We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-1 