Three-dimensional acoustic Chern insulators with arbitrary Chern vectors

The Chern vector is a vectorial generalization of the scalar Chern number, being able to characterize the topological phases of three-dimensional (3D) Chern insulators. Such a vectorial generalization extends the applicability of Chern-type bulk-boundary correspondence from one-dimensional edge states to two-dimensional surface states, whose unique features, such as robust chiral surface states and forming nontrivial torus knots or links in the surface Brillouin zone, have been demonstrated recently in 3D photonic crystals. However, to date, the Chern vectors have been limited to the simplest form with only one nonzero component, resulting in the chiral surface states only propagating along four side surfaces parallel to the Chern vector, and the surface-state torus knots or links emerging, 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 with multiple nonzero components that allow the chiral surface states to propagate along all six surfaces and 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 have potential applications in designing topological acoustic devices.