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) over cap T), charge conjugation ((C) over cap), and/or U(1) ((N) over cap) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as (T) over cap(N) over cap(T) over cap (-1) = (N) over cap or (T) over cap(N) over cap(T) over cap (-1) = -(N) over cap. 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 S-z spin-rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the S-z 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) over cap). The full symmetry group is actually a projective symmetry group.