Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, Puebla, 72570, Mexico.
Department of Physics, Yeshiva University, New York City, New York 10016, USA.
Phys Rev E. 2019 Aug;100(2-1):022142. doi: 10.1103/PhysRevE.100.022142.
This work shows that dynamical features typical of full random matrices can be observed also in the simple finite one-dimensional (1D) noninteracting Anderson model with nearest-neighbor couplings. In the thermodynamic limit, all eigenstates of this model are exponentially localized in configuration space for any infinitesimal on-site disorder strength W. But this is not the case when the model is finite and the localization length is larger than the system size L, which is a picture that can be experimentally investigated. We analyze the degree of energy-level repulsion, the structure of the eigenstates, and the time evolution of the finite 1D Anderson model as a function of the parameter ξ∝(W^{2}L)^{-1}. As ξ increases, all energy-level statistics typical of random matrix theory are observed. The statistics are reflected in the corresponding eigenstates and also in the dynamics. We show that the probability in time to find a particle initially placed on the first site of an open chain decays as fast as in full random matrices and much faster than when the particle is initially placed far from the edges. We also see that at long times, the presence of energy-level repulsion manifests in the form of the correlation hole. In addition, our results demonstrate that the hole is not exclusive to random matrix statistics, but emerges also for W=0, when it is in fact deeper.
这项工作表明，在具有最近邻耦合的简单一维（1D）非相互作用安德森模型中，也可以观察到全随机矩阵的典型动力学特征。在热力学极限下，对于任何微小的局域无序强度 W，该模型的所有本征态在构型空间中都是指数局域的。但当模型是有限的且局域长度大于系统大小 L 时，情况并非如此，这是一个可以进行实验研究的情况。我们分析了能级排斥的程度、本征态的结构以及有限 1D 安德森模型作为参数 ξ∝(W^{2}L)^{-1}的函数的时间演化。随着 ξ 的增加，观察到了随机矩阵理论的所有典型能级统计数据。这些统计数据反映在相应的本征态中，也反映在动力学中。我们表明，最初放置在开链第一个位置的粒子随时间找到的概率衰减得与完全随机矩阵一样快，比粒子最初放置在远离边缘时快得多。我们还看到，随着时间的推移，能级排斥的存在表现为相关孔的形式。此外，我们的结果表明，该孔不仅限于随机矩阵统计，而且当 W=0 时也会出现，实际上此时孔更深。
