Universality and phase diagram around half-filled Landau levels

Gated $\mathrm{GaAs}/{\mathrm{Al}}_x\mathrm{Ga}{\mathrm{As}}_{1-x}$ heterostructures were used to determine the low-temperature behavior of the two-dimensional electron gas near filling factor $\nu =\frac{1}{2}$ in the disorder-magnetic-field plane. We identify a line on which ${\sigma }_{\mathrm{xy}}$ is temperature independent, has value ${\sigma }_{\mathrm{xy}}=0.5\mathrm{}\pm 0.02\left(\frac{e^2}{h}\right)$, and a distinct line on which ${\rho }_{\mathrm{xy}}=2\mathrm{}\pm 0.04\left(\frac{h}{e^2}\right)$. The phase boundaries between the Hall insulator and the principal quantum Hall liquids at $\nu =1\mathrm{} \mathrm{and} \frac{1}{3}$ show levitation of the delocalized states of the first Landau levels for electrons and composite fermions. Finally, the data suggest that there is no true metallic phase around $\nu =\frac{1}{2}$.