  Recently, higher-order topological phases have endowed materials many exotic
topological phases. For three-dimensional (3D) higher-order topologies, it
hosts topologically protected 1D hinge states or 0D corner states, which extend
the bulk-boundary correspondence of 3D topological phases. Meanwhile, the
enrichment of group symmetries with exploration of projective symmetry algebras
redefined the fundamentals of nontrivial topological matter with artificial
gauge fields, leading to the discovery of new topological phases in classical
wave systems. In this Letter, we construct an acoustic topological semimetal
characterized by both the first and the second Stiefel-Whitney (SW) topological
charges by utilizing the projective symmetry. Different from conventional
high-order topologies with multiple bulk-boundary correspondences protected by
different class topological invariants, acoustic high-order Stiefel-Whitney
semimetal (HOSWS) has two different bulk-edge correspondences protected by only
one class (SW class) topological invariant. Two types of topological hinge and
surface states are embedded in bulk bands at the same frequency, featuring
similar characteristics of bound states in the continuum (BICs). In
experiments, we demonstrate the existence of high-quality surface state and
hinge state at the interested frequency window with polarized intensity field
distributions.

Mesoscale and Nanoscale Physics