  Recently, higher-order topological insulators have been attracting extensive
interest. Unlike the conventional topological insulators that demand bulk gap
closings at transition points, the higher-order band topology can be changed
without bulk closure and exhibits as an obstruction of higher-dimensional
boundary states. Here, we report the first experimental realization of
three-dimensional surface-obstructed topological insulators with using acoustic
crystals. Our acoustic measurements demonstrate unambiguously the emergence of
one-dimensional topological hinge states in the middle of the bulk and surface
band gaps, as a direct manifestation of the higher-order band topology.
Together with comparative measurements for the trivial and
phase-transition-point insulators, our experimental data conclusively evidence
the unique bulk-boundary physics for the surface-obstructed band topology. That
is, the topological phase transition is determined by the closure of surface
gap, rather than by closing the bulk gap. Our study might spur on new
activities to deepen the understanding of such elusive topological phases.
