Research ArticlePHYSICS

Structure and topology of band structures in the 1651 magnetic space groups

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Science Advances  03 Aug 2018:
Vol. 4, no. 8, eaat8685
DOI: 10.1126/sciadv.aat8685
  • Fig. 1 Magnetic reSM.

    (A) Example tight-binding model for reSM. There are two sublattices per unit cell (shaded) due to an antiferromagnetic order mx, producing a total of four bands. On-site potential J stands for the exchange coupling Embedded Image. In addition to the standard nearest-neighbor hopping t, a spin-dependent hopping ± tJσx is included, which can be viewed as originating from an exchange coupling to a magnetic moment in the middle of the nearest-neighbor bonds (section S8). (B) Dispersion relation at J/t = 1 and tJ/t = 1/4. In this case, η in Eq. 4 is +1, and the dispersion is gapped between the second and the third bands. (C) Dispersion relation at J/t = 1 and tJ/t = 3/4. Now, η = −1, and a pair of Dirac nodes exist as predicted. (D) Fermi surface of the 3D version of the reSM. The two Weyl points on the k3 = 0 (k3 = π) plane have the chirality +1 (–1), indicating a huge Fermi arc on some 2D surfaces. The signs on the TRIMs indicate the product of the C2z rotation eigenvalues of occupied bands in this model.

  • Fig. 2 Magnetic feSMs.

    (A and B) Symmetries of a magnetic order can prohibit AIs at odd-site fillings. (A) The magnetic point group symmetries of a ferromagnetic arrangement are compatible with nondegenerate local energy levels. (B) However, those of the depicted hedgehog defect force all the energy levels to exhibit even degeneracies, which forbids AIs when a lone electron is localized to the purple site. (C) When hedgehog and antihedgehog defects are arranged into a diamond lattice, the previous argument suggests that no AI is allowed whenever the site fillings at the defect cores are odd. (D) The hypothetical magnetic structure in (C) could be realizable in spinel structures if the diamond sites are occupied by atoms with odd atomic numbers (purple) and the magnetic atoms (blue) at the pyrochlore sites exhibit an all-in-all-out magnetic order. (E to H) feSMs arising from the magnetically ordered Fu-Kane-Mele model (Eq. 6). (E) When the magnetically modulated hopping tJ is weak compared to the spin-orbit coupling λ, the fermiology is governed by rings of gap closing (circled in red), growing out from the original Dirac points at X when tJ = 0. (F) The positions of the rings in the Brillouin zone are shown in red. (G and H) For Embedded Image, the nodal rings become connected at the momentum W. Thin lines indicate copies of the gapless momenta in the repeated zone scheme, included to illustrate the connectivity of the rings.

  • Fig. 3 Symmetry indicators for antiferromagnetic topological insulators.

    (A) Example of the inversion parity combination of valence bands in the Embedded Image nontrivial phase of MSG 2.7. (B) Realization of the Embedded Image nontrivial phase by staggered stacking of Chern insulators. The red (blue) disks represent a Chern insulator with C = +1 (C = −1). (C) Breaking the Embedded Image symmetry (the half translation in z, followed by the TR) leads to a “higher-order” state with a 1D equatorial chiral mode on the surface (6271).

  • Table 1 Characterization of BSs in a MSG (excerpt from tables S1 to S6).
    MSG*dXBSνBS§
    2.4I9(2, 2, 2, 4)1
    2.7IV5(2)2
    3.4IV3(2)2
    209.51IV3(1)2a

    *MSG number in the BNS notation, followed by a Roman numeral I, …, IV indicating its type.

    †Number of linearly independent BSs.

    ‡Symmetry-based indicator of band topology, which takes the form Embedded Image, denoted by the collection of positive integers (n1, n2, ⋯).

    §For most of the MSGs, the set of physical BS fillings {ν}BS and the set of AI fillings {ν}AI agree with each other, and they take the form Embedded Image. The superscript letter a indicates violation to this rule, as detailed in table S8.

    • Table 2 Characterization of MLGs through the corresponding MSG (excerpt from table S7).
      MSG*dXBSvBS
      2.5 (1)II5(2)2
      2.5 (2)II5(2)2
      2.5 (3)II5(2)2
      3.4 (1)IV3(2)2
      3.4 (2)IV2(1)2

      *The numbers in parentheses label the different ways to project the MSG down to 2D planes (section S7).

