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DYNAMICAL MEAN-FIELD THEORY FOR CORRELATED LATTICE FERMIONS

Krzysztof Byczuk

Published Jun 1, 2008• 1 citation
Publisher

Abstract

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.

DOI
10.1142/9789812709455_0001

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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.
We review recent progress in our theoretical understanding of strongly correlated fermion systems in the presence of disorder. Results were obtained by the application of a powerful nonperturbative approach, the dynamical mean-field theory (DMFT), to interacting disordered lattice fermions. In particular, we demonstrate that DMFT combined with geometric averaging over disorder can capture Anderson localization and Mott insulating phases on the level of one-particle correlation functions. Results are presented for the ground state phase diagram of the Anderson–Hubbard model at half-filling, both in the paramagnetic phase and in the presence of antiferromagnetic order. We find a new antiferromagnetic metal which is stabilized by disorder. Possible realizations of these quantum phases with ultracold fermions in optical lattices are discussed.
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.
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.
In the last one and a half decade, the Dynamical Mean Field Theory (DMFT) allowed to get insight into the Mott metal-insulator transition (MMIT) and many long-standing questions about the nature of the transition could be answered, at least in the limit of infinite dimensions. In this thesis, the adaption of the DMFT to inhomogeneous systems is studied, opening the possibility to study a rich variety of systems and challenging physical questions. Firstly, the basic problem is studied, how a metal penetrates into a Mott insulator, where analytically found power-laws at the critical point can be confirmed with numerics. Secondly, the phase separation between metal and insulator is investigated, which can occur since the MMIT is a first order transition. The spontaneous build up of domains is observed in numerical simulations and the domain walls are examined. Finally, the MMIT of fermionic atoms in an optical lattice is studied.
The DOS, the dynamic (optical) conductivity, and the phase diagram of a strongly correlated and strongly disordered paramagnetic Anderson-Hubbard model are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are taken into account by the DMFT, and disorder is taken into account via an appropriate generalization of the self-consistent theory of localization. The DMFT effective single-impurity problem is solved by a numerical renormalization group (NRG); we consider the three-dimensional system with a semielliptic DOS. The correlated metal, Mott insulator, and correlated Anderson insulator phases are identified via the evolution of the DOS and dynamic conductivity, demonstrating both the Mott-Hubbard and Anderson metal-insulator transition and allowing the construction of the complete zero-temperature phase diagram of the Anderson-Hubbard model. Rather unusual is the possibility of a disorder-induced Mott insulator-to-metal transition.
We study the Anderson-Hubbard model to examine the interplay between the disordered potential and the Hubbard interactions. We use the dynamical mean-field theory (DMFT) with an exact diagonalization as an impurity solver to effectively deal with the quantum fluctuations due to the local interactions. In order to characterize the paramagnetic metal-insulator transition in disordered systems, we compute the typical local density of states, as well as the arithmetically-averaged local density of states. Our approach is found to reproduce the characteristic features of the metal-insulator transition in infinite dimensions, such as the existence of the disorder-driven metallic phase for moderate values of disorder strength and interactions, as observed in earlier studies. In two dimensions, the system turns out to exhibit more insulating behavior when we increase the cluster size in the cellular DMFT. The metallic phase, which is induced from weak Mott insulators by the introduction of a disordered potential, is not observed in two dimensions.