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User:EverettYou/Free Fermion SPT

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Clifford Algebra

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p\q 0 1 2 3 4 5 6 7
0 R C H 2H H(2) C(4) R(8) 2R(8)
1 2R R(2) C(2) H(2) 2H(2) H(4) C(8) R(16)
2 R(2) 2R(2) R(4) C(4) H(4) 2H(4) H(8) C(16)
3 C(2) R(4) 2R(4) R(8) C(8) H(8) 2H(8) H(16)
4 H(2) C(4) R(8) 2R(8) R(16) C(16) H(16) 2H(16)
5 2H(2) H(4) C(8) R(16) 2R(16) R(32) C(32) H(32)
6 H(4) 2H(4) H(8) C(16) R(32) 2R(32) R(64) C(64)
7 C(8) H(8) 2H(8) H(16) C(32) R(64) 2R(64) R(128)

Fermion σ Models

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Dirac Models

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General form of the Dirac Hamiltonian

and are Hermitian matrices that square to 1.

Possible realizations:

Majorana Models

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General form of the Majorana Hamiltonian

and are Hermitian matrices that square to 1. Additional requirement from Majorana fermion: must be transpose symmetric and must be transpose antisymmetric.

Possible realizations:

  • Chiral Majorana edge state of p+ip superconductor
  • Bulk p+ip superconductor
  • He3 B Phase

Topological Numbers of Bulk Fermion

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Classifications

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The classification table of complex fermion SPT.

Cartan Symmetry d=
TRS PHS SLS 0 1 ...
A × × × Z 0 ...
AIII × × 0 Z ...

The classification table of real fermion SPT.

Cartan Symmetry d=
TRS PHS SLS 0 1 2 3 4 5 6 7 ...
D × +1 × Z2 Z2 Z 0 0 0 Z 0 ...
DIII −1 +1 0 Z2 Z2 Z 0 0 0 Z ...
AII −1 × × Z 0 Z2 Z2 Z 0 0 0 ...
CII −1 −1 0 Z 0 Z2 Z2 Z 0 0 ...
C × −1 × 0 0 Z 0 Z2 Z2 Z 0 ...
CI +1 −1 0 0 0 Z 0 Z2 Z2 Z ...
AI +1 × × Z 0 0 0 Z 0 Z2 Z2 ...
BDI +1 +1 Z2 Z 0 0 0 Z 0 Z2 ...

References

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  • Alexei Kitaev. Periodic table for topological insulators and superconductors. arXiv:0901.2686
  • Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig. Topological insulators and superconductors: ten-fold way and dimensional hierarchy. arXiv:0912.2157
  • Michael Stone, Ching-Kai Chiu, Abhishek Roy. Symmetries, Dimensions, and Topological Insulators: the mechanism behind the face of the Bott clock. arXiv:1005.3213