Sub-symmetry protected topological states

Abstract

A hallmark of symmetry-protected topological phases (SPTs) are topologically protected boundary states, which are immune to perturbations that respect the protecting symmetry. It is commonly believed that any perturbation that destroys an SPT phase simultaneously destroys the boundary states. However, by introducing and exploring a weaker sub-symmetry (SubSy) requirement on perturbations, we find that the nature of boundary state protection is in fact more complex. We demonstrate that the boundary states are protected by only the SubSy using prototypical Su-Schrieffer-Heeger (SSH) and breathing Kagome lattice (BKL) models, even though the overall topological invariant and the SPT phase are destroyed by SubSy preserving perturbations. By employing judiciously controlled symmetry breaking in photonic lattices, we experimentally demonstrate such SubSy protection of topological states. Furthermore, we introduce a long-range hopping symmetry in BKLs, which resolves a debate on the topological nature of their corner states. Our results apply to other systems beyond photonics, heralding the possibility of exploring the intriguing properties of SPT phases in the absence of full symmetry in different physical contexts.

Main text & supplementary material

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• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Sub-symmetry Protected Topology in Topological Insulators and Superconductors
• A new class of higher-order topological insulators that localize energy at arbitrary multiple sites
• Higher-Order Topological Systems and Their Sub-Symmetry-Protected Topology
• Topology reconstruction for asymmetric systems by isomorphic mapping or perturbation approximation
• Antiferromagnetic topological insulators in heavy-fermion systems
• Scattering description of edge states in Aharonov-Bohm triangle chains
• Nanoimprinted topological laser in the visible