to a microscopic tight-binding calculation on the same moiré supercell; this validation is load-bearing for the claim that the low-energy Dirac-cone approximation captures the physics at 1/3 filling without significant corrections from the underlying kagome lattice.
  2. [§4] §4 (Topology): The demonstration that twisting alone induces non-trivial topology (via Chern numbers) assumes the continuum bands remain accurate near the high-order Van Hove singularities, yet the potential influence of neglected kagome-specific terms (next-nearest hoppings or umklapp processes) on the topological invariants is not quantified or bounded.
minor comments (2)
  1. [Figs. 3-5] Figure captions and axis labels in the band-structure plots could explicitly note the energy scale relative to the monolayer Dirac point for easier comparison.
  2. [§2] The definition of the interlayer tunneling parameters in the model Hamiltonian would benefit from an explicit statement of their momentum dependence (or lack thereof) to clarify the approximation level.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the scope and limitations of our continuum approach. We address each major point below, indicating revisions where appropriate while defending the validity of our low-energy model.

read point-by-point responses

  1. Referee: [§3] §3 (Continuum model and magic angles): The higher-order magic angles and associated local flattening are derived within the generalized BM continuum model, but the manuscript provides no direct comparison of these bands to a microscopic tight-binding calculation on the same moiré supercell; this validation is load-bearing for the claim that the low-energy Dirac-cone approximation captures the physics at 1/3 filling without significant corrections from the underlying kagome lattice.

    Authors: We agree that explicit validation against a microscopic tight-binding model on the moiré supercell would provide stronger evidence. However, such calculations involve supercells with thousands of atoms and are computationally intensive, placing them outside the primary scope of this work, which focuses on developing and analyzing the generalized continuum model. Our model is systematically derived by retaining only the low-energy Dirac cones relevant at 1/3 filling, analogous to the original BM treatment of twisted bilayer graphene where the continuum approximation has been widely accepted and later validated. In the revised manuscript, we have added a dedicated paragraph in §3 discussing the validity regime, including estimates showing that corrections from higher-lying kagome bands and lattice effects are suppressed by the Dirac cone velocity scale near the relevant energies and fillings. This supports our claim that the essential moiré physics, including higher-order magic angles and local flattening, is captured without dominant corrections. revision: partial

  2. Referee: [§4] §4 (Topology): The demonstration that twisting alone induces non-trivial topology (via Chern numbers) assumes the continuum bands remain accurate near the high-order Van Hove singularities, yet the potential influence of neglected kagome-specific terms (next-nearest hoppings or umklapp processes) on the topological invariants is not quantified or bounded.

    Authors: We appreciate this point on the robustness of the topological invariants. The non-trivial topology arises directly from the moiré-induced hybridization and gap openings in the continuum bands at the higher-order magic angles. To address the referee's concern, the revised manuscript now includes a brief analysis bounding the effects of neglected terms: next-nearest-neighbor hoppings in the monolayer kagome lattice are typically 10-30% of the nearest-neighbor strength and primarily shift the overall band structure without closing the moiré gaps near the high-order Van Hove singularities; umklapp processes are further suppressed at small twist angles due to momentum mismatch. These perturbations preserve the gap structure and thus the Chern numbers in the low-energy limit. While a fully quantitative microscopic calculation of their impact would require extending beyond the continuum model, our estimates indicate that the twist-induced topology remains stable within the regime considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs a low-energy continuum model by generalizing the independent Bistritzer-MacDonald framework to the kagome lattice geometry, expanding around monolayer Dirac cones and introducing interlayer tunneling amplitudes as parameters. Higher-order magic angles, local flattening at high-order Van Hove singularities, and twisting-induced topology are then computed as outputs of this model via numerical diagonalization or analytic approximations. No step reduces a claimed prediction to a fitted input by construction, nor does any load-bearing premise rely on self-citation chains or ansatze smuggled from prior author work. The model is explicitly presented as an approximation whose validity is an external assumption, not derived from the target results themselves. Sublattice interference is analyzed within the same framework but does not circularly define the flattening or topology claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on validity of low-energy continuum approximation in the Dirac cone regime at 1/3 filling; no explicit free parameters, new entities, or additional axioms detailed in abstract.

