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As the counterpart of Hermitian nodal structures, the geometry formed byexceptional points (EPs), such as exceptional lines (ELs), entails intriguingspectral topology. We report the experimental realization of order-3exceptional lines (EL3) that are entirely embedded in order-2 exceptionalsurfaces (ES2) in a three-dimensional periodic synthetic momentum space. TheEL3 and the concomitant ES2, together with the topology of the underlyingspace, prohibit the evaluation of their topology in the eigenvalue manifold byprevailing topological characterization methods. We resolve this issue bydefining a winding number that associates with the resultants of theHamiltonian. This resultant winding number detects EL3 but ignores the ES2,allowing the diagnosis of the topological currents carried by the EL3, whichenables the prediction of their evolution under perturbations. Our resultsexemplify unprecedented topology of higher-order exceptional geometries and mayinspire new non-Hermitian topological applications.

Realization and Topological Properties of Third-Order Exceptional Lines Embedded in Exceptional Surfaces