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

TL;DR

This paper analytically verifies Chalker's dynamical scaling hypothesis for critical multifractal eigenstates in a strong multifractality regime, confirming power law behaviors and deriving an expression for the fractal dimension.

Contribution

It provides an analytical verification of Chalker’s scaling for strong multifractality and derives an explicit formula for the fractal dimension $d_2$.

Findings

Power law behavior of return probability confirmed in different limits.

Exponents for power laws are equal up to order $b^2$.

Analytical expression for fractal dimension $d_2$ derived.

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 $\tau/N\rightarrow\infty$ and $N/\tau\rightarrow\infty$, 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.