Hall conductivity (${\sigma }_{xy}$)

Hall conductivity is the transverse part of the electrical response in which an applied electric field drives a perpendicular current density. In two dimensions, one convention writes $j_x={\sigma }_{xx}{E}_x+{\sigma }_{xy}{E}_y$ and $j_y=-{\sigma }_{xy}{E}_x+{\sigma }_{xx}{E}_y$.

The sign depends on the magnetic-field orientation and chosen axes. In a quantum Hall effect plateau, the longitudinal response is strongly suppressed and the Hall response takes quantized values. Electrical conductivity and electrical resistivity are inverse tensors; taking the reciprocal of a single component is generally incorrect when longitudinal and transverse responses coexist.