Models and methods for complex quantum nanostructures

Since their conceptualization by P. W. Anderson, quantum impurity models have been a driving force in the development of powerful theoretical tools and the understanding of anomalous behavior of solids. Although the fundamental physics question about the basic effect of magnetic impurities in metals has been solved through the development of the numerical renormalization group (NRG), technological and theoretical advances have kept the field relevant. In particular the utilization of the Kondo effect in single molecule electronics is a promising technology to drive the miniaturization of electronic components even further, while presenting the field of quantum impurity physics with new challenges. As the experimental realization of molecular electronics proceeds with ever more elaborate techniques to improve stability and reproducibility, the theoretical study of molecular electronics struggles with the complexity of combining non-equilibrium physics, molar numbers of atoms and strong interactions. In this thesis we develop a suite of theoretical methods to aid the study of molecular electronics and we present case studies to showcase our techniques. To tackle the challenge of realistically studying the Kondo effect in molecular electronics systems we divide the problem in two parts: first, the derivation of a simple Kondo-type model representing the molecule and secondly the conductance calculation using state-of-the-art NRG methods. A novel solution to the first problem is achieved by formulating the derivation of accurate effective models for quantum impurity systems as a machine learning problem. We also apply machine learning ideas to the NRG algorithm itself to derive an exact differentiable version of NRG. To address the conductance calculations in general impurity models we introduce a novel algorithm, based on the linear conductance Kubo formula, which we demonstrate to improve over the regular implementation of the Kubo formula in NRG. Finally, we apply these methods to a molecular electronics device based on benzene and a quantum dot device based on graphene.