Research ArticleCONDENSED MATTER PHYSICS

Refined symmetry indicators for topological superconductors in all space groups

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Science Advances  01 May 2020:
Vol. 6, no. 18, eaaz8367
DOI: 10.1126/sciadv.aaz8367
  • Fig. 1 Illustration of the equivalence relation in Eq. 21.

    Electron-like states are colored in red, and hole-like states are colored in blue. States outside of the dashed box represent trivial DOFs included in the deformation process. See the “Refined symmetry indicators for superconductors” section for the definition of vectors in this figure.

  • Fig. 2 Examples of wire construction for P4/m realizing higher-order TSCs with Majorana corner modes (circled by green ellipses).

    (A) (νx1D,z8)=(1,4) phase in 2D. Orange marks represent translation-breaking perturbations on the edge, gapping out pairs of zero modes. (B) (z8, kz = 0, z8, kz = π, z16) = (4,4,8) phase in 3D.

  • Fig. 3

    Quasiparticle spectrum (top), symmetry representations of E < 0 states (middle), and the surface band structure (bottom) of HkBdG with Eqs. 72 and 73. In (A), each bands are doubly degenerate due to the inversion and the time-reversal symmetry. For (B) and (C), inversion-breaking perturbations in Eq. 76 are included. Symmetry representations colored in red is for HkBdG, and those in blue is for Hvac. In the surface band structure, states localized near edges are colored in red.

  • Table 1 Settings used in the calculation for refined SIs and the corresponding AZ symmetry classes.

    SpinTime-reversal symmetryAZ classes
    Spinful−1DIII
    Spinless+1BDI, CI
    Spinful0D
    Spinless0D, C
  • Table 2 List of XBdG for class DIII systems with P1̄ (20), P4, and P4/m symmetry in each spatial dimension.

    SG (rep of Δk)1D2D3D
    P1̄ (Au)2(ℤ2)2 × ℤ4(ℤ2)3 × (ℤ4)3 × ℤ8
    P4 (B)122
    P4/m (Au)(ℤ2)22 × ℤ8(ℤ2)4 × ℤ4 × ℤ8 × ℤ16
    P4/m (Bu)12 × ℤ82 × (ℤ4)2 × ℤ8
    P4/m (Bg)1(ℤ2)2(ℤ2)3
  • Table 3 List of XWTSC for class DIII systems with P1̄, P4, and P4/m symmetry in each spatial dimension.

    List of XWTSC for class DIII systems with P1̄, P4, and P4/m symmetry in each spatial dimension..

    SG (rep of Δk)1D2D3D
    P1̄ (Au)2(ℤ2)3(ℤ2)7
    P4 (B)1(ℤ2)2(ℤ2)2
    P4/m (Au)(ℤ2)2(ℤ2)2(ℤ2)7
    P4/m (Bu)1(ℤ2)2(ℤ2)3
    P4/m (Bg)122

Supplementary Materials

  • Supplementary Materials

    Refined symmetry indicators for topological superconductors in all space groups

    Seishiro Ono, Hoi Chun Po, Haruki Watanabe

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    • Sections S1 to S5
    • Tables S1 to S9

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