随机矩阵的Jacobi交叉系综与透射特征值的统计
Jacobi Crossover Ensembles of Random Matrices and Statistics of Transmission Eigenvalues
摘要：
研究了时间反演对称性破缺时混沌介观腔电导性质的转变。我们考虑两种类型的转换，即，传输本征值的布朗运动模型，正交么正和辛么正交叉取决于电子的自旋-旋转对称性的存在与否。在这两种情况下，交叉由布朗运动参数τ控制，该参数测量时间反演对称性破缺的程度。结果表明，所得结果对应于随机矩阵的Jacobi交叉系综。我们导出了传输本征值的高阶能级密度和关联函数。对于任意数目的模（通道），我们还得到了平均电导、平均散粒噪声功率和电导方差随τ的精确表达式。此外，我们给出了散粒噪声功率的方差的渐近结果的交叉，确切的结果是太长。在τ \rightarrow 0和τ \rightarrow \infty极限中，再现了正交（或辛）和酉系综的已知结果。在弱时间反演对称性破缺区域，我们的结果与半经典预言一致。
英文摘要：
We study the transition in conductance properties of chaotic mesoscopic cavities as time-reversal symmetry is broken. We consider the Brownian motion model for transmission eigenvalues for both types of transitions, viz., orthogonal-unitary and symplectic-unitary crossovers depending on the presence or absence of spin-rotation symmetry of the electron. In both cases the crossover is governed by a Brownian motion parameter τ, which measures the extent of time-reversal symmetry breaking. It is shown that the results obtained correspond to the Jacobi crossover ensembles of random matrices. We derive the level density and the correlation functions of higher orders for the transmission eigenvalues. We also obtain the exact expressions for the average conductance, average shot-noise power and variance of conductance, as functions of τ, for arbitrary number of modes (channels) in the two leads connected to the cavity. Moreover, we give the asymptotic result for the variance of shot-noise power for both the crossovers, the exact results being too long. In the τ \rightarrow 0 and τ \rightarrow \infty limits the known results for the orthogonal (or symplectic) and unitary ensembles are reproduced. In the weak time-reversal symmetry breaking regime our results are shown to be in agreement with the semiclassical predictions.
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