Periodically driven systems are ubiquitously found in both classical and quantum regimes. In the field of photonics, these Floquet systems have begun to provide insight into how time periodicity can extend the concept of spatially periodic photonic crystals and metamaterials to the time domain. However, despite the necessity arising from the presence of nonreciprocal coupling between states in a photonic Floquet medium, a unified non-Hermitian band structure description remains elusive. We experimentally reveal the unique Bloch-Floquet and non-Bloch band structures of a photonic Floquet medium emulated in the microwave regime with a one-dimensional array of time-periodically driven resonators. These non-Hermitian band structures are shown to be two measurable distinct subsets of complex eigenfrequency surfaces of the photonic Floquet medium defined in complex momentum space.

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Line excitations in topological phases are a subject of particular interest because their mutual linking structures encode robust topological information of matter. It has been recently shown that the linking and winding of complex eigenenergy strings can classify one-dimensional non-Hermitian topological matter. However, in higher dimensions, bundles of linked strings can emerge such that every string is mutually linked with all the other strings. Interestingly, despite being an unconventional topological structure, a non-Hermitian Hopf bundle has not been experimentally clarified. Here, we make the first attempt to explore the non-Hermitian Hopf bundle by visualizing the global linking structure of spinor strings in the momentum space of a two-dimensional electric circuit. By exploiting the flexibility of reconfigurable couplings between circuit nodes, we can study the non-Hermitian topological phase transition and gain insight into the intricate structure of the Hopf bundle. Furthermore, we find that the emergence of a higher-order skin effect in real space is accompanied by the linking of spinor strings in momentum space, revealing a bulk-boundary correspondence between the two domains. The proposed non-Hermitian Hopf bundle platform and visualization methodology pave the way to design new topologically robust non-Hermitian phases of matter.

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Eigenstate coalescence in non-Hermitian systems is widely observed in diverse scientific domains encompassing optics and open quantum systems. Recent investigations have revealed that adiabatic encircling of exceptional points (EPs) leads to a nontrivial Berry phase in addition to an exchange of eigenstates. Based on these phenomena, we propose in this work an exhaustive classification framework for EPs in non-Hermitian physical systems. In contrast to previous classifications that only incorporate the eigenstate exchange effect, our proposed classification gives rise to finer Z2 classifications depending on the presence of a π Berry phase after the encircling of the EPs. Moreover, by mapping arbitrary one-dimensional systems to the adiabatic encircling of EPs, we can classify one-dimensional non-Hermitian systems characterized by topological phase transitions involving EPs. Applying our exceptional classification to various one-dimensional models, such as the non-reciprocal Su–Schrieffer–Heeger (SSH) model, we exhibit the potential for enhancing the understanding of topological phases in non-Hermitian systems. Additionally, we address exceptional bulk-boundary correspondence and the emergence of distinct topological boundary modes in non-Hermitian systems.