Observation of Embedded Topology in a Trivial Bulk via Projective Crystal Symmetry

Bulk-boundary correspondence is the foundational principle of topological physics, first established in the quantum Hall effect, where a D-dimensional topologically nontrivial bulk gives rise to (D-1)-dimensional boundary states. The advent of higher-order topology has generalized this principle to a hierarchical chain, enabling topological states to appear at (D-2) or even lower-dimensional boundaries. To date, all known realizations of topological systems must require a topologically nontrivial bulk to initiate the chain of action for bulk-boundary correspondence. Here, in an acoustic crystal platform, we experimentally demonstrate an exception to this paradigm-embedded topology in a trivial bulk-where the bulk-boundary correspondence originates from a trivial bulk. Rather than relying on global symmetries, we employ projective crystal symmetry, which induces nontrivial topology not at the outset in the D-dimensional bulk, but midway through the correspondence hierarchy in lower-dimensional boundaries. We further realize a three-dimensional system exhibiting embedded topology that supports zero-dimensional topological states, achieving the longest possible chain of action for such an unconventional bulk-boundary correspondence in physical space. Our Letter experimentally establishes a new form of bulk-boundary correspondence initiated from a trivial bulk, opening additional degrees of freedom for the design of robust topological devices.