The Locality Cost of Fully Flat Hopf Insulators

Abstract

Hopf topology permits a strictly finite-range Hamiltonian with one exactly flat topological band. We prove, however, that extending flatness to the complete two-band spectrum necessarily sacrifices strict locality or the gap: any gapped, Hermitian, translationally invariant two-band Hamiltonian with strictly finite-range hopping and two exactly flat bands has vanishing Hopf invariant. Equivalently, within this two-band setting, a Hopf band admits no compactly supported, translation-covariant, orthonormal Wannier generator. For factorized one-flat-band Hopf parents, the unavoidable partner dispersion equals the Gram symbol of translated compact localized states and encodes their nonorthogonality. Full flattening converts this dispersion into exponentially decaying but infinitely supported hopping. Model-independent bounds provide a sufficient criterion for finite-range approximants to retain the Hopf phase. An explicit model yields the axial decay length $ξ_z/a=1/\ln 2$, parameter-free hopping tails, and residual bandwidths testable in circuit and photonic lattices. Hopf topology therefore does not prohibit a flat band but forces the locality--flatness cost to appear as either partner-band dispersion or nonlocal hopping.