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

Abstract

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 $Φ_h$ be a free fermionic interaction, possibly including pairing terms, whose infinite-volume Majorana coefficient operator $h$ has no spectrum in $(-g,g)$, and let $\mathcal V$ be an even self-adjoint many-body interaction. For sufficiently small $|λ|$, we construct pure ground states $ω_λ$ of the interacting bulk system $Φ_λ=Φ_h+λ\mathcal V$ and quasi-local automorphisms $β_{0,λ}$ such that $ω_λ=ω_0\circβ_{0,λ}$, where $ω_0$ is the free ground state of $Φ_h$. Each $ω_λ$ has a GNS Hamiltonian with gap at least $g-C_*|λ| \|\mathcal V\|_*$, where $\|\cdot\|_*$ 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 $Γ\subset\mathbb{R}^d$, a sufficient condition is decay as $(1+|x-y|)^{-\varpi}$ with $\varpi>2d$, 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 $\mathcal V$ 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.

The paper asks whether a one-particle bulk gap in a free fermion system survives weak, extensive, spatially long-range interactions, and whether quasi-local spectral flow connects the free and interacting ground states without assuming uniformly gapped finite-volume realizations. The setting includes pairing, nonperiodic geometries, and topologically nontrivial free systems. For $\Phi_\lambda=\Phi_h+\lambda\mathcal V$ with $\operatorname{spec}(h)\cap(-g,g)=\varnothing$, it constructs pure ground states $\omega_\lambda=\omega_0\circ\beta_{0,\lambda}$ whose GNS Hamiltonians have one-dimensional kernels and gaps at least $g-C_*|\lambda|\|\mathcal V\|_*$. The guaranteed interval is $|\lambda|\le g/(4C_*\|\mathcal V\|_*)$. For two-site terms on uniformly discrete subsets of $\mathbb R^d$, pointwise decay $(1+r)^{-\varpi}$ with $\varpi>2d$ suffices for both free coefficients and perturbations. Polynomial transport preserves the weighted spatial exponent $p>d$ and is smooth in coupling. Stretched-exponential input of order $\theta_{\rm in}$ yields transport of every order $\theta<\theta_{\rm in}$. The proof combines opposite-Hamiltonian doubling, a quasi-local Bogoliubov transformation to a product vacuum, and an excitation-number lower bound. Differentiating a centered Hamiltonian in the spectral-flow frame produces vacuum-annihilating terms; uniform locality estimates close a gap bootstrap before removing regional perturbation cutoffs. Polynomial propagation estimates come from a companion paper. A further theorem treats interacting product-state references satisfying an excitation-number coercivity condition. The polynomial theorem requires a geometric cell-partition condition and an exponential support-size weight, automatic for bounded-body interactions. The results do not establish uniqueness among all algebraic ground states; that conclusion is deferred to companion work on a potentially smaller coupling interval.