Finite-size scaling analysis of localization transition for scalar waves in a three-dimensional ensemble of resonant point scatterers

Abstract

We use the random Green's matrix model to study the scaling properties of the localization transition for scalar waves in a three-dimensional (3D) ensemble of resonant point scatterers. We show that the probability density p(g) of normalized decay rates of quasi-modes g is very broad at the transition and in the localized regime and that it does not obey a single-parameter scaling law for finite system sizes that we can access. The single-parameter scaling law holds, however, for the small-g part of p(g) which we exploit to estimate the critical exponent ν of the localization transition. Finite-size scaling analysis of small-q percentiles gq of p(g) yields an estimate ν 1.55 ± 0.07. This value is consistent with previous results for Anderson transition in the 3D orthogonal universality class and suggests that the localization transition under study belongs to the same class.

Search for Anderson localization of light by cold atoms in a static electric field

“…By analogy with previous studies of scalar waves [29], light [14,16,18] and elastic waves [30], we expect the eigenvectors of G to be extended at low number densities ρ of atoms (ρ = N/V , where V = 4πR 3 /3 is the volume in which the atoms are distributed). This is indeed confirmed by the calculation of the inverse participation ratio (IPR) of eigenvectors which quantifies the degree of eigenvector localization:…”

“…where Ψ αjm denotes the m-th component of the eigenvector Ψ α on the atom j and we assumed that the eigenvectors are normalized: of the eigenvalue distribution, where very few eigenvalues are found for a given atomic configuration. The latter branches correspond to eigenvectors localized on pairs of closely located atoms and have been previously shown to exist for all types of waves [14,16,29,30]. Analytic expressions of these branches can be readily obtained, see Eqs.…”

Intensity of Waves Inside a Strongly Disordered Medium

“…We have extensively studied Anderson localization in the model defined by Eqs. (1)-(3) in our previous works [26,27]. In particular, we have found that spatially localized modes appear for scatterer number densities ρ = N/V k 3 0 /4π in a narrow frequency band between two density-dependent mobility edges ω I c = ω I c (ρ/k 3 0 ) and ω II c = ω II c (ρ/k 3 0 ) slightly above the resonance frequency ω 0 .…”

Aperiodic Photonics of Elliptic Curves

“…The Green's matrix (6) is a non-Hermitian matrix. As a consequence, it has complex eigenvalues Λ n (n ∈ 1, 2, · · · , 3N ) with [65][66][67][68]. Moreover, it is important to realize that the Green's matrix method is an eigenvalue method that captures the fundamental physics of multiple scattering of vector waves for any assembly of electric scattering point dipoles.…”