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arXiv 1208.4789cond-mat.dis-nnmath-phmath.MP
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一维安德森局域化模型中E=0带中心的非局域态统计

Statistics of anomalously localized states at the center of band E=0 in the one-dimensional Anderson localization model

V. E. Kravtsov, V. I. Yudson

备注：
25 pages, 9 figures
发表信息：
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL v.46, 025001 (2013)
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摘要：

本文研究了具有弱不相关在位无序的一维紧束缚链（一维Anderson模型）中带中心（E=0）反常的本征函数振幅的分布函数$P(|ψ|^{2})$.重点讨论了$|ψ|^{2}$远大于逆典型局域化长度$\ell_{0}$的强局域化态（ALS）的概率。利用最近在我们的工作中发现的生成函数$Φ_{an}(u,ϕ)$的解，我们发现ALS概率分布$P(|ψ|^{2})$在$|ψ|^{2}\ell_{0} >> 1$。作为一个辅助的初步步骤，我们发现在$u >> 1$的生成函数$Φ_{an}(u,ϕ)$的渐近形式，可以用来计算其他统计性质的中心带异常。我们发现，在适度大的值的$|ψ|^{2}\ell_{0}$，ALS的概率在E=0小于在远离异常的能量。然而，当$|ψ|^{2}\ell_{0}$的值非常大时，趋势是相反的：在E=0处比在远离异常的能量处更容易指数地产生非常强的局域态。我们还发现了领先的长期在小$|ψ|^{2}<< \ell_{0}^{-1}$的$P(|ψ|^{2})$的行为，并表明，它是一致的指数本地化对应的李雅普诺夫指数发现较早的Kappus和Wegner和德里达和加德纳。

英文摘要：

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 $\ell_{0}$. Using the solution to the generating function $Φ_{an}(u,ϕ)$ found recently in our works we find the ALS probability distribution $P(|ψ|^{2})$ at $|ψ|^{2}\ell_{0} >> 1$. As an auxiliary preliminary step we found the asymptotic form of the generating function $Φ_{an}(u,ϕ)$ 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 $|ψ|^{2}\ell_{0}$, the probability of ALS at E=0 is smaller than at energies away from the anomaly. However, at very large values of $|ψ|^{2}\ell_{0}$, 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 $|ψ|^{2}<< \ell_{0}^{-1}$ and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner and Derrida and Gardner.

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