Phase diagram of the weak-magnetic-field quantum Hall transition quantified from classical percolation

Abstract

We consider magnetotransport in high-mobility two-dimensional electron gas ${\ensuremath{\sigma}}_{xx}\ensuremath{\gg}1$ in a nonquantizing magnetic field. We employ a weakly chiral network model to test numerically the prediction of the scaling theory that the transition from an Anderson to a quantum Hall insulator takes place when the Drude value of the nondiagonal conductivity ${\ensuremath{\sigma}}_{xy}$ is equal to $1/2$ (in the units of ${e}^{2}/h$). The weaker the magnetic field, the harder it is to locate a delocalization transition using quantum simulations. The main idea of this study is that the position of the transition does not change when a strong local inhomogeneity is introduced. Since the strong inhomogeneity suppresses interference, transport reduces to classical percolation. We show that the corresponding percolation problem is bond percolation over two sublattices coupled to each other by random bonds. Simulation of this percolation allows us to access the domain of very weak magnetic fields. Simulation results confirm the criterion ${\ensuremath{\sigma}}_{xy}=1/2$ for values ${\ensuremath{\sigma}}_{xx}\ensuremath{\sim}10$, where they agree with earlier quantum simulation results. However, for larger ${\ensuremath{\sigma}}_{xx}$, we find that the transition boundary is described by ${\ensuremath{\sigma}}_{xy}\ensuremath{\sim}{\ensuremath{\sigma}}_{xx}^{\ensuremath{\kappa}}$ with $\ensuremath{\kappa}\ensuremath{\approx}0.5$, i.e., the transition takes place at higher magnetic fields. The strong inhomogeneity limit of magnetotransport in the presence of a random magnetic field, pertinent to composite fermions, corresponds to a different percolation problem. In this limit, we find for the delocalization transition boundary ${\ensuremath{\sigma}}_{xy}\ensuremath{\sim}{\ensuremath{\sigma}}_{xx}^{0.6}$.