The forward approximation as a mean field approximation for the Anderson and Many Body Localization transitions

TL;DR

This paper demonstrates that the forward approximation acts as a mean field theory for Anderson and many-body localization transitions, providing accurate critical disorder estimates especially in high dimensions, and compares well with exact numerical results.

Contribution

It introduces the forward approximation as a mean field approach for localization transitions, showing its accuracy and computational efficiency in high-dimensional models.

Findings

Critical disorder estimates are within 1% of numerical values in 5D.

Critical exponent ν=1 for both Anderson and many-body localized phases.

The approximation's accuracy improves with increasing dimensionality.

Abstract

In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-) or many-body localized phase. This approximation, which consists in summing over the amplitudes of only the shortest paths in the locator expansion, is known to over-estimate 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 $h_c=4.0\pm 0.3$ is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent $\nu=1$. 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.