      †Defined as in Table 1.

      Supplementary Materials

      • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/8/eaat8685/DC1

        Section S1. Tables for spinful electrons

        Section S2. feSMs in stoichiometric compounds

        Section S3. Notations

        Section S4. Compatibility relations

        Section S5. Details of AIs

        Section S6. Details of Formulab

        Section S7. Magnetic layer groups

        Section S8. Tight-binding model for reSM

        Section S9. Tables for spinless electrons

        Table S1. Characterization of MSGs in the triclinic family for spinful electrons.

        Table S2. Characterization of MSGs in the monoclinic family for spinful electrons.

        Table S3. Characterization of MSGs in the orthorhombic family for spinful electrons.

        Table S4. Characterization of MSGs in the tetragonal family for spinful electrons.

        Table S5. Characterization of MSGs in the hexagonal family for spinful electrons.

        Table S6. Characterization of MSGs in the cubic family for spinful electrons.

        Table S7. Characterization of the projections of MSGs that correspond to MLG, assuming spinful electrons.

        Table S8. MSGs for which spinful electrons exhibit exceptional filling patterns.

        Table S9. MSGs that can host feSMs with nodal-point Fermi surfaces pinned at high-symmetry momenta.

        Table S10. MSGs that can host feSMs with movable nodal Fermi surfaces.

        Table S11. MSGs that can host filling-enforced metals.

        Table S12. Effect of turning on spin-orbit coupling on the fermiology of TR-symmetric filling-enforced (semi)metals.

        Table S13. Characterization of MSGs in the triclinic family for spinless electrons.

        Table S14. Characterization of MSGs in the monoclinic family for spinless electrons.

        Table S15. Characterization of MSGs in the orthorhombic family for spinless electrons.

        Table S16. Characterization of MSGs in the tetragonal family for spinless electrons.

        Table S17. Characterization of MSGs in the hexagonal family for spinless electrons.

        Table S18. Characterization of MSGs in the cubic family for spinless electrons.

        Table S19. MSGs for which spinless electrons exhibit exceptional filling patterns.

      • Supplementary Materials

        This PDF file includes:

        • Section S1. Tables for spinful electrons
        • Section S2. feSMs in stoichiometric compounds
        • Section S3. Notations
        • Section S4. Compatibility relations
        • Section S5. Details of AIs
        • Section S6. Details of T~b
        • Section S7. Magnetic layer groups
        • Section S8. Tight-binding model for reSM
        • Section S9. Tables for spinless electrons
        • Table S1. Characterization of MSGs in the triclinic family for spinful electrons.
        • Table S2. Characterization of MSGs in the monoclinic family for spinful electrons.
        • Table S3. Characterization of MSGs in the orthorhombic family for spinful electrons.
        • Table S4. Characterization of MSGs in the tetragonal family for spinful electrons.
        • Table S5. Characterization of MSGs in the hexagonal family for spinful electrons.
        • Table S6. Characterization of MSGs in the cubic family for spinful electrons.
        • Table S7. Characterization of the projections of MSGs that correspond to MLG, assuming spinful electrons.
        • Table S8. MSGs for which spinful electrons exhibit exceptional filling patterns.
        • Table S9. MSGs that can host feSMs with nodal-point Fermi surfaces pinned at high-symmetry momenta.
        • Table S10. MSGs that can host feSMs with movable nodal Fermi surfaces.
        • Table S11. MSGs that can host filling-enforced metals.
        • Table S12. Effect of turning on spin-orbit coupling on the fermiology of TR-symmetric filling-enforced (semi)metals.
        • Table S13. Characterization of MSGs in the triclinic family for spinless electrons.
        • Table S14. Characterization of MSGs in the monoclinic family for spinless electrons.
        • Table S15. Characterization of MSGs in the orthorhombic family for spinless electrons.
        • Table S16. Characterization of MSGs in the tetragonal family for spinless electrons.
        • Table S17. Characterization of MSGs in the hexagonal family for spinless electrons.
        • Table S18. Characterization of MSGs in the cubic family for spinless electrons.
        • Table S19. MSGs for which spinless electrons exhibit exceptional filling patterns.

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