Dynamics of elastic hyperbolic lattices

Abstract

Hyperbolic lattices tessellate the hyperbolic space, which affords the opportunity for an infinite number of regular tessellations. Thus, hyperbolic lattices significantly extend the design space typically associated with lattices in Euclidean space, and potentially provide access to unexplored wave phenomena. We here investigate the dynamic behavior of hyperbolic tessellations governed by interactions whose strength depends upon the distance of neighboring nodes. The exploration of their spectral characteristics illustrates a rich dynamic behavior, which is characterized by eigenstates that are primarily localized either at the center, or towards the boundary of the Poincare circle. The variation of the spectrum resulting from a hyperbolic translation of the lattice is evaluated in the context of phason dynamics. Specifically, the translation is considered as a phason of the hyperbolic lattice, which leads to a family of tessellations whose spectra lead to variations in eigenstates. Such variations identify modes that are strongly localized either at the center or at the boundary. The variation of these modes in terms of the translation identify an adiabatic transformation which is associated with the edge-to-edge pumping of selected eigenmodes.