Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions
Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions
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前向近似作为安德森和多体定位转变的平均场近似
DOI：
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发表时间：
2015
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通讯作者：
A. Scardicchio
中科院分区：
文献类型：
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作者：
Francesca Pietracaprina;V. Ros;A. Scardicchio
本文分析了前向近似在某些具有安德森（单体）或多体定域相的模型中的预言。这种近似，它包括在定位器扩展的最短路径的振幅求和，是已知的高估的临界值的障碍，确定的发病的本地化阶段。然而，近似提供的结果变得越来越准确的局部协调（维数）的图形，定义的跳跃矩阵，是更大的。在这个意义上，前向近似可以看作是无限维中安德森跃迁的平均场理论。的总和可以有效地计算使用传输矩阵技术，并与最精确的精确对角化的结果进行比较。对于安德森问题，我们发现一个临界值的混乱，这是0.9%的最精确的可用数值已经在5个空间维度，而多体局域相的海森堡模型与随机场的临界无序h（c）= 4.0 +/- 0.3是惊人的接近最近的结果通过精确对角化。在这两种情况下，我们得到一个临界指数nu = 1。在安德森的情况下，后者并不显示依赖于维数，因为它是常见的平均场近似。我们讨论的相关性的最短路径的单和多体问题，并评论我们的结果与问题的定向聚合物在随机介质中的连接。
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 h(c) = 4.0 +/- 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.