In this thesis we study the strongly-correlated-electron physics of the longstanding H-Tc-superconductivity problem using a non-perturbative method, the Dynamical Mean Field Theory (DMFT), capable to go beyond standard perturbation-theory techniques. DMFT is by construction a local theory which neglects spatial correlation. Experiments have however shown that the latter is a fundamental property of cuprate materials. In a first step, we approach the problem of spatial correlation in the normal state of cuprate materials using a phenomenological Fermi-Liquid-Boltzmann model. We then introduce and develop in detail an extension to DMFT, the Cellular Dynamical Mean Field Theory (CDMFT), capable of considering short-ranged spatial correlation in a system, and we implement it using the exact diagonalization algorithm . After benchmarking CDMFT with the exact one-dimensional solution of the Hubbard Model, we employ it to study the density-driven Mott metal-insulator transition in the two-dimensional Hubbard Model, focusing in particular on the anomalous properties of the doped normal state close to the Mott insulator. We finally study the superconducting state. We show that within CDMFT the one-band Hubbard Model supports a d-wave superconductive state, which strongly departs from the standard BCS theory. We conjecture a link between the instabilities found in the normal state and the onset of superconductivity.
In this thesis we study the strongly-correlated-electron physics of the longstanding H-Tc-superconductivity problem using a non-perturbative method, the Dynamical Mean Field Theory (DMFT), capable to go beyond standard perturbation-theory techniques. DMFT is by construction a local theory which neglects spatial correlation. Experiments have however shown that the latter is a fundamental property of cuprate materials. In a first step, we approach the problem of spatial correlation in the normal state of cuprate materials using a phenomenological Fermi-Liquid-Boltzmann model. We then introduce and develop in detail an extension to DMFT, the Cellular Dynamical Mean Field Theory (CDMFT), capable of considering short-ranged spatial correlation in a system, and we implement it using the exact diagonalization algorithm . After benchmarking CDMFT with the exact one-dimensional solution of the Hubbard Model, we employ it to study the density-driven Mott metal-insulator transition in the two-dimensional Hubbard Model, focusing in particular on the anomalous properties of the doped normal state close to the Mott insulator. We finally study the superconducting state. We show that within CDMFT the one-band Hubbard Model supports a d-wave superconductive state, which strongly departs from the standard BCS theory. We conjecture a link between the instabilities found in the normal state and the onset of superconductivity.
Understanding the impact of electronic correlations onto the properties of cuprate superconductors is one of the most prominent tasks of solid state physics. In particular, the intermingling of short range strong correlations, driving the Mott or charge-transfer insulator in the parent compound, and the long range antiferromagnetic scattering processes in the highly doped regime poses severe difficulties to the theoretical description. This thesis studies the impact of short and intermediate ranged strong correlations on the electronic properties of the 2D Hubbard model in the half-filled and doped cases by means of cellular dynamical mean-field theory (CDMFT), i.e. real-space clusters embedded in a dynamical mean-field. Specifically, the effect of incorporating gradually longer ranged correlations onto the metal-insulator transition is analysed and dynamic next-neighbour spin correlations are identified as the cause of the insulating state. Furthermore, by introducing the Bethe-Salpeter scheme for embedded clusters, spin, charge and pairing susceptibilities are computed, incorporating the short ranged strong correlations exactly via the effective interaction of CDMFT and at the same time treating long-ranged fluctuations in a perturbative manner. Firstly, this allows to study the charge response at the Mott transition, identifying the non-local extended s-wave character of the transition. Secondly, in the hole-doped case, the pseudogap, the Knight-shift, a d-wave superconducting instability and unilateral charge instabilities with a periodicity of 4 lattice sites in the s-wave and d-wave channel are observed, all of which strongly depend on next-neighbour spin singlet-like correlations contained in the effective interaction of CDMFT. This work paves the way for computing the conductivity at the boundary between the pseudogap regime and the Fermi liquid. This doctoral thesis was conducted at the Max Planck Institute for Solid State Research and the University of Stuttgart.
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.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
Quantum mechanical lattice models, such as the famous Hubbard model, were originally intended as simplified descriptions of the behavior of electrons in real solids. Today it is clear that even such simple models can present remarkably rich physics. At the same time, obtaining reliable solutions for correlated quantum systems is difficult and requires the development of both computing resources and new algorithms and approximations. Parallel with the theoretical developments, applications of lattice models have also expanded to the study of ultracold atoms in optical lattices. The hallmark of these experiments is the possibility to design different kinds of lattices and to tune the interactions between the atoms, making it possible to do real experiments on theoretical "toy models". \n \nIn this thesis we have used dynamical mean-field theory (DMFT) and its cluster extensions to study superfluidity and topological phenomena in lattice models. The DMFT combines the exact solution of a small reference system with a self-consistent mean-field treatment of the rest of the lattice. After a short introduction to the physical background of the lattice models, we review the formal definition and practical aspects of the DMFT method. We discuss the Baym-Kadanoff-Luttinger-Ward functional formalism that can be used to derive the DMFT equations, and review the numerical exact diagonalization and continuous-time quantum Monte-Carlo algorithms that were used in the implementation of the method. \n \nPublications I and VI discuss superfluid states induced by correlation effects that cannot easily be treated within a static mean-field theory. In Publication I we study a bilayer system where two different types of superfluids can be found depending on the densities of particles in the layers. Publication VI has a focus in the d-wave superconductivity and non-uniform, striped magnetism of the two-dimensional Hubbard model, which are thought to be connected with the high-Tc superconductivity in cuprate materials. The stripe state along with non-Fermi-liquid properties visible only in a beyond-mean-field treatment were observed also in the study of the Hubbard model on the Lieb lattice in Publication V. \n \nTopological and geometric properties of lattice systems, such as the Berry curvature and Chern numbers, play a central role in Publications II, III and IV. In Publication II we used exact diagonalization, DMFT and mean-field theory to study the topological phase diagram of an interacting Haldane model. The methods consistently predict a phase diagram with an interme-diate topological phase not found in the well-known non-interacting case. In Publications III and IV mean-field theory is used to study the connection between Berry curvature and the superfluid weight in lattice models, and DMFT calculations were used to support the mean-field results. Topological lattice systems have recently been realized in ultracold gas experiments and it is hoped that the calculations presented here inspire this line of research.
