  To date, the vast majority of architected materials have leveraged two
physical principles to control wave behavior, namely Bragg interference and
local resonances. Here, we describe a third path: structures that accommodate a
finite number of delocalized zero-energy modes, leading to anomalous dispersion
cones that nucleate from extreme spatial dispersion at 0 Hz. We explain how to
design such zero-energy modes in the context of elasticity and show that many
of the landmark wave properties of metamaterials can also be induced at an
extremely subwavelength scale by the associated anomalous cones, without
suffering from the same bandwidth limitations. We then validate our theory
through a combination of simulations and experiments. Finally, we present an
inverse design method to produce anomalous cones at desired locations in
momentum space.

Applied Physics