Interacting One-Dimensional Fermionic Symmetry-Protected Topological Phases

Abstract

In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z[subscript 2], or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S[subscript z] spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z[subscript 4]. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z[subscript n] discrete S[subscript z] spin rotation and time-reversal symmetries are classified by Z[subscript 4] when n is even and Z[subscript 2] when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.