On the generalized dimensions of multifractal eigenstates

Abstract

Recently, based on heuristic arguments, it was conjectured that an intimate relation exists between any multifractal dimensions, $D_q$ and $D_{q^{\prime }}$, of the eigenstates of critical random matrix ensembles: $D_{q^{\prime }}\approx q{D}_q[q^{\prime }+(q-q^{\prime })D_q{]}^{-1}$, $1\le q,q^{\prime }\le 2$. 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, $\bra τ_{\tbox W}^{-q} \ket \propto N^{-σ_q}$ 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.

12 pages, 15 figures. Minor corrections made. arXiv admin note: text overlap with arXiv:1201.6353