Algebraic law of local correlations in a driven Rydberg atomic system

Understanding the mechanism behind the buildup of inner correlations is crucial for studying nonequilibrium dynamics in complex, strongly interacting many-body systems. Here we investigate both analytically and numerically the buildup of antiferromagnetic (AF) correlations in a dynamically tuned Ising model with various geometries, realized in a Rydberg atomic system. Through second-order Magnus expansion (ME), we demonstrate quantitative agreement with numerical simulations for diverse configurations including $2 \times n$ lattice and cyclic lattice with a star. We find that the AF correlation magnitude at fixed Manhattan distance obeys a universal superposition principle: It corresponds to the algebraic sum of contributions from all shortest paths. This superposition law remains robust against variations in path equivalence, lattice geometries, and quench protocols, establishing a new paradigm for correlation propagation in quantum simulators.