Classifying topological floppy modes in the continuum

TL;DR

This work develops a continuum theory of topological floppy modes by augmenting linear elasticity with $n_w$ soft-mode fields, enabling robust edge states and Weyl points to arise from purely continuum invariants. A continuum Maxwell criterion $n_K = d + n_w$ and a determinant-based invariant for the effective compatibility matrix establish a lattice-independent classification of edge modes and Weyl zero modes, with additional fields encoding local soft lattice modes. The authors demonstrate, via homogenization of ball-and-spring lattices, how these continuum moduli relate to microscopic structure and present efficient PDE-based methods to predict coarse-grained deformations without full lattice simulations. In 2D, they show that $n_w\ge1$ yields topological edge polarization (governed by a discriminant $\Delta$ and polarization directions) and that $n_w\ge2$ enables Weyl points, with a phase diagram obtained from a distorted kagome-based lattice. This framework provides design principles for mechanical metamaterials and offers scalable, continuum-level tools to analyze and engineer topological states of matter in elasticity.

Abstract

In floppy mechanical lattices, robust edge states and bulk Weyl modes are manifestations of underlying topological invariants. To explore the universality of these phenomena independent of microscopic detail, we formulate topological mechanics in the continuum. By augmenting standard linear elasticity with additional fields of soft modes, we define a continuum version of Maxwell counting, which balances degrees of freedom and mechanical constraints. With one additional field, these augmented elasticity theories can break spatial inversion symmetry and harbor topological edge states. We also show that two additional fields are necessary to harbor Weyl points in two dimensions, and define continuum invariants to classify these states. In addition to constructing the general form of topological elasticity based on symmetries, we derive the coefficients based on the systematic homogenization of microscopic lattices. By solving the resulting partial differential equations, we efficiently predict coarse-grained deformations due to topological floppy modes without the need for a detailed lattice-based simulation. Our discovery formulates novel design principles and efficient computational tools for topological states of matter, and points to their experimental implementation in mechanical metamaterials.

Figures (8)

