Abstract
Topology has emerged as a powerful framework for controlling wave propagation and band degeneracies, yet conventional topological phases are insufficient for multiband and open systems involving dissipation, gain, and non-reciprocity. This thesis advances non-Abelian and non-Hermitian topology in photonic crystals and lattices through two complementary directions. First, it investigates PT-symmetry-protected real triple-degenerate points (RTDs), characterized by the Euler number. An extended framed Poincaré–Hopf relation is established between the π₂ Euler charge of an RTD and the signed sum of π₁ Abelian and non-Abelian quaternion charges carried by its penetrating nodal lines. This correspondence imposes global connectivity constraints and a no-go theorem requiring Euler-charge neutrality across the Brillouin zone. Symmetry-allowed compounds of type-I and type-II RTDs are identified and experimentally verified by microwave near-field mapping in a metallic photonic meta-crystal. The thesis also introduces non-Hermitian directed-graph networks supporting pure-decay modes with smooth, monotonic exponential profiles protected by graph geometry rather than exceptional points or balanced gain and loss. These modes obey a universal power-partition rule and define quantized integer or half-integer decay charges. Microwave experiments confirm their robustness against moderate disorder. Building on this mechanism, synthetic gauge fields are used to select single, paired, or multiple pure-decay modes while preserving their amplitude profiles and enabling independent multidimensional spectral control. These results deepen the understanding of real and complex topology and provide practical routes toward stable lasers, directional amplifiers, energy funnels, open-system simulators, and multiband topological photonic circuits.
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