In strongly correlated materials, quantum fluctuations taking place at the level of atomic sites play a central role. These local processes are captured nonperturbatively in dynamical mean field theory (DMFT). Nonlocal mechanisms, however, are also suspected to be essential for the proper description of some of these materials, including the well-known high-temperature cuprate superconductors and the more exotic systems of atoms adsorbed on semiconducting surfaces.In this thesis, the effect of nonlocal, potentially long-range fluctuations beyond the DMFT framework is investigated from two angles.First, long-ranged interactions in adatom systems, computed from first principles within the constrained random-phase approximation (cRPA), are treated within the extended DMFT (EDMFT) framework and its combination with the diagrammatic GW method (GW +EDMFT). They are shown to explain some material trends observed in these compounds. In addition, thanks to the spatial and temporal information delivered by GW+EDMFT, they solve existing puzzles raised by different experimental conclusions drawn from different probes.Second, a new approach to the Hubbard model including nonlocal correlation effects is proposed. Formally based on the local expansion of the triply-irreducible functional, this approach, dubbed TRILEX, approximates the renormalized three-leg electron-boson vertex by a local but frequency-dependent vertex computed from an impurity model with retarded density-density and spin-spin interactions. This local vertex is used to construct frequency- and momentum-dependent self-energies capturing nonlocal correlation effects. By construction, this method reconciles two theoretical pictures of cuprate superconductors, namely the spin fluctuation theory at weak interaction strengths and the doped-Mott-insulator picture at strong coupling.This novel approach is applied to the single-band Hubbard model. It is shown to reduce to DMFT at strong coupling and high temperatures, and to fluctuation-exchange theories at weak coupling, while substantial deviations from both limits are observed at intermediate coupling. There, upon doping, antiferromagnetic spin fluctuations are observed to lead to a momentum-differentiated spectral intensity on the Fermi surface, which is reminiscent of the formation of Fermi arcs in photoemission measurements on cuprate superconductors.These two methodological developments are intrinsically linked to algorithmic aspects which are reviewed in the last part of this work. In particular, an open-source library dedicated to massively parallel computations in the field of quantum many-body physics, TRIQS, is presented, with a focus on error estimation through the binning and jackknife methods. The hybridization-expansion continuous-time quantum Monte-Carlo algorithm in the segment picture -- central to the TRILEX method -- and its state-of-the-art implementation are explained. Finally, as a further illustration, a new lightweight method relying on this algorithm and designed for realistic systems, the combined screened-exchange and dynamical DMFT method (SEx+DDMFT), is presented together with an application to the BaCo2As2 compound.
This review is devoted to generalization of dynamical mean-field theory (DMFT) for strongly correlated electronic systems towards the account of different types of additional interactions, necessary for correct physical description of many experimentally observed phenomena in such systems. As additional interactions we consider: (1) interaction of electrons with antiferromagnetic (or charge) fluctuations of order parameter in high-Tc superconductors leading to the formation of pseudogap state, (2) scattering of electrons on static disorder and its role in general picture of Anderson-Hubbard metal-insulator transition, (3) electron-phonon interaction and corresponding anomalies of electronic spectra in strongly correlated systems. Proposed DMFT+Sigma approach is based on taking into account above mentioned interactions by introducing additional self-energy Sigma (in general momentum dependent) into conventional DMFT scheme and calculated in a self-consistent way within the standard set of DMFT equations. Here we formulate general scheme of calculation of both one-particle (spectral functions and densities of states) and two-particle (optical conductivity) properties. We examine the problem of pseudogap formation, including the Fermi arc formation and partial destruction of the Fermi surface, metal-insulator transition in disordered Anderson-Hubbard model, and general picture of kink formation within electronic spectra in strongly correlated systems. DMFT+Sigma approach is generalized to describe realistic materials with strong electron-electron correlations based on LDA+DMFT method. General scheme of LDA+DMFT method is presented together with some of its applications to real systems. The LDA+DMFT+Sigma approach is employed to modelling of pseudogap state of electron and hole doped high-T_c cuprates. Comparison with variety of ARPES experiments is given.