Sum Rule of the Hall Conductance in Random Quantum Phase Transition

The Hall conductance $\sigma_{xy}$ of two-dimensional lattice electrons with random potential is investigated. The change of $\sigma_{xy}$ due to randomness is focused on. It is a quantum phase transition where the sum rule of $\sigma_{xy}$ plays an important role. By the string (anyon) gauge, numerical study becomes possible in sufficiently weak magnetic field regime. It is essential to discuss the floating scenario in the continuum model. Topological objects in the Bloch wavefunctions, charged vortices, are obtained explicitly. The anomalous plateau transitions ($\Delta \sigma_{xy}= 2,3,... >1$) and the trajectory of delocalized states are discussed.