Universal Relation between Longitudinal and Transverse Conductivities in Quantum Hall Effect

We show that any critical transition region between two adjacent Hall plateaus in either integer or fractional quantum Hall effect is characterized by a universal "semicircle" relationship between the longitudinal and transverse conductivities ${\sigma }_{\mathrm{xx}}$, ${\sigma }_{\mathrm{xy}}$, provided the sample is homogeneous and isotropic on a large scale. This is demonstrated both for the phase-coherent quantum transport as well as for the incoherent transport.