Special solutions of the Hartree-Fock (HF) problem for Coulomb interacting electrons, being described by a simple model of the Cu-O planes in La2CuO4, are presented. One of the mean field states obtained, is able to predict some of the basic properties of this material, such as its insulator character and the antiferromagnetic order. The natural appearance of pseudogaps in some states of this compound is also indicated by another of the HF states obtained. These surprising results follow after eliminating spin and crystal symmetry restrictions which are usually imposed on the single particle HF orbitals, by means of employing a rotational invariant formulation of the HF scheme which was originally introduced by Dirac. Therefore, it is exemplified how, up to now being considered strong correlation effects, can be described by improving the HF solution of the physical systems. In other words, defining the correlation effects as such ones shown by the physical system and which are not predicted by the best HF (lowest energy) solution, allows to explain currently assumed as strong correlation properties, as simple mean field ones. The discussion also helps to clarify the role of the antiferromagnetism and pseudogaps in the physics of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures for the description of the transition metal oxides and other strongly correlated electron systems.
Special solutions of the Hartree-Fock (HF) problem for Coulomb interacting electrons, being described by a simple model of the Cu-O planes in La2CuO4, are presented. One of the mean field states obtained, is able to predict some of the basic properties of this material, such as its insulator character and the antiferromagnetic order. The natural appearance of pseudogaps in some states of this compound is also indicated by another of the HF states obtained. These surprising results follow after eliminating spin and crystal symmetry restrictions which are usually imposed on the single particle HF orbitals, by means of employing a rotational invariant formulation of the HF scheme which was originally introduced by Dirac. Therefore, it is exemplified how, up to now being considered strong correlation effects, can be described by improving the HF solution of the physical systems. In other words, defining the correlation effects as such ones shown by the physical system and which are not predicted by the best HF (lowest energy) solution, allows to explain currently assumed as strong correlation properties, as simple mean field ones. The discussion also helps to clarify the role of the antiferromagnetism and pseudogaps in the physics of the HTSC materials and indicates a promising way to start conciliating the Mott and Slater pictures for the description of the transition metal oxides and other strongly correlated electron systems.
Effective Hamiltonians offer a realistic and practical approach to the representation of low-lying electronic excitations in strongly correlated charge transfer insulators, such as the high-Tc parent La2CuO4. Not only do such Hamiltonians minimize the required number of orbital degrees of freedom, but it is suggested that the fundamental problem of charged excitations in these materials may be folded into renormalized Hamiltonian parameters which already reflect solid state screening, and that such parameters may easily be obtained from constrained local density functional calculations. Beneficial consequences of the renormalization include reasonable insulating gaps already from the Hartree-Fock one-electron spectra, and a restriction of configuration interaction to those essential correlations associated with the spin degrees of freedom. In large periodic clusters where states have well characterized crystal momentum, such spin correlations in the vicinity of particles doped into the parent insulator are still reminiscent of the total spin eigenstates found in impurity model solutions.
The concept of electronic correlations plays an important role in modern condensed matter physics. It refers to interaction effects which cannot be explained within a static mean‐field picture as provided by Hartree‐Fock theory. Electronic correlations can have a very strong influence on the properties of materials. For example, they may turn a metal into an insulator (Mott‐Hubbard metal‐insulator transition). In these lecture notes I (i) introduce basic notions of the physics of correlated electronic systems, (ii) discuss the construction of mean‐field theories by taking the limit of high lattice dimensions, (iii) explain the simplifications of the many‐body perturbation theory in this limit which provide the basis for the formulation of a comprehensive mean‐field theory for correlated fermions, the dynamical mean‐field theory (DMFT), (v) derive the DMFT self‐consistency equations, and (vi) apply the DMFT to investigate electronic correlations in models and materials.
High temperature superconductivity in copper oxides (cuprates) represents one of the most striking phenomena in condensed matter physics. These materials host superconducting states with the highest known critical temperatures, emerging upon doping antiferromagnetic insulator compounds. The antiferromagnetic (AF) insulating and superconducting (SC) phases defy standard descriptions in terms of band picture and conventional phonon-mediated superconductivity. In fact, these phenomena are genuine signatures of strong electronic correlations which hinder effective single-particle descriptions of the many-electron system. After 40 years since their discovery, it is widely accepted that the Hubbard model on the square lattice provides a minimal description of the low-energy electronic properties of cuprates. The model describes the competition between the electronic kinetic energy, modelled by nearest-neighbours hopping t, and the Coulomb repulsion, modelled by a purely local interaction potential U. At half-filling, this competition results in the so-called "Mott localisation" phenomenon, which distinguishes the metallic phase for U/t « 1 from an insulating phase for U/t » 1. The latter is characterised by the almost complete localisation of electrons due to the high energetic cost of hopping towards occupied sites. Interestingly, this basic competition is intimately related to AF ordering and, upon doping, to the formation of the SC state with characteristic d-wave symmetry. The fact that a purely repulsive interaction U is responsible for the formation of a SC state is remarkable, being superconductivity typically associated with an effective attractive interaction. In fact, a d-wave SC state can also be obtained, in general, by considering an attractive potential between electrons of neighbouring sites. Motivated by this observation, in this thesis, we study a modified version of the Hubbard model, dubbed as the "extended Hubbard model", in which we include a nearest-neighbours attractive potential -V. The general goal of this work is to explore the