Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices

Alexander D. Mirlin, Yan V. Fyodorov, Frank-Michael Dittes, Javier Quezada, and Thomas H. Seligman
Phys. Rev. E 54, 3221 – Published 1 October 1996
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Abstract

We study statistical properties of the ensemble of large N×N random matrices whose entries Hij decrease in a power-law fashion Hij∼|i-j|α. Mapping the problem onto a nonlinear σ model with nonlocal interaction, we find a transition from localized to extended states at α=1. At this critical value of α the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at α=1, parametrized by the value of the coupling constant of the σ model. At α>1 all states are expected to be localized with integrable power-law tails. At the same time, for 1<α<3/2 the wave packet spreading at a short time scale is superdiffusive: 〈|r|〉∼t1/(2α1), which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2<α<1 the statistical properties of eigenstates are similar to those in a metallic sample in d=(α-1/2)1 dimensions. Finally, the region α<1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (α=0). The theoretical predictions are compared with results of numerical simulations. © 1996 The American Physical Society.

  • Received 26 April 1996

DOI:https://doi.org/10.1103/PhysRevE.54.3221

©1996 American Physical Society

Authors & Affiliations

Alexander D. Mirlin, Yan V. Fyodorov, Frank-Michael Dittes, Javier Quezada, and Thomas H. Seligman

  • Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, 76128 Karlsruhe, Germany
  • Fachbereich Physik, Universität Gesamthochschule Essen, 45117 Essen, Germany
  • Forschungszentrum Rossendorf, Institut für Kern- und Hadronenphysik, 01314 Dresden, Germany
  • Tecnologico de Monterey, Guadalajara Campus, Guadalajara, Mexico
  • Instituto de Fisica, Universidad Nacional Autónoma de México Laboratorio de Cuernavaca, 62191 Cuernavaca, Morelos, Mexico

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Issue

Vol. 54, Iss. 4 — October 1996

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