In this chapter, we present a bird’s-eye view of dynamical mean field theory (DMFT) and its extensions to highly correlated systems. We first demonstrate the elementary introduction to DMFT on a single site with detailed derivations and then a solution of the simple interacting models: the Hubbard model and the Falicov–Kimball model on infinite dimensional Bethe lattice using DMFT. Thereafter, we focus on two kinds of non-local extensions of single-site DMFT - (a) cluster dynamical mean field theory (CDMFT) on real space and (b) dynamical cluster approximation (DCA) on momentum space. Finally, we conclude the chapter with the application of DMFT combining with density functional theory (DFT) in the real-material systems (LDA+DMFT).
In this chapter, we present a bird’s-eye view of dynamical mean field theory (DMFT) and its extensions to highly correlated systems. We first demonstrate the elementary introduction to DMFT on a single site with detailed derivations and then a solution of the simple interacting models: the Hubbard model and the Falicov–Kimball model on infinite dimensional Bethe lattice using DMFT. Thereafter, we focus on two kinds of non-local extensions of single-site DMFT - (a) cluster dynamical mean field theory (CDMFT) on real space and (b) dynamical cluster approximation (DCA) on momentum space. Finally, we conclude the chapter with the application of DMFT combining with density functional theory (DFT) in the real-material systems (LDA+DMFT).
Numerical studies on strongly correlated fermionic systems are very complicated and still provide essential problems. The main reason is the exponential growth of the un- derlying Hilbert state space with the system size and the fermionic sign problem for Monte Carlo studies. Among the most widely employed numerical techniques for study- ing two-dimensional quantum many-body systems are cluster extensions of the dynamical mean-field theory (DMFT), e.g. dynamical cluster approximation (DCA). They map an infinitely large multi-dimensional lattice problem to a one-dimensional impurity problem. In 2015 it was shown that the density matrix renormalisation group (DMRG) used as an impurity solver for DMFT (DMFT+DMRG) on the imaginary-frequency axis allows to solve multi-site and multi-band problems extremely fast compared to other solvers. Within this thesis, we further develop this DMRG+DMFT approach to apply the method on real material settings. The step from artificial, completely degenerate multi-band mod- els with simple dispersion relations on a Bethe lattice, studied in 2015, to systems with realistic band structures and lifted degeneracies involves more challenges than originally suspected. In this thesis, we will first recapitulate relevant methods for our approach like matrix prod- uct states, the density matrix renormalisation group and several time evolution methods. In this context we will present several improvements ranging from optimised time evo- lutions to entanglement based optimisations of tensor networks. Second, we will present a very detailed description of the dynamical mean field theory. We will focus on both methodological aspects and implementation details. This chapter is intended to allow other researcher to implement their own DMFT code using DMRG as an impurity solver. Third, we will discuss three different models to show the extent of problems DMRG+ DMFT is able to solve. We will focus on multi-site DCA calculations in the case of the two-dimensional Hubbard model and show that DMRG allows to tackle systems with intermediate interaction strengths at low temperatures, which are unsolvable with other solvers. In the second case, the real material Sr2VO4, we will show the first two-site DCA results for a realistic three-band model. In contrast to assumptions, partly reintroducing the momentum dependence of the self-energy does not improve agreement between exper- imental observations and theoretical results. Finally, we will move on to another realistic three-band model, which describes Sr2RuO4, to show how to deal with the influence of spin-orbit coupling on DMFT. We will present the first low-temperature results for this material and will confirm previous results of simplified model calculations.
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 the previous chapters DMFT has been developed as a methodology for calculation of dynamical and thermodynamic properties of many-body systems, with applications to model systems. In order for DMFT to be a general method for quantitative calculations, there must be systematic procedures to derive all the ingredients from first principles. In this chapter we consider the most prominent examples, materials with partially filled 3 d , 4 f , or 5 f localized, atomic-like states, and set up the characteristic form of the hamiltonian. This is the basis for DFT+DMFT methods discussed here and used in the examples in Ch. 20, and for the combination of DMFT with many-body perturbation theory in Ch. 21. Simpler techniques, including self-interaction correction, DFT+U, and DFT+Gutzwiller, are also useful and are summarized in the last sections of this chapter. Up to this point, dynamical mean-field theory has been developed for model systems with Hubbard-type hamiltonians, where the hopping matrix elements and interactions are considered as parameters. For problems with strong interactions, DMFT leads to renormalized bands, satellites in the one-particle spectra, magnetic phase transitions and local moments, metal–insulator transitions, and many other phenomena observed in real materials, as brought out in Ch. 2. The purpose of this chapter is to provide the framework for a quantitative theory of these materials. These are difficult problems, experimentally, theoretically, and computationally, and the first step is to identify the aspects of the materials that must be treated explicitly as the local strongly interacting system, and the “rest” that are treated by a different method. This division of the problem is useful because a real material involves many degrees of freedom, i.e., the electronic states must be described in a basis sufficiently large to describe all the relevant states. However, the part that can be treated by non-perturbative solvers (Ch. 18) is limited by the computational cost that scales rapidly with the number of states included in the embedded site or cluster. Thus one must choose a small number of localized orbitals that inevitably are non-unique. This may seem like a step backward for quantitative methods; however, it opens a window for methods to treat these interesting problems and it presents a challenge to overcome the limitations.
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.
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.
In this lecture we give an introduction to theoretical description of strongly correlated materials based on dynamical mean-¿eld theory (DMFT) and its extensions.The goal of this theoretical construction is to retain the many-body aspects of local atomic physics within the extended solid.The effects of short-range non-local correlations within cluster extensions of the DMFT scheme, as well as long-range Àuctuations within the fully renormalized dual-fermion perturbation scheme are discussed extensively.Recent progress in the numerical solution of the DMFT effective quantum impurity problem within the recently developed continuous-time Quantum Monte Carlo schemes is reviewed.We then describe realistic extensions of this approach which combine the accuracy of ¿rst-principle Density-Functional Theory with the treatment of local many-body effects within DMFT.