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

TL;DR

This paper demonstrates experimentally that a trivial bulk can host embedded topological states through projective crystal symmetry, challenging the traditional bulk-boundary correspondence paradigm and enabling new topological device designs.

Contribution

It introduces the concept of embedded topology originating from a trivial bulk via projective crystal symmetry, expanding the understanding of topological states beyond nontrivial bulk conditions.

Findings

Embedded topology observed in a trivial bulk in an acoustic crystal

Realization of a 3D system with zero-dimensional topological states

Longest chain of bulk-boundary correspondence initiated from a trivial bulk

Abstract

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 work 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.