发表机构
Eastern Institute of Technology; Department of Physics, Shaoxing University（宁波东方理工大学; 绍兴大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明，将Hopf拓扑的单平带扩展为两带全平带时，会因严格有限范围哈密顿量的局域性要求而消失Hopf不变量，代价体现为伴带色散或非局域跃迁，还给出了显式模型与可测试的残余带宽。
AI 中文摘要
Hopf拓扑允许具有一个精确平带的严格有限范围哈密顿量。然而我们证明，将平带扩展到完整的两带谱必然会牺牲严格局域性或能隙：任何具有严格有限范围跃迁、两个精确平带的带隙、厄米、平移不变两带哈密顿量都具有消失的Hopf不变量。等价地，在该两带设置中，Hopf带不具有紧支撑、平移协变、正交的Wannier生成元。对于因子化的单平带Hopf母本，不可避免的伴带色散等于平移紧局域态的Gram符号，并编码它们的非正交性。完全平化将该色散转换为指数衰减但无限支撑的跃迁。与模型无关的界为有限范围近似量保留Hopf相提供了充分标准。一个显式模型给出轴向衰减长度ξ_z/a=1/ln2、无参数跃迁尾，以及可在电路和光子晶格中测试的残余带宽。因此，Hopf拓扑并不禁止平带，而是迫使局域性-平带代价以伴带色散或非局域跃迁的形式出现。
英文摘要
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.
Comments4 pages, 3 figures; Supplemental Material included
