  We investigate the origin of the violation of the Stokes-Einstein (SE)
relation in two-dimensional Yukawa liquids. Using comprehensive molecular
dynamics simulations, we identify the time scales supporting the violation of
the SE relation $D\propto (\eta/T)^{-1}$, where $D$ is the self-diffusion
coefficient and $\eta$ is the shear viscosity. We first compute the
self-intermediate scattering function $F_s(k,t)$, the non-Gaussian parameter
$\alpha_2$, and the autocorrelation function of the shear stress $C_{\eta}(t)$.
The timescales obtained from these functions are included the structural
relaxation time $\tau_{\alpha}$, the peak time of the non-Gaussian parameter
$\tau_{\alpha_2}$, and the shear stress relaxation time $\tau_{\eta}$. We find
that $\tau_{\eta}$ is coupled with $D$ for all temperatures indicating the SE
preservation, however, $\tau_{\alpha}$ and $\tau_{\alpha_2}$ are decoupled with
$D$ at low temperatures indicating the SE violation. Surprisingly, we find that
the origins of this violation are related to the non-exponential behavior of
the autocorrelation function of the shear stress and non-Gaussian behavior of
the distribution function of particle displacements. These results confirm
dynamic heterogeneity that occurs in two-dimensional Yukawa liquids that
reflects the presence of regions in which dust particles move faster than the
rest when the liquid cools to below the phase transition temperature.
