Dynamical Mean-Field Theory (DMFT) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperatures. By merging density functional theory with DMFT a powerful method for the calculation of the properties of correlated electron materials has become available, which is applicable to bulk systems and heterostructures, including topological states of matter. The inclusion of non-local correlations into DMFT makes it possible to explore unconventional superconductivity and the critical behavior at thermal or quantum phase transitions. By generalizing DMFT to non-equilibrium states also the real-time dynamics of correlated systems can be investigated. In this brief review the foundations and current status of DMFT are discussed along with characteristic physical insights obtained with this approach.
Dynamical Mean-Field Theory (DMFT) has opened new perspectives for the investigation of strongly correlated electron systems and greatly improved our understanding of correlation effects in models and materials. In contrast to Hartree-Fock-type approximations the mean field of DMFT is dynamical, whereby local quantum fluctuations are fully taken into account. DMFT becomes exact in the limit of high spatial dimensions or coordination number. Using DMFT the dynamics of correlated electron systems can be investigated non-perturbatively at all interaction strengths, electron densities and temperatures. By merging density functional theory with DMFT a powerful method for the calculation of the properties of correlated electron materials has become available, which is applicable to bulk systems and heterostructures, including topological states of matter. The inclusion of non-local correlations into DMFT makes it possible to explore unconventional superconductivity and the critical behavior at thermal or quantum phase transitions. By generalizing DMFT to non-equilibrium states also the real-time dynamics of correlated systems can be investigated. In this brief review the foundations and current status of DMFT are discussed along with characteristic physical insights obtained with this approach.
Abstract Electronic correlations strongly influence the properties of matter. For example, they can induce a discontinuous transition from conducting to insulating behavior. In this paper basic terms of the physics of correlated electrons are explained. In particular, I describe some of the steps that led to the formulation of a comprehensive, non‐perturbative many‐body approach to correlated quantum many‐body systems, the dynamical mean‐field theory (DMFT). The DMFT becomes exact in the limit of high lattice dimensions (d → ∞) and allows one to go beyond the investigation of simple correlation models and thereby better understand, and even predict, the properties of electronically correlated materials.
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.
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.
The study of nonequilibrium phenomena in correlated lattice systems has developed into one of the most active and exciting branches of condensed matter physics. This research field provides rich new insights that could not be obtained from the study of equilibrium situations, and the theoretical understanding of the physics often requires the development of new concepts and methods. On the experimental side, ultrafast pump-probe spectroscopies enable studies of excitation and relaxation phenomena in correlated electron systems, while ultracold atoms in optical lattices provide a new way to control and measure the time evolution of interacting lattice systems with a vastly different characteristic time scale compared to electron systems. A theoretical description of these phenomena is challenging because, first, the quantum-mechanical time evolution of many-body systems out of equilibrium must be computed and second, strong-correlation effects which can be of a nonperturbative nature must be addressed. This review discusses the nonequilibrium extension of the dynamical mean field theory (DMFT), which treats quantum fluctuations in the time domain and works directly in the thermodynamic limit. The method reduces the complexity of the calculation via a mapping to a self-consistent impurity problem, which becomes exact in infinite dimensions. Particular emphasis is placed on a detailed derivation of the formalism, and on a discussion of numerical techniques, which enable solutions of the effective nonequilibrium DMFT impurity problem. Insights gained into the properties of the infinite-dimensional Hubbard model under strong nonequilibrium conditions are summarized. These examples illustrate the current ability of the theoretical framework to reproduce and understand fundamental nonequilibrium phenomena, such as the dielectric breakdown of Mott insulators, photodoping, and collapse-and-revival oscillations in quenched systems. Furthermore, remarkable novel phenomena have been predicted by the nonequilibrium DMFT simulations of correlated lattice systems, including dynamical phase transitions and field-induced repulsion-to-attraction conversions.
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.
Abstract. The dynamical mean-field theory (DMFT) is a widely applicable approx-imation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particu-lar, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the foundations of the DMFT, derive the underlying self-consistency equa-tions, and present several applications which have provided important insights into the properties of correlated matter.