Numerical investigations of Anderson and Quantum Hall transitions:logarithmic conformal invariance, criticality without fine-tuning, multifractal dynamics

无序可以阻碍，事实上，完全停止波的传播，而是被限制在有限扩展的体积内。p·w·安德森在60多年前发现了这种被称为安德森定位的效应。通过调整外部控制参数，例如，可以驱动将局域相与无序金属的其他相连接起来的无序强度转变。各自相变的基本理论方面是本提案的主题。重点将放在量子霍尔（QH）跃迁上，它是特殊的，因为它们连接了由拓扑（第一陈氏）指数不同的局域相。该建议追求三个密切相关的目标。(i)量子场论的最新成果预测了临界波函数振幅第q阶矩的系统尺寸缩放与矩（q* - q）之间的对称关系；具体来说，x_q = x_{q*-q}表示相应的缩放指数。数字q*取决于手头特定跃迁的对称类。此外，在局部保形不变性的情况下，预测了x_q对q的抛物线依赖性。引人注目的是，最近的数值工作排除了c类跃迁的抛物线依赖性，很可能也排除了整数QH效应，即a类。因此，现有的关于后一类跃迁的临界理论的建议——属于wessw - zumino - novikov - witten类型——被排除了。相反，我们建议调查实现了什么。从场论的角度来看，剩下的候选是对数共形场论。我们的第一个目标是提供对数项的数值证据以及相应前因子的定量估计。（ii）一个美国小组最近发表的数值数据表明，典型的QH转变状态出现在拓扑绝缘体的无序板的表面。如果这一发现是正确的，那将是非常惊人的，因为它暗示了在没有微调控制参数的情况下存在QH临界性。然而，现有的计算缺乏足够的精度要求，无法自信地做出这样一个强有力的声明；此外，对这些观察结果还没有理论上的理解。我们的第二个目标是改进我们自己的有希望的，但仍然是初步的数据对A类的影响，并提供一个令人信服的数字证据。此外，在与场理论的同事密切合作下，我们的目标是对这一现象有更深入的了解。（iii）拓扑绝缘体表面存在的QH临界可以为一种全新的实验和理论（全光学）诊断工具开辟道路，即由激光照射驱动的高次谐波产生。提案的第三个目标是探索各自的潜力并提出相应的实验。

Disorder can hinder and, in fact, entirely stop the propagation of waves that instead become confined to a volume of finite extension. P. W. Anderson discovered this effect called Anderson localization more than sixty years ago. By tuning external control parameters, e.g., the disorder strength transitions can be driven that connect localized phases with other phases of disordered metals. Fundamental theoretical aspects of the respective phase transitions are the topic of this proposal. An emphasis will be on Quantum Hall (QH) transitions, which are special in the sense that they connect localized phases that differ by a topological (first Chern) index. The proposal pursues three closely related objectives. (i) Recent achievements in quantum field theories predict a symmetry relation that connects the system size scaling of the q-th moment of critical wavefunction amplitude with the moment (q* - q); specifically, x_q = x_{q*-q} for the respective scaling exponents. The number q* depends on the symmetry class of the specific transition at hand. Moreover, a parabolic dependency on q is predicted for x_q in the presence of local conformal invariance. Strikingly, the most recent numerical work rules out such a parabolic dependency for the class-C transition and very likely also for the integer QH effect, i.e. class A. Consequently, the existing proposals for the critical theory of the latter transitions – being of the Wess-Zumino-Novikov-Witten type - are ruled out. We propose to investigate what is realized instead. From a field-theoretic perspective the remaining candidates are logarithmic conformal field theories. Our first objective is to provide numerical evidence for log-terms together with a quantitative estimate of the corresponding prefactor. (ii) A US-group has recently published numerical data that suggests that states typical of the QH transition appear at surfaces of disordered slabs of a topological insulators. This finding, if correct, is most amazing because it implies the existence of QH criticality without fine-tuning a control parameter. However, the existing computations are lacking behind the precision requirements needed in order to make such a strong statement with confidence; moreover, there is no theoretical understanding of these observations. Our second objective is to improve upon our own promising, but still preliminary data on the effect for class A and provide a compelling numerical evidence. Further, in close collaboration with colleagues from field-theory we are aiming also at a deeper understanding of this phenomenon. (iii) The existence of QH criticality at surfaces of topological insulators could open the route towards an entirely new experimental and theoretical (all-optical) diagnostic tool, i.e., the high-harmonics generation driven by laser irradiation. The third objective of the proposal is to explore the respective potential and propose corresponding experiments.