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

Abstract

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 ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}$ (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 ${Z}_{2}^{f}\ifmmode\times\else\texttimes\fi{}{G}_{b}$. We find that the $(d+1)$-dimensional $[(d+1)\mathrm{D}]$ fermionic SPT phases with bosonic symmetry ${G}_{b}$ and from fermion decoration construction can be described in a compact way using higher group homomorphism: $\mathcal{B}{G}_{b}\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({Z}_{2},2;{Z}_{2},d)$. In fact, the fermion symmetry is more precisely described by the structure ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}}$ (or ${Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}}$ with time-reversal symmetry). In this case the $(d+1)\mathrm{D}$ fermionic SPT phases are better described by $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}S{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}(S{O}_{\ensuremath{\infty}},1;{Z}_{2},d)$ [or $\mathcal{B}({Z}_{2}^{f}\ensuremath{\leftthreetimes}{G}_{b}\ensuremath{\leftthreetimes}{O}_{\ensuremath{\infty}})\stackrel{\ensuremath{\varphi}}{\ensuremath{\rightarrow}}\mathcal{B}({O}_{\ensuremath{\infty}},1;{Z}_{2},d)]$.