通过纽结图进行拓扑分类：费米面色散与Lifshitz转变
Topological classification through knotted graphs: Fermi surface dispersions and Lifshitz transitions
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中文总结 AI 辅助
本研究提出基于纽结图的统一拓扑框架，通过Yamada集与序列完整分类真实材料费米面色散及Lifshitz转变，并扩展至非厄米例外面。
中文摘要 AI 辅助
纽结理论为能带结构提供了丰富的拓扑分类体系，但其适用范围存在根本性限制：纽结不变量仅能对能隙闭合处的一维节线进行分类，无法编码真实材料的完整色散或丰富的费米面结构。在此我们证明，纽结图（在三维空间中允许图状交点的纽结）——迄今在凝聚态文献中尚未被探索——为分类整个能带色散乃至某些情境下的本征态拓扑提供了统一的拓扑语言。我们提出一个超越现有Yamada多项式的新框架，能够完全拓扑刻画真实费米面的复杂性，关键包括其多个不连通部分如何嵌套。这产生了Yamada集，一种边界解析的扩展，将跨能量的完整拓扑演化组织成Yamada序列：一个紧凑的色散级指纹，直接与Lifshitz转变的实验特征相关联。我们的框架通过真实材料的基于DFT的能带结构得到验证。超越色散级分类，该框架可扩展到非厄米例外面，其中Berry曲率通量进一步为纽结图骨架配备有向阿贝尔边缘流，该边缘流也捕捉本征态拓扑。
英文摘要
Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.
发表机构
- National University of Singapore（新加坡国立大学）
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