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We report on the experimental observation of a gapped nontrivial Eulertopology in an acoustic metamaterial. The topological invariant in this case isthe Euler class, which characterizes the two-dimensional geometry of theacoustic Bloch wave function of multiple bands in the Brillouin zone -- afeature that cannot be captured by conventional topological classificationschemes. Starting from a carefully designed acoustic system, we retrieve a richphase diagram that includes the gapped Euler phase on top of non-Abeliantopological nodal phases. In particular we find that the gapped phase ischaracterized by a meron (half-skyrmion) type topology in which thecancellation of the orbital Zak phases by Zak phases of the bands contributesin forming the characterizing winding number of the acoustic bands. Usingpump-probe techniques, we are able to extract the wave functions of theacoustic Bloch bands and demonstrate the non-trivial Euler topology via adirect observation of the meron geometry of these bands in the Brillouin zone,in addition to the detection of the acoustic bulk and edge dispersions. Wefinally observe a topological edge state in the gapped Euler phase which is adirect manifestation of the underlying cancellation of the Zak phases. Theseexperimental features reveal the unprecedented topological Euler phase as wellas setting a benchmark for probing acoustic wave function geometry in theBrillouin zone.

Experimental observation of meronic topological acoustic Euler insulators