Comment on “Scaling behavior of classical wave transport in mesoscopic media at the localization transition”

Abstract

We emphasize the importance of the position dependence of the diffusion coefficient D(r) in the self-consistent theory of localization and argue that the scaling law T ∝ ln L/L 2 obtained by Cheung and Zhang [Phys. Rev. B 72, 235102 (2005)] for the average transmission coefficient T of a disordered slab of thickness L at the localization transition is an artifact of replacing D(r) by its harmonic mean. The correct scaling T ∝ 1/L 2 is obtained by properly treating the position dependence of D(r). PACS numbers: 42.25.DdIn a recent paper [1] Cheung and Zhang (CZ) apply the self-consistent (SC) theory of localization to study the transmission of waves through a slab of disordered medium at the Anderson localization transition. The SC theory is a powerful tool to deal with the phenomenon of Anderson localization, but its application to disordered media of finite size requires some care. In the original papers by Vollhardt and Wölfle [2], the size L of disordered sample was acknowledged using a lower cut-off in the integration over momentum. Despite the obvious crudeness of this approach, it was sufficient to recover the main results of the scaling theory of localization [3] and added a great physical insight into the phenomenon of disorder-induced localization. Later on, Van Tiggelen et al. [4] argued that in a medium of finite size the SC theory naturally leads to a position dependence of the diffusion coefficient D(r). This adapted SC theory was successfully applied to study coherent backscattering [4] and dynamics [5,6] of localized waves. Microscopic justifications for position dependence of D have been recently presented based on the diagrammatic [7] and field-theoretic [8] calculations.CZ propose a way of overcoming technical difficulties caused by the position dependence of D(r) [1] (see also [9]). They average the equation for 1/D(r), their Eq. (1), over the sample volume, thus replacing D(r) by its harmonic meanD. In this Comment we argue that although such an approach can be justified in the weak localization regime [9], it is not adequate at the mobility edge and in the Anderson localization regime. In particular, our calculations that properly treat the position dependence of D(r), do not confirm the scaling law T ∝ ln L/L 2 found by CZ for the transmission coefficient T of a disordered slab of thickness L at the mobility edge. Instead, we find T ∝ 1/L 2 in agreement with the scaling theory of localization [3].To study the scaling of the average transmission coefficient T with the thickness L of disordered slab, we solve the two equations of SC theory -Eqs. (1) and (2) of Ref.[6] with Ω = 0 (stationary regime) and k = 1 (mobility edge) [10] -numerically. We use the same boundary FIG. 1: Average transmission coefficient T of a disordered slab of thickness L at the Anderson localization transition. Circles were obtained from the self-consistent theory of localization with a position-dependent diffusion coefficient D(z) by numerical solution [6]. The solid red line is a fit to the numerical results using Eq....

Self-Consistent Theory of Anderson Localization: General Formalism and Applications

“…The scaling of the transmission coefficient for classical waves through a disordered madium near the Anderson transition was considered within the position dependent self-consistent theory in Ref. 76 The transmission of microwave pulses through quasi-one-dimensional samples has been measured recently and was analyzed in terms of the self-consistent theory. 77 It was found that while the self-consistent theory can account very well for the propagation at intermediate times, it fails at longer times when the transport occurs by hopping between localized regions.…”

Anderson localization as position-dependent diffusion in disordered waveguides

“…The new ingredient is the position dependence of the renormalized diffusion coefficient D(r) that accounts for a stronger impact of interference effects in the bulk of the disordered sample as compared to the regions adjacent to boundaries. This position dependence is crucial in open disordered media 16 . D(r) also appears in the supersymmetry approach to wave transport 17 , which confirms that this concept goes beyond a particular technique (diagrammatic or supersymmetry methods) used in the calculations.…”