Elementary derivation of the stacking rules of invertible fermionic topological phases in one dimension

Invertible fermionic topological (IFT) phases are gapped phases of matter with nondegenerate ground states on any closed spatial manifold. When open boundary conditions are imposed, nontrivial IFT phases support gapless boundary degrees of freedom. Distinct IFT phases in one-dimensional space with an internal symmetry group G_f have been characterized by a triplet of indices ([(ν,ρ)],[μ]). IFT phases of matter form an Abelian group structure under the operation of ''stacking''. In this talk, I will first give an operational definition of the indices ([(ν,ρ)],[μ]) from the perspective of the boundary. I will then show an elementary derivation of the stacking rules of IFT phases with any symmetry group G_f , i.e., I will provide an explicit formula for ([(ν_Λ,ρ_Λ)],[μ_Λ]) that is obtained from stacking two IFT phases characterized by the triplets of boundary indices ([(ν₁,ρ₁)],[μ₁]) and ([(ν₂,ρ₂)],[μ₂]).