SCALING THEORY OF THE HALL-COEFFICIENT NEAR THE METAL-INSULATOR-TRANSITION, A RENORMALIZATION-GROUP APPROACH

We investigate the critical properties of the Hall conductivity in wea k magnetic fields near the disorder driven metal-insulator transition. We derive an effective lagrangian governing the interactions of the d iffusive modes in the presence of time-reversal and particle-hole symm etry breaking and identify a new operator relevant to the scaling beha vior of the weak-field Hall conductivity. A Wilson-Polyakov renormaliz ation-group analysis is carried out and the flow equations are obtaine d near two dimensions. We show that the perturbative behavior of the H all 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 renorma lization-group equations close to the zero-field fixed point, our theo ry predicts that the Hall number vanishes at the mobility edge with an exponent g = 1 in agreement with several experimental observations. W e clarify the connection between the field-theoretic approach and the conventional perturbative approach and elucidate why perturbation in t he disorder, which is valid in the weak localization regime fails to p redict the correct critical behavior.