2011/01/31 by Cécile Monthus, Cecile Monthus, Thomas Garel · 47 citations
Mathematics · Physics and Astronomy · #Anderson localization #Criticality #Measure (data warehouse) #Moment (physics) #Multifractal system #Quantum and electron transport phenomena #Random lasers and scattering media #Scattering #Second moment of area #Singularity #Spectral Theory in Mathematical Physics #Tree (set theory) #cond-mat.dis-nn
paper · pdf · doi:10.1088/1751-8113/44/14/145001
published in Journal of Physics A Mathematical and Theoretical 44(14), 145001 (Institute of Physics) · v2=final version (16 pages)
openalex publication_date 2011/03/08 · arxiv created 2011/03/09 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/08
In contrast to finite dimensions where disordered systems display multifractal statistics only at criticality, the tree geometry induces multifractal statistics for disordered systems also off criticality. For the Anderson tight-binding localization model defined on a tree of branching ratio K = 2 with N generations, we consider the Miller–Derrida scattering geometry (1994 J. Stat. Phys. 75 357), where an incoming wire is attached to the root of the tree, and where K N outcoming wires are attached to the leaves of the tree. In terms of the K N transmission amplitudes t j , the total Landauer transmission is T ≡ ∑ j | t j | 2 , so that each channel j is characterized by the weight w j = | t j | 2 / T . We numerically measure the typical multifractal singularity spectrum f (α) of these weights as a function of the disorder strength W and we obtain the following conclusions for its left termination point α + ( W ). In the delocalized phase W < W c , α + ( W ) is strictly positive α + ( W ) > 0 and is associated with a moment index q + ( W ) > 1. At criticality, it vanishes α + ( W c ) = 0 and is associated with the moment index q + ( W c ) = 1. In the localized phase W > W c , α + ( W ) = 0 is associated with some moment index q + ( W ) < 1. We discuss the similarities with the exact results concerning the multifractal properties of the directed polymer on the Cayley tree.