Generalised Langevin Equation Formulation for Anomalous Diffusion in the Ising Model at the Critical Temperature

We consider the two- (2D) and three-dimensional (3D) Ising model on a square lattice at the critical temperature $T_c$, under Monte-Carlo spin flip dynamics. The bulk magnetisation and the magnetisation of a tagged line in the 2D Ising model, and the bulk magnetisation and the magnetisation of a tagged plane in the 3D Ising model exhibit anomalous diffusion. Specifically, their mean-square displacement increases as power-laws in time, collectively denoted as $\sim t^c$, where $c$ is the anomalous exponent. We argue that the anomalous diffusion in all these quantities for the Ising model stems from time-dependent restoring forces, decaying as power-laws in time — also with exponent $c$ — in striking similarity to anomalous diffusion in polymeric systems. Prompted by our previous work that has established a memory-kernel based Generalised Langevin Equation (GLE) formulation for polymeric systems, we show that a closely analogous GLE formulation holds for the Ising model as well. We obtain the memory kernels from spin-spin correlation functions, and the formulation allows us to consistently explain anomalous diffusion as well as anomalous response of the Ising model to an externally applied magnetic field in a consistent manner.