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Topological invariants and topological charges in photonic systems

Topology in photonics comes in two distinct flavors: global and local. Global topology considers invariants that are obtained by integrating over the energy band, whereas local topology considers defects, typically vortices, in the far-field emission. These topologies are described by a wide range of models built in both real and momentum space, which are connected only by computationally expensive numerical methods that lack physical intuition. Here we propose a general framework based on a real-space Hamiltonian capable of describing electric field as a vector in both near- and far fields, allowing us to bridge between topological defects in the far-field and global topological invariants. The proposed Hamiltonian is constructed from the symmetry-representations of the lattice, is deformable to both atomic localized-mode (tight-binding) and photonic delocalized-mode (long-range) limits, and allows for independent control over the energies of eigenmodes of different symmetries at high-symmetry points of the Brillouin zone. This symmetry-based approach enables the design of structures with almost arbitrary topological properties and is not limited to photonic systems, but could apply to any system with engineered real-space couplings.