Unsupervised learning of topological non-Abelian braiding in non-Hermitian bands

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. The topological classification of complex-energy bands has uncovered various topological phases beyond Hermitian systems. Long and colleagues exploit unsupervised learning to fully identify the non-Abelian braiding topology of non-Hermitian bands.