[en] We present a full description of the nonergodic properties of wave functions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths: the largest one describes localization along rare branches and diverges with a critical exponent ν∥ = 1 at the transition. The second length, which describes localization along typical branches, reaches at the transition a finite universal value (which depends only on the connectivity of the graph), with a singularity controlled by a new critical exponent ν⊥ = 1/2. We show numerically that these two localization lengths control the finite-size scaling properties of key observables: wave-function moments, correlation functions, and spectral statistics. Our results are identical to the theoretical predictions for the typical localization length in the many-body localization transition, with the same critical exponent. This strongly suggests that the two transitions are in the same universality class and that our techniques could be directly applied in this context.
Research Center/Unit :
CESAM - Complex and Entangled Systems from Atoms to Materials - ULiège
Disciplines :
Physics
Author, co-author :
García-Mata, Ignacio; Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), CONICET–UNMdP, Funes 3350, B7602AYL Mar del Plata, Argentina
Martin, John ; Université de Liège - ULiège > Département de physique > Optique quantique
Dubertrand, Rémy; Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany and Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne NE1 8ST, United Kingdom
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Here, (Equation presented) has been determined numerically through the exponential increase of the number (Equation presented) of sites at distance (Equation presented).
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