Transport coefficients close to the mobility edge and nonlinear -model composite operators

We have expressed the conductivity and the ultrasonic attenuation coefficient in terms of nonlinear -model composite operators. The conductivity is represented by the sum of local and nonlocal terms. The nonlocal part is given by a correlation function of operators involving derivatives. It is expected to vanish to three-loop order in the orthogonal case but gives a nonzero result at two-loop order in the unitary case. The ultrasonic attenuation coefficient is represented by a local term only. We use this result to derive the scaling behavior of the ultrasonic attenuation coefficient close to the mobility edge in both the orthogonal and the unitary case. © 1987 The American Physical Society.