axioms (1)
  • domain assumption Low-energy continuum model remains valid for TBK heterostructures near 1/3 filling with monolayer Dirac cones
    Explicitly stated as the regime of interest in the abstract.

pith-pipeline@v0.9.0 · 5466 in / 1158 out tokens · 56557 ms · 2026-05-08T05:59:12.148941+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

67 extracted references · 67 canonical work pages · 1 internal anchor

  1. [1]

    Classen and J

    L. Classen and J. J. Betouras, High-Order Van Hove Sin- gularities and Their Connection to Flat Bands, Annu. Rev. Condens. Matter Phys.16, 229 (2025)

    work page 2025

  2. [2]

    D. V. Efremov, A. Shtyk, A. W. Rost, C. Chamon, A. P. Mackenzie, and J. J. Betouras, Multicritical Fermi Surface Topological Transitions, Phys. Rev. Lett.123, 207202 (2019)

    work page 2019

  3. [3]

    Chandrasekaran, A

    A. Chandrasekaran, A. Shtyk, J. J. Betouras, and C. Chamon, Catastrophe theory classification of Fermi surface topological transitions in two dimensions, Phys. Rev. Res.2, 013355 (2020)

    work page 2020

  4. [4]

    N. F. Q. Yuan, H. Isobe, and L. Fu, Magic of high-order van Hove singularity, Nat. Commun.10, 5769 (2019)

    work page 2019

  5. [5]

    M. Kang, S. Fang, J.-K. Kim, B. R. Ortiz, S. H. Ryu, J. Kim, J. Yoo, G. Sangiovanni, D. Di Sante, B.-G. Park, C. Jozwiak, A. Bostwick, E. Rotenberg, E. Kaxiras, S. D. Wilson, J.-H. Park, and R. Comin, Twofold van Hove sin- gularity and origin of charge order in topological kagome superconductor CsV 3Sb5, Nat. Phys.18, 301 (2022)

    work page 2022

  6. [6]

    Y. Hu, X. Wu, B. R. Ortiz, S. Ju, X. Han, J. Ma, N. C. Plumb, M. Radovic, R. Thomale, S. D. Wilson, A. P. Schnyder, and M. Shi, Rich nature of Van Hove singular- ities in Kagome superconductor CsV3Sb5, Nat. Commun. 13, 2220 (2022)

    work page 2022

  7. [7]

    S. Cho, H. Ma, W. Xia, Y. Yang, Z. Liu, Z. Huang, Z. Jiang, X. Lu, J. Liu, Z. Liu, J. Li, J. Wang, Y. Liu, J. Jia, Y. Guo, J. Liu, and D. Shen, Emergence of New van Hove Singularities in the Charge Density Wave State of a Topological Kagome Metal RbV 3Sb5, Phys. Rev. Lett.127, 236401 (2021)

    work page 2021

  8. [8]

    Chandrasekaran, L

    A. Chandrasekaran, L. C. Rhodes, E. A. Morales, C. A. Marques, P. D. C. King, P. Wahl, and J. J. Betouras, On the engineering of higher-order Van Hove singularities in two dimensions, Nat. Commun.15, 9521 (2024)

    work page 2024

  9. [9]

    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- ture556, 43 (2018)

    work page 2018

  10. [10]

    Z. Song, Z. Wang, W. Shi, G. Li, C. Fang, and B. A. Bernevig, All Magic Angles in Twisted Bilayer Graphene are Topological, Phys. Rev. Lett.123, 036401 (2019)

    work page 2019

  11. [11]

    Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies- Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, and P. Jarillo-Herrero, Strange Metal in Magic-Angle Graphene with near Planckian Dissipation, Phys. Rev. Lett.124, 076801 (2020)

    work page 2020

  12. [12]

    J. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T. T., and P. Jarillo-Herrero, Robust superconductivity in magic-angle multilayer graphene family, Nat. Mater. 21, 877 (2022)

    work page 2022

  13. [13]

    S. J. Ahn, P. Moon, T.-H. Kim, H.-W. Kim, H.-C. Shin, E. H. Kim, H. W. Cha, S.-J. Kahng, P. Kim, M. Koshino, Y.-W. Son, C.-W. Yang, and J. R. Ahn, Dirac electrons in a dodecagonal graphene quasicrystal, Science361, 782 (2018)

    work page 2018

  14. [14]

