On the generalized dimensions of multifractal eigenstates

TL;DR

This paper verifies a conjectured relation between multifractal dimensions of eigenstates in critical random matrix ensembles, extends its applicability, and links scattering properties to level compressibility, offering new insights into multifractality and level correlations.

Contribution

The study confirms a conjectured relation between multifractal dimensions, extends it to new models and parameter ranges, and connects scattering delay times to level compressibility.

Findings

Validated the relation between $D_q$ and $D_{q'}$ for various models.

Extended the relation to $q<1/2$ and deterministic models.

Linked inverse moments of Wigner delay times to level compressibility.

Abstract

Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, $D_q$ and $D_{q'}$, of the eigenstates of critical random matrix ensembles: $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$, $1\le q, q' \le 2$. Here, we verify this relation by extensive numerical calculations on critical random matrix ensembles and extend its applicability to $q<1/2$ but also to deterministic models producing multifractal eigenstates and to generic multifractal structures. We also demonstrate, for the scattering version of the power-law banded random matrix model at criticality, that the scaling exponents $\sigma_q$ of the inverse moments of Wigner delay times, $\bra \tau_{\tbox W}^{-q} \ket \propto N^{-\sigma_q}$ where $N$ is the linear size of the system, are related to the level compressibility $\chi$ as $\sigma_q\approx q(1-\chi)[1+q\chi]^{-1}$ for a limited range of $q$; thus providing a way to probe level correlations by means of scattering experiments.