Sum Rule of the Hall Conductance in Random Quantum Phase Transition

TL;DR

This paper investigates how randomness affects the Hall conductance in two-dimensional lattice electrons, revealing a sum rule during quantum phase transitions and exploring topological and delocalized state behaviors.

Contribution

It introduces a numerical approach using the string gauge to study the sum rule of Hall conductance in weak magnetic fields, highlighting topological objects and anomalous plateau transitions.

Findings

Sum rule of Hall conductance during quantum phase transition

Explicit identification of topological vortices in wavefunctions

Observation of anomalous plateau transitions with large conductance jumps

Abstract

The Hall conductance $\sigma_{xy}$ of two-dimensional {\it 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 {\it sum rule} of $\sigma_{xy}$ plays an important role. By the {\it string} (anyon) gauge, numerical study becomes possible in sufficiently weak magnetic field regime which 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.