Symmetry-protected topological phases in noninteracting fermion systems

Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal (̂ \^T), charge conjugation (̂ \^C), and/or $U(1)$ (̂ \^N) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as ̂ \^T ̂ \^N ̂ \^T^-1= ̂ \^N or ${ ̂ \^{}{}}{T}{ ̂ \^{}{}}{N}{{ ̂ \^{}{}}{T}}^{{-}1}={-}{ ̂ \^{}{}}{N}$. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free-fermion phases for those possible symmetry groups. In particular, we can view the $U(1)$ as a spin rotation. We find that superconductors with the Sz spin-rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the Sz spin-rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free-fermion phases can be realized by electron systems with certain symmetries. We also point out that, to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (-)^ ̂ \^N. The full symmetry group is actually a projective symmetry group.