  We numerically study shear-induced diffusion of soft athermal particles in
two dimensions. The Green-Kubo (GK) relation is applicable to diffusion
coefficient of the particles near jamming, where both mean squared particle
velocities and relaxation time included in the GK formula are well explained by
critical scaling. We show that auto-correlation functions of the transverse
velocities are stretched exponential if the system is below jamming or shear
rate is large enough. However, if the system is above jamming and the shear
rate is sufficiently small, the auto-correlations exhibit long-time tails such
that time integral in the GK formula diverges in two dimensions. We propose
empirical scaling relations for the critical exponents and demonstrate that the
long-time tails cause finite size effects on the shear-induced diffusion
coefficient.
