Fig. 2: Acoustic realization of elementary braiding model.

a A photo of our square-lattice acoustic metamaterial (with \(t=2\)). The lattice constant \(a=40\) mm. Inset: an enlarged view of the unit cell with two mirrors labelled. b Unit-cell structure of our acoustic metamaterial, highlighted with three variable geometry parameters. c Time evolution of the geometric parameters. The discrete spheres correspond to our acoustic metamaterials at eight moments, and the dashed lines are fitted piecewise linear functions. d Band structures at different moments plotted along the diagonals of the square BZ (depicted in the inset of the \(t=0\) pannel). The blue and red arrows indicate the moving directions of the band nodes. In the cases of \(t=0\), \(6\) and 10, the eigenvalues of the mirrors \({{{{{{\mathscr{M}}}}}}}_{1\bar{1}}\) and \({{{{{{\mathscr{M}}}}}}}_{11}\) are labeled with \(\pm\) at the HSPs, \(\Gamma\) and \({{{{{\rm{M}}}}}}\). e Eigenfield distributions of the HSP states at \(t=6\). The vanishing pressure fields (\(p=0\), white) in certain cavities exhibit a nature of sublattice polarization.