Delocalization and ergodicity of the Anderson model on Bethe lattices
Delocalization and ergodicity of the Anderson model on Bethe lattices
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Bethe 格子上 Anderson 模型的离域性和遍历性
DOI：
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发表时间：
2018
期刊：
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通讯作者：
M. Tarzia
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文献类型：
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作者：
G. Biroli;M. Tarzia
本文综述了贝特格上安德森模型的离域非遍历状态的研究现状。我们还利用信念传播给出了新的结果，它包括直接在给定样本上求解格林函数的自洽递归关系。这使我们能够在数值上研究非常大的系统尺寸，并直接访问与特征函数和能级统计相关的可观测值。与最近的工作一致，我们在Cayley树上建立了一个离域非遍历相的存在性。在随机正则图上，我们的结果表明，当系统大小大于交叉尺度$N_c (W)$时，遍历性恢复，该尺度以指数方式快速发散，接近局部化过渡。这个尺度对应于平均能级间距小于Thouless能量$E_{Th} (W)$的大小。这种能量标度在接近局域跃迁时呈指数级迅速消失，在此标度以下，能级统计中的遍历性在热力学极限下得以恢复。值得注意的是，在$N_c (W)$以下的随机正则图的行为与在无循环无限Cayley树的根附近发现的行为一致，{\it =}只有在$N_c (W)$以上才出现循环的影响，随机正则图的行为与Cayley树不同。结果表明，在随机正则图的热力学极限下，遍历性得以恢复。然而，所有探测体积小于$N_c(W)$和倍于$\hbar/E_{Th} (W)$的可观测值都表现得好像存在中间阶段。考虑到$N_c(W)$和$\hbar/E_{Th} (W)$的散度非常快，这些非遍历效应在定位转变之前的一个大区域非常明显，它们可能与Cayley树上存在的中间相有关。
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.