Scaling of the Static Conductivity in the Quantum Hall Effect

We performed a numerical study of the static diagonal conductivity ${\sigma }_{\mathrm{xx}}$ in the lowest Landau level for a disordered two-dimensional system in a magnetic field with short range impurity potentials. We find scaling of the conductivity peak at a single critical energy which is governed by both the localization length exponent $\nu =2.37\mathrm{}\pm 0.05$ and an exponent ${\eta }^{\prime }=1.63\mathrm{}\pm 0.03$. We argue that ${\eta }^1$ can be identified with the fractal dimension $D\left(2\right)\mathrm{}$. For the value of the critical conductivity we obtained $\frac{(0.5\mathrm{}\pm 0.02)e^2}{h}$ in agreement with the hypothesis of universality.