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

TL;DR

This paper studies the critical dynamics at the 3D Anderson transition, verifying multifractal relations, analyzing wave packet behavior, and predicting experimental implications including scattering rates and temperature dependence.

Contribution

It provides numerical verification of the relation between multifractal eigenstates and critical exponents, and explores the impact on electron scattering and dynamics at the transition.

Findings

Verified the relation η = d - D_2 with η ≈ 1.3

Predicted a change in temperature dependence of electron-phonon scattering at low T

Found electron-electron scattering rate is linear in T and depends on conductance

Abstract

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.