Dirac cones (DCs) play a pivotal role in various unique phenomena ranging from massless electrons in graphene to robust surface states in topological insulators (TIs). Recent studies have theoretically revealed a full Dirac hierarchy comprising an eightfold bulk DC, a fourfold surface DC, and a twofold hinge DC, associated with a hierarchy of topological phases including first-order to third-order three-dimensional (3D) topological insulators, using the same 3D base lattice. Here, we report the first experimental observation of the Dirac hierarchy in 3D acoustic TIs. Using acoustic measurements, we unambiguously reveal that lifting of multifold DCs in each hierarchy can induce two-dimensional topological surface states with a fourfold DC in a first-order 3D TI, one-dimensional topological hinge states with a twofold DC in a second-order 3D TI, and zero-dimensional topological corner states in a third-order 3D TI. Our Letter not only expands the fundamental research scope of Dirac physics, but also opens up a new route for multidimensional robust wave manipulation.

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Topological acoustic interface states in one-dimensional (1D) acoustic topological insulators (ATIs) are zero-dimensional (0D) topological states localized at an interface. Unlike topological edge states that can propagate to deliver information in acoustic waveguides, the 0D topological interface states generally cannot serve as information carriers to deliver information from one location to another due to their intrinsic localization. Here, we design and demonstrate a 1D ATI with a movable interface, enabling the 0D topological acoustic interface states to deliver information from one location to another. The ATI design is based on two types of elemental building blocks—denoted as “1” and “0”—which are programmable. These elements of 1 and 0, when periodically arranged, can form topologically distinct crystals, whose interface hosts acoustic topological interface states in two bandgaps simultaneously. Since these two types of elements can switch from each other with external control, a programmable 1D dual-band ATI can be constructed. By programming coding sequences of 1 and 0 elements, we can observe dynamically movable 0D topological interface states riding on a moving interface along the 1D ATI in both bandgaps. Our work opens an avenue to develop topological acoustic devices with programmable and dynamic functions, which may have a variety of potential applications in the fields of energy trapping, topological pumping, information processing, and sound communication.