Slow Relaxation in Anderson Critical Systems

We study the single particle dynamics in disordered systems with long range hopping, focusing on the critical cases, i.e., the hopping amplitude decays as $1/r^d$ in $d$-dimension. We show that with strong on-site potential disorder, the return probability of the particle decays as power-law in time. As on-site potential disorder decreases, the temporal profile smoothly changes from a simple power-law to the sum of multiple power-laws with exponents ranged from $0$ to $\nu_\textrm{max}$. We analytically compute the decay exponents using a simple resonance counting argument, which quantitatively agrees with exact numerical results. Our result implies that the dynamics in Anderson Critical systems are dominated by resonances.