Altshuler B L, Cuevas E, Ioffe L B, Kravtsov V E
Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA.
Departamento de Física, Universidad de Murcia, E30071 Murcia, Spain.
Phys Rev Lett. 2016 Oct 7;117(15):156601. doi: 10.1103/PhysRevLett.117.156601. Epub 2016 Oct 6.
We combine numerical diagonalization with semianalytical calculations to prove the existence of the intermediate nonergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson, and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered random regular graphs (RRG) of N sites with the connectivity K=2. By extrapolation of the results of both approaches to N→∞ we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamics exponent D(W) with the accuracy sufficient to claim that they are nontrivial in the broad interval of disorder strength W_{E}<W<W_{c}. The thorough analysis of the exact diagonalization results for RRG with N>10^{5} reveals a singularity in D_{1,2}(W) dependencies which provides clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}≈10.0. We discuss the implications of these results for quantum and classical nonintegrable and many-body systems.
我们将数值对角化与半解析计算相结合，以证明在无序分层晶格上的安德森模型中存在中间非遍历但非局域化相。我们提出了一种新的广义布居动力学，它能够在阿布 - 查克拉、安德森和索利斯递归方案中检测非局域化态的遍历性违反情况。通过对具有连通性(K = 2)的(N)个位点的无序随机正则图（RRG）集合上的安德森模型进行精确对角化所提取的随机波函数统计，对这一结果进行了补充。通过将两种方法的结果外推到(N→∞)，我们获得了分形维数(D_{1}(W))和(D_{2}(W))以及布居动力学指数(D(W))，其精度足以声称它们在无序强度(W_{E}<W<W_{c})的宽区间内是非平凡的。对(N>10^{5})的RRG的精确对角化结果的深入分析揭示了(D_{1,2}(W))依赖关系中的一个奇点，这为在(W_{E}≈10.0)时RRG上两个非局域化相之间的一阶转变提供了明确证据。我们讨论了这些结果对量子和经典非可积及多体系统的影响。
