Singular topological states for compact subwavelength analog mechanical computers

Abstract

As digital computers (DCs) approach their performance limits, further advancement in artificial intelligence (AI) demands a paradigm shift toward more powerful computing architectures. Recent developments in metamaterials have positioned wave-based analog mechanical computers (AMCs) as a promising alternative, offering rapid operation, low power consumption, and the ability to perform complex tasks. However, AMCs still face critical challenges, most notably, their high sensitivity to imperfections and noise, and their limited ability to operate at low frequencies in a compact footprint. In this work, we address these limitations by introducing a new family of topological states referred to as singular topological states which emerge from tailored spatial modulations in locally resonant metamaterials (LRMMs). This approach leverages the ability of LRMMs to achieve subwavelength functionalities in compact structures. We begin by analyzing a one-dimensional (1D) LRMM chain with embedded local resonators and demonstrate the existence of edge state modes (ESMs). To verify that these modes exhibit the same topological protection observed in conventional Bragg-based metamaterials, we extend our study to two-dimensional (2D) waveguiding configurations. Our results reveal the remarkable capacity of these modes to suppress backscattering and maintain robustness against imperfections, even though classical calculations yield a zero Chern number, apparently contradicting traditional definitions of topological protection. Finally, we harness these modes to design a topological mechanical solver capable of solving first-order differential equations. The resulting system demonstrates strong accuracy and robustness, offering a promising path toward practical, compact, low-frequency, and noise-resilient analog mechanical computing that can interface effectively with its surrounding environment.