Sub-symmetry-protected topological states

Abstract

Abstract A hallmark of symmetry-protected topological phases are topological boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys such a topological phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker sub-symmetry requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. Here we demonstrate that the boundary states are protected by only the sub-symmetry, using Su–Schrieffer–Heeger and breathing kagome lattice models, even though the overall topological invariant and the associated topological phase can be destroyed by sub-symmetry-preserving perturbations. By precisely controlling symmetry breaking in photonic lattices, we experimentally demonstrate such sub-symmetry protection of topological states. Furthermore, we introduce a long-range hopping symmetry in breathing kagome lattices, which resolves a debate on the higher-order topological nature of their corner states. Our results apply beyond photonics and could be used to explore the properties of symmetry-protected topological phases in the absence of full symmetry in different physical contexts.