• Figure 1: Classification of topological elasticity in two dimensions. (a) We consider a continuum system finite in the $x_2$-direction and periodic along the $x_1$-direction. Such a system could represent the continuum limit of a ball-and-spring network, but our results are independent of the microscopic realization. We indicate the displacement magnitudes of the two continuum edge floppy modes [defined in Eq. (\ref{['eq:def_continuum_edge_mode']})] using color gradients. (b--d) We construct an elasticity theory by adding $n_w$ kinematic fields arising from local soft modes to the displacement field. Then, we classify the resulting phenomenology according to $n_w$. A theory with $n_w = 0$ corresponds to standard linear elasticity, which is unable to capture topological phenomena. For $n_w \geq 1$, topological polarization is present if $\Delta$ [defined in Eq. (\ref{['eq:Delta_definition']})] is positive, and is characterized by the polarization directions $\mathbf{p}_1, \mathbf{p}_2$. Theories with $n_w \geq 2$ can exhibit even richer phenomenology by describing systems with Weyl points. (e) Representative displacement magnitudes associated with the continuum floppy modes are shown in columns I -- III. Floppy modes localized on the same edge are displayed side-by-side for clarity. Column II illustrates how both floppy modes are localized on the same edge if the normal $\mathbf{n}$ to the strip edges makes an acute angle with both $\mathbf{p}_1$ and $\mathbf{p}_2$. In column III, as $q_1$ passes through Weyl point coordinate $q_1^W$, the blue floppy mode switches from being localized at the bottom edge to the top edge, becoming a bulk mode at the transition point. (f) The physical phenomena associated with topological mechanics in the continuum are illustrated using indentation tests in columns I -- III. Edges with no localized floppy modes are more rigid. The presence of a Weyl point, shown in column III, results in a wavevector-dependent edge rigidity. The deformations shown here have been exaggerated for clarity.
• Figure 2: Physical interpretation of the scaling relationship $\varepsilon \sim \mathcal{O}(\delta)$. This choice of scaling relationship links two ratios. The grid on the right represents the many unit cells present in a system with length scale $L$. The box on the left represents a single unit cell and the perturbation $\Delta r$ applied to the site positions in order to obtain a gapped lattice from a gapless configuration. The relationship $\varepsilon \sim \mathcal{O}(\delta)$ connects a length scale associated with the geometry of a single unit cell to the characteristic number of unit cells over which spatial variations occur in the continuum. In the continuum theory, this scaling relationship ensures that we consider elastic terms of the same order in $\delta$, leading to elasticity with broken spatial inversion symmetry.
• Figure 3: Universal dependence of continuum floppy modes on lattice perturbations. We compare analytical predictions with numerical results from strips of $\varepsilon$-perturbed kagome lattices (defined in Fig. \ref{['fig:distorted_kagome']}) periodic along the $\mathbf{b}_1$-direction and finite along the $\mathbf{b}_2$-direction. We investigate how the wavenumber component $\mathrm{Re}(\bar{q}_2)$ and inverse decay length $\mathrm{Im}(\bar{q}_2)$ normal to the strip edge depend on the geometrical perturbation $\varepsilon$ to the lattice and on the wavenumber component $q_1$ parallel to the edge. For both the (a) unpolarized lattice and the (b) polarized lattice, we find that (i) $\mathrm{Re}(\bar{q}_2) / \bar{q}_1$ and (ii) $\mathrm{Im}(\bar{q}_2) / \varepsilon$ depend only on the ratio $\varepsilon / \bar{q}_1$, provided $\varepsilon$ and $\bar{q}_1$ are sufficiently small. The plots show two sets of curves: the thin solid curves correspond to $\bar{q}_1 = 10^{-6}$ and the thick dashed curves correspond to $\bar{q}_1 = 0.1$. Despite the large difference between the $\bar{q}_1$ values, the solid and dashed curves are nearly indistinguishable. The curves are colored according to the floppy modes they represent. In (a)(ii), the inverse decay lengths $\mathrm{Im}(\bar{q}_2)$ have opposite signs, corresponding to floppy modes localized on opposite edges, while (b)(ii) exhibits continuum floppy modes with $\mathrm{Im}(\bar{q}_2)$ of the same sign. The $\bar{q}_r$ values shown are dimensionless, as defined in Sec. \ref{['sec:site_displacements_and_compatibility_matrix']}. The rectangles on the right are strips showing representative displacement magnitudes for the continuum floppy modes.
• Figure 4: Constructing distorted kagome lattices. The positions of sites in the kagome unit cell are shifted along directions parallel to unit vectors $\mathbf{c}_1, \mathbf{c}_2, \mathbf{c}_3$ by distances $w_1, w_2, w_3$ respectively. The blue arrows represent the changes in site position due to the geometrical distortion. The undistorted unit cell has three sites placed at the vertices of an equilateral triangle. The distorted unit cell shown corresponds to $w_1 = 1, w_2 = 1.5, w_3 = 1.75$.
• Figure 5: How to obtain continuum fields from a discrete lattice. (a) A gapless lattice is geometrically perturbed, shifting the site positions by distances proportional to $\varepsilon a$, where $a$ is the unit-cell size. (b) The resulting gapped lattice has a compatibility matrix given by Eq. (\ref{['eq:perturbed_compatibility_matrix']}), which acts on the site displacements to give the bond extensions. (c) The lattice degrees of freedom are the $n_s d$ site displacements, which are projected onto three subspaces, as shown in Eq. (\ref{['eq:phi_decomposition']}). (d) The subspaces are spanned by the eigenvectors of $\mathbf{C}(\mathbf{0})^{\mathrm{T}}\mathbf{K}\mathbf{C}(\mathbf{0})$, classified according to whether they are in the null spaces of both $\mathbf{C}(\mathbf{0})$ and $\mathbf{C}_w(\mathbf{0})$ [translations $\Phi_u$], in only the null space of $\mathbf{C}(\mathbf{0})$ [soft modes $\Phi_w$], or in neither of these null spaces [high-frequency modes $\Phi_v$]. These eigenvectors are natural ways to express the lattice degrees of freedom. The equations of motion are used to express $\Phi_v$ in terms of $\Phi_u$ and $\Phi_w$, given by Eq. (\ref{['eq:pseudo-inversion_Phi_v']}). (e) The continuum fields arise naturally from the degrees of freedom in (d). The translations $\Phi_u$ give rise to strain, and the soft modes $\Phi_w$ give rise to scalar fields and their gradients. These continuum fields satisfy the equations of motion Eqs. (\ref{['eq:continuum_eq']}) and the constitutive relations Eq. (\ref{['eq:constitutive']}). In taking the continuum limit, the $n_s d$ lattice degrees of freedom, where $d$ is the number of spatial dimensions, have been reduced to $d + n_w$ continuum fields, where $n_w$ is the number of local soft modes.