Fermion decoration construction of symmetry-protected trivial order for fermion systems with any symmetry and in any dimension

We use higher-dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry-protected trivial (SPT) orders (which are also known as symmetry-protected topological orders) in any dimensions and for generic fermion symmetries G₅, which can be a nontrivial Z₂^f extension Z₂^fG₁ (where Z₂^f is the fermion-number-parity symmetry and G₁ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv: 1201. 2648), where G₅ is assumed to be Z₂^fG₁. We find that the (d+1) -dimensional (d+1) D fermionic SPT phases with bosonic symmetry G₁ and from fermion decoration construction can be described in a compact way using higher group homomorphism: BG₁B (Z₂, 2;Z₂, d). In fact, the fermion symmetry is more precisely described by the structure Z₂^fG₁O_ (or Z₂^fG₁O_ with time-reversal symmetry). In this case the (d+1) D fermionic SPT phases are better described by B (Z₂^fG₁O_) B (SO_, 1;Z₂, d) or B (Z₂^fG₁O_) B (O_, 1;Z₂, d).