  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.
