Statistics of the two-point transmission at Anderson localization transitions

Abstract

At Anderson critical points, the statistics of the two-point transmission $T_L$ for disordered samples of linear size $L$ is expected to be multifractal with the following properties [Janssen {\it et al} PRB 59, 15836 (1999)] : (i) the probability to have $T_L\sim 1/L^{\kappa }$ behaves as $L^{\Phi (\kappa )}$, where the multifractal spectrum $\Phi (\kappa )$ terminates at $\kappa =0$ as a consequence of the physical bound $T_L\le 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q}\sim 1/L^{X(q)}$ become frozen above some threshold: $X(q\ge q_{sat})=-\Phi (\kappa =0)$, i.e. all moments of order $q\ge q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($\kappa =0$). In the present paper, we test numerically these predictions for the ensemble of $L\times L$ power-law random banded matrices, where the random hopping $H_{i,j}$ decays as a power-law $(b/|i-j|{)}^a$. This model is known to present an Anderson transition at $a=1$ between localized ($a>1$) and extended ($a<1$) states, with critical properties that depend continuously on the parameter $b$. Our numerical results for the multifractal spectra ${\Phi }_b(\kappa )$ for various $b$ are in agreement with the relation $\Phi (\kappa \ge 0)=2[f(\alpha =d+\frac{\kappa }{2})-d]$ in terms of the singularity spectrum $f(\alpha )$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation ${\kappa }_{typ}(b)=2({\alpha }_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.

v2=final version with two new appendices with respect to v1; 12 pages, 10 figures

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• A critical Dyson hierarchical model for the Anderson localization transition
• Random elastic networks : strong disorder renormalization approach
• The Fock-space landscape of many-body localisation
• Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry
• Directional Localization in Disordered 2D Tight-Binding Systems: Insights from Single Particle Entanglement Measures
• Statistical properties of two-particle transmission at Anderson transition