Anomalous diffusion in the Ising-like models

Abstract

In this thesis, we first report that at the critical point, anomalous diffusion is a common phenomenon for both $2D$ and $3D$ Ising models with single-spin flip dynamics. Specifically, we numerically find that anomalous diffusion in Ising-like systems follows the Generalised Langevin Equation description. We also find that for temperatures around $T_c$, the anomalous diffusion exponent flows away from the critical value of the true exponent on both sides, which indicates that the results could be treated as a method to identify phase transition in Ising systems. Secondly, we report that the anomalous diffusion exponents can be used to measure the dynamical exponents for Ising-like systems. With this new method to calculate the dynamical exponent, we numerically confirm that the $\phi^4$ model shares its dynamical exponent with the $2D$ Ising model, i.e., they belong to the same universality class. We also extend this method to numerically obtain the dynamical exponent of the $2D$ bond-diluted Ising model, from which we find that the dynamical exponent increases to infinity when the bond concentration approaches the percolation threshold; we refer to this as "super slowing down" behavior. Finally, by treating the Fourier modes of magnetization as approximate dynamical eigenmodes, we analytically derive the mean square displacement and autocorrelation function for magnetization in the Ising model with Kawasaki dynamics.