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

安德森和量子霍尔跃迁的数值研究：对数共形不变性、无需微调的临界性、多重分形动力学

• 批准号: 499345971
• 负责人: Professor Dr. Ferdinand Evers
• 金额: --
• 依托单位: Institut für Theoretische Physik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/499345971?language=en
• 关键词: Numerical; investigations; Anderson; Quantum; Hall

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.

无序可以阻碍，事实上，完全停止波的传播，而不是成为有限的扩展体积。P. W.六十多年前，安德森发现了这种被称为安德森局域化的效应。通过调整外部控制参数，例如，可以驱动无序强度转变，其将局部相与无序金属的其它相连接。各自的相变的基本理论方面的主题，这一建议。重点将放在量子霍尔（QH）跃迁上，它们的特殊之处在于它们连接了因拓扑（第一陈氏）指数而不同的局部相。该提案追求三个密切相关的目标。(i)量子场论的最新成果预言了一个对称关系，它将临界波函数振幅的q阶矩与矩（q *-q）的系统尺寸标度联系起来;具体地说，对于相应的标度指数，x_q = x_{q *-q}。数q * 取决于手头特定跃迁的对称类。此外，在局部共形不变性的存在下，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，也很可能排除了整数QH效应，即A类跃迁。因此，现有的建议，批判理论的后过渡-是韦斯-祖米诺-诺维科夫-威滕类型-被排除在外。我们建议研究实现了什么。从场论的角度来看，剩下的候选者是对数共形场论。我们的第一个目标是提供数值证据的对数项连同相应的前因子的定量估计。 (ii)一个美国小组最近发表的数值数据表明，典型的QH跃迁的状态出现在拓扑绝缘体的无序板的表面。这个发现，如果正确的话，是最令人惊讶的，因为它意味着存在的QH临界没有微调的控制参数。然而，现有的计算缺乏所需的精度要求，以便有信心地做出这样一个强有力的声明;此外，对这些观测没有理论上的理解。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，在与场论同事的密切合作中，我们的目标也是更深入地理解这一现象。(iii)拓扑绝缘体表面QH临界性的存在可以开辟一条通往全新实验和理论（全光学）诊断工具的道路，即，由激光照射驱动的高次谐波产生。该提案的第三个目标是探索各自的潜力并提出相应的实验。