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2D Theoretically Twistable Material Database

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arxiv 2411.09741 v1 pith:NWDTG5VZ submitted 2024-11-14 cond-mat.mtrl-sci cond-mat.mes-hall

Yi Jiang , Urko Petralanda , Grigorii Skorupskii , Qiaoling Xu , Hanqi Pi , Dumitru Călugăru , Haoyu Hu , Jiaze Xie
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Rose Albu Mustaf Peter Höhn Vicky Haase Maia G. Vergniory Martin Claassen Luis Elcoro Nicolas Regnault Jie Shan Kin Fai Mak Dmitri K. Efetov Emilia Morosan Dante M. Kennes Angel Rubio Lede Xian Claudia Felser Leslie M. Schoop B. Andrei Bernevig
This is my paper · ORCID
classification cond-mat.mtrl-scicond-mat.mes-hall
keywords materialsmoirtwistedtwistablestructuressystemstheoreticallytopological
verification ladder T0 review T1 audit T2 compute T3 formal

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The study of twisted two-dimensional (2D) materials, where twisting layers create moir\'e superlattices, has opened new opportunities for investigating topological phases and strongly correlated physics. While systems such as twisted bilayer graphene (TBG) and twisted transition metal dichalcogenides (TMDs) have been extensively studied, the broader potential of a seemingly infinite set of other twistable 2D materials remains largely unexplored. In this paper, we define "theoretically twistable materials" as single- or multi-layer structures that allow for the construction of simple continuum models of their moir\'e structures. This excludes, for example, materials with a "spaghetti" of bands or those with numerous crossing points at the Fermi level, for which theoretical moir\'e modeling is unfeasible. We present a high-throughput algorithm that systematically searches for theoretically twistable semimetals and insulators based on the Topological 2D Materials Database. By analyzing key electronic properties, we identify thousands of new candidate materials that could host rich topological and strongly correlated phenomena when twisted. We propose representative twistable materials for realizing different types of moir\'e systems, including materials with different Bravais lattices, valleys, and strength of spin-orbital coupling. We provide examples of crystal growth for several of these materials and showcase twisted bilayer band structures along with simplified twisted continuum models. Our results significantly broaden the scope of moir\'e heterostructures and provide a valuable resource for future experimental and theoretical studies on novel moir\'e systems.

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Forward citations

Cited by 8 Pith papers

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

  1. Organizing Principles for Moir\'e Quantum Matter

    cond-mat.mtrl-sci 2026-07 accept novelty 7.0 of 10

    Parent valley momentum, orbital content and moiré symmetry jointly organize emergent flat-band Hubbard, topological and quasi-1D models across all 2D lattice classes.

  2. Robustness of real-space topology in moir\'e systems

    cond-mat.mes-hall 2025-06 conditional novelty 7.0 of 10

    The real-space Chern number of ensembles of Bloch states is robust and symmetry-forced to be nonzero in twisted TMDs and twisted bilayer graphene.

  3. Catalog of phonon emergent particles

    cond-mat.mtrl-sci 2024-11 conditional novelty 7.0 of 10

    A symmetry-only mapping from atomic Wyckoff positions to phonon irreps shows that nearly all emergent particles at high-symmetry points are enforced by space-group symmetry, yielding a catalog of 20,516,167 phonon eme...

  4. Extended s-wave superconductivity in M-point twisted bilayer SnSe2

    cond-mat.supr-con 2026-07 conditional novelty 6.0 of 10

    FRG simulations predict that AB-stacked twisted bilayer SnSe2 hosts spin-fluctuation-mediated extended s-wave superconductivity upon doping an antiferromagnetic state at half filling.

  5. A Catalogue of Topological Moir\'{e} Bands in Twisted Semiconductors

    cond-mat.mtrl-sci 2026-07 conditional novelty 6.0 of 10

    Parent valley character plus stacking symmetry, not chemistry, organizes bandwidth scaling and topological bands across more than 1,000 twisted semiconductor moiré structures.

  6. Helical Domain-Wall-Ring Networks Reshape Superconducting Correlations

    cond-mat.mes-hall 2026-06 unverdicted novelty 6.0 of 10

    In helical domain-wall-ring networks, self-consistent finite-size calculations show inter-ring phase locking is strongly suppressed even where infinite-size RG predicts strong coupling, while the SC scaling dimension ...

