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 ${\sigma }_{xy}^{\text{AH}}\propto {\sigma }_{xx}^{1.40\sim 1.75}$ 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 ${\sigma }_{xy}^{\text{AH}}\sim {\sigma }_{xx}^{\gamma }$ with $\gamma$ predicted to be $1.33⩽\gamma ⩽1.76$, 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.

• Received 12 September 2011

DOI:https://doi.org/10.1103/PhysRevB.84.165304