Theoretical analysis of glide-Z₂ magnetic topological photonic crystals

Gapped systems with glide symmetry can be characterized by a Z₂ topological invariant. We study the magnetic photonic crystal with a gap between the second and third lowest bands, which is characterized by the nontrivial glide-Z₂ topological invariant that can be determined by symmetry-based indicators. We show that under the space group No. 230 (I a3¯d), the topological invariant is equal to a half of the number of photonic bands below the gap. Therefore, the band gap between the second and third lowest bands is always topologically nontrivial, and to realize the topological phase, we need to open a gap for the Dirac point at the P point by breaking time-reversal symmetry. With staggered magnetization, the photonic bands are gapped and the photonic crystal becomes topological, whereas with uniform magnetization, a gap does not open, which can be attributed to the minimal band connectivity exceeding two in this case. By introducing the notion of Wyckoff positions, we show how the topological characteristics are determined from the structure of the photonic crystals.