  7. Emergent Interacting Phases in the Strong Coupling Limit of Twisted M-Valley Moir\'e Systems: Application to SnSe${}_2$

    cond-mat.str-el 2025-08 conditional novelty 6.0 of 10

    Twisted SnSe2 realizes quasi-1D triangular (AA) and kagome (AB) interacting models with predicted dimer, valence-bond-solid, and frustrated spin-liquid phases.

  8. Movable Dirac Points with Ferroelectrics: Kink States and Berry Curvature Dipoles

    cond-mat.mes-hall 2025-06 conditional novelty 6.0 of 10

    Movable Dirac points in ferroelectric 2D materials control topological kink-state conductance and switch the Berry-curvature-dipole-induced second-harmonic Hall conductivity.

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Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 8 Pith papers

  1. [1]

    S1: 1.2.104

    Γ-SOC, computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2Hf2_SOC, SEBR FIG. S1: 1.2.104

    work page

  2. [2]

    S2: 1.3.213

    Γ-SOC, computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2Zr2_SOC, SEBR FIG. S2: 1.3.213

    work page

  3. [3]

    S3: 5.1.3 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Si2_SOC, SEBR FIG

    K, experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 80, C2_SOC, AccidentalFermi FIG. S3: 5.1.3 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Si2_SOC, SEBR FIG. S4: 5.1.2 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ge2_SOC, SEBR FIG. S5: 1.1.15

    work page

  4. [4]

    HSP TABLE S81: Experimental materials with valley type: rectangular- HSP

    Rectangular lattice a. HSP TABLE S81: Experimental materials with valley type: rectangular- HSP. Formula LG ID Gap Database Lattice VBM valley CBM valley Twist scoreTopology Bulk Mat. type P 42 3.1.7 0.91 C2DB rectangular Γ Γ 0.62 OAI Yes Exp.M.Exfo ZrS3 46 6.1.6 1.18 C2DB rectangular / Γ 0.41 LCEBR Yes Exp.W.Exfo ZrSe3 46 6.1.7 0.41 C2DB rectangular Γ / ...

    work page

  5. [5]

    HSP TABLE S95: Computationally exfoliable materials with valley type: oblique-HSP

    Oblique lattice a. HSP TABLE S95: Computationally exfoliable materials with valley type: oblique-HSP. Formula LG ID Gap Database Lattice VBM valley CBM valley Twist scoreTopology Bulk Mat. type Na2TiH4O5 2 6.2.125 1.69 MC2D oblique Γ B 0.43 LCEBR Yes Comp.Exfo CuI 2 6.2.97 1.98 MC2D oblique Γ Γ 0.30 LCEBR Yes Comp.Exfo VH2O3 2 3.2.50 0.40 MC2D oblique / Γ...

    work page

  6. [6]

    S6: 1.1.17

    K-SOC, experimental Band plots of twistable semimetals GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Sn2_SOC, SEBR FIG. S6: 1.1.17

    work page

  7. [7]

    S7: 1.2.121 B

    K-SOC, computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Hg2Pt4Se6_SOC, SEBR FIG. S7: 1.2.121 B. Square lattice C. Rectangular lattice

    work page

  8. [8]

    S8: 1.2.29

    nHSP , computationally exfoliable Y1 GM Y S4 3 2 1 0 1 2 3 4 E Ef [eV] LG 18, Hg3S2_SOC, SEBR FIG. S8: 1.2.29

    work page

Show all 47 references
  1. [9]

    S9: 1.3.24 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, Mo2S4_SOC, NLC FIG

    nHSP-SOC, experimental Band plots of twistable semimetals GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, S4W2_SOC, NLC FIG. S9: 1.3.24 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, Mo2S4_SOC, NLC FIG. S10: 1.1.1

  2. [10]

    S11: 6.2.307 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 17, Hg8O4_SOC, NLC FIG

    nHSP-SOC, computationally exfoliable Y1 GM Y S4 3 2 1 0 1 2 3 4 E Ef [eV] LG 10, Hg4O2_SOC, LCEBR FIG. S11: 6.2.307 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 17, Hg8O4_SOC, NLC FIG. S12: 1.2.25 Band plots of twistable insulators IX. BAND PLOTS OF TWISTABLE INSULA TORS A. Hexag...