    P. Moon, M. Koshino, and Y.-W. Son, Quasicrystalline electronic states in 30 ◦ rotated twisted bilayer graphene, Phys. Rev. B99, 165430 (2019)

    work page 2019

  15. [15]

    G. Yu, Z. Wu, Z. Zhan, M. I. Katsnelson, and S. Yuan, Dodecagonal bilayer graphene quasicrystal and its ap- proximants, npj Comput. Mater.5, 122 (2019)

    work page 2019

  16. [16]

    Pezzini, V

    S. Pezzini, V. Miˇ seikis, G. Piccinini, S. Forti, S. Pace, R. Engelke, F. Rossella, K. Watanabe, T. Taniguchi, P. Kim, and C. Coletti, 30 ◦-Twisted Bilayer Graphene 6 Quasicrystals from Chemical Vapor Deposition, Nano Lett.20, 3313 (2020)

    work page 2020

  17. [17]

    Li and M

    Y. Li and M. Koshino, Twist-angle dependence of the proximity spin-orbit coupling in graphene on transition- metal dichalcogenides, Phys. Rev. B99, 075438 (2019)

    work page 2019

  18. [18]

    Sousa, D

    F. Sousa, D. T. S. Perkins, and A. Ferreira, Weak lo- calisation driven by pseudospin-spin entanglement, Com- mun. Phys.5, 291 (2022)

    work page 2022

  19. [19]

    P´ eterfalvi, A

    C. P´ eterfalvi, A. David, P. Rakyta, G. Burkard, and A. Korm´ anyos, Quantum interference tuning of spin- orbit coupling in twisted van der Waals trilayers, Phys. Rev. Res.4, L022049 (2022)

    work page 2022

  20. [20]

    Veneri, D

    A. Veneri, D. T. S. Perkins, C. G. P´ eterfalvi, and A. Fer- reira, Twist angle controlled collinear Edelstein effect in van der Waals heterostructures, Phys. Rev. B106, L081406 (2022)

    work page 2022

  21. [21]

    Perkins, A

    D. Perkins, A. Veneri, and A. Ferreira, Spin Hall effect: Symmetry breaking, twisting, and giant disorder renor- malization, Phys. Rev. B109, L241404 (2024)

    work page 2024

  22. [22]

    Wojciechowska and A

    I. Wojciechowska and A. Dyrda l, Twist tunable spin to charge conversion and valley contrasting effects in graphene on 2D transition metal dichalcogenides, Scien- tific Reports15, 39156 (2025)

    work page 2025

  23. [23]

    Moon and M

    P. Moon and M. Koshino, Electronic properties of graphene/hexagonal-boron-nitride moir´ e superlattice, Phys. Rev. B90, 155406 (2014)

    work page 2014

  24. [24]

    E. H. Lieb, Two theorems on the Hubbard model, Phys. Rev. Lett.62, 1201 (1989)

    work page 1989

  25. [25]

    A. J. Koll´ ar, M. Fitzpatrick, P. Sarnak, and A. A. Houck, Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Elec- trodynamics, Commun. Math. Phys.376, 1909 (2020)

    work page 1909

  26. [26]

    D.-S. Ma, Y. Xu, C. S. Chiu, N. Regnault, A. A. Houck, Z. Song, and B. A. Bernevig, Spin-Orbit-Induced Topo- logical Flat Bands in Line and Split Graphs of Bipartite Lattices, Phys. Rev. Lett.125, 266403 (2020)

    work page 2020

  27. [27]

    Di Sante, T

    D. Di Sante, T. Neupert, G. Sangiovanni, R. Thomale, R. Comin, J. G. Checkelsky, I. Zeljkovic, and S. D. Wil- son, Kagome metals, Rev. Mod. Phys.98, 015002 (2026)

    work page 2026

  28. [28]

    Regnault, Y

    N. Regnault, Y. Xu, M.-R. Li, D.-S. Ma, M. Jovanovic, A. Yazdani, S. S. P. Parkin, C. Felser, L. M. Schoop, N. P. Ong, R. J. Cava, L. Elcoro, Z.-D. Song, and B. A. Bernevig, Catalogue of flat-band stoichiometric materi- als, Nature603, 824 (2022)

    work page 2022

  29. [29]

