Signatures of Topological Phonons in Superisostatic Lattices

Soft topological surface phonons in idealized ball-and-spring lattices with coordination number $z=2d$ in $d$ dimensions become finite-frequency surface phonons in physically realizable superisostatic lattices with $z>2d$. We study these finite-frequency modes in model lattices with added next-nearest-neighbor springs or bending forces at nodes with an eye to signatures of the topological surface modes that are retained in the physical lattices. Our results apply to metamaterial lattices, prepared with modern printing techniques, that closely approach isostaticity.

Figure 1

(a) Unit cell of the KL with NN bonds (black) and additional NNN bonds (blue). (b) Unit cell of the KL with bending energies (blue arcs). (c) The $X_{nt}$ and (d) the $X_t$ conformation. The thick black bonds mark our bottom surface, and the thick dashed line marks a possible cut to liberate a top surface. The green arrow indicates the topological polarization $R_T$ of $X_t$.

Figure 2

Low-frequency mode structure for (a) $X_{nt}$, $v=0.1$; (b) $X_{nt}$, $v=0.001$; (c) $X_t$, $v=0.1$; (d) $X_t$, $v=0.001$. The gray areas are the projected bulk bands, and the black curves within these bands are bulk mode frequencies as a function of $q=q_x$ for different values of $q_y$, $q_y=0,\pi /10,\pi /5,\dots$. Note the strongly nonmonotonic behavior of these modes for $X_{\mathrm{t}}$ at $v=0.001$, a consequence of the lobes in density plots of the lowest mode at $v=0$ with $\omega \sim (q_y^2-\beta q_x^2)$ with $\beta >0$ determined by $X_t$ [1, 2]. The color codes for curves in (a)–(d) are red for bottom surface, blue for top surface, orange for longitudinal and transverse bulk sound modes at $q_y=0$, and green for Rayleigh waves predicted by elasticity theory. Each surface mode is a linear combination of four modes that decay with $y$. The two smallest $\kappa$ (largest penetration depths) are plotted in (e) [(f)] for each surface mode shown in (a) and (b) [(c) and (d)]. The color codes for these curves are red (blue) for bottom (top) surface modes at $v=0.1$, purple for bottom and top acoustic modes and for the optical mode at $v=0.001$, and dashed green for elastic theory results. Each of the purple curves in (e) consists of a nearly degenerate pair. The gray ellipsis in (f) highlights a hard-to-see onset of the acoustic mode after which the upper purple curve consists of a nearly degenerate pair. (g) $c_R$ (gray) for $X_{nt}$ and $c_R$ (black), $\mathrm{\Delta }\omega$ (magenta), and $q_0$ (brown) for $X_t$. The lines in the corresponding colors are power-law fits with $c_R\sim \mathrm{\Delta }\omega \sim v^{0.5}$ and $q_0\sim v^{0.25}$, in agreement with our crude estimates. Note that the surface modes for $v=0.001$ remain small and nearly flat throughout the SBZ implying low-energy pointlike surface invaginations such as observed in Ref. [24].