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We report on the experimental observation of a gapped nontrivial Euler topology in an acoustic metamaterial. The topological invariant in this case is the Euler class, which characterizes the two-dimensional geometry of the acoustic Bloch wave function of multiple bands in the Brillouin zone -- a feature that cannot be captured by conventional topological classification schemes. Starting from a carefully designed acoustic system, we retrieve a rich phase diagram that includes the gapped Euler phase on top of non-Abelian topological nodal phases. In particular we find that the gapped phase is characterized by a meron (half-skyrmion) type topology in which the cancellation of the orbital Zak phases by Zak phases of the bands contributes in forming the characterizing winding number of the acoustic bands. Using pump-probe techniques, we are able to extract the wave functions of the acoustic Bloch bands and demonstrate the non-trivial Euler topology via a direct observation of the meron geometry of these bands in the Brillouin zone, in addition to the detection of the acoustic bulk and edge dispersions. We finally observe a topological edge state in the gapped Euler phase which is a direct manifestation of the underlying cancellation of the Zak phases. These experimental features reveal the unprecedented topological Euler phase as well as setting a benchmark for probing acoustic wave function geometry in the Brillouin zone., Comment: 7+5 pages, 4+3 figures

We design a two-dimensional acoustic crystal (AC) to obtain topologically protected edge states for sound waves. The AC is composed of a triangular array of a complex unit cell consisting of two identical triangle-shaped steel rods arranged in air. The steel rods are placed on the vertices of the hexagonal unit cell so that the whole lattice possesses the C6v symmetry. We show that by simply rotating all triangular rods around their respective centers by 180 degrees, a topological phase transition can be achieved, and more importantly, such a transition is accomplished with no need of changing the fill ratios or changing the positions of the rods. Interestingly, the achieved topologically nontrivial band gap has a very large frequency width, which is really beneficial to future applications. The topological properties of the AC are rooted in the spatial symmetries of the eigenstates. It is well known that there are two doubly-degenerate eigenstates at the point for a C6v point group, and they are usually called the p and d states in electronic system. By utilizing the spatial symmetries of the p and d states in the AC, we can construct the pseudo-time reversal symmetry which renders the Kramers doubling in this classical system. We find pseudospin states in the interface between topologically trivial and nontrivial ACs, where anticlockwise (clockwise) rotational behaviors of time-averaged Poynting vectors correspond to the pseudospin-up (pseudospin-down) orientations of the edge states, respectively. These phenomena are very similar to the real spin states of quantum spin Hall effect in electronic systems. We also develop an effective Hamiltonian for the associated bands to characterize the topological properties of the AC around the Brillouin zone center by the kp perturbation method. We calculate the spin Chern numbers of the ACs, and reveal the inherent link between the band inversion and the topological phase transition. With full-wave simulations, we demonstrate the one-way propagation of sound waves along the interface between topologically distinct ACs, and demonstrate the robustness of the edge states against different types of defects including bends, cavity and disorder. Our design provides a new way to realize acoustic topological effects in a wide frequency range spanning from infrasound to ultrasound. Potential applications and acoustic devices based on our design are expected, so that people can manipulate and transport sound waves in a more efficient way.