Topological bound modes in phononic lattices with nonlocal interactions

Topological protection has opened new possibilities for unconventional wave guiding, object cloaking, improved energy transport, as well as surface, edge, or corner mode localization. In elasticity, these phenomena have largely been explored and exemplified through discrete models having nearest neighbor couplings. Interactions beyond the nearest neighbors in one-dimensional studies, on the other hand, have recently shown great potential for topological wave phenomena. In this work we investigate the topological modes of a two-dimensional mass–spring hexagonal lattice with connections between both nearest and third nearest neighboring masses. We show that non-nearest connections allow for (i) the formation of additional Dirac cones and (ii) a migration in their location in the reciprocal space as a function of the relative stiffness between nearest and third nearest neighbor connections. These additional Dirac cones are linked to a corresponding increase in the number of topological edge modes, which hybridize and result in bound modes at interfaces between lattices that are inverted copies of each other. Explicit expressions for the mode shapes and frequencies of these bound modes are derived and their topological origin is elucidated. We also demonstrate that by varying the relative stiffness between nearest and third nearest neighbor connections, bound modes lying in the band gap can be achieved. While in the case of only nearest neighbor connections the bound modes are at a fixed frequency in the pass band, varying the stiffness of a single nonlocal spring can shift their frequency and isolate them as desired within a band gap. Transient numerical simulations conducted on a finite lattice allow to quantify the confinement along the transverse direction as a wave propagates in a waveguide with sharp turns, reporting negligible backscattering. Finally, a possible realization of hexagonal unit cells with third nearest neighbor connections is proposed. The concepts here presented open novel avenues for topological wave guiding and confinement leveraging bound modes to design waveguides with superior energy localization potential.