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 traveling-wave modulations whose velocities can be either lower (subluminal) or higher (superluminal) than the speed of light, enabling the study of their topological properties and the prediction of spatiotemporal interface states. For each modulation regime, we use a Lorentz transformation to a frame in which the modulation depends on only one of the transformed variables. Then, we identify a conserved joint parity-time-reversal symmetry in the new variables that enforces the quantization of a spatiotemporal Zak phase, elevating it to a $Z_2$ topological invariant. Finally, we calculate the associated interface states and uncover unique features arising from time-varying effects, including selective directional amplification, propagation along subluminal and superluminal boundaries, frequency- and momentum-converted replicas, and broadband amplification even in the absence of momentum gaps. Our framework holds for spatiotemporal modulations of any velocity, providing a unified description that encompasses photonic time crystals and clarifies their topological origin.