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

Abstract

© 2019 American Physical Society. 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 Gf, which can be a nontrivial Z2f extension Z2Gb (where Z2f is the fermion-number-parity symmetry and Gb is the bosonic symmetry). This generalizes the previous results from group supercohomology of Gu and Wen (arXiv:1201.2648), where Gf is assumed to be Z2f×Gb. We find that the (d+1)-dimensional [(d+1)D] fermionic SPT phases with bosonic symmetry Gb and from fermion decoration construction can be described in a compact way using higher group homomorphism: BGb→φB(Z2,2;Z2,d). In fact, the fermion symmetry is more precisely described by the structure Z2f Gb SO∞ (or Z2f Gb O∞ with time-reversal symmetry). In this case the (d+1)D fermionic SPT phases are better described by B(Z2f Gb SO∞)→φB(SO∞,1;Z2,d) [or B(Z2f Gb O∞)→φB(O∞,1;Z2,d)].