Nature Machine Intelligence ( IF 18.8 ) Pub Date : 2024-07-12 , DOI: 10.1038/s42256-024-00871-1 Yang Long , Haoran Xue , Baile Zhang
The topological classification of energy bands has laid the foundation for the discovery of various topological phases of matter in recent decades. While previous work focused on real-energy bands in Hermitian systems, recent studies have shifted attention to the intriguing topology of complex-energy, or non-Hermitian, bands, freeing them from the constraint of energy conservation. For example, the spectral winding of complex-energy bands can give rise to unique topological structures such as braids, holding substantial promise for advancing quantum computing. However, discussions of complex-energy braids have been predominantly limited to the Abelian braid group \({{\mathbb{B}}}_{2}\) owing to its relative simplicity. Identifying topological non-Abelian braiding remains challenging, as it lacks a universally applicable topological invariant for characterization. Here we present a machine learning algorithm for the unsupervised identification of non-Abelian braiding within multiple complex-energy bands. We demonstrate that the results are consistent with Artin’s well-known topological equivalence conditions in braiding. Inspired by these findings, we introduce a winding matrix as a topological invariant for characterizing braiding topology. The winding matrix also reveals the bulk-edge correspondence of non-Hermitian bands with non-Abelian braiding. Finally, we extend our approach to identify non-Abelian braiding topology in two-dimensional and three-dimensional exceptional semimetals and address the unknotting problem in an unsupervised manner.
中文翻译:
非厄米带中拓扑非阿贝尔编织的无监督学习
能带的拓扑分类为近几十年来物质各种拓扑相的发现奠定了基础。虽然之前的工作主要集中在埃尔米特系统中的真实能带,但最近的研究已将注意力转移到复杂能量或非埃尔米特带的有趣拓扑上,使它们摆脱了能量守恒的约束。例如,复杂能带的光谱缠绕可以产生独特的拓扑结构,例如辫子,为推进量子计算带来了巨大的希望。然而,由于其相对简单,对复杂能量辫子的讨论主要局限于阿贝尔辫子群 \({{\mathbb{B}}}_{2}\)。识别拓扑非阿贝尔编织仍然具有挑战性,因为它缺乏普遍适用的拓扑不变量来表征。在这里,我们提出了一种机器学习算法,用于在多个复杂能带内无监督地识别非阿贝尔编织。我们证明了结果与 Artin 著名的编织拓扑等效条件一致。受这些发现的启发,我们引入了缠绕矩阵作为拓扑不变量来表征编织拓扑。缠绕矩阵还揭示了非厄米特带与非阿贝尔编织的体边对应关系。最后,我们扩展了我们的方法来识别二维和三维特殊半金属中的非阿贝尔编织拓扑,并以无监督的方式解决打结问题。