Engineering Topological Phases with a Traveling‐Wave Spacetime Modulation

Time‐variant systems have recently garnered considerable attention due to their unique potential in manipulating electromagnetic waves. Here, a novel class of topological spacetime crystals is introduced, with a traveling‐wave modulation that mimics certain aspects of physical motion. Challenging intuition, our findings reveal that, even though such systems rely on a linear momentum bias, it is feasible to engineer an internal angular momentum and non‐trivial topological phases by leveraging the symmetry of its structural elements. Furthermore, these platforms exhibit a gauge degree of freedom associated with the arbitrariness in the choice of the coordinate transformation that eliminates the time dependence of the system's Hamiltonian. The system's topology is intricately governed by a synthetic magnetic potential whose field lines can be controlled by manipulating material anisotropy. Remarkably, the proposed spacetime crystals host an unconventional class of scattering‐immune edge states, whose oscillation frequency adapts continuously along the propagation path, shaped by the geometric attributes of the path itself.