Observation of Dispersion Anomalies by Design

Band structures encode electronic, optical, and acoustic properties of matter and can serve as an essential tool in material discovery and design. Dispersion anomalies- sharp, non-standard features in the frequency-wavenumber relation- have been historically correlated with phonon-electron coupling or long-range interaction. Here, through a combination of experimental, numerical, and analytical methods, it is shown how magnetic couplings can induce negative stiffness and sculpt dispersion relations to support zero-frequency phonon anomalies at arbitrary, non-zero wavenumbers. The approach enables the realization of complete wavenumber bandgaps without time-modulation, electron-phonon coupling, or long-range interactions. The conditions under which non-differentiable zero-frequency phonons exist away from the high-symmetry points are identified. The framework generalizes across monoatomic and diatomic lattices, locally resonant metamaterials, non-local systems, as well as higher dimensional crystals. In addition, the first- passive or active- experimental observation of wavenumber bandgaps in higher dimensions is reported. This work establishes a new paradigm in dispersion engineering and provides means for understanding wave-matter interaction in both the frequency and wavenumber domains.