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 $G_f$, which can be a nontrivial $Z_2^f$ extension $Z_2^f⋋G_b$ (where $Z_2^f$ is the fermion-number-parity symmetry and $G_b$ is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where $G_f$ is assumed to be $Z_2^f\times G_b$. We find that the $(d+1)$-dimensional $\left[\right(d+1)\mathrm{D}]$ fermionic SPT phases with bosonic symmetry $G_b$ and from fermion decoration construction can be described in a compact way using higher group homomorphism: $\mathcal{B}{G}_b\stackrel{\phi }{\to }\mathcal{B}(Z_2,2;Z_2,d)$. In fact, the fermion symmetry is more precisely described by the structure $Z_2^f⋋G_b⋋S{O}_{\infty }$ (or $Z_2^f⋋G_b⋋O_{\infty }$ with time-reversal symmetry). In this case the $(d+1)\mathrm{D}$ fermionic SPT phases are better described by $\mathcal{B}(Z_2^f⋋G_b⋋S{O}_{\infty })\stackrel{\phi }{\to }\mathcal{B}(S{O}_{\infty },1;Z_2,d)$ [or $\mathcal{B}(Z_2^f⋋G_b⋋O_{\infty })\stackrel{\phi }{\to }\mathcal{B}(O_{\infty },1;Z_2,d)]$.

• Received 20 February 2019
• Revised 15 October 2019

DOI:https://doi.org/10.1103/PhysRevB.100.235141