Higher-order
topological semimetals (HOTSMs) represent a novel
type of gapless topological phase, hosting boundary states with dimensions
at least two lower than those of their bulk geometry. Such nontrivial
boundary states have been predicted and observed in three-dimensional
(3D) gapless topological systems, representing features of the HOTSMs.
However, their two-dimensional (2D) analogs, represented especially
by the corner states in monolayer graphene-like structures, have thus
far remained only a theoretical exploration. Here, we experimentally
demonstrate nontrivial corner states in specially tailored photonic
graphene hosting Dirac points, manifesting the HOTSM-like property
in a 2D photonic setting. Such corner states in the otherwise gapless
system exhibit distinct phase structures depending on the lattice
corner and edge geometry, and are completely degenerate with the zero-energy
edge states. Remarkably, we find that these “gapless”
corner states remain intact at zero-energy even in a finite-sized
graphene lattice, protected by chiral symmetry. Unlike corner states
in higher-order topological insulators or topological crystalline
insulators with certain rotational symmetry, these corner states are
localized exclusively to one corner without any coupling to the bulk
or other corners, despite long-distance propagation.