Hall Resistivity and Dephasing in the Quantum Hall Insulator

The long-standing problem of the Hall resistivity ${\rho }_{\mathrm{xy}}$ in the Hall insulator phase is addressed using four-lead Chalker-Coddington networks. Electron interaction effects are introduced via a finite dephasing length. In the quantum coherent regime, we find that ${\rho }_{\mathrm{xy}}$ scales with the longitudinal resistivity ${\rho }_{\mathrm{xx}}$, and they both diverge exponentially with the dephasing length. In the Ohmic limit (where the dephasing length is shorter than the Hall puddle size), ${\rho }_{\mathrm{xy}}$ remains quantized and independent of ${\rho }_{\mathrm{xx}}$. This suggests a new experimental probe for dephasing processes.