Electronic correlations in topological materials

The interest in topological materials has flourished in recent years due to their topological protection of remarkable physical properties under weak perturbations, which ensures the realisation of robust exotic features in realistic materials. However, many of the theoretical achievements of topology in condensed matter, such as the landmark relationship between Hall conductance and the first Chern number discovered by Thouless, Kohmoto, Nightingale and den Nijs, inherently assume a single-particle description of the many-body wave function, so that the role of interactions and correlations in topological materials is not yet fully understood. This work aims to partially fill this gap, covering a wide range of topological phenomena induced or supported by electronic interactions. First, the topological character of spin excitons and their condensation in quantum spin Hall insulators will be investigated, with particular emphasis on their effect on the physics of electronic chiral edge states. Second, the joint effects of flat optical moiré phonons and Coulomb repulsion on the phase diagram of twisted bilayer graphene are studied, showing that their interplay can stabilise topological insulators and superconductors in agreement with experimental evidence. Third, the exact role of the Green's function zeros in the topological invariants for strongly correlated systems is elucidated, showing that the zeros contribute on an equal footing with the Green's function poles. In particular, in the case of a topological Mott insulator, it is shown that the topology is carried entirely by the Green's function zeros and not by the poles, a genuine strongly correlated effect that lacks any non-interacting counterpart. In the last part, a new explanation for the peculiar Fermi liquid properties of topological Kondo insulators is proposed, motivated by recent findings that zeros of the Green's function could directly correspond to the presence of neutral quasiparticles. In particular, a new realistic model for these strongly interacting topological insulators is introduced and then studied by means of cluster dynamical mean field calculations, confirming the presence of ingap Green's function zeros.