Quantum information insights into strongly correlated electrons

Recent theoretical and experimental advances have opened new perspectives on the characterization of strongly correlated phases of matter, adding a new layer of understanding based on concepts and tools borrowed from quantum information theory. We join the effort by investigating, within dynamical mean-field theory (DMFT) and its cluster extension, the local and quasilocal (i.e. short-range) quantum information content of salient landmarks in the phase diagram of the two-dimensional Hubbard model, the cornerstone model for the description of the physics of cuprate materials, and of a closely related model for interacting topological states on the honeycomb lattice. In the first part of the thesis, we focus on the DMFT solution of the Kane-Mele-Hubbard model, at zero temperature. In the thermodynamic limit we find evidence of an interaction-driven discontinuous quantum phase transition between the Z2 quantum spin-Hall insulator and anisotropic antiferromagnetic solutions, with an intermediate regime of coexistence of nontrivial topology and magnetic order. A clear-cut interpretation of these results is given in terms of a rigorous quantification of quantum and classical correlations contained in a single local orbital (which we refer to as intra-orbital mutual information), providing a notion of "statistical distance" from the Hartree-Fock description of the system. The resulting analysis complements the established Green's function based understanding of the relationship between dynamical and static mean-field theories. In particular, we find the magnetic solutions of dynamical mean-field theory to asymptotically approach the corresponding uncorrelated Hartree-Fock states, in the strong coupling limit, in stark contrast with the paramagnetic Mott-Hubbard solution, which in turn reveals a maximally correlated local spin-orbital pair at large interaction strength. Furthermore, these findings cast some light on the relationship between Mott localization and the possible development of nonlocal entanglement. In the second part of the thesis, we provide an alternative view of both the interaction-driven and density-driven paramagnetic Mott transitions in the two-dimensional Hubbard model, in terms of rigorous measures of entanglement and correlation between two spatially separated electronic orbitals, with no contribution from their environment. A space-resolved analysis of cluster dynamical mean-field theory results elucidates the prominent role of the nearest-neighbor entanglement in probing Mott localization: two traditional upper bounds and two recently introduced lower bounds for its magnitude sharply increase at the metal-insulator transition, in contrast with the moderate variation found at all interaction strengths that are sufficiently far from the transition point. At half-filling, the two-site entanglement beyond nearest neighbors is shown to be quickly damped as the inter-site distance is increased within the cluster, suggesting that Mott-Hubbard insulators may follow an area law. However, the size of the simulated clusters does not allow a quantitative analysis of the decay, so that a precise classification of the spatial entanglement properties of the system is left for future study. In the presence of hole-doping, we show how the pseudogap metal separating the Mott-Hubbard insulator from the hole-dominated Fermi liquid features quasilocal entanglement properties that are strikingly similar to the localized Mott phase, while it is separated from the low-entangled Fermi liquid by a discontinuous jump in all the computed entanglement and correlation measures. All the presented results ultimately resolve a conundrum of previous analyses based on the single-site von Neumann entropy, which has been found to monotonically decrease when the interaction is increased, defeating the purpose of capturing and understanding strong electronic correlations with the aid of quantum information concepts. Both the intra-orbital mutual information and the quasilocal two-site entanglement, on the other hand, recover instead the distinctive character of Mott insulators and pseudogap metals as strongly correlated many-body states, demonstrating its central role in future advancements in the field of quantum materials.