Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires
Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires
复制标题
布朗运动系综和传输特征值的参数相关性：在耦合量子台球和无序线中的应用
DOI：
10.1051/jp1:1995111
复制
发表时间：
1995
期刊：
影响因子：
--
通讯作者：
J. Pichard
中科院分区：
文献类型：
--
作者：
K. Frahm;J. Pichard
假设两个不同的布朗运动系综，计算 N 通道量子散射体的传输特征值 Ti 的参数相关性。第一个是 Dyson 引入的原始系综，并假设 S 矩阵具有各向同性扩散。我们推导了传输特征值的相应 Fokker-Planck 方程，该方程可以针对酉情况映射到一维虚数时间的 N 个非相互作用费米子的精确可解问题上。在一定限度内，我们为 T i 恢复了与最近获得的能级相同的通用参数相关性。作为一种应用，我们考虑在施加磁场时通过两个由 n 通道点接触弱耦合的混沌腔进行传输。假设每个混沌腔的 S 矩阵属于戴森圆形酉系综 (CUE)，并且当 n 增加时，它具有 2 x CUE → 1 CUE 交叉。我们计算传输特征值 Ti 的所有类型的相关函数，并得到平均电导及其方差的精确有限 N 个结果，作为参数 n 的函数。第二布朗运动系综假设传递矩阵 M 是由乘法组合律产生的各向同性扩散。众所周知，该模型描述了长度为 L 的无序线，并给出了另一个描述 Ti 的 L 相关性的 Fokker-Planck 方程。 Beenakker 和 Rejaei 最近获得了该方程在酉情况下的精确解，给出了它们的 L 相关联合概率分布。利用这个结果，我们展示了如何计算任意 L 和 N 的所有类型的相关函数。这使我们能够得到平均电导的积分表达式，该表达式在极限 N → ∞ 内与 Zirnbauer 等人获得的微观非线性模型结果一致，从而建立了两种方法的等效性。我们回顾了通过两个弱耦合量子点和通过无序线的传输之间的定性差异，并讨论了两个布朗运动模型的福克-普朗克方程之间的数学类比。
The parametric correlations of the transmission eigenvalues T i of a N-channel quantum scatterer are calculated assuming two different Brownian motion ensembles. The first one is the original ensemble introduced by Dyson and assumes an isotropic diffusion for the S-matrix. We derive the corresponding Fokker-Planck equation for the transmission eigenvalues, which can be mapped for the unitary case onto an exactly solvable problem of N non-interacting fermions in one dimension with imaginary time. We recover for the T i the same universal parametric correlation than the ones recently obtained for the energy levels, within certain limits. As an application, we consider transmission through two chaotic cavities weakly coupled by a n-channel point contact when a magnetic field is applied. The S-matrix of each chaotic cavity is assumed to belong to the Dyson circular unitary ensemble (CUE) and one has a 2 x CUE → one CUE crossover when n increases. We calculate all types of correlation functions for the transmission eigenvalues T i and we get exact finite N results for the averaged conductance and its variance , as a function of the parameter n. The second Brownian motion ensemble assumes for the transfer matrix M an isotropic diffusion yielded by a multiplicative combination law. This model is known to describe a disordered wire of length L and gives another Fokker-Planck equation which describes the L-dependence of the T i . An exact solution of this equation in the unitary case has recently been obtained by Beenakker and Rejaei, which gives their L-dependent joint probability distribution. Using this result, we show how to calculate all types of correlation functions, for arbitrary L and N. This allows us to get an integral expression for the average conductance which coincides in the limit N → ∞ with the microscopic non linear -model results obtained by Zirnbauer et al., establishing the equivalence of the two approaches. We review the qualitative differences between transmission through two weakly coupled quantum dots and through a disordered line and we discuss the mathematical analogies between the Fokker-Planck equations of the two Brownian motion models.