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1Dynamical Mean-Field Theory

D. Vollhardt, Krzysztof Byczuk, Marcus Kollar

Published Aug 17, 2016• 18 citations
OpenAlex

Abstract

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.

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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.

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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.

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Dynamical mean-field theory studies of superfluidity and topological phases in lattice models

Tuomas I. Vanhala

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.

•Jan 1, 2018•0 citations
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.
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.
Since the first investigation of the Hubbard model in the limit of infinite dimensions by Metzner and Vollhardt, dynamical mean-field theory (DMFT) has become a very powerful tool for the investigation of lattice models of correlated electrons. In DMFT the lattice model is mapped on an effective quantum impurity model in a bath which has to be determined self-consistently. This approach led to significant progress in our understanding of typical correlation problems such as the Mott transition; furthermore, the combination of DMFT with ab-initio methods now allows for a realistic treatment of correlated materials. The focus of these lecture notes is on the relation between quantum impurity physics and the physics of lattice models within DMFT. Issues such as the observability of impurity quantum phase transitions in the corresponding lattice models are discussed in detail.
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.
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.
Dynamical mean-field theory (DMFT) is a successful method to investigate interacting lattice fermions. In these lecture notes we present an introduction into the DMFT for lattice fermions with interaction, disorder and external inhomogeneous potentials. This formulation is applicable to electrons in solids and to cold fermionic atoms in optical lattices. We review here our investigations of the Mott-Hubbard and Anderson metal-insulator transitions in correlated, disordered systems by presenting selected surprising results. 1.
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.