Concise networks consisting of nodes and bonds have been extensively developed to capture the properties of various systems, from DNA mobility¹ to allosteric behaviors of proteins^2,3, epithelial morphogenesis⁴, and jamming in granular materials⁵. Mechanical networks (lattices) provide a platform to explore the mechanical counterparts of topological insulators, due to the correspondence between classic phonon modes and quantum electronic states. A class of lattices on the verge of mechanical instability, i.e., Maxwell lattices⁶, have balanced numbers of degrees of freedom and constraints in the bulk and can host zero modes at their free boundaries. Interestingly, infinitesimal zero modes can become finitely deformable and propagate in one-dimensional topological lattices in the form of solitons⁷. However, nonlinear excitations in topological mechanical systems with multiple zero modes remain elusive.

To extend the concept of solitons in topological lattices described by a scalar field theory, we report vector solitons in a topological mechanical lattice, illustrating the interaction of dual degrees of freedom. Our inspiration for this study derives from the coupled double chains of indium atomic wires with sublattice symmetry breaking^8,9. Different from the long-range interactions of atomic chains, our topological lattice is comprised of interconnected bonds and hinged rhombic linkages pivoted to the laboratory frame, showcasing glide-reflection symmetry (Figure 1a). The balanced numbers of length constraints and degrees of freedom in the bulk demonstrate that the mechanical lattice is a Maxwell lattice. Cutting interconnected bonds at two ends unleashes two additional degrees of freedom in the lattice with open boundaries. We are curious about what nonlinear excitations correspond to different zero modes.

To address this, we calculate the two orthogonal zero modes in the absence of self-stress using the Maxwell-Calladine index theorem¹⁰(Figure 1 a and b). Then, we conform the lattice along these modes by gradually nudging the node situated at the polarized boundary. On one hand, the transformed configuration along the horizontal direction exhibits a domain wall, displaying a classical kink profile for the horizontal nodal positions, while the vertical components remain zero (Figure 1c). On the other hand, the deformed nodes along the vertical direction demonstrate co-propagating kink and antikink across the lattice (Figure 1d). More interestingly, the topological mechanical lattice shows four transformable homogeneous configurations and domain walls that interpolate different states among them

To unravel the physics underlying these nonlinear responses, we build a continuum model and derive the coupled equations. We find that these equations resonate with realms known for multiferroic phase transitions, nonlinear optics, and multispecies condensates¹¹, supporting vector soliton solutions. The term ’vector solitons’ here describes the nodal positions of zero-energy configurations, triggered with minimal energy input, different from the elastic pulses prevalent in monostable elastic metamaterials with coupled translational-rotational components and amplitude gaps^12,13. Furthermore, we find that the vector soliton supports superposed sub-solitons, a unique phenomenon in soliton physics, shedding light on soliton collisions. To observe sub-solitons that approach or depart from each other, we demonstrate the evolution of a domain structure along linearly superposed modes (Figure 1e). This unveils three distinct phases separated by two domain walls (Figure 1f), with the vector soliton’s components emerging as the sum or the difference of two kinks, which reflect the transfer of quantized degrees of freedom from the boundary into the bulk of the lattice.

The coexistence of multiple phases in the domain structure resembles the stationary domain wall in a rotating-squares structure with embedded magnets, showcasing a multistable energy landscape¹⁴. Further analysis broadens our perspective, revealing that the trivial solutions correspond to distinct ground states of the system and the soliton solutions represent transition states on an effective energy surface (Figure 1h). Leveraging the continuum model, we extend the concept of multiple degenerate configurations and vector soliton solutions to any distorted lattice. Our analytical predictions offer a convenient method for identifying polarization shifts based on geometric parameters, bypassing the need to calculate phonon bands. Moreover, although we primarily focus on the static character of vector solitons, dynamic responses also showcase the superposition of sub-solitons under a boost.

Our research serves as a catalyst for developing additional models applicable to macroscopic lattices that are analogous to crystal structures. These lattices involve fixed boundaries and connected rotating blocks. This conceptual framework has potential applications in exploring other topological properties experimentally. Additionally, our study reveals the connection between topological lattices and multistable systems, presenting challenges in describing transformable topological systems through geometric insights. Lastly, solitons, which act as robust information carriers, hold promise for topologically protected information processing. The observation of multiple soliton interactions indicates the feasibility of crafting functional devices such soliton-based logic gates, diodes, and algebraic operations in mechanical computing systems.

References: 

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