Numerical investigations of ensembles of Boguliubov-deGennes Hamiltonians

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The physics of disordered metals and semiconductors continues to hold surprises and therefore is still one of the important active fields in contemporary condensed matter physics. In recent years, the interplay between disorder and interaction has come into focus again, e.g., in disordered superconductors in connection with the superconductor-insulator junction. Despite intensive efforts, the theoretical understanding of such systems is still limited, among other reasons because analytical calculations in the most interesting parameter regime can hardly be carried out in a systematically controlled manner, i.e. the regime in which disorder, interaction and kinetic energy of similar strength, This is where the project proposed here comes into play.In the first funding period, a code could be developed that allows for an extremely efficient self-consistent numerical treatment of disorder and interaction within the framework of the attractive Hubbard model and Boguliubov-deGennes theory. Large ensembles of self-consistent random matrices can be generated and systematically evaluated with regard to their statistical properties. It was thus possible for the first time to numerically access the important intermediate parameter regime and to confirm the long suspected formation of superconducting islands there. For the second funding period, we propose to exploit this success and study physical observables that are to be determined in current experiments, some of which are still in planning stage. Specifically, a strong interest is in the form of the gap in the local spectral function and in understanding the apparent “gapless superconductivity”, which was recently observed experimentally. The explanatory hypothesis we will test involves free magnetic moments, which inevitably arise due to the disorder and whose existence our calculations are able to demonstrate. Another subject of investigation will be the superconducting stiffness, as well as its dependence on magnetic field and temperature, which recently became accessible also to high precision experiments.Finally, as last stage of the project a phase diagram should be created with axes of magnetic field, temperature and disorder strength. On the one hand, the diagram is about identifying a regime in which the putative "failed superconductors" could reside. On the other hand, an American-Indian team recently proposed a phase diagram, which, however, could did not account sufficiently accurately for the formation of local moments. Specifically, the importance of repulsive interactions has not been recognized. In the follow-up project, these investigations are to be deepened and expanded to include an extended model with attractive and repulsive interaction.