Evolution and global charge conservation for polarization singularities emerging from non-Hermitian degeneracies

Core concepts in singular optics, especially the polarization singularities, have rapidly penetrated the surging fields of topological and non-Hermitian photonics. For open photonic structures with non-Hermitian degeneracies in particular, polarization singularities would inevitably encounter another sweeping concept of Berry phase. Several investigations have discussed, in an inexplicit way, connections between both concepts, hinting at that nonzero topological charges for far-field polarizations on a loop are inextricably linked to its nontrivial Berry phase when degeneracies are enclosed. In this work, we reexamine the seminal photonic crystal slab that supports the fundamental two-level non-Hermitian degeneracies. Regardless of the invariance of nontrivial Berry phase (concerning near-field Bloch modes defined on the momentum torus) for different loops enclosing both degeneracies, we demonstrate that the associated far polarization fields (defined on the momentum sphere) exhibit topologically inequivalent patterns that are characterized by variant topological charges, including even the trivial scenario of zero charge. Moreover, the charge carried by the Fermi arc actually is not well defined, which could be different on opposite bands. It is further revealed that for both bands, the seemingly complex evolutions of polarizations are bounded by the global charge conservation, with extra points of circular polarizations playing indispensable roles. This indicates that although not directly associated with any local charges, the invariant Berry phase is directly linked to the globally conserved charge, physical principles underlying which have all been further clarified by a two-level Hamiltonian with an extra chirality term. Our work can potentially trigger extra explorations beyond photonics connecting Berry phase and singularities.