Subsymmetry-protected compact edge states

Abstract: Sub-symmetry (SubSy) protected topological states represent a concept that goes beyond the conventional framework of symmetry-protected topological (SPT) phases, demonstrating that topological boundary states can remain robust even when the pertinent symmetry holds only in a subset of Hilbert space. Typical SPT and SubSy boundary states decay exponentially into the bulk, which means they are not confined in just few lattice sites close to the boundary. Here, we introduce topologically compact edge states protected by SubSy, featuring extreme two-site localization at boundaries of a lattice, without any decay into the bulk. The compactness arises from local destructive interference at the boundary, while topological protection is ensured by SubSy, characterized by quantized winding numbers. Experimentally, we observe compact edge states in laser-written photonic lattices with engineered rhombic-like unit cells, confirming their robustness against perturbations under both chiral symmetry and SubSy conditions. Our results highlight the potential of SubSy protection for achieving topological confinement of light, paving the way for applications in compact waveguides, lasers, and high-sensitivity photonic sensors.