Soft topological modes protected by symmetry in rigid mechanical metamaterials

Topological mechanics can realize soft modes in mechanical metamaterials in which the number of degrees of freedom for particle motion is finely balanced by the constraints provided by interparticle interactions. However, solid objects are generally hyperstatic (or overconstrained). Here, we show how symmetries may be applied to generate topological soft modes even in overconstrained, rigid systems. To do so, we consider non-Hermitian topology based on nonsquare matrices, and design a hyperstatic material in which low-energy modes protected by topology and symmetry appear at interfaces. Our approach presents a novel way of generating softness in robust scale-free architectures suitable for miniaturization to the nanoscale.

Figure 1

Hyperstatic 1D rotor chain with generalized inversion symmetry. (a) Right-leaning rotor chain with positive rotor angle $\theta$ (measured from the red rotor head). (b) The winding number of $\vec{R}\left(k\right)$ around the origin is 0 for $\theta >0$. (c) Left-leaning rotor chain with negative $\theta$. (d) The winding number of $\vec{R}\left(k\right)$ around the origin is 1 for $\theta <0$.

Figure 2

Topologically protected localized modes at a sharp interface in a hyperstatic lattice with generalized inversion symmetry. (a) A sharp interface between right- and left-leaning hyperstatic rotor chains. (b) The localized mode (red), the lattice parameter c (yellow) at the interface. (c) Density of states showing two topologically protected localized modes, one at each interface. (d) The region of parameter space (light blue) for localized modes to exist at an interface between a lattice with $c=c_L$ on the left, and $c=c_R$ on the right, with $c_L+c_R=2$ in deep blue.

Figure 3

Topologically protected localized modes at a smooth interface in a hyperstatic lattice with generalized inversion symmetry. (a) A smooth interface between right- and left-leaning hyperstatic rotor chains. (b) For smoothly varying parameter $c$, the localized interface modes can be mapped to bound states in an effective potential (blue) for the Schrödinger equation. (c) The two lowest localized modes (red) at the interface. (d) Density of states showing the two topologically protected modes at the two interfaces. (e) The region of parameter space (blue) in which localized modes exist at a smooth interface with $c=c_L$ on the left and $c=c_R$ on the right, calculated in the continuum approximation (see Ref. [60] and Eq. (32) in the SM [59]) for different interface widths $W$. $c_L+c_R=2$ is shown in deep blue. The grey regions correspond to the nontopological interface, when localized interface modes are not guaranteed to exist.