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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >Statistics of anomalously localized states at the center of band e = 0 in the one-dimensional Anderson localization model
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Statistics of anomalously localized states at the center of band e = 0 in the one-dimensional Anderson localization model

机译：一维安德森定位模型中位于e = 0波段中心处的异常定位状态的统计量

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摘要

We consider the distribution function P(|ψ|~2) of the eigenfunction amplitude at the center-of-band (E = 0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with |ψ|~2 much larger than the inverse typical localization length ?0. Using the recently found solution for the generating function Φan(u, φ) we obtain the ALS probability distribution P(|ψ|~2) at |ψ|~2?0 ? 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Φan(u, φ) at u ? 1 which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of |ψ|~2?0, the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of |ψ|~2?0, the tendency is inverted: it is exponentially easier to create a very strongly localized state at E = 0 than at energies away from the anomaly. We also found the leading term in the behavior of P(|ψ|~2) at small |ψ|~2 ? ?~(-1)0 and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.
机译：我们考虑一维紧密结合链中具有弱的不相关现场障碍的一维紧束链（E = 0）的特征函数振幅在带中心（E = 0）异常处的分布函数P（|ψ| ~~ 2）维安德森模型）。特别强调的是|ψ|〜2的异常局域化状态（ALS）的概率远大于逆向典型局域化长度λ0。使用最近发现的生成函数Φan（u，φ）的解，我们获得|ψ|〜2？0？处的ALS概率分布P（|ψ|〜2）。 1.作为辅助的初步步骤，我们在u？处找到了生成函数Φan（u，φ）的渐近形式。 1，可用于计算频带中心异常处的其他统计属性。我们表明，在|ψ|〜2？0的适度大值下，E = 0时ALS的概率小于远离异常处的能量。然而，在|ψ|〜2？0的非常大的值时，这种趋势就相反了：在E = 0处比在远离异常处的能量处创建非常强烈的局部状态要容易得多。我们还发现在|ψ|〜2？小时P（|ψ|〜2）的行为中的主导项。并证明它与Kappus和Wegner较早发现的Lyapunov指数对应的指数局部化是一致的。

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