Berry bands and pseudo-spin of topological photonic phases

Realising photonic analogues of the robust, unidirectional edge states of electronic topological insulators would improve our control of light on the nanoscale and revolutionise the performance of photonic devices. Here we show that new symmetry protected topological phases can be detected by reformulating energy eigenproblems as Berry curvature eigenproblems. The "Berry bands" span the same eigenspace as the original valence energy bands, but separate into pseudo-spinful and pseudo-spinless subspaces in $\mathrm{C}_2\mathcal{T}$-symmetric crystals. We demonstrate the method on the well-known case of Wu & Hu [Phys. Rev. Lett. 114, 223901 (2015)] and a recently discovered fragilely topological crystal, and show that both crystals belong to the same $\mathrm{C}_2\mathcal{T}$-protected $\mathbb{Z}_2$ topological phase. This work helps unite theory and numerics, and is useful in defining and identifying new symmetry-protected phases in photonics and electronics.