arXiv 2103.157292021-03physics.opticscond-mat.mes-hallquant-ph

对称性与高阶例外点

Symmetry and Higher-Order Exceptional Points

Ipsita Mandal, Emil J. Bergholtz

备注：: minor typos fixed
发表信息：: Phys. Rev. Lett. 127, 186601 (2021)
DOI：: 10.1103/PhysRevLett.127.186601

摘要：

例外点（EPs），在这两个特征值和特征向量合并，是普遍存在的和独特的非厄米特系统的功能。二阶EP由于其丰富性而成为迄今为止研究最多的，仅需要调整两个实参数，这比一般找到普通埃尔米特本征值简并所需的三个参数要少。高阶EP通常需要更多的微调，因此被认为扮演的角色要小得多。然而，在这里，我们阐明了物理相关的对称性，使高阶EP显着更丰富，概念更丰富。更突出的是，三阶EP一般只需要两个实调谐参数的宇称时间（PT）对称性或广义手征对称性的存在。值得注意的是，我们发现，这些不同的对称性产生拓扑不同类型的EP。我们说明了我们的研究结果在简单的模型，并显示如何与通用的$\sim k^{1/3}$色散的三阶EP的保护PT对称性，而与$\sim k^{1/2}$色散的三阶EP的保护出现在非厄米Lieb晶格模型的手征对称性。更一般地说，我们确定了稳定的，弱的，脆弱的方面，保护的高阶EP，梳理出其伴随的现象。

英文摘要：

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine-tuning, and are thus assumed to play a much less prominent role. Here, however, we illuminate how physically relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. More saliently, third-order EPs generically require only two real tuning parameters in the presence of either a parity-time (PT) symmetry or a generalized chiral symmetry. Remarkably, we find that these different symmetries yield topologically distinct types of EPs. We illustrate our findings in simple models, and show how third-order EPs with a generic $\sim k^{1/3}$ dispersion are protected by PT symmetry, while third-order EPs with a $\sim k^{1/2}$ dispersion are protected by the chiral symmetry emerging in non-Hermitian Lieb lattice models. More generally, we identify stable, weak, and fragile aspects of symmetry-protected higher-order EPs, and tease out their concomitant phenomenology.

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