Topological flatband loop states in fractal-like photonic lattices

Abstract: Noncontractible loop states (NLSs) are recently realized topological entity in flatband lattices, arising typically from band touching at a point where a flat band intersects one or more dispersive bands. There exists also band touching across a plane, where one flat band overlaps another all over the Brillouin zone without crossing a dispersive band. Such isolated plane-touching flat bands remain largely unexplored. For example, what are the topological features associated with such flatband degeneracy? Here, we demonstrate for the first time to our knowledge nontrivial NLSs and robust boundary modes in a system with such degeneracy. Based on a tailored photonic lattice constructed from the well-known fractal Sierpinski gasket, we theoretically analyze the wavefunction singularities and the conditions for the existence of the NLSs. We show that the NLSs can exist in both singular and nonsingular flat bands, as a direct reflection of the real-space topology. Experimentally, we observe directly such flatband NLSs in a laser-written Corbino-shaped fractal-like lattice. This work not only leads to a deep understanding of the mechanism behind the nontrivial flatband states, but also opens up new avenues to explore fundamental phenomena arising from the interplay of flatband degeneracy, fractal structures and band topology.