Sixteen-Fold Way for Fermionic Topological Orders

Abstract

Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a ${\mathbb{Z}}_2$ one-form symmetry, generated in each theory by a single nontrivial ${\mathbb{Z}}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single ${\mathbb{Z}}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the ${\mathbb{Z}}_2$ one-form symmetry, which acquires a ${\mathbb{Z}}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.

17 pages, 4 figures. Fixed the description of the lattice Hamiltonian