Statistics of the two-point transmission at Anderson localization transitions

TL;DR

This paper investigates the multifractal statistics of two-point transmission at Anderson localization transitions using power-law random banded matrices, confirming theoretical predictions and exploring behavior across different phases.

Contribution

It numerically tests multifractal predictions for two-point transmission statistics at Anderson transitions in a power-law random banded matrix model, relating spectra to eigenfunction singularities.

Findings

Numerical results agree with multifractal spectrum relations.

Typical exponents relate to eigenfunction singularity spectrum.

Statistics in localized and delocalized phases are discussed.

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 $T_L \leq 1$; (ii) the exponents $X(q)$ that govern the moments $\overline{T_L^q} \sim 1/L^{X(q)}$ become frozen above some threshold: $X(q \geq q_{sat}) = - \Phi(\kappa=0)$, i.e. all moments of order $q \geq q_{sat}$ 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 $\Phi(\kappa \geq 0) = 2 [ f(\alpha= d+ \frac{\kappa}{2}) -d ]$ 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.