The zero-index characterization of stable short-range entanglement for free-fermion states

Let $\mathcal H=\ell^2(\mathbb Z^2)$ and let $P$ be the Fermi projection of a gapped Hamiltonian $H$ satisfying exponential locality. Let $\omega_P$ be the associated quasi-free state, and suppose that $P$ has a well-defined topological index $\mathcal C(P)$. Zero-index stable SRE conjecture. Then

$$\mathcal C(P)=0 \quad\Longleftrightarrow\quad \omega_P\text{ is stably SRE}.$$

The preceding discussion notes that the implication from vanishing Chern number to stable SRE is known under translation invariance, while extending the argument to general exponentially local Hamiltonians requires an appropriate many-body notion of locality and a corresponding Lieb–Robinson bound. Thus the stated equivalence remains open in the generality given.