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 $T_L \sim 1/L^κ$ behaves as $L^{Φ(κ)}$, where the multifractal spectrum $Φ(κ)$ terminates at $κ=0$ 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: $X(q \geq q_{sat}) = - Φ(κ=0)$, i.e. all moments of order $q \geq q_{sat}$ are governed by the measure of the rare samples having a finite transmission ($κ=0$). 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 $Φ_b(κ)$ for various $b$ are in agreement with the relation $Φ(κ\geq 0) = 2 [ f(α= d+ \fracκ{2}) -d ]$ in terms of the singularity spectrum $f(α)$ of individual critical eigenfunctions, in particular the typical exponents are related via the relation $κ_{typ}(b)= 2 (α_{typ}(b)-d)$. We also discuss the statistics of the two-point transmission in the delocalized phase and in the localized phase.