Hermitian Hamiltonians are commonly used to describe physical systems due to their entirely real-valued eigenvalues which are suitable for representing steady states. However, recent theoretical developments on non-Hermitian Hamiltonians have inspired physicists to rethink the formalism by using non-Hermitian Hamiltonians to study systems that incorporate richer physical processes, such as loss and/or gain. Such systems are commonplace in optics and classical waves. Therefore, classical waves, such as light and acoustic waves, are quickly becoming the hotspot for research in non-Hermitian physics.

Non-Hermitian systems live on Riemannian manifolds, which give rise to properties that have no counterpart in Hermitian systems. In particular, branch-point singularities known as “exceptional points” can emerge on these manifolds. Many intriguing phenomena can be realized by leveraging the unique properties of the exceptional points. In this talk, I will present our recent works on the topological properties associated with trajectories of exceptional points on the Riemannian manifolds. First, we focus on a two-level system that can produce an exceptional parabola. Two different types of non-Hermitian topological invariants, respectively associated with eigenvalues and geometric phase of eigenvectors, are used to arrive at a holistic picture of the topological properties induced by the existence of the exceptional parabola. Second, we extend the discussion to a three-level system in multi-dimensional phase space, in which multi-valued fractional/integer winding numbers are identified. These systems are realized and their properties verified in acoustic-wave experiments.
 

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