  Monte Carlo simulations have been performed to analyze the sub-diffusion
dynamics of a tagged monomer in self-avoiding polymerized membranes in the flat
phase. By decomposing the mean square displacement into the out-of-plane
($\parallel$) and the in-plane ($\perp$) components, we obtain good data
collapse with two distinctive diffusion exponents $2 \alpha_{\parallel} = 0.36
\pm 0.01$ and $2 \alpha_{\perp} = 0.21 \pm 0.01$, and the roughness exponents
$\zeta_{\parallel} = 0.6 \pm 0.05$ and $\zeta_{\perp} = 0.25 \pm 0.05 $,
respectively for each component. Their values are consistent with the relation
from the rotational symmetry. We derive the generalized Langevin equations to
describe the sub-diffusional behaviors of a tagged monomer in the intermediate
time regime where the collective effect of internal modes in the membrane
dominate the dynamics to produce negative memory kernels with a power-law. We
also briefly discuss how the long-range hydrodynamic interactions alter the
exponents.
