Disorder and interaction effects in topological insulators

Topological phases of matter are conventionally characterized at the non-interacting level through global invariants obtained by tracking the evolution of occupied single-particle states across the Brillouin zone. This paradigm, however, becomes inadequate in the presence of disorder, which breaks translational symmetry, or strong electron-electron interactions, which invalidate a single-particle description. In these regimes, new theoretical and computational tools are required to properly characterize topological properties. At the same time, the interplay between band topology, disorder, and correlations can give rise to a rich phenomenology, including disorder-induced phases and exotic states emerging from interaction. This thesis investigates the effects of disorder and strong electronic correlations in topological insulators, combining methodological developments with numerical simulations. A central theme is the extension of topological concepts beyond their standard formulation, with particular emphasis on real-space formulations and many-body diagnostics. The first part focuses on disordered systems, introducing a suite of techniques to detect topological phases in non-crystalline systems. We consider single-point topological invariants defined within the framework of very large supercells, together with topological local markers applicable to both finite and extended systems. A key conceptual result is the demonstration that spatially resolved markers can act as bona fide local order parameters, challenging the view that topological phase transitions can only be described through global quantities. Using paradigmatic tight-binding models, we show that these markers exhibit meaningful spatial fluctuations and correlation functions. At criticality, such fluctuations become long-ranged, and the associated correlation functions display critical exponents that appear universal across different types of disorder, while remaining sensitive to the nature of the transition and to the symmetry class of the model. These methods are then applied to realistic topological systems, focusing on a monolayer of jacutingaite with vacancies, using effective Wannier Hamiltonians derived from first-principles calculations. The second part investigates strong electron-electron interactions in topological insulators within the variational Monte Carlo approach. The topology of interacting systems is characterized using many-body markers inspired by the modern theory of polarization. This framework is first applied to a one-dimensional two-orbital fermionic model with on-site intra-orbital Hubbard interaction U and nearest-neighbor repulsion V, described using a multi-orbital Jastrow–Slater variational wavefunction. The results show that correlations do not merely renormalize the non-interacting picture, but can generate genuinely new phases. In particular, at finite U, the transition between topological and trivial insulators is no longer direct: an intermediate insulating phase emerges, characterized by signatures of gapless spin excitations. A small nearest-neighbor repulsion further stabilizes this state, whereas sufficiently large V gives rise to two distinct charge density wave insulators. The analysis is then extended to the two-dimensional Bernevig-Hughes-Zhang model for quantum spin Hall insulators, in the presence of both intra- and inter-orbital interactions. In this setting, strong intra-orbital interactions stabilize antiferromagnetic order, while the inclusion of inter-orbital coupling provides evidence for the emerging of an excitonic insulating phase. Taken together, these results highlight the richness of correlation-driven phenomena in topological matter and establish the Jastrow-Slater variational Ansatz as a reliable and versatile tool for investigating correlated topological phases.