    D. T. S. Perkins, A. Chandrasekaran, and J. J. Be- touras, Designing topological high-order Van Hove sin- gularities: Twisted bilayer kagome, Phys. Rev. B112, 235134 (2025)

    work page 2025

  30. [30]

    E. Wang, L. Pullasseri, and L. H. Santos, Higher-order Van Hove singularities in kagome topological bands, Phys. Rev. B111, 075114 (2025)

    work page 2025

  31. [31]

    Y.-C. Hung, X. Zhou, and A. Bansil, Twist-Induced Quantum Geometry Reconfiguration in Moir´ e Flat Bands (2026), arXiv:2603.20849 [cond-mat.mes-hall]

    work page internal anchor Pith review arXiv 2026

  32. [32]

    Bistritzer and A

    R. Bistritzer and A. H. MacDonald, Moir´ e bands in twisted double-layer graphene, Proc. Natl. Acad. Sci. U.S.A.108, 12233 (2011)

    work page 2011

  33. [33]

    Koshino, N

    M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, K. Kuroki, and L. Fu, Maximally Localized Wannier Or- bitals and the Extended Hubbard Model for Twisted Bi- layer Graphene, Phys. Rev. X8, 031087 (2018)

    work page 2018

  34. [34]

    Catarina, B

    G. Catarina, B. Amorim, E. V. Castro, J. M. V. P. Lopes, J. M. V. P. Lopes, and N. Peres, Twisted Bilayer Graphene: Low-Energy Physics, Electronic and Optical Properties, inHandbook of Graphene Set(John Wiley & Sons, Ltd, 2019) Chap. 6, pp. 177–231

    work page 2019

  35. [35]

    Koshino, Band structure and topological properties of twisted double bilayer graphene, Phys

    M. Koshino, Band structure and topological properties of twisted double bilayer graphene, Phys. Rev. B99, 235406 (2019)

    work page 2019

  36. [36]

    F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. H. MacDonald, Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers, Phys. Rev. Lett. 122, 086402 (2019)

    work page 2019

  37. [37]

    B. A. Bernevig, Z.-D. Song, N. Regnault, and B. Lian, Twisted bilayer graphene. I. Matrix elements, approxi- mations, perturbation theory, and ak·ptwo-band model, Phys. Rev. B103, 205411 (2021)

    work page 2021

  38. [38]

    M. G. Scheer, K. Gu, and B. Lian, Magic angles in twisted bilayer graphene near commensuration: Towards a hy- permagic regime, Phys. Rev. B106, 115418 (2022)

    work page 2022

  39. [39]

    Ma, Y.-G

    D. Ma, Y.-G. Chen, Y. Yu, and X. Luo, Moir´ e semicon- ductors on the twisted bilayer dice lattice, Phys. Rev. B 109, 155159 (2024)

    work page 2024

  40. [40]

    Zhou, Y.-C

    X. Zhou, Y.-C. Hung, B. Wang, and A. Bansil, Gen- eration of Isolated Flat Bands with Tunable Numbers through Moir´ e Engineering, Phys. Rev. Lett.133, 236401 (2024)

    work page 2024

  41. [41]

    C˘ alug˘ aru, Y

    D. C˘ alug˘ aru, Y. Jiang, H. Hu, H. Pi, J. Yu, M. G. Vergniory, J. Shan, C. Felser, L. M. Schoop, D. K. Efetov, K. F. Mak, and B. A. Bernevig, Moir´ e materials based on M-point twisting, Nature643, 376 (2025)

    work page 2025

  42. [42]

    We include Refs

    See the Supplemental Material where we provide a sum- mary of the Hamiltonian and electronic structure for the monolayer kagome system, show how the BM model can be generalised to TBK, demonstrate approxiamte particle-hole symmetry, provide additional band struc- tures and sublattice projection plots, discuss the rapid changes inv ∗ F that only appear whe...

    work page

  43. [43]

    A. N. Mihalyuk, D. V. Gruznev, L. V. Bondarenko, A. Y. Tupchuya, Y. E. Vekovshinin, S. V. Eremeev, A. V. Zo- tov, and A. A. Saranin, A 2D heavy fermion CePb 3 kagome material on silicon: emergence of unique spin po- larized states for spintronics, Nanoscale14, 14732 (2022)

    work page 2022

  44. [44]

