Gravitational anomaly of 3+1 dimensional Z₂ toric code with fermionic charges and fermionic loop self-statistics

Quasiparticle excitations in $3+1$ dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in $3+1$ dimensional topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics, that distinguishes two bosonic topological orders that both superficially resemble $3+1$ d ${\mathbb{Z}}_2$ gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic ${\mathbb{Z}}_2$ gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl), and, as we argue, can only exist at the boundary of a non-trivial 4+1d invertible bosonic phase, stable without any symmetries, i.e. it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable $4+1$ d Walker-Wang model and computing the loop self-statistics in the fermionic ${\mathbb{Z}}_2$ gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a non-trivial ${\mathbb{Z}}_2$ classified $4+1$ d invertible phase with action $S=\frac{1}{2} \int w_2 w_3$.