Multifractal dimensions for critical random matrix ensembles

TL;DR

This paper proposes and numerically verifies a relation between eigenfunction multifractal dimensions and level compressibility in critical random matrix ensembles, extending understanding of the Anderson transition.

Contribution

It introduces a new conjectured relation between eigenfunction multifractal dimensions and level compressibility, supported by extensive numerical evidence.

Findings

Verified the relation between $D_q$ and $D_{q'}$ numerically.

Established a unified relation between $D_q$ and level compressibility $hi$.

Extended the understanding of multifractality at the Anderson transition.

Abstract

Based on heuristic arguments we conjecture that an intimate relation exists between the eigenfunction multifractal dimensions $D_q$ of the eigenstates of critical random matrix ensembles $D_{q'} \approx qD_q[q'+(q-q')D_q]^{-1}$, $1\le q \le 2$. We verify this relation by extensive numerical calculations. We also demonstrate that the level compressibility $\chi$ describing level correlations can be related to $D_q$ in a unified way as $D_q=(1-\chi)[1+(q-1)\chi]^{-1}$, thus generalizing existing relations with relevance to the disorder driven Anderson--transition.