Second Chern crystals with inherently nontrivial topology


 Chern insulators have been generalized to many classical wave systems and thereby lead to many potential applications such as robust waveguide, quantum computation and high-performance laser. However, the band structure of a material can be either topologically trivial or nontrivial, depending on how the crystal structure is designed. Here, we propose a second Chern crystal in a four-dimensional parameter space by introducing two extra synthetic translation dimensions. Since topology of the bulk bands in the synthetic translation space is intrinsically nontrivial, our proposed four-dimensional crystal is guaranteed to be topologically nontrivial regardless of the crystal's detailed configuration. We derive the topologically protected modes on the lower dimensional boundaries of such a crystal via dimension reduction. Remarkably, we observe the one-dimensional gapless dislocation modes and confirm their robustness in experiments. Our findings provide novel perspectives on the topologically nontrivial crystals and may inspire the designs of classical wave devices.