Non-Hermitian chiral surface waves in disordered odd solids

Abstract

Chiral surface waves are surface-localized modes that propagate unidirectionally along a boundary, enabling directed transport and minimal back-scattering. While first identified in quantum systems, they were recently shown to emerge in classical metamaterials in the presence of `odd elasticity'. Owing to the non-reciprocality of odd elasticity, these waves exhibit growing amplitudes during propagation, reminiscent of the non-Hermitian skin effect. To date, studies of odd elastic systems have mainly focused on ordered structures. Whether structurally-disordered materials can host non-Hermitian chiral surface waves (NHCSW) remains unexplored. We address this question using a minimal model of torque-driven disordered odd solids. Such solids are abundant, from biological gels such as the cytoskeleton driven by motor-proteins to synthesized systems such as magnetic colloidal gels. We find that torque-driven disordered odd solids have unique NHCSW with stronger surface localization and stable boundary velocity, in contrast to previous lattice models of odd solids. These distinct features stem from an intrinsic interplay between boundary torques and odd elasticity in torque-driven odd solids. Our results offer a new strategy to control NHCSW using active torques.

Figures (4)

• Figure 1: (a) Illustration of an elastic material composed of rigid rod-like particles. Importantly, our model lee2025 applies for any other complex rigid particles (granules, colloids, fiber composites, etc.). (b) Coarse-graining at position $\bm{R}$ in the deformed/real space. We consider a locally-disordered, isotropic elastic material, in the presence of local active torque $\tau^\alpha$ ($\alpha$ being the particle index). The various fields ${\bm X}({\bm R})$ are the average of the particle's ${\bm X}^\alpha$ within the coarse-graining volume. (c) Particle displacement $\bm{u}^\alpha$ and internal rotation $\theta^\alpha$ away from the rest position and rest orientation (blue line) in the undeformed state. The individual rest orientations have no universal direction due to the disordered nature.
• Figure 2: Dynamic boundary $h(x,t)$ moves with vertical displacement $u_y(x,0,t)$ (black arrows), measured from the flat undeformed boundary (bottom back dashed). $u_y$ contains a static part $u_y^\text{st}$ (gray dashed arrow) due to static deformations balancing the pre-stress, and a dynamic part $u_y^\text{dyn}$ (red dotted arrow) related to wave excitations. The local normal director $\hat{\bm{N}}$ is determined by the boundary tilt $\nabla_x h = \nabla_x u_y$.
• Figure 3: (a) (top) Sketch of a non-Hermitian chiral surface wave (for $\tilde{K}^o<0$ as an example) that propagates unidirectionally along the boundary with growing amplitude. The enhancement is $\tilde{q}_x = - {\rm Im}(\tilde{k}_x)$ where $\tilde{k}_x \equiv 2 k_x ({{\mu}/{\rho}\omega^2})^{1/2}$. This surface wave is comprised of two eigenmodes with the same propagation velocity along the boundary, but with distinct penetration depths $\delta^{(n)}$. Here, $1/\tilde{q}_y^{(n)}$ are the intrinsic penetration depths. (bottom) Color maps of $\delta^{(n)}$ versus $\tilde{K}^o = K^o/\mu$ and the boundary position $\tilde{x} \equiv x({\rho\omega^2/4\mu})^{1/2}$. Along the enhancement direction ($\tilde{q}_x,\tilde{x}>0$ for $\tilde{K}^o<0$ and vice versa), $\delta^{(n)}$ increases and the wave penetrates deeper. (b) The enhancement $\tilde{q}_x$ and the wave velocity $\tilde{v}_x = 1/\text{Re}(\tilde{k}_x)$ (bottom inset) versus $\tilde{K}^o$. Notice the matching sign of $\tilde{q}_x$ and $\tilde{v}_x$ and their behaviors at large $|\tilde{K}^o|$ (insets): vanishing enhancement (after initial linear growth), and almost constant absolute wave velocity. (c) Intrinsic inverse bulk decay lengths $\tilde{q}_y^{(n)}$ of the two modes and their competition with the enhancement, quantified by the localization ratios $|\tilde{q}_x|/\tilde{q}_y^{(n)}$ (inset). The bulk decays always dominates over the enhancement, leading to small and vanishing $|\tilde{q}_x|/\tilde{q}_y^{(n)}$ at large $|\tilde{K}^o|$.
• Figure 4: (a) Color maps of the enhancement $\tilde{q}_x$ versus odd elasticity $\tilde{K}^o$ and the boundary torque $\tilde{\tau}^\circ$ (top)/the proportionality $\alpha\equiv \tau^\circ/4\tilde{K}^o$ (bottom). All contour curves use the same color scheme across panels. Note that the contour curves are constructed by scanning over $\tilde{\tau}^\circ$ (or $\alpha$) [details in the main text]. We first focus on the role of $\tilde{\tau}^\circ$. (b) For large $|\tilde{\tau}^\circ|$, the dominant effect of $\tilde{\tau}^\circ$ is approximately an horizontal shift of the zero torque curve ($\tilde{\tau}^\circ=0$, purple; gay from left to right: $\tau^\circ=$-16, -8, 8, 16). (c) The shifting effects of $\tilde{\tau}^\circ$ leads to the universal asymptotic slope $d\tilde{\tau}^\circ/d\tilde{K}^o=4$ for all the contours at large $\tilde{\tau}^\circ$. In particular, the contour $\tilde{q}_x=0$ (green) has the asymptote $\tilde{\tau}^\circ = 4 \tilde{K}^o$, corresponding to our disordered odd solid. (d) A schematic intersection analysis from geometry illustrates the origin of the discrete transition in the max/min contours at $\alpha=1$ (our odd solid, blue dashed) at the bottom of panel a.