Bulk gap stability and spectral flow for fermions with weak long-range interactions

We prove that weak long-range many-body interactions preserve the bulk gap of a free fermion system and that a quasi-local spectral flow transports the free ground state to ground states of the interacting system. Let Φₕ be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator h has no spectrum in $(-g,g)$, and let V be an even self-adjoint many-body interaction. For sufficiently small $|λ|$, we construct pure ground states ω_λ of the interacting bulk system Φ_λ=Φₕ+λ V and quasi-local automorphisms β0,λ such that ω_λ=ω₀∘β0,λ, where ω₀ is the free ground state of Φₕ. Each ω_λ has a GNS Hamiltonian with gap at least g-C_*|λ| \| V\|_*, where \|·\|_* is the relevant interaction norm. Our result allows polynomial decay of both the free coefficients and the many-body interaction. For two-site interactions on a uniformly discrete point set Γᵈ, a sufficient condition is decay as (1+|x-y|)⁻ with >2d, for both the hopping and pairing coefficients and the interaction terms. The spectral-flow generator preserves the spatial exponent of the polynomial interaction norm, and its automorphisms satisfy locality bounds at that exponent. If h and V have stretched-exponential decay, so does the spectral-flow generator. The proof constructs the spectral flow and establishes the gap through an excitation-number comparison, using polynomial propagation bounds for evolved interactions proved in a companion paper.