用局域景观理论计算极低能量下本征态局域长度
Computing the eigenstate localisation length at very low energies from Localisation Landscape Theory
摘要：
虽然安德森本地化在很大程度上是很好理解的，但它的描述传统上相当繁琐。一个最近发展起来的理论-本地化景观理论（LLT）-具有无与伦比的实力和优势，无论是计算和概念，替代方法。首先，我们证明了本地化长度不能方便地计算直接从确切的本征态，从而激发需要的LLT方法。然后，我们确认，与LLT的有效势的哈密顿量具有非常相似的低能本征态的物理势，证明了有效势在我们的新方法中起着至关重要的作用。我们继续使用LLT来计算非常低能量的最大局部化本征态的局部化长度，如本征态指数衰减的长度尺度所定义的，（手动）测试我们的发现对精确对角化。然后，我们描述了几种机制，本征态传播在更高的能量在有效势的图片击穿的隧道，并明确表明，我们的方法不再适用于这种制度。我们把我们的计算方案的背景下，通过解释连接到更一般的多维隧道问题，并讨论所涉及的近似。我们的方法计算的局域化长度可以适用于（几乎）任意的无序，连续的潜力在非常低的能量。
英文摘要：
While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -- Localisation Landscape Theory (LLT) -- has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential, justifying the crucial role the effective potential plays in our new method. We proceed to use LLT to calculate the localisation length for very low-energy, maximally localised eigenstates, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We then describe several mechanisms by which the eigenstates spread out at higher energies where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. Our method of calculating the localisation length can be applied to (nearly) arbitrary disordered, continuous potentials at very low energies.
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