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

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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.