Exceptional Topological Signatures of Non-Hermitian Photonic Hopf-Link Braids

This paper investigates the topological properties of non-Hermitian photonic lattices, specifically focusing on how competing sources of non-Hermiticity—such as nonreciprocal hopping and gain-loss potentials—influence the formation and stability of complex energy Hopf-link braids. The author seeks to understand how higher-order interactions and non-Abelian gauge fields expand the topological landscape of these systems.

To explore these phenomena, the author constructs a one-dimensional tight-binding lattice based on the Hatano-Nelson model. This model incorporates non-Abelian SU(2) gauge phases and nonreciprocal hopping amplitudes up to the third-nearest-neighbor (NNNN) level, alongside staggered onsite gain-loss potentials. The study employs the braiding index to quantify the topological structure of the energy bands and utilizes biorthogonal eigenvector tracking to map the evolution of exceptional phase boundaries. The Petermann factor is used as a diagnostic tool to identify the degree of non-orthogonality and the coalescence of eigenstates at exceptional points (EPs).

The research reveals that mixing multiple nonreciprocal channels drives the system into highly intricate, nested Hopf-link braids. These braids are characterized by braiding indices (ν = ±6) that signify opposite handedness in the spectral topology. The study identifies an EP-mediated topological phase transition where the linking number changes abruptly. Furthermore, the author demonstrates that increasing gain-loss non-Hermiticity drives a topological crossover; as the gain-loss parameter exceeds a critical value, the exceptional contours contract into isolated regimes and eventually disappear, causing the energy bands to un-braid into a topologically trivial, gapped phase.

The identification of EP-mediated Hopf-link braiding provides a new mechanism for controlling topological phases in non-Hermitian systems. Because these structures are robust and can be implemented in existing experimental platforms like photonic waveguide arrays and integrated laser chips, they offer promising avenues for developing fault-tolerant communication channels and advanced quantum computing architectures.