Topological invariant of non-Hermitian space-time-modulated photonic crystals

We propose a medium transformation approach to formulate the adjoint system of space-time-modulated photonic crystals (STMPCs), essential for the bi-orthogonal Berry connection when calculating a topological invariant. We show that the non-Abelian Zak phase of STMPCs comprising stacked photonic time crystals and dielectrics is quantized to 0 or 1 for both entangled and isolated bands. We find that eigenmodes at the center and edge of the Brillouin zone differ in symmetry for the band with nontrivial Zak phases, while they share the same symmetry for trivial Zak phases. In addition, topological phase transitions owing to band inversion are observed. The proposed medium transformation method may serve as an alternative approach to exploring more intricate topological phenomena in non-Hermitian systems and potentially can be incorporated with the non-Bloch band theory.