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Two critical localization lengths in the Anderson transition on random graphs

I. García-Mata, J. Martin, R. Dubertrand, O. Giraud, B. Georgeot, and G. Lemarié
Phys. Rev. Research 2, 012020(R) – Published 21 January 2020

Abstract

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.

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  • Received 24 April 2019
  • Revised 29 August 2019

DOI:https://doi.org/10.1103/PhysRevResearch.2.012020

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsGeneral PhysicsNetworks

Authors & Affiliations

I. García-Mata1,2, J. Martin3, R. Dubertrand4, O. Giraud5, B. Georgeot6, and G. Lemarié6,*

  • 1Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), CONICET–UNMdP, Funes 3350, B7602AYL Mar del Plata, Argentina
  • 2Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET), Argentina
  • 3Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, Université de Liège, Bât. B15, B-4000 Liège, Belgium
  • 4Institut 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
  • 5LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
  • 6Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, CNRS, UPS, France

  • *Corresponding author: lemarie@irsamc.ups-tlse.fr

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Vol. 2, Iss. 1 — January - March 2020

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