  3. [11]

    S13: 6.1.78 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ge2H2_SOC, OAI FIG

    Γ, experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, GaN_SOC, LCEBR FIG. S13: 6.1.78 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ge2H2_SOC, OAI FIG. S14: 3.1.18 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, PbI2_SOC, LCEBR FIG. S15: 6.1.54 GM K M GM4 3 2 1 0 1 2 3 4 E Ef ...

    1963

  4. [12]

    S23: 6.2.1129 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, STl2_SOC, LCEBR FIG

    Γ, computationally exfoliable Band plots of twistable insulators GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, PbT e_SOC, LCEBR FIG. S23: 6.2.1129 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, STl2_SOC, LCEBR FIG. S24: 6.2.1302 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, NaPSn_SOC...

  5. [13]

    S33: 6.3.2619 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2Hg_SOC, LCEBR FIG

    Γ, computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, BrCu_SOC, LCEBR FIG. S33: 6.3.2619 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2Hg_SOC, LCEBR FIG. S34: 6.3.2320 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ag2I2_SOC, LCEBR FIG. S35: 6.1.17 GM K M GM4 3 2...

  6. [14]

    S43: 6.1.13 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Sb2_SOC, OAI FIG

    Γ-SOC, experimental Band plots of twistable insulators GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, BiIT e_SOC, LCEBR FIG. S43: 6.1.13 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Sb2_SOC, OAI FIG. S44: 3.1.25 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, As2_SOC, OAI FIG. S45: 3....

    1963

  7. [15]

    S49: 6.2.1108 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, PbT e_SOC, LCEBR FIG

    Γ-SOC, computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, GeT e_SOC, LCEBR FIG. S49: 6.2.1108 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, PbT e_SOC, LCEBR FIG. S50: 6.2.1129 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, AsSb_SOC, LCEBR FIG. S51: 6.2.1090

  8. [16]

    S52: 6.1.41 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ag2I2_SOC, LCEBR FIG

    Γ-SOC, computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Cu2I2_SOC, LCEBR FIG. S52: 6.1.41 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ag2I2_SOC, LCEBR FIG. S53: 6.1.17 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 71, Al2Si2T e6_SOC, OAI FIG. S54: 3.3.304 GM K M GM...

  9. [17]

    S56: 6.3.2323 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Cl2N2Zr2_SOC, LCEBR FIG

    K, experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2N2Zr2_SOC, LCEBR FIG. S56: 6.3.2323 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Cl2N2Zr2_SOC, LCEBR FIG. S57: 6.3.2408 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, MoS2_SOC, OAI FIG. S58: 3.1.39 GM K M GM4 3 2 1 0 1...

  10. [18]

    S63: 6.2.1287 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 66, P2S6Sn2_SOC, OAI FIG

    K, computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, O6PbSe2_SOC, LCEBR FIG. S63: 6.2.1287 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 66, P2S6Sn2_SOC, OAI FIG. S64: 3.2.152

  11. [19]

    S65: 3.3.495 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 71, P2S6Sn2_SOC, OAI FIG

    K, computationally stable Band plots of twistable insulators GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, Cl2Zr_SOC, OAI FIG. S65: 3.3.495 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 71, P2S6Sn2_SOC, OAI FIG. S66: 3.3.356 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Br2Hf2N2_SOC, LC...

  12. [20]

    S73: 3.1.46 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, WS2_SOC, OAI FIG

    K-SOC, experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, WSe2_SOC, OAI FIG. S73: 3.1.46 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, WS2_SOC, OAI FIG. S74: 3.1.45 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, MoS2_SOC, OAI FIG. S75: 3.1.39 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [...

  13. [21]

    S80: 6.2.1106 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, BrNZr_SOC, LCEBR FIG

    K-SOC, computationally exfoliable Band plots of twistable insulators GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, Ga2Se2_SOC, LCEBR FIG. S80: 6.2.1106 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, BrNZr_SOC, LCEBR FIG. S81: 6.2.1099

  14. [22]

    S82: 3.1.33

    K-SOC, computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, Ga2Se2_SOC, OAI FIG. S82: 3.1.33

  15. [23]

    S83: 6.1.75 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, HfS2_SOC, LCEBR FIG

    M , experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, SnS2_SOC, LCEBR FIG. S83: 6.1.75 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, HfS2_SOC, LCEBR FIG. S84: 6.1.43 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Se2Sn_SOC, LCEBR FIG. S85: 6.1.71 GM K M GM4 3 2 1 0 1 2 3 4 E...

