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Although anomalous diffusion is a ubiquitous phenomenon in physics, biology, and many other research fields, it is only recently that the phenomenon was observed in the two-dimensional Ising model at its critical temperature.

In this thesis, Zhong reports that at the critical point, anomalous diffusion is a common phenomenon for both two- (2D) and three-dimensional (3D) Ising models with single-spin flip dynamics. Zhong has numerically shown that the Generalized Langevin Equation is a proper model to describe the anomalous diffusion in Ising-like systems. For temperatures around the critical temperature, Zhong finds that the diffusion exponent is flowing away from the critical value of the true exponent on both sides. This indicates that these results could be treated as a method to identify the phase transition in Ising systems.

Besides, Zhong finds that the anomalous exponent can be used to measure the dynamical exponent for Ising-like systems. With this new method to calculate the dynamical exponent, Zhong confirms that the phi^4 model shares its dynamical exponent with the 2D Ising model. In other words, they belong to the same universality class. Zhong then extends this method to measure the dynamical exponent of the 2D bond-diluted Ising model. Here Zhong finds that the dynamical exponent increases to infinity when the bond concentration approaches the percolation threshold; scientists refer to this as "super slowing down" behavior. Finally, by treating the Fourier modes of the magnetization as the dynamical eigenmodes, Zhong derives the mean-square displacement and autocorrelation function of the magnetization in the Ising model with Kawasaki dynamics analytically in approximation. The results are consistent with results from the simulations.