A New Class of Topological Elastic Lattices with Disorders & Unconventional Boundaries

Topological lattices exhibit robust wave localization at their boundaries, a property traditionally linked to bulk-edge correspondence. Recently, a new class of elastic lattices has been identified that hosts edge states on unconventional boundaries, driven by hidden topological characteristics of the dynamical matrix. Here, we extend this framework to a broader class of elastic lattices, incorporating disorder and multiple degrees of freedom. We emphasize the significance of rank deficiency in local stiffness matrices, which enables a unique factorization of the global stiffness matrix. This insight reveals a novel family of disordered elastic lattices with a direct mapping to disordered analogs of known topological systems. We illustrate this through disordered variants of the finite-frequency Su-Schrieffer-Heeger (SSH) and Kitaev models under free-free, fixed-fixed, and mixed boundary conditions. Our approach provides a resilient strategy for designing topological lattices adaptable to diverse boundary conditions and disorders. The underlying mathematical framework is versatile and applicable to topological lattices in other fields, including photonics, acoustics, and electronic circuits.