Determined surface modes of three-dimensional classical-wave crystals under a synthetic translation dimension

Synthetic dimensions have emerged as a powerful framework for exploring topological phenomena beyond real space. Here, we investigate the evolution of surface modes in three-dimensional periodic crystals by introducing a synthetic translation dimension associated with unit-cell translation. By combining the surface-parallel Bloch wave vector with the cyclic translation parameter, we construct a minimal two-dimensional synthetic parameter space that fully captures boundary spectral evolution. Remarkably, this surface-mode evolution persists even when multiple bulk bands are involved, extending beyond the single-band scenarios. We demonstrate that the existence and spectral flow of surface modes are independent of microscopic details and rely only on the topology of bulk bands. Our results are validated in both three-dimensional acoustic and photonic crystal platforms, underscoring the universality of the mechanism across classical-wave systems. These findings establish synthetic translation dimensions as a general framework for predicting and controlling surface-mode evolution in three-dimensional periodic media.