Anderson transitions : multifractal or non-multifractal statistics of the transmission as a function of the scattering geometry

Abstract

The scaling theory of Anderson localization is based on a global conductance $g_L$ that remains a random variable of order O(1) at criticality. One realization of such a conductance is the Landauer transmission for many transverse channels. On the other hand, the statistics of the one-channel Landauer transmission between two local probes is described by a multifractal spectrum that can be related to the singularity spectrum of individual eigenstates. To better understand the relations between these two types of results, we consider various scattering geometries that interpolate between these two cases and analyse the statistics of the corresponding transmissions. We present detailed numerical results for the power-law random banded matrices (PRBM model). Our conclusions are : (i) in the presence of one isolated incoming wire and many outgoing wires, the transmission has the same multifractal statistics as the local density of states of the site where the incoming wire arrives; (ii) in the presence of backward scattering channels with respect to the case (i), the statistics of the transmission is not multifractal anymore, but becomes monofractal. Finally, we also describe how these scattering geometries influence the statistics of the transmission off criticality.

12 pages, 9 figures

References in corpus (9)

• Anderson Transitions
• Exact relations between multifractal exponents at the Anderson transition
• Boundary multifractality at the integer quantum Hall plateau transition: implications for the critical theory
• $f(\alpha )$ Multifractal spectrum at strong and weak disorder
• Boundary multifractality in critical 1D systems with long-range hopping
• Scattering at the Anderson transition: Power--law banded random matrix model
• Statistics of the two-point transmission at Anderson localization transitions
• Two-level correlation function of critical random-matrix ensembles
• Critical properties in long-range hopping Hamiltonians

Cited by in corpus (2)

• Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums
• A critical Dyson hierarchical model for the Anderson localization transition