We establish that the Fourier modes of the magnetization serve as the dynamical eigenmodes for the two-dimensional Ising model at the critical temperature with local spin-exchange moves, i.e., Kawasaki dynamics. We obtain the dynamical scaling properties for these modes, and use them to calculate the time evolution of two dynamical quantities for the system, namely the autocorrelation function and the mean-square deviation of the line magnetizations. At intermediate times $1 lesssim t lesssim L^{z_c}$, where $z_c=4-eta=15/4$ is the dynamical critical exponent of the model, we find that the line magnetization undergoes anomalous diffusion. Following our recent work on anomalous diffusion in spin models, we demonstrate that the Generalized Langevin Equation (GLE) with a memory kernel consistently describes the anomalous diffusion, verifying the corresponding fluctuation-dissipation theorem with the calculation of the force autocorrelation function.
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