Interface States in Space-Time Photonic Crystals: Topological Origin, Propagation and Amplification

Studying the topology of spatiotemporal media poses a fundamental challenge: their remarkable properties stem from breaking spatial and temporal symmetries, yet this same breaking obscures their topological characterization. Here, we show that space-time symmetries persist in crystals with travelling-wave modulation, enabling the study of their topological properties and the prediction of spatiotemporal interface states. Using a Lorentz transformation to the frame comoving with the modulation, we identify a conserved joint parity-time-reversal symmetry in the new variables that enforces the quantization of the Zak phase, elevating it to a $\mathbb{Z}_2$ topological invariant. We then calculate the associated interface states and uncover unique features arising from time-varying effects, including selective directional excitation, propagation along moving boundaries, frequency-converted replicas, and broadband amplification even in the absence of momentum gaps.