近日，美国斯坦福大学的范汕洄及其研究团队取得一项新进展。经过不懈努力，他们对作为黎曼曲面的一维非厄米带结构进行研究。相关研究成果已于2024年7月10日在国际知名学术期刊《物理评论A》上发表。

该研究团队通过将一维非厄米带结构看作黎曼曲面，证明了从穿孔复平面的基本群到作用于黎曼片上有序排列的置换群的群同态的单态表示，实质上构成了系统的一个拓扑不变量。这一单态表示与实验可观测效应之间的紧密联系，是通过基础多值函数所引发的分支割来建立的。开放边界谱被解释为连接某些分支点的分支割，其与单态表示的一致性严重限制了其可能的形态。

沿闭环的编织词由环路内分支点的数目和排列标记控制，其相交数由判别式的缠绕数给出。单态表示作为拓扑不变量的重要性以及对黎曼表面几何形状的分析(由带结构定义)可以用来产生关于非厄米系统物理行为的重要见解。

据悉，非厄米周期系统具有其厄米周期系统所没有的独特性质，包括开边界条件谱中的非厄米集肤效应和体带结构的非平凡编织。

附：英文原文

Title: One-dimensional non-Hermitian band structures as Riemann surfaces

Author: Heming Wang, Lingling Fan, Shanhui Fan

Issue&Volume: 2024/07/10

Abstract: Non-Hermitian periodic systems possess unique properties not found in their Hermitian counterparts, including non-Hermitian skin effects in the open-boundary-condition spectrum and nontrivial braiding of the bulk band structure. Here, by viewing one-dimensional non-Hermitian band structures as Riemann surfaces, we show that the monodromy representation, a group homomorphism from the fundamental group of a punctured complex plane to the permutation group operating on the ordering of the Riemann sheets, serves as a topological invariant of the system. The connection between monodromy representations and the experimental observable effects is established through the branch cuts induced by the underlying multivalued functions. An open-boundary spectrum is interpreted as branch cuts connecting certain branch points, and its consistency with the monodromy representation severely limits its possible morphology. A braid word along a closed loop is controlled by the number and permutation labels of branch points within the loop, and its crossing number is given by the winding number of the discriminant. The importance of the monodromy representation as a topological invariant and the analysis of the Riemann surface geometry as defined by the band structure can be used to generate important insights about the physical behaviors of non-Hermitian systems.

DOI: 10.1103/PhysRevA.110.012209

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.110.012209