Probing the eigenfunction fractality with a stop watch

TL;DR

This paper investigates the fractal properties of eigenfunctions in a critical PBRM model by analyzing scattering phase distributions and delay times, revealing how these properties depend on system parameters and channel position.

Contribution

It introduces a numerical study linking eigenfunction multifractality to scattering phase and delay time distributions at criticality in the PBRM model.

Findings

Distribution of scattering phases becomes uniform with increasing bandwidth.

Inverse moments of delay times scale with system size and multifractal dimensions.

Scaling law for delay times depends on the channel position.

Abstract

We study numerically the distribution of scattering phases ${\cal P}(\Phi)$ and of Wigner delay times ${\cal P}(\tau_W)$ for the power-law banded random matrix (PBRM) model at criticality with one channel attached to it. We find that ${\cal P}(\Phi)$ is insensitive to the position of the channel and undergoes a transition towards uniformity as the bandwidth $b$ of the PBRM model increases. The inverse moments of Wigner delay times scale as $<\tau_W^{-q} >\sim L^{- q D_{q+1}}$, where $D_q$ are the multifractal dimensions of the eigenfunctions of the corresponding closed system and $L$ is the system size. The latter scaling law is sensitive to the position of the channel.