Non-Hermitian systems are known for their intriguing topological properties, which underpin various exotic physical phenomena. Exceptional points, in particular, play a pivotal role in fine-tuning these systems for optimal device functionality and material characteristics. These points can give rise to exceptional surfaces with embedded lower-dimensional non-isolated singularities. Here we introduce a topological classification for non-defective intersection lines of exceptional surfaces, where exceptional surfaces intersect transversally. We achieve this classification by constructing a quotient space of an order-parameter space under equivalence relations of eigenstates. We unveil that the fundamental group of these gapless structures is a non-Abelian group on three generators. This classification not only reveals a unique form of non-Hermitian gapless phases featuring a chain of non-defective intersection lines but also predicts the unexpected existence of topological edge states in one-dimensional lattice models protected by the intersection singularities. Our classification opens avenues for realizing robust topological phases.

Topologically protected edge states have been extensively studied in systems characterized by the topological invariants in band gaps (also called line gaps). In this study, we unveil a whole new form of edge states supported by non-Hermitian systems that transcends the established paradigms of band-gap topology. In contrast to the traditional stable edge states in topological insulators with specific band gaps, the one-dimensional systems we investigate are inherently gapless with the Brillouin zones being mapped to the loops encircling hypersurface singularities in a higher-dimensional space with parity-time symmetry. These hypersurface singularities are nonisolated degeneracies embedded entirely on exceptional surfaces, rendering the energy gaps in our systems inevitably closed at the intersections of the Brillouin zone loop and the exceptional surfaces. Unexpectedly, such gapless systems still afford topologically protected edge states at system boundaries, challenging the conventional understanding based on band gaps. To elucidate the existence of these edge states in the absence of a band-gap-based invariant, we propose a theoretical framework based on eigenframe rotation and deformation that incorporates non-Bloch band theory. Finally, we experimentally demonstrate this new form of topological edge states with nonreciprocal circuits for the first time. Our work extends topological edge states from gapped phases to gapless phases, offering new insights into topological phenomena.

Exceptional degeneracies, unique to open systems, are important in non-Hermitian topology. While bulk-Fermi-arcs connecting second-order exceptional points (EP2s) have been observed, the existence of bulk-Fermi-arcs linking higher-order exceptional points remains unexplored. Here, we introduce an unconventional bulk-Fermi-arc in systems with parity-time and pseudo-Hermitian symmetries, which links paired third-order exceptional points (EP3s), where three eigenvalues share identical real parts but distinct imaginary parts. We realize these systems using topological circuits and experimentally demonstrate this unconventional bulk-Fermi-arc. A winding number defined from resultant vector shows that the bulk-Fermi-arc is stabilized by the exchange of Riemannian sheets. Furthermore, analysis via eigenframe deformation and rotation reveals that the EP3 pair is topologically nontrivial and equivalent to a single defective triple point. The EP3s can split from the triple point by varying system parameters, with this splitting protected by topological equivalence. This finding offers insights into non-Hermitian topology with potential applications in wave engineering.

In artificial-intelligence-aided signal processing, existing deep learning models often exhibit a black-box structure. The integration of topological methods serves a dual purpose of making models more interpretable as well as extracting structural information from time-dependent data for smarter learning. Here, we provide a transparent and broadly applicable methodology, TopCap, to capture topological features inherent in time series for machine learning. Rooted in high-dimensional ambient spaces, TopCap is capable of capturing features rarely detected in datasets with low intrinsic dimensionality. Compared to prior approaches, we obtain descriptors which probe finer information such as the vibration of a time series. This information is then vectorised and fed to multiple machine learning algorithms. Notably, in classifying voiced and voiceless consonants, TopCap achieves an accuracy exceeding 96%, significantly outperforming traditional convolutional neural networks in both accuracy and efficiency, and is geared towards designing topologically enhanced convolutional layers for deep learning speech and audio signals.