  16. [24]

    S91: 6.2.1302 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, CCl2Lu2_SOC, LCEBR FIG

    M , computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, STl2_SOC, LCEBR FIG. S91: 6.2.1302 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, CCl2Lu2_SOC, LCEBR FIG. S92: 6.2.1199 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 80, C6B2_SOC, OAI FIG. S93: 3.2.206 GM K M GM...

  17. [25]

    S101: 6.3.2510 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, CCl2Sc2_SOC, LCEBR FIG

    M , computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, I2Zn_SOC, LCEBR FIG. S101: 6.3.2510 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, CCl2Sc2_SOC, LCEBR FIG. S102: 6.3.2355 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, In2Se3_SOC, LCEBR FIG. S103: 6.3.2527

  18. [26]

    S104: 3.1.35

    M -SOC, experimental Band plots of twistable insulators GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 78, Ga2T e2_SOC, OAI FIG. S104: 3.1.35

  19. [27]

    S105: 3.1.25 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, As2_SOC, OAI FIG

    nHSP , experimental GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Sb2_SOC, OAI FIG. S105: 3.1.25 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, As2_SOC, OAI FIG. S106: 3.1.11 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, P2_SOC, OAI FIG. S107: 3.1.23 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [...

  20. [28]

    S113: 6.2.1108 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 66, As2S6SnTl2_SOC, LCEBR FIG

    nHSP , computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, GeT e_SOC, LCEBR FIG. S113: 6.2.1108 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 66, As2S6SnTl2_SOC, LCEBR FIG. S114: 6.2.1065 Band plots of twistable insulators

  21. [29]

    S115: 6.3.2546 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ga2S2_SOC, OAI FIG

    nHSP , computationally stable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, O2Pt_SOC, LCEBR FIG. S115: 6.3.2546 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, Ga2S2_SOC, OAI FIG. S116: 3.3.432 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 72, NiO2_SOC, LCEBR FIG. S117: 6.3.2540 GM K M GM...

  22. [30]

    S120: 6.2.1090 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, NaPSn_SOC, LCEBR FIG

    nHSP-SOC, computationally exfoliable GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, AsSb_SOC, LCEBR FIG. S120: 6.2.1090 GM K M GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 69, NaPSn_SOC, LCEBR FIG. S121: 6.2.1128 B. Square lattice

  23. [31]

    S122: 6.2.944 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 61, Ba3Br2In2O5_SOC, LCEBR FIG

    Γ, computationally exfoliable Band plots of twistable insulators X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 61, Ag4Cs4O4_SOC, LCEBR FIG. S122: 6.2.944 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 61, Ba3Br2In2O5_SOC, LCEBR FIG. S123: 6.2.946 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 61, C4...

  24. [32]

    S132: 6.3.1738 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, S2Sn_SOC, LCEBR FIG

    Γ, computationally stable X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Br2Cu2_SOC, LCEBR FIG. S132: 6.3.1738 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, S2Sn_SOC, LCEBR FIG. S133: 6.3.1685 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Ag2I2_SOC, LCEBR FIG. S134: 6.3.1724 X GM M X4 3...

  25. [33]

    S139: 6.2.970 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Cu2Na2T e2_SOC, LCEBR FIG

    Γ-SOC, computationally exfoliable Band plots of twistable insulators X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Ag2K2T e2_SOC, LCEBR FIG. S139: 6.2.970 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Cu2Na2T e2_SOC, LCEBR FIG. S140: 6.2.1008

  26. [34]

    S141: 6.3.1724 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Cu2I2_SOC, LCEBR FIG

    Γ-SOC, computationally stable X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Ag2I2_SOC, LCEBR FIG. S141: 6.3.1724 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Cu2I2_SOC, LCEBR FIG. S142: 6.3.1761 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, HgI2_SOC, LCEBR FIG. S143: 6.3.1669

