Statistics of wave functions in mesoscopic systems
Statistics of wave functions in mesoscopic systems
复制标题
介观系统中波函数的统计
DOI：
10.1063/1.531685
复制
发表时间：
1996
影响因子：
1.3
通讯作者：
K. Efetov
中科院分区：
文献类型：
--
作者：
V. Fal’ko;K. Efetov
我们回顾了最近关于受限混沌系统中波函数涨落的研究结果。波动可能是由于随机电势或墙壁混沌散射的结果。在各种情况下计算波函数局部振幅的整个分布函数f 1 和联合两点分布。该计算是使用超对称技术进行的，并采用了非线性超对称 σ 模型的简化版本的研究，该模型专为研究混沌量子系统离散谱中单个本征态的性质而开发。对于不是很大的振幅，可以使用 σ 模型的零维近似来实现完整的描述。在各种对称类的极限下计算的分布函数显示了称为波特-托马斯统计的普遍行为，并且远处点的波动不相关。在系综之间的交叉区域中，局部振幅的分布表现出更复杂的行为：这种情况下的波动在超过平均自由程的距离上相关。对于受局域化影响最大的状态（我们称之为预局域化）产生的大振幅，零维近似不再有效。相反，它们的波函数的统计量是由简化的 σ 模型的非平凡真空决定的，该模型与共形场论中已知的刘维尔模型非常相似。特别是，简化的 σ 模型的真空态遵循刘维尔方程，这表明在二维中，预局域态具有近乎关键的性质：我们用指数谱 μ<1 证明了它们的多重分形和幂律统计平均包络线 |φ(r)|2∝r -2μ 在局域长度以下的中间距离范围内，并获得了分布函数 f 1 的对数正态尾部。我们还发现了长度短于局域化长度的准一维导线中的预局域态：它们的统计平均包络具有幂律渐近性|φ(x)|2∝x -2，并且分布函数的尾部类似于描述无限导线中的局域态。
We review the results of a recent study of fluctuations of wave functions in confined chaotic systems. The fluctuations can be due to a random potential or be a consequence of a chaotic scattering by the walls. The entire distribution function of the local amplitudes of the wave functions,f 1, and the joint two‐point distribution are calculated in various situations. The computation is performed using the supersymmetry technique and employs the studies of a reduced version of the non‐linear supersymmetric σ‐model developed especially for investigating the properties of a single eigenstate in a discrete spectrum of a chaotic quantum system. For not very large amplitudes, the complete description can be achieved using the zero‐dimensional approximation of the σ‐model. The distribution function calculated in the limit of various symmetry classes shows the universal behavior known as the Porter‐Thomas statistics, and fluctuations at distant points do not correlate. In the crossover regime between the ensembles, the distribution of local amplitudes shows a somewhat more sophisticated behavior: the fluctuations in this case are correlated over distances exceeding the mean free path. For large amplitudes generated by the states the most affected by the localization (we call them prelocalized), the zero‐dimensional approximation is no longer valid. Instead, the statistics of their wave functions is determined by nontrivial vacua of the reduced σ‐model which is quite similar to the Liouville model known in conformal field theory. In particular, the vacuum state of the reduced σ‐model obeys the Liouville equation, which indicates that in two dimensions the prelocalized states have nearly critical properties: we prove their multifractality and power‐law statistically averaged envelope |φ(r)|2∝r −2μ at the intermediate range of distances below the localization length with a spectrum of exponents μ<1, as well as obtain a logarithmically‐normal tail of the distribution functionf 1. We also find an evidence of prelocalized states in quasi‐one‐dimensional wires with the length shorter than the localization length: their statistically averaged envelope has power‐law asymptotics, |φ(x)|2∝x −2, and the tail of the distribution function is similar to that describing localized states in the infinite wires.
影响因子：
8.6
作者：
ABRAHAMS, E;ANDERSON, PW;RAMAKRISHNAN, TV
通讯作者：
RAMAKRISHNAN, TV