Continuum Dirac Theory and Nonlinear Extensions of a Chain of Alternating non-linear springs: an SSH version of the Fermi-Pasta-Ulam-Tsingou model
A mechanical version of the SSH model, made of equal masses joined by two alternating non-linear springs is studied in the continuum limit. Such a non-linear dimerized mechanical chain generates a nonlinear, dispersive, two-field hyperbolic system of Boussinesq type of differential equations, describing coupled acoustic and gapped optical fields. The corresponding Lagrangian density is found, revealing that the acoustic and optical degrees of freedom are coupled through a structure that is formally analogous to a synthetic spin-orbit coupling, where the spatial motion is locked to the internal sublattice degree of freedom. An emergent non-Abelian Berry connection term also appears. Some particular solitary-wave solutions are found using variational theory, as the effective optical field obeys an undamped and unforced cubic-quintic Duffing equation. Near the Brillouin-zone edge, the system becomes a nonlinear Dirac field theory of the Soler type, providing a direct mechanical realization of interacting relativistic solitons. By factoring out fast spatial oscillations, an effective Jackiw-Rebbi Hamiltonian is obtained, with topological zero modes at domain walls. These modes are akin to the electronic flat bands in moire graphene systems associated with superconductivity and charge fractionalization. Finally, we discuss the mechanical rigidity transition and its effects on the fields as the system evolves toward the fully dimerized limit, in which the zero modes are floppy modes.