Dynamical scaling for critical states: is Chalker's ansatz valid for strong fractality?

Abstract

The dynamical scaling for statistics of critical multifractal eigenstates proposed by Chalker is analytically verified for the critical random matrix ensemble in the limit of strong multifractality controlled by the small parameter $b\ll 1$. The power law behavior of the quantum return probability $P_N(\tau )$ as a function of the matrix size $N$ or time $\tau$ is confirmed in the limits $\tau /N\to \infty$ and $N/\tau \to \infty$, respectively, and it is shown that the exponents characterizing these power laws are equal to each other up to the order $b^2$. The corresponding analytical expression for the fractal dimension $d_2$ is found.

4 pages, 1 figure

References in corpus (3)

• Anderson Transitions
• Direct observation of Anderson localization of matter-waves in a controlled disorder
• Eigenfunction fractality and pseudogap state near superconductor-insulator transition

Cited by in corpus (5)

• Return probability and scaling exponents in the critical random matrix ensemble
• Levy flights and multifractality in quantum critical diffusion and in classical random walks on fractals
• Statistical properties of the Green function in finite size for Anderson Localization models with multifractal eigenvectors
• A critical Dyson hierarchical model for the Anderson localization transition
• Virial expansion of the non-linear sigma model in the strong coupling limit