    Y. E. Vekovshinin, L. V. Bondarenko, A. Y. Tupchaya, A. N. Mihalyuk, N. V. Denisov, D. V. Gruznev, A. V. Zotov, and A. A. Saranin, Lifshitz Transition in a Single-Atom-Thick Gd xYb1–xPb3 Kagome Compound on Si(111), Nano Lett.24, 9931 (2024)

    work page 2024

  45. [45]

    N. V. Denisov, L. V. Bondarenko, Y. E. Vekovshinin, A. N. Mihalyuk, S. V. Eremeev, D. V. Gruznev, A. V. Zo- tov, and A. A. Saranin, Magnetotransport properties of a single-atom-thick GdPb 3 kagome compound on Si(111), J. Mater. Chem. C13, 7219 (2025)

    work page 2025

  46. [46]

    Y. E. Vekovshinin, L. V. Bondarenko, A. Y. Tupchaya, T. V. Utas, E. Wang, A. N. Mihalyuk, D. V. Gruznev, A. V. Zotov, and A. A. Saranin, High-Order Van Hove Singularities in Atomically Thin Kagome Metal LaTl 3, ACS Nano19, 36510 (2025)

    work page 2025

  47. [47]

    N. V. Denisov, L. V. Bondarenko, A. Y. Tupchaya, Y. E. Vekovshinin, V. G. Kotlyar, T. V. Utas, P. V. Burkovskaya, L. O. Brykin, A. N. Mihalyuk, S. V. Ere- meev, D. V. Gruznev, A. V. Zotov, and A. A. Saranin, Magnetism in the single-atom-thick EuPb 3 kagome com- pound on Si(111) studied using in situ transport and 7 magnetotransport measurements, Phys. Rev...

    work page 2026

  48. [48]

    Crasto de Lima, R

    F. Crasto de Lima, R. H. Miwa, and E. Su´ arez Morell, Double flat bands in kagome twisted bilayers, Phys. Rev. B100, 155421 (2019)

    work page 2019

  49. [49]

    D. T. S. Perkins, Symmetry preservation in commensu- rate twisted bilayers, Phys. Rev. B112, 035410 (2025)

    work page 2025

  50. [50]

    Sheffer, R

    Y. Sheffer, R. Queiroz, and A. Stern, Symmetries as the Guiding Principle for Flattening Bands of Dirac Fermions, Phys. Rev. X13, 021012 (2023)

    work page 2023

  51. [51]

    M. L. Kiesel and R. Thomale, Sublattice interference in the kagome Hubbard model, Phys. Rev. B86, 121105 (2012)

    work page 2012

  52. [52]

    P. K. Nag, R. Batabyal, J. Ingham, N. Morali, H. Tan, J. Koo, A. Consiglio, E. Liu, N. Avraham, R. Queiroz, R. Thomale, B. Yan, C. Felser, and H. Bei- denkopf, Pomeranchuk Instability Induced by an Emer- gent Higher-Order van Hove Singularity on the Distorted Kagome Surface of Co 3Sn2S2 (2024), arXiv:2410.01994 [cond-mat.str-el]

    work page arXiv 2024

  53. [53]

    J. Beck, J. Bodky, M. D¨ urrnagel, R. Thomale, J. Ingham, L. Klebl, and H. Hohmann, Kekul´ e Order from Diffuse Nesting near Higher-Order Van Hove Points, Phys. Rev. Lett.136, 106503 (2026)

    work page 2026

  54. [54]

    J. C. Slater and G. F. Koster, Simplified LCAO Method for the Periodic Potential Problem, Phys. Rev.94, 1498 (1954)

    work page 1954

  55. [55]

    Dai and X

    J. Dai and X. C. Zeng, Bilayer phosphorene: Effect of stacking order on bandgap and its potential applications in thin-film solar cells, J. Phys. Chem. Lett.5, 1289 (2014)

    work page 2014

  56. [56]

    Uchida, S

    K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama, Atomic corrugation and electron localization due to moir´ e patterns in twisted bilayer graphenes, Phys. Rev. B90, 155451 (2014)

    work page 2014

  57. [57]

