Acoustic Three-dimensional Chern Insulators with Arbitrary Chern Vectors

The Chern vector is a vectorial generalization of the scalar Chern number, being able to characterize the topological phase of three-dimensional (3D) Chern insulators. Such a vectorial generalization extends the applicability of Chern-type bulk-boundary correspondence from one-dimensional (1D) edge states to two-dimensional (2D) surface states, whose unique features, such as forming nontrivial torus knots or links in the surface Brillouin zone, have been demonstrated recently in 3D photonic crystals. However, since it is still unclear how to achieve an arbitrary Chern vector, so far the surface-state torus knots or links can emerge, not on the surface of a single crystal as in other 3D topological phases, but only along an internal domain wall between two crystals with perpendicular Chern vectors. Here, we extend the 3D Chern insulator phase to acoustic crystals for sound waves, and propose a scheme to construct an arbitrary Chern vector that allows the emergence of surface-state torus knots or links on the surface of a single crystal. These results provide a complete picture of bulk-boundary correspondence for Chern vectors, and may find use in novel applications in topological acoustics.