  27. [35]

    S144: 6.2.956 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 57, O4PTl_SOC, LCEBR FIG

    M , computationally exfoliable X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 61, F2Zn_SOC, LCEBR FIG. S144: 6.2.956 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 57, O4PTl_SOC, LCEBR FIG. S145: 6.2.931

  28. [36]

    S146: 6.3.1660 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, S2Sn_SOC, LCEBR FIG

    X, computationally stable Band plots of twistable insulators X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, GeS2_SOC, LCEBR FIG. S146: 6.3.1660 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 59, S2Sn_SOC, LCEBR FIG. S147: 6.3.1685

  29. [37]

    S148: 6.2.1010

    nHSP , computationally exfoliable X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Cu2Se2Tl2_SOC, LCEBR FIG. S148: 6.2.1010

  30. [38]

    S149: 6.3.1807 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Pb2T e2_SOC, LCEBR FIG

    nHSP , computationally stable X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Pb2S2_SOC, LCEBR FIG. S149: 6.3.1807 X GM M X4 3 2 1 0 1 2 3 4 E Ef [eV] LG 64, Pb2T e2_SOC, LCEBR FIG. S150: 6.3.1809 C. Rectangular lattice

  31. [39]

    S151: 3.1.7 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, S6Zr2_SOC, LCEBR FIG

    HSP , experimental Band plots of twistable insulators GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 42, P4_SOC, OAI FIG. S151: 3.1.7 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, S6Zr2_SOC, LCEBR FIG. S152: 6.1.6

  32. [40]

    S153: 6.2.416 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, C2N4Pb2_SOC, LCEBR FIG

    HSP , computationally exfoliable GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, K2O2Tl2_SOC, LCEBR FIG. S153: 6.2.416 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, C2N4Pb2_SOC, LCEBR FIG. S154: 6.2.365 GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 15, S6Tl4Zr2_SOC, LCEBR FIG. S155:...

  33. [41]

    S163: 6.3.580 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, Br2In2O2_SOC, LCEBR FIG

    HSP , computationally stable Band plots of twistable insulators GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 14, Au2Se2_SOC, LCEBR FIG. S163: 6.3.580 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, Br2In2O2_SOC, LCEBR FIG. S164: 6.3.1439 Y1 GM Y S4 3 2 1 0 1 2 3 4 E Ef [eV] LG 22, ...

  34. [42]

    S173: 6.1.1

    nHSP , experimental GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 32, Ge2Se2_SOC, LCEBR FIG. S173: 6.1.1

  35. [43]

    S174: 3.2.134 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, N4_SOC, OAI FIG

    nHSP , computationally exfoliable Band plots of twistable insulators Y1 GM Y S4 3 2 1 0 1 2 3 4 E Ef [eV] LG 47, Hg4I2O2_SOC, OAI FIG. S174: 3.2.134 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 46, N4_SOC, OAI FIG. S175: 3.2.132 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 40, Hg4O4...

  36. [44]

    S181: 6.3.580 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 42, As4_SOC, OAI FIG

    nHSP , computationally stable GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 14, Au2Se2_SOC, LCEBR FIG. S181: 6.3.580 GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 42, As4_SOC, OAI FIG. S182: 3.3.152

  37. [45]

    S183: 6.1.2 Band plots of twistable insulators

    nHSP-SOC, experimental GM X S Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 32, S2Sn2_SOC, LCEBR FIG. S183: 6.1.2 Band plots of twistable insulators

  38. [46]

    S184: 6.2.313 D

    nHSP-SOC, computationally exfoliable GM Y S X GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 11, PbS2Sn_SOC, LCEBR FIG. S184: 6.2.313 D. Oblique lattice

  39. [47]

    S185: 6.2.125 GM B A Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 1, AgNO2_SOC, LCEBR FIG

    HSP , computationally exfoliable GM B A Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 2, H4Na2O5Ti_SOC, LCEBR FIG. S185: 6.2.125 GM B A Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 1, AgNO2_SOC, LCEBR FIG. S186: 6.2.3 GM B A Y GM4 3 2 1 0 1 2 3 4 E Ef [eV] LG 2, I2Sb2T e2_SOC, LCEBR FIG. S187: 6.2.148 96

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