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.
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.
Abstract This article is a short introduction to the modern computational techniques used to tackle the many-body problem in materials. The aim is to present the basic ideas, using simple examples to illustrate strengths and weaknesses of each method. We will start from density-functional theory (DFT) and the Kohn–Sham construction—the standard computational tools for performing electronic structure calculations. Leaving the realm of rigorous density-functional theory, we will discuss the established practice of adopting the Kohn–Sham Hamiltonian as approximate model. After recalling the triumphs of the Kohn–Sham description, we will stress the fundamental reasons of its failure for strongly-correlated compounds, and discuss the strategies adopted to overcome the problem. The article will then focus on the most effective method so far, the DFT+DMFT technique and its extensions. Achievements, open issues and possible future developments will be reviewed. The key differences between dynamical (DFT+DMFT) and static (DFT+ U ) mean-field methods will be elucidated. In the conclusion, we will assess the apparent dichotomy between first-principles and model-based techniques, emphasizing the common ground that in fact they share.
The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations.
Lecture notes of the Autumn School on Correlated Electrons 2018:Dieter Vollhardt: From Infinite Dimensions to Real Materials; Ole Andersen: From Materials to Models: Deriving Insight from Bands; Ferdi Aryasetiawan: Effective Electron-Electron Interaction in Many-Electron Systems; Marcus Kollar: The Foundations of Dynamical Mean-Field Theory; Michael Potthoff: Cluster Extensions of Dynamical Mean-Field Theory; Frank Lechermann: Charge Self-Consistency in Correlated Electronic Structure Calculations; Eva Pavarini: LDA+DMFT: Multi-Orbital Hubbard Models; Hao Tjeng: Determining Orbital Wavefunctions using Core-Level Non-Resonant Inelastic X-Ray Scattering; Hans Gerd Evertz: DMRG for Multiband Impurity Solvers; Fakher Assaad: Quantum Monte Carlo Methods for Fermion-Boson Problems; Erik Koch: Analytic Continuation of Quantum Monte Carlo Data; Hartmut Hafermann: Introduction to Diagrammatic Approaches; Thomas Maier: Dynamical Mean-Field and Dynamical Cluster Approximation Based Theory of Superconductivity; Karsten Held: Quantum Criticality and Superconductivity in Diagrammatic Extensions of DMFT; Martin Eckstein: Correlated Electrons out of Equilibrium: Non-Equilibrium DMFT
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.
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 materials with strong electron correlation the proper treatment of local atomic physics described by orbital occupations is crucial. Reduced density-matrix functional theory is a natural extension of density functional theory for systems that are dominated by orbital physics. We review the current state of reduced density-matrix functional theory (RDMFT). For atomic structure relaxations or ab-initio molecular dynamics the combination of density functional theory (DFT) and dynamical mean-field theory (DMFT) possesses a number of disadvantages, like the cumbersome evaluation of forces. We therefore describe a method, DFT+RDMFT, that combines many-particle effects based on reduced density-matrix functional theory with a density functional-like framework. A recent development is the construction of density-matrix functionals directly from many-particle theory such as methods from quantum chemistry or many-particle Green’s functions. We present the underlying exact theorems and describe current progress towards quantitative functionals.