Non-Hermitian Topology and Exceptional-Point Geometries

Abstract

Non-Hermitian theory is a theoretical framework that excels at describing open systems. It offers a powerful tool in the characterization of both the intrinsic degrees of freedom (DOFs) of a system and the interactions with the external environment. The non-Hermitian framework consists of mathematical structures that are fundamentally different from those of Hermitian theories. These structures not only underpin novel approaches for precisely tailoring non-Hermitian systems for applications but also give rise to topologies not found in Hermitian systems. In this paper, we comprehensively review non-Hermitian topology by establishing its relationship with the behaviors of complex eigenvalues and biorthogonal eigenvectors. Special attentions are given to exceptional points - branch-point singularities on the complex eigenvalue manifolds that exhibit non-trivial topological properties. We also discuss recent developments in non-Hermitian band topology, such as the non-Hermitian skin effect and non-Hermitian topological classifications

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• Experimental Realization of Geometry-Dependent Skin Effect in a Reciprocal Two-Dimensional Lattice
• Chiral exceptional point and coherent suppression of backscattering in silicon microring with low loss Mie scatterer
• Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems
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• Exceptional entanglement phenomena: non-Hermiticity meeting non-classicality
• Scale-tailored localization and its observation in non-Hermitian electrical circuits
• Realization and Topological Properties of Third-Order Exceptional Lines Embedded in Exceptional Surfaces
• Geometry-dependent skin effects in reciprocal photonic crystals
• Resonant Excitation of Quasinormal Modes of Black Holes
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• Braid Protected Topological Band Structures with Unpaired Exceptional Points
• Non-Markovian Quantum Exceptional Points
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• Measuring topological invariants for higher-order exceptional points in quantum three-mode systems
• Emergence of tunable exceptional points in altermagnet-ferromagnet junctions
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• Quantum entanglement and non-Hermiticity in free-fermion systems
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• Novel Topology and Manipulation of Scattering Singularities in Complex non-Hermitian Systems
• Tweaking Spectral Topology and Exceptional Points by Nonlinearity in Non-Hermitian Polariton Systems
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• Point-Gap Bound States in Non-Hermitian Systems
• Construction of Non-Hermitian Parent Hamiltonian from Matrix Product States
• Polariton vortex Chern insulator
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• Anomalous dynamical response of non-Hermitian topological phases
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• Observation of Braid-Protected Unpaired Exceptional Points
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• Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials
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