Corrections to the Bethe lattice solution of Anderson localization

TL;DR

This paper develops a numerical method to compute corrections to the Bethe lattice solution of Anderson localization, revealing how finite-size effects and loops influence critical behavior in various lattice structures.

Contribution

It introduces a new approach to calculate corrections to the Bethe lattice solution, accounting for finite-size and loop effects in RRGs and Euclidean lattices, and predicts the destruction of critical behavior in finite dimensions.

Findings

$1/N$ corrections diverge exponentially near criticality.

Finite loops destroy the Bethe lattice's critical behavior.

Supports the idea that the upper critical dimension is infinity.

Abstract

We study numerically Anderson localization on lattices that are tree-like except for the presence of one loop of varying length $L$. The resulting expressions allow us to compute corrections to the Bethe lattice solution on i) Random-Regular-Graph (RRG) of finite size $N$ and ii) euclidean lattices in finite dimension. In the first case we show that the $1/N$ corrections to to the average values of observables such as the typical density of states and the inverse participation ratio have prefactors that diverge exponentially approaching the critical point, which explains the puzzling observation that the numerical simulations on finite RRGs deviate spectacularly from the expected asymptotic behavior. In the second case our results, combined with the $M$-layer expansion, predict that corrections destroy the exotic critical behavior of the Bethe lattice solution in any finite dimension, strengthening the suggestion that the upper critical dimension of Anderson localization is infinity. This approach opens the way to the computation of non-mean-field critical exponents by resumming the series of diverging diagrams through the same recipes of the field-theoretical perturbative expansion.