Fluctuations of the inverse participation ratio at the Anderson transition

Statistics of the inverse participation ratio (IPR) at the critical point of the localization transition is studied numerically for the power-law random banded matrix model. It is shown that the IPR distribution function is scale-invariant, with a power-law asymptotic ``tail''. This scale invariance implies that the fractal dimensions $D_q$ are non-fluctuating quantities, contrary to a recent claim in the literature. A recently proposed relation between $D_2$ and the spectral compressibility $\chi$ is violated in the regime of strong multifractality, with $\chi\to 1$ in the limit $D_2\to 0$.Comment: 4 pages, 3 eps figure

Quantum criticality and minimal conductivity in graphene with long-range disorder

We consider the conductivity $\sigma_{xx}$ of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class $\sigma$-model including a topological term with $\theta=\pi$. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of $e^2/h$. When the effective time reversal symmetry is broken, the symmetry class becomes unitary, and $\sigma_{xx}$ acquires the value characteristic for the quantum Hall transition.Comment: 4 pages, 1 figur