Sum Rule of Hall Conductance in a Random Quantum Phase Transition

Abstract

The Hall conductance ${\sigma }_{\mathrm{xy}}$ of two-dimensional lattice electrons in a random potential is investigated. The randomness induces a quantum phase transition where the sum rule of ${\sigma }_{\mathrm{xy}}$ plays an important role. Using the string (anyon) gauge, numerical study becomes possible in sufficiently weak magnetic fields essential to the floating scenario in the continuum model. Topological objects in the Bloch wave functions (charged vortices) are obtained explicitly. Anomalous plateau transitions $(\Delta {\sigma }_{\mathrm{xy}}=2,3,\dots >1\mathrm{})$ and the trajectory of delocalized states are discussed as well.