  Kagome lattices represent an archetype of intriguing physics, attracting a
great deal of interest in different branches of natural sciences, recently in
the context of topological crystalline insulators. Here, we demonstrate two
distinct classes of corner states in breathing Kagome lattices (BKLs) with
"bearded" edge truncation, unveiling their topological origin. The in-phase
corner states are found to exist only in the topologically nontrivial regime,
characterized by a nonzero bulk polarization. In contrast, the out-of-phase
corner states appear in both topologically trivial and nontrivial regimes,
either as bound states in the continuum or as in-gap states depending on the
lattice dimerization conditions. Furthermore, the out-of-phase corner states
are highly localized, akin to flat-band compact localized states, and they
manifest both real- and momentum-space topology. Experimentally, we observe
both types of corner states in laser-written photonic bearded-edge BKLs,
corroborated by numerical simulations. Our results not only deepen the current
understanding of topological corner modes in BKLs, but also provide new insight
into their physical origins, which may be applied to other topological BKL
platforms beyond optics.
