Non-Hermitian Photonic Spin Hall Insulators

Photonic platforms invariant under parity ($\mathcal{P}$), time-reversal($\mathcal{T}$), and duality ($\mathcal{D}$) can support topological phasesanalogous to those found in time-reversal invariant ${\mathbb{Z}_2}$ electronicsystems with conserved spin. Here, we demonstrate the resilience of theunderlying spin Chern phases against non-Hermitian effects, notably materialdissipation. We identify that non-Hermitian,$\mathcal{P}\mathcal{D}$-symmetric, and reciprocal photonic insulators fallinto two topologically distinct classes. Our analysis focuses on the topologyof a $\mathcal{P}\mathcal{D}$-symmetric and reciprocal parallel-plate waveguide(PPW). We discover a critical loss level in the plates that marks a topologicalphase transition. The Hamiltonian of the$\mathcal{P}\mathcal{T}\mathcal{D}$-symmetric system is found to consist of aninfinite direct sum of Kane-Mele type Hamiltonians with a common band gap. Thisstructure leads to the topological charge of the waveguide being an ill-definedsum of integers due to the particle-hole symmetry. Each component of thisseries corresponds to a spin-polarized edge state. Our findings present aunique instance of a topological photonic system that can host an infinitenumber of edge states in its band gap.