User:EverettYou/Free Fermion SPT

Clifford Algebra

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

Dirac Models

General form of the Dirac Hamiltonian

$H=c^{\dagger }(i\partial _{i}\alpha ^{i}+n_{j}\beta ^{j})c.$

$\alpha _{i}$ and $\beta _{j}$ are Hermitian matrices that square to 1.

Possible realizations:

• Chiral edge state of integer quantum Hall
• Metallic surface of topological insulator
• Dirac fermion in graphene (= 2 copies)
• Weyl fermion (1 copy)
• Weyl semimetal (= 2n copies)

Majorana Models

General form of the Majorana Hamiltonian

$H={\frac {1}{2}}\chi ^{\intercal }(i\partial _{i}\alpha ^{i}+n_{j}\beta ^{j})\chi .$

$\alpha _{i}$ and $\beta _{j}$ are Hermitian matrices that square to 1. Additional requirement from Majorana fermion: $\alpha _{i}$ must be transpose symmetric and $\beta _{j}$ must be transpose antisymmetric.

Possible realizations:

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

Classifications

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

• 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