物理学进展

物理学进展 ›› 2025, Vol. 45 ›› Issue (5): 209-222.doi: 10.13725/j.cnki.pip.2025.05.001

• •    下一篇

非厄米系统对称性与拓扑分类研究进展

李冠良, 赵宇军   

  1. 华南理工大学物理与光电学院，广州 510640 
  • 出版日期:2025-10-20 发布日期:2025-10-24

Advances in symmetry and topological classification of non-Hermitian systems

LI Guanliang, ZHAO Yujun   

  1. School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641 , China 
  • Online:2025-10-20 Published:2025-10-24

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摘要：

近年来，非厄米系统的研究突破了传统量子力学中厄米算符的框架，揭示了复数本征值与 非厄米对称性所蕴含的新的物理规律。相较于厄米体系，非厄米系统通过复本征值的实–虚二元结 构实现了对动力学演化的统一描述，并涌现出临界点、非正交本征态等新概念和非厄米趋肤效应 等独特现象。这些特性源于其哈密顿量的非厄米性，如伪厄米性对实数谱的约束、Jordan 块结构 的代数–几何重数分离，以及广义布里渊区理论对体边关联的重构。Hatano-Nelson 模型和非厄米 Su-Schrieffer-Heeger 模型展示了非互易跃迁与复能谱的拓扑响应，为理解非厄米趋肤效应和能带 奇异性提供了范例。在对称性与拓扑分类方面，非厄米系统扩展了传统 Altland-Zirnbauer 十重 分类，形成 Bernard-LeClair 38 类对称框架，涵盖伪厄米性、手性对称性及复共轭/转置操作的 组合效应。拓扑分类通过 K 理论和同伦理论双轨推进，前者将非厄米系统映射至厄米框架，后者 则解析能带复流形的连续形变特性。未来研究需攻克高维系统普适理论、晶体对称性对分类的影 响及实验平台实现等挑战，推动非厄米拓扑理论在开放量子系统、非平衡态物理和新型器件设计 中的应用。 

关键词: 非厄米系统, 伪厄米性, Hatano-Nelson 模型, 非厄米 SSH 模型, 对称性, 拓扑相

Abstract:

Recent years have witnessed groundbreaking developments in non-Hermitian systems, which transcend the framework of Hermitian operators in conventional quantum mechanics to reveal new physical laws inherent in complex eigenvalues and non-Hermitian symmetries. Unlike Hermitian systems, non-Hermitian systems achieve unified description of dynamical evolution through the real-imaginary dual structure of complex eigenvalues, manifesting unique phenomena including exceptional points, non-orthogonal eigenstates, and the non-Hermitian skin effect (NHSE). These distinctive properties originate from the non-Hermitian nature of their Hamiltonians, such as pseudo-Hermiticity constraining real spectra, the separation of algebraic-geometric multiplicities in Jordan block structures, and the generalized Brillouin zone theory reconstructing bulk-boundary correspondence. Prototypical models like the HatanoNelson model and non-Hermitian Su-Schrieffer-Heeger (SSH) model demonstrate nonreciprocal hopping and topological responses in complex energy spectra, providing paradigmatic platforms for understanding NHSE and energy band singularities. In symmetry and topological classification, non-Hermitian systems extend the traditional Altland-Zirnbauer tenfold classification to the 38-fold Bernard-LeClair symmetry classes, encompassing pseudo-Hermiticity, chiral symmetry, and combined conjugation-transposition operations. Topological classification progresses through dual approaches: K-theory mapping to Hermitian frameworks and homotopy theory analyzing deformation characteristics of complex band manifolds. Future challenges involve developing universal theories for high-dimensional systems, elucidating crystalline symmetry impacts on classification, and implementing experimental platforms, ultimately advancing applications in open quantum systems, nonequilibrium physics, and novel device engineering.

Key words:  non-Hermitian systems, pseudo-Hermiticity, Hatano-Nelson model, nonHermitian SSH model, symmetry, topological phase

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