Free and Interacting SRE Phases of Fermions: Beyond the Ten-Fold Way

It is well-known that sufficiently strong interactions can destabilize some SPT phases of free fermions, while others remain stable even in the presence of interactions. It is also known that certain interacting phases cannot be realized by free fermions. We systematically study both of these phenomena in low dimensions and determine the map from free to interacting SPT phases for an arbitrary unitary symmetry G. In particular, in dimension zero and one we describe precisely which SPT phases can be realized by free fermions. We show that in dimension three there are no non-trivial free fermionic SPT phases with a unitary symmetry. We also describe how to compute invariants characterizing interacting phases for free band Hamiltonians with symmetry G (in any dimension) using only representation theory.