Local determination of low-energy states in gapped fermion systems with weak long-range interactions

We consider bulk-gapped fermion systems whose infinite-volume bulk is obtained by adding sufficiently weak even many-body interactions to a free Hamiltonian with spectral gap $g>0$, allowing polynomial spatial decay of both parts. We prove quantitative local determination of low-energy states: whenever another interaction agrees with the bulk interaction inside a finite or infinite region $U$, with bulk terms crossing its boundary unchanged or removed, all its states with excitation energies in $[0,3g/4]$ have local expectations close to those of the bulk ground state deep inside $U$. The error decays with the distance to the complement of $U$, independently of the interaction terms supported outside $U$ and of the chosen low-energy state. Quantitative local determination has strong consequences. (i) The bulk interaction has a unique algebraic ground state. (ii) A strong "local perturbations perturb locally" (LPPL) principle holds: arbitrary distant defects change local ground-state expectations only by a spatially decaying error. (iii) Low-energy excitations in finite samples remain localized near the boundary for all times, up to spatially decaying tails, uniformly over arbitrary even boundary terms. (iv) Correlations from deep inside $U$ to outside $U$ are small, with quantitative bounds in terms of spatial separation and distance to $U^c$. To prove local determination of low-energy states, we combine quasi-adiabatic transport of the free ground state to the interacting bulk ground state with a new condition, local excitation detectability (LED). A companion paper establishes gap stability and constructs the transport for weak interactions, while we prove LED at sufficiently weak coupling. The abstract theorem is nonperturbative and applies whenever the transport and LED hypotheses hold.