Anderson Transitions and Quantum Criticality in Novel Materials: Topological Insulators, Graphene and Friends

Under construction.

Classification Scheme and Critical Fixed Points: Anderson Transitions

Quantum interference is sensitive to basic symmetries, most notably time reversal, spin rotations and other, less obvious ones, like "chiral" symmetries. Therefore, many different kinds of Anderson-insulators exist the most famous one being the quantum-Hall insulator. Even though in the last years the "periodic table" of dirty-metal phases has been completed, the properties of these phases and in particular the possible transitions between them are still largely unexplored and often poorly understood. Our numerical effort is devoted to shed more light into this intricate and fascinating world of localization-induced quantum criticality. Many different methods are developed and employed for this purpose in our group. They include transfer-matrices, wave-function propagation with Krylov-space methods and Lanczos-type of techniques and many others...

Aspects of Multifractality at the Metal-Insulator-Transition

Wavefunction of a graphene-lattice with vacancies. The logarithmic color coding exhibits the large amplitude fluctuations that are the hallmark of (nearly "frozen") multifractality in this system. (Diploma project T. Mayer)

One of the particularly beautiful aspects of the critical state relates to the amplitude distribution of the associated wavefunctions. Since such wavefunctions live right at the boundary between a metal and an insulator, they have very peculiar properties. In particular, they exhibit an unusual type of self-similarity that is called "multi-fractality". A fractal structure has the property that it fills only a certain fraction of the embedding space, so that it carries a fractal dimension that is lower than the dimension of the embedding. Wavefunctions at the Anderson-transition are multi-fractal, because different moments of their amplitude each carry their own specific fractal dimension. We study the associated set of exponents, the multi-fractal spectrum, because it contains essential information about the nature and classification of the transition -- but also, because current experimental efforts are promising so that hopefully such spectra
can also be measured, soon. Our most current example is graphene, that exhibits multifractality in the presence of vacancy-disorder (if treated on a tight binding level).