User:EverettYou/Free Fermion SPT

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. The classification table of real fermion SPT.