Multifractality and interaction at localization transitions between topological phases in superconducting systems

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This is a DFG-RSF cooperation project between the KIT group and the Landau Institute for Theoretical Physics (Russia) in the framework of the call for joint German-Russian project proposals. Cooperation with the Russian team which is lead by Prof. Igor Burmistrov and consists of world-leading experts in Anderson localization and interactions in disordered systems (including topological insulators and disordered superconductors) is crucial for success of the project. Disordered interacting systems of fermions can have rich phase diagrams with topological phases separated by Anderson localization-delocalization transitions. The hallmark of Anderson criticality is multifractality. The primary example of such situation is the integer quantum Hall effect. There are two close cousins of the quantum Hall effect that occur in two dimensions: spin and thermal quantum Hall effects. These phenomena take place in the presence of particle-hole symmetry, i.e., in systems with superconducting order---topological superconductors. Topological materials represent currently one of most active research directions in condensed matter physics. The interest to topological superconductors is additionally enhanced by the fact that corresponding excitations---Majorana fermions---may find applications in quantum information processing. In this project we plan to develop a comprehensive picture of generalized multifractality in the superconducting symmetry classes C and D in the absence of interactions and, then, armed with the obtained insights to shed light on possible topological phases in the presence of electron-electron interaction. The results obtained in the project are expected to be relevant for d-wave disordered superconductors, granular superconducting films in a magnetic field, triplet odd-parity superconductors, non-abelian fractional quantum Hall systems, spin liquids, and surfaces of 3D topological superconductors.The work program consists of four interconnected tasks. In Task 1 we will develop a theory of generalized multifractality in class C. In Tasks 2 and 3 we will explore the interplay of multifractality and interactions in the spin quantum Hall effect. Specifically, Task 2 will address real inelastic-scattering processes leading to dephasing and to thermal broadening of quantum phase transitions. The goal of Task 3 is to investigate renormalization phenomena induced by interplay of disorder and interaction and, on this basis, the phase diagrams and transitions between phases in the presence of interaction. Task 4 will be devoted to investigation of multifractality, interactions, phase diagram, and phase transitions in systems of class D hosting the thermal quantum Hall effect.