Direct Measurement of Topological Properties of an Exceptional Parabola

Non-Hermitian systems can produce branch singularities known as exceptional points (EPs). The topological properties of an EP can involve either the winding of eigenvalues that produces a discriminant number (DN) or the eigenvector holonomy that generates a Berry phase, making them richer than typical degenerate points in Hermitian systems. However, the multiplicity of topological invariants also makes them challenging to investigate experimentally. Here, we study a trajectory of square-root EPs with both theory and acoustic experiments. We obtain both the DNs and Berry phases through the measurement of both eigenvalues and eigenfunctions, which are found distinct for non-homotopic paths. Our findings not only shed light on exotic non-Hermitian topology but also provide a route for the experimental characterization of non-Hermitian topological invariants.