[論文レビュー] Delocalization and ergodicity of the Anderson model on Bethe lattices

Giulio Biroli, Marco Tarzia|arXiv (Cornell University)|Oct 17, 2018

Theoretical and Computational Physics参考文献 1被引用数 41

ひとこと要約

本論文は、Bethe lattices 上の Anderson model の delocalized non-ergodic および ergodic regime を Belief Propagation を用いて分析し、Cayley trees 上に delocalized non-ergodic 相を、random regular graphs (RRG) における disorder-dependent なスケール N_c(W) を超えた ergodic クロスオーバーを明らかにします。

ABSTRACT

We review the state of the art on the delocalized non-ergodic regime of the Anderson model on Bethe lattices. We also present new results using Belief Propagation, which consists in solving the self-consistent recursion relations for the Green's functions directly on a given sample. This allows us to numerically study very large system sizes and to directly access observables related to the eigenfunctions and energy level statistics. In agreement with recent works, we establish the existence of a delocalized non-ergodic phase on Cayley trees. On random regular graphs instead our results indicate that ergodicity is recovered when the system size is larger than a cross-over scale $N_c (W)$, which diverges exponentially fast approaching the localization transition. This scale corresponds to the size at which the mean-level spacing becomes smaller than the Thouless energy $E_{Th} (W)$. Such energy scale, which vanishes exponentially fast approaching the localization transition, is the one below which ergodicity in the level statistics is restored in the thermodynamic limit. Remarkably, the behavior of random regular graphs below $N_c (W)$ coincides with the one found close to the root of loop-less infinite Cayley trees, {\it i.e.} only above $N_c (W)$ the effects of loops emerge and random regular graphs behave differently from Cayley trees. Our results indicate that ergodicity is recovered in the thermodynamic limit on random regular graph. However, all observables probing volumes smaller than $N_c(W)$ and times smaller than $\hbar/E_{Th} (W)$ are expected to behave as if there were an intermediate phase. Given the very fast divergence of $N_c(W)$ and $\hbar/E_{Th} (W)$ these non-ergodic effects are very pronounced in a large region preceding the localization transition, and they can be related to the intermediate phase present on Cayley trees.

研究の動機と目的

• Investigate the delocalized non-ergodic regime of the Anderson model on Bethe lattices.
• Differentiate ergodic behavior on random regular graphs from non-ergodic behavior on Cayley trees.
• Develop and apply Belief Propagation to solve self-consistent Green’s function equations on large finite samples.
• Access observables related to eigenfunctions and energy level statistics, such as level compressibility and eigenstate overlaps.
• Characterize how system size and disorder influence ergodicity and multifractality in these lattices.

提案手法

• Solve self-consistent recursion relations for Green’s functions on large finite samples using Belief Propagation (BP).
• Compare BP results on random regular graphs (RRG) and Cayley trees to reveal boundary/loop effects.
• Compute level statistics via adjacent-gap ratio r and mutual overlap q between successive eigenvectors.
• Measure inverse participation ratio (IPR) and the support set to assess multifractality and ergodicity.
• Analyze the spectrum of fractal dimensions f(α) and its edge α_- and α_1 to characterize multifractality.
• Identify a crossover size N_c(W) where ergodicity is restored in RRG and relate it to Thouless energy E_Th(W).

実験結果

リサーチクエスチョン

• RQ1 Does a delocalized non-ergodic phase exist on Cayley trees and/or random regular graphs?
• RQ2 Is ergodicity restored in the thermodynamic limit on random regular graphs, and what is the crossover scale N_c(W)?
• RQ3 How do level statistics and eigenfunction statistics behave across disorder strengths and system sizes on Bethe lattices?
• RQ4 How do boundary and loop effects distinguish Cayley trees from random regular graphs in this context?
• RQ5 What is the role of multifractality in the delocalized regime and how does it relate to observable quantities like IPR and f(α)?

主な発見

• On Cayley trees, there is a genuine delocalized non-ergodic phase with multifractal eigenfunctions.
• On random regular graphs, ergodicity is recovered in the thermodynamic limit, with a crossover scale N_c(W) that diverges exponentially near the localization transition.
• Level statistics on RRG are GOE-like in the delocalized phase, but finite-size effects cause non-monotonic behavior of observables with N, characterized by N_c(W).
• The mean-level spacing crossing the Thouless energy E_Th(W) defines the energy scale below which ergodicity re-emerges in the thermodynamic limit.
• The fractal exponents D_1 and D_2 flow with N and W, indicating non-monotonic finite-size effects tied to the crossover region.
• The singularity spectrum f_N(α) shows non-monotonic evolution with system size near the crossover, linking Cayley-tree multifractality to RRG non-ergodic features.