Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions

Francesca Pietracaprina, Valentina Ros, and Antonello Scardicchio
Phys. Rev. B 93, 054201 – Published 1 February 2016

Abstract

In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-body) or many-body localized phase. This approximation, which consists of summing over the amplitudes of only the shortest paths in the locator expansion, is known to overestimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean-field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is 0.9% off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder hc=4.0±0.3 is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent ν=1. In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean-field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.

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  • Received 25 September 2015
  • Revised 7 January 2016

DOI:https://doi.org/10.1103/PhysRevB.93.054201

©2016 American Physical Society

Authors & Affiliations

Francesca Pietracaprina and Valentina Ros

  • SISSA - International School for Advanced Studies, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

Antonello Scardicchio

  • Abdus Salam ICTP, Strada Costiera 11, 34151 Trieste, Italy; INFN Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy; and Dipartimento di Fisica, Università degli Studi di Bari “Aldo Moro”, I-70126 Bari, Italy

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Issue

Vol. 93, Iss. 5 — 1 February 2016

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