Scaling of the anomalous Hall effect in the insulating regime

We develop the theory of the scaling of the anomalous Hall effect (AHE) in the insulating regime, which is observed in experiments to relate the anomalous Hall and diagonal conductivities by σ_xy^AH∝σ_xx^1.40∼1.75 for a large range of materials. This scaling is qualitatively different from that observed in metals. Basing our theory on the phonon-assisted hopping mechanism and percolation theory in random networks, we derive a general formula for the anomalous Hall conductivity, in which the percolation theory averaging of the random-linked triad clusters is a key aspect that captures the correct observed physics. We show that it scales with the longitudinal conductivity as σ_xy^AH∼σ_xx^γ with γ predicted to be 1.33⩽γ⩽1.76, quantitatively in agreement with the experimental observations. Our theory predicts that this scaling remains similar regardless of whether the hopping process is of long-range (variable-range hopping) or short-range type (activation hopping), or is influenced by interactions, i.e., the Efros-Shklovskii regime. Our theory completes the understanding of the AHE phase diagram in the insulating regime.