Dynamical scaling for critical states: Validity of Chalker’s ansatz for strong fractality

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 $b⪡1$. The power-law behavior of the quantum return probability $P_N\left(\tau \right)$ 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.

Figure 1

Duality relation verified by numerical diagonalization of RMT, Eq. (6). The deviation from Eq. (24) is less than 1% for all values of $B=\left(\pi b\right)2^{1/4}$.