Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ( PT ) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in PT -symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous PT symmetry breaking is captured by Chern–Euler and Chern–Stiefel–Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

中文翻译：

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同伦、对称和非厄米能带拓扑


非厄米矩阵在自然描述中无处不在，从经典耗散系统（包括光学、电学和机械超材料）到波散射和开放量子多体系统。使用 K 理论对非厄米系统进行开创性的线隙和点隙分类加深了对许多物理现象的理解。然而，大量的系统仍然超出了这个描述。参考点和线通常不能区分多个非厄米能带是否表现出有趣的异常点、光谱辫状和交叉。为了解决这个问题，我们考虑两个不同的概念：非厄米带隙和分离间隙，它们至关重要地涵盖了广泛的多频带场景，从而能够描述具有对称性的通用能带结构。有了这些概念，我们在存在物理相关的奇偶时间的情况下，为间隙系统和节点系统提供了统一且全面的分类（ PT ）和使用同伦理论的伪厄米对称性。这揭示了源自特征值和波函数的新的稳定拓扑，并且还显着地暗示了不同的脆弱拓扑相。特别是，我们揭示了不同的阿贝尔相和非阿贝尔相PT -对称系统，由框架和编织拓扑描述。 相应的不变量对于不会引起（异常）简并的保对称扰动具有鲁棒性，并且它们还预测节点相位的变形规则。我们进一步证明自发PT陈-欧拉和陈-施蒂费尔-惠特尼描述捕获了对称性破缺，这是以前被忽视的前所未有的非厄米拓扑的指纹。这些结果为在广泛的物理平台上对各种新颖的拓扑现象进行理论和实验探索打开了大门。