Path-dependent correlations in dynamically tuned Ising models and its short-time behavior: Application of Magnus expansion

We study both analytically and numerically the buildup of antiferromagnetic (AF) correlation in the dynamically tuned Ising models. In short-time scale, we apply Magnus expansion (ME) to derive the high-order analytic expression of the connected correlation functions and compare it with exactly numerical results for the different lattice geometries, e.g., 1D chain, 2×n lattice, and n×n lattice. It is shown that the high-order expansion is required to describe accurately the buildup of AF correlation in the quench dynamics. Moreover, through a 2D square lattice, we find that the magnitude of AF correlation for the same Manhattan distance is proportional to the number of the shortest paths in a sufficiently long time until long and distinct paths are involved significantly. Finally, we propose an applicable experimental setup to realize our findings.