标题： 多重例外点的阿贝尔谱拓扑 显示英文标题

标题： Abelian Spectral Topology of Multifold Exceptional Points

摘要： 非厄米物理学的兴起丰富了拓扑相的数量，包括没有厄米对应物的现象。尽管多重 exceptional point（特异点）的拓扑性质是被研究得最广泛的独特的非厄米特征之一，但直到最近，这些具有$n$-重能谱退化的特异点（EP$n$），即对应的特征向量也发生聚合的点，才以拓扑结果环绕数和相关的阿贝尔加倍定理的形式揭示出来。然而，关于 EP$n$和它们的拓扑本质更数学根本性的描述仍然是一个开放的问题。为了填补这一空白，本文重新审视了普通系统和具有局域对称性的系统中 EP$n$的拓扑分类，并以更数学上易于处理的（局域）相似关系来推广它，同时扩展到包括所有此类相似关系以及非局域对称性。通过结果向量——其分量由相应的特征多项式及其导数的结果式给出——拓扑环绕数的本质可以通过多种方式理解：i) 作为（厄米）拓扑物质的十倍分类框架；ii) Mayer--Vietoris 序列框架；iii) 向量丛的分类。该分类方案进一步预测了由非局域对称性诱导的、由$\mathbb{Z}_2$不变量保护的拓扑体费米弧的存在，被称为$\mathbb{Z}_2$保护的费米弧。我们的工作揭示了 EP$n$拓扑性质所依赖的数学基础，丰富了对非厄米谱特征的理论理解，因此将在经典和量子物理的现代实验中找到广泛应用。 显示英文摘要

摘要： The advent of non-Hermitian physics has enriched the plethora of topological phases to include phenomena without Hermitian counterparts. Despite being among the most well-studied uniquely non-Hermitian features, the topological properties of multifold exceptional points, $n$-fold spectral degeneracies (EP$n$s) at which also the corresponding eigenvectors coalesce, were only recently revealed in terms of topological resultant winding numbers and concomitant Abelian doubling theorems. Nevertheless, a more mathematically fundamental description of EP$n$s and their topological nature has remained an open question. To fill this void, in this article, we revisit the topological classification of EP$n$s in generic systems and systems with local symmetries, generalize it in terms of more mathematically tractable (local) similarity relations, and extend it to include all such similarities as well as non-local symmetries. Through the resultant vector, whose components are given in terms of the resultants between the corresponding characteristic polynomial and its derivatives, the topological nature of the resultant winding number is understood in several ways: in terms of i) the tenfold classification of (Hermitian) topological matter, ii) the framework of Mayer--Vietoris sequence, and iii) the classification of vector bundles. The classification scheme further predicts the existence of topological bulk Fermi arcs protected by a $\mathbb{Z}_2$-invariant, induced by non-local symmetries, dubbed $\mathbb{Z}_2$-protected Fermi arcs. Our work reveals the mathematical foundations on which the topological nature of EP$n$s resides, enriches the theoretical understanding of non-Hermitian spectral features, and will therefore find great use in modern experiments within both classical and quantum physics.