Engineering Topological Phases with a Traveling-Wave Spacetime Modulation

Time-variant systems have recently garnered considerable attention due to their unique potentials in manipulating electromagnetic waves. Here, we introduce a novel class of topological systems that rely on spacetime crystals with a traveling-wave modulation that emulates 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, we establish that the proposed 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 Hamiltonian that describes the system. The topology is intricately governed by a synthetic magnetic potential whose field lines can be controlled by manipulating material anisotropy. Remarkably, we demonstrate that the proposed spacetime crystals host an unconventional class of scattering-immune edge states. The oscillation frequency of the edge states adapts continuously along the propagation path, shaped by the geometric attributes of the path itself.