Fluctuation of the Correlation Dimension and the Inverse Participation Number at the Anderson Transition

TL;DR

This paper investigates how the correlation dimension fluctuates at the Anderson transition, revealing a fixed point in its distribution linked to the inverse participation number's scaling, thus connecting eigenstate fluctuations to critical phenomena.

Contribution

It demonstrates a numerical link between the fluctuation of the correlation dimension and the scaling properties of the inverse participation number at the Anderson transition.

Findings

Distribution of correlation dimension has a fixed point at criticality.

State-to-state fluctuations are connected to eigenstate scaling properties.

Fixed point corresponds to the typical inverse participation number value.

Abstract

The distribution of the correlation dimension in a power law band random matrix model having critical, i.e. multifractal, eigenstates is numerically investigated. It is shown that their probability distribution function has a fixed point as the system size is varied exactly at a value obtained from the scaling properties of the typical value of the inverse participation number. Therefore the state-to-state fluctuation of the correlation dimension is tightly linked to the scaling properties of the joint probability distribution of the eigenstates.