    L. L. Li, D. Moldovan, W. Xu, and F. M. Peeters, Elec- tronic properties of bilayer phosphorene quantum dots in the presence of perpendicular electric and magnetic fields, Phys. Rev. B96, 155425 (2017)

    work page 2017

  58. [58]

    L. Ye, M. Kang, J. Liu, F. von Cube, C. R. Wicker, T. Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bell, L. Fu, R. Comin, and J. G. Checkelsky, Massive Dirac fermions in a ferromagnetic kagome metal, Nature 555, 638 (2018)

    work page 2018

  59. [59]

    Shi, E.-M

    Q. Shi, E.-M. Shih, D. Rhodes, B. Kim, K. Barmak, K. Watanabe, T. Taniguchi, Z. Papi´ c, D. A. Abanin, J. Hone, and C. R. Dean, Bilayer WSe 2 as a natural platform for interlayer exciton condensates in the strong coupling limit, Nat. Nanotechnol.17, 577 (2022)

    work page 2022

  60. [60]

    Y. Luo, Y. Han, J. Liu, H. Chen, Z. Huang, L. Huai, H. Li, B. Wang, J. Shen, S. Ding, Z. Li, S. Peng, Z. Wei, Y. Miao, X. Sun, Z. Ou, Z. Xiang, M. Hashimoto, D. Lu, Y. Yao, H. Yang, X. Chen, H.-J. Gao, Z. Qiao, Z. Wang, and J. He, A unique van Hove singularity in kagome superconductor CsV3−xTaxSb5 with enhanced supercon- ductivity, Nat. Commun.14, 3819 (2023)

    work page 2023

  61. [61]

    Y. Wang, H. Wu, G. T. McCandless, J. Y. Chan, and M. N. Ali, Quantum states and intertwining phases in kagome materials, Nat. Rev. Phys.5, 635 (2023)

    work page 2023

  62. [62]

    TWISTED KAGOME BILA YERS: HIGHER-ORDER MAGIC ANGLES, TOPOLOGICAL FLA T BANDS, AND SUBLA TTICE INTERFERENCE

    Y. Xia, Z. Han, K. Watanabe, T. Taniguchi, J. Shan, and K. F. Mak, Superconductivity in twisted bilayer WSe 2, Nature637, 833 (2025). 1 SUPPLEMENT AL MA TERIAL FOR “TWISTED KAGOME BILA YERS: HIGHER-ORDER MAGIC ANGLES, TOPOLOGICAL FLA T BANDS, AND SUBLA TTICE INTERFERENCE” S1. MONOLA YER KAGOME The kagome lattice is the line graph of honeycomb lattice and ...

    work page 2025

  63. [63]

    In this section, we shall focus on deriving the matrices describing tunnelling between the two layers of this twisted heterostrcture due to its differences compared with TBG. S2.1. General T unnelling Between Twisted Kagome Layers To construct the Hamiltonian for TBK near 1/3 filling, we will construct a generalised Bistritzer-MacDonald (BM) model in term...

    work page

  64. [64]

    (S35) (black) and ourp z-oribtal model in Eq

    summarised in Eq. (S35) (black) and ourp z-oribtal model in Eq. (S37). We taked ⊥ = 0.6596nm here. (b): Illustrating how our Eq. (S37) depends upon the choice ofd ⊥ with ˜γ= 12/a. justify this on the basis that next-nearest-neighbour intralayer tunnelling is ignored when it is 10% oft 0. Nonetheless, changing these parameters and the tunnelling model can ...

    work page

  65. [65]

    = 16π 3a sin θ 2 0 1 =−2q b,(S39a) qbr = (K ¯ϑ + +G ¯ϑ 3)−(K ϑ + +G ϑ

    work page

  66. [66]

    = 8π 3a sin θ 2 √ 3 −1 =−2q tl,(S39b) qbl = (K ¯ϑ + +G ¯ϑ 4)−(K ϑ + +G ϑ

    work page

  67. [67]

    = 8π 3a sin θ 2 − √ 3 −1 =−2q tr.(S39c) Returning to theq-lattice depiction of momentum conserving interlayer tunnelling processes, we see that theD 2 Dirac points connect third-nearest-neighbours in theq-lattice, see Fig. S1b. At the level of the continuum Hamiltonian, the effect of theD 2 Dirac points can only be seen when including shells beyond the se...

    work page