interplay between the purely local repulsion and the non-local attraction. This is not only motivated by the fact that both interactions are, in principle, responsible for d-wave superconductivity, but also by recent experimental observations which strongly suggest that the term -V needs to be included in order to accurately describe the magnetic spectra of some specific compounds. In this thesis, we study the extended Hubbard model using the Hartree-Fock method. Namely, we approximate the ground state by implementing a variational search in the subspace of the many-body Hilbert space of gaussian states. Despite its simplicity, the Hartree-Fock method provides a powerful framework to study competition among different phases, originating from different symmetry breaking mechanisms. In particular, we study the competition between three different phases: the normal phase, characterised by no symmetry breaking, the AF phase, characterised by a lowering of SU(2) symmetry, and the SC phase, characterised by the breaking of U(1) charge conservation. This thesis work led to a number of original results, which can be summarised by the general observation that the non-local attraction -V acts cooperatively with the local repulsion U for the stabilisation of the various phases hosted by the Hubbard model. For the normal phase, we find that the non-local attraction acts as a source of Mott localisation, leading to an effective reduction of the electronic bandwidth. In particular, for values of V larger than a critical value and close to half-filling, we find a strong Mott localisation characterised by the complete flattening of the electronic bands. Interestingly, this result shows an example of Mott localisation that can be described at the level of the Hartree-Fock approximation, whereas Mott localisation in the conventional Hubbard model usually requires more sophisticated approximation schemes. For the AF phase, we restrict our analysis to the half-filled case. We find that the pairing-induced Mott localisation leads to a significant enhancement of the magnetisation, primarily originating from the local repulsion U. Finally, we focus on the superconducting phase and derive self-consistency equations in both singlet and triplet Cooper pairing channels, showing how Cooper pairing can be mediated simultaneously by both the local repulsion and the non-local attraction. Therefore, we numerically investigate the singlet channel and find evidence of "symmetry mixing", namely, the formation of a superconducting gap that is neither symmetric nor anti-symmetric under rotations of π/2 in the plane. We show that symmetry mixing is connected to the emergence of a finite nematic order parameter, that signals the breaking of planar rotational symmetry. This nematic symmetry breaking results in the fact that the renormalisation of bands becomes anisotropic, producing a difference in electronic kinetics in the x and y directions. To summarise, the original results obtained in this thesis show that the non-local pairing provides a different route towards the formation of key strongly correlated phenomena, including Mott localisation and nematic superconductivity, and call for future investigation using more sophisticated methods such as, for example, the Dynamical Mean Field Theory.
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.
Since the genesis of condensed matter physics, strongly correlated fermionic systems have shown a variety of fascinating properties and remain a vital topic in the field. Such systems arise through electronic interaction, and despite decades of intensive research, no holistic approach to solving this problem has been found. During that time, physicists have compiled a wealth of individual experimental and theoretical results, which together give an invaluable insight into these materials, and, in some instances, can explain correlated phenomena. However, there are several systems that stubbornly refuse to fall completely in line with current theoretical descriptions, among them the high-Tc cuprates and heavy fermion compounds. Although the two material classes have been around for the better part of the last 50 years, large portions of their respective phase diagram are still under intensive debate. Recent experiments in several electron-doped cuprates compounds, e.g. neodymium cerium copper oxide (Nd2xCexCuO4), reveal a charge ordering about an antiferromagnetic ground state. So far, it has not been conclusively clarified how this intertwining of charge and spin polarization comes about and how it can be reconciled with a rigorous theoretical description. The heavy-fermion semimetals, on the other hand, have enjoyed renewed scientific interest with the discovery of topological Kondo insulators, a new material class offering a unique interface of topology, symmetry breaking, and correlated phenomena. In this context, samarium hexaboride (SmB6) has emerged as a prototypical system, which may feature a topological ground state. In this thesis, we present a spin rotational invariant auxiliary particle approach to investigate the propensities of interacting electrons towards forming new states of order. In particular, we study the onset of spin and charge order in high-Tc cuprate systems and Kondo lattices, as well as the interplay of magnetism and topology. To that end, we use a sophisticated mean-field approximation of bosonic auxiliary particles augmented by a stability analysis of the saddle point via Gaussian fluctuations. The latter enables the derivation of dynamic susceptibilities, which describe the response of the system under external fields and offer a direct comparison to experiments. Both the mean-field and fluctuation formalisms require a numerical tool that is capable of extremizing the saddle point equations, on the one hand, and reliably solving a loop integral of the susceptibility-type, on the other. A full, from scratch derivation of the formalism tailored towards a software implementation, is provided and pedagogically reviewed. The auxiliary particle method allows for a rigorous description of incommensurate magnetic order and compares well to other established numerical and analytical techniques. Within our analysis, we employ the two-dimensional one-band Hubbard as well as the periodic Anderson model as minimal Hamiltonians for the high-Tc cuprates and Kondo systems, respectively. For the former, we observe a regime of intertwined charge- and spin-order in the electron-doped regime, which matches recent experimental observations in the cuprate material Nd2xCexCuO4. Furthermore, we localize the emergence of a Kondo regime in the periodic Anderson model and establish the magnetic phase diagram of the two-band model for topological Kondo insulators. The emerging antiferromagnetic ground state can be characterized by its topological properties and shows, for a non-trivial phase, topologically protected hinge modes.
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.