Semicircle: An exact relation in the integer and fractional quantum Hall effect

We present experimental results on the quantized Hall insulator in two dimensions. This insulator, with vanishing conductivities, is characterized by the quantization (within experimental accuracy) of the Hall resistance in units of the quantum unit of resistance, h/e². The measurements were performed in a two-dimensional hole system, confined in a Ge/SiGe quantum well, when the magnetic field is increased above the ν = 1 quantum Hall state. This quantization leads to a nearly perfect semicircle relation for the diagonal and Hall conductivities. Similar results are obtained with a higher-mobility n-type modulation-doped GaAs/AlGaAs sample, when the magnetic field is increased above the ν = 1/3 fractional quantum Hall state.