Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions

TL;DR

This paper develops a method using higher dimensional bosonization and fermion decoration to construct exactly solvable models of fermionic symmetry protected trivial (SPT) orders across all dimensions and for general fermion symmetries, extending previous work.

Contribution

It introduces a generalized fermion decoration construction for SPT orders applicable to any fermion symmetry group $G_f$, including non-trivial $Z_2^f$ extensions, and describes these phases using higher groups.

Findings

Constructed exactly solvable fermion models for SPT orders in any dimension.

Extended previous group supercohomology results to more general fermion symmetries.

Described SPT phases using higher group structures.

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 non-trivial $Z_2^f$ extension (where $Z_2^f$ is the fermion-number-parity symmetry). This generalizes the previous results from group superconhomology of Gu and Wen (arXiv:1201.2648), where $G_f$ is assumed to be a trivial $Z_2^f$ extension. We find that the SPT phases from fermion decoration construction can be described in a compact way using higher groups.