  Spectral gaps in the vibrational modes of disordered solids are key design
elements in the synthesis and control of phononic metamaterials that exhibit a
plethora of novel elastic and mechanical properties. However, reliably
producing these gaps often require a high degree of network specificity through
complex control optimization procedures. In this work, we present as an
additional tool to the existing repertoire, a numerical scheme that rapidly
generates sizeable spectral gaps in absence of any fine tuning of the network
structure or elastic parameters. These gaps occur even in disordered
polydisperse systems consisting of relatively few particles ($N \sim
10^2-10^3$). Our proposed procedure exploits sticky potentials that have
recently been shown to suppress the formation of soft modes, thus effectively
recovering the linear elastic regime where band structures appear, at much
shorter length scales than in conventional models of disordered solids. Our
approach is relevant to design and realization of gapped spectra in a variety
of physical setups ranging from colloidal suspensions to 3D-printed elastic
networks.
