Critical dynamics and multifractal exponents at the Anderson transition in 3d disordered systems

We investigate the dynamics of electrons in the vicinity of the Anderson transition in $d=3$ dimensions. Using the exact eigenstates from a numerical diagonalization, a number of quantities related to the critical behavior of the diffusion function are obtained. The relation $\eta = d-D_{2}$ between the correlation dimension $D_{2}$ of the multifractal eigenstates and the exponent $\eta$ which enters into correlation functions is verified. Numerically, we have $\eta\approx 1.3$. Implications of critical dynamics for experiments are predicted. We investigate the long-time behavior of the motion of a wave packet. Furthermore, electron-electron and electron-phonon scattering rates are calculated. For the latter, we predict a change of the temperature dependence for low $T$ due to $\eta$. The electron-electron scattering rate is found to be linear in $T$ and depends on the dimensionless conductance at the critical point.