Scaling theory of the Hall coefficient near the metal-insulator transition, a renormalization-group approach

Abstract

We investigate the critical properties of the Hall conduxtivity in weak magetic fields near the disorder driven metal-insulator transition. We derive an effective lagrangian governing the interactions of the diffusive modes in the presence of time-reversal and particle-hole symmetry breaking and identify a new operator relevant to the scaling behavior of the weak-field Hall conductivity. A Wilson-Polyakov renormalization-group analysis is carried out and the flow equations are obtained near two dimensions. We show that the perturbative behavior of the Hall conductivity in the weakly localized regime, in two dimensions, is not simply related to the critical behavior of the Hall conductivity, due to the role played by a new scaling variable. Solving the renormalization-group equations close to the zero-field fixed point, our theory predicts that the Hall number vanishes at the mobility edge with an exponent g = 1 in agreement with several experimental observations. We clarify the connection between the field-theoretic approach and the conventional perturbative approach and elucidate why perturbation in the disorder, which is valid in the weak localization regime fails to predict the correct critical behavior.