Photonic $\mathbb{Z}_2$ topological Anderson insulators

That disorder can induce nontrivial topology is a surprising discovery in topological physics. As a typical example, Chern topological Anderson insulators (TAIs) have been realized in photonic systems, where the topological phases exist without symmetry protection. In this work, by taking TM and TE polarizations as pseudo-spin degrees of freedom, we theoretically propose a scheme to realize disorder-induced symmetry-protected topological (SPT) phase transitions in two-dimensional photonic crystals (PCs) with a combined time-reversal, mirror and duality symmetry $\mathcal{T}_f=\mathcal{T}M_z\mathcal{D}$. In particular, we demonstrate that the disorder-induced SPT phase persists even without pseudo-spin conservation, thereby realizing a photonic $\mathbb{Z}_2$ TAI, in contrast to a $\mathbb{Z}$-classified quantum spin Hall (QSH) TAI with decoupled spins. By formulating a new scattering approach, we show that the topology of both the QSH and $\mathbb{Z}_2$ TAIs can be manifested by the accumulated spin rotations of the reflected waves from the PCs. Using a transmission structure, we also illustrate the trivialization of a disordered QSH phase with an even integer topological index caused by spin coupling.