  Topological insulators with unique gapless edge states have revolutionized
the understanding of electronic properties in solid materials. These gapless
edge states are dictated by the topological invariants associated with the
quantization of generalized Berry phases of the bulk energy bands through the
bulk-edge correspondence, a paradigm that can also be extended to acoustic and
photonic systems. Recently, high-order topological insulators (HOTIs) are
proposed and observed, where the bulk topological invariants result in gapped
edge states and in-gap corner or hinge states, going beyond the conventional
bulk-edge correspondence. However, the existing studies on HOTIs are restricted
to tight-binding models which cannot describe the energy bands of conventional
sonic/photonic crystals that are due to multiple Bragg scatterings. Here, we
report theoretical prediction and experimental observation of acoustic
second-order topological insulators (SOTI) in two-dimensional (2D) sonic
crystals (SCs) beyond the tight-binding picture. We observe gapped edge states
and degenerate in-gap corner states which manifest bulk-edge correspondence in
a hierarchy of dimensions. Moreover, topological transitions in both the bulk
and edge states can be realized by tuning the angle of the meta-atoms in each
unit-cell, leading to various conversion among bulk, edge and corner states.
The emergent properties of the acoustic SOTIs open up a new route for
topological designs of robust localized acoustic modes as well as topological
transfer of acoustic energy between 2D, 1D and 0D modes.
