Fig. 3: Complex-valued phonon band structure and frozen evanescent phonon modes.

a Numerically calculated phonon band structure for elastic-wave propagation in the metamaterial beam (\(N=3\)) defined in Fig. 2 for wave propagation along the \(z\)-direction. The representation is as in Fig. 1b, except that only real-valued frequencies are depicted, \({{\rm{Im}}}\left(\omega \right)=0\). Out of many modes (gray), two are highlighted. The blue modes correspond to longitudinal waves, and the red modes to twist waves. Out of the corresponding local minima in the green plane, evanescent modes emerge (cf. Fig. 1b) that touch the \(\omega=0\) plane at the positions of the colored dots. For the longitudinal mode (blue dot) relevant to the below experiments, we find the complex-valued wavenumber \({k}_{z}\approx \left(0.666-0.026\,{{\rm{i}}}\right)\,\pi /{a}_{z}\). Similar band structures are shown in Supplementary Fig. 1 for \(N=2\) and \(N=4\). b Illustration of this frozen mode. The axial component of the displacement vector, \({u}_{z}\), is depicted in a false-color representation. The static spatial oscillation period of \(p=2\pi /{\mathrm{Re}}\left({k}_{z}\right)\approx {Na_z}={3a_z}\) is clearly visible. The mode exponentially decays with decay length \(l=1/|{{\rm{Im}}}({k}_{z})|\). c Same as panel (b), but for the zero-frequency twist mode (red dot in a), with an azimuthal component of the displacement vector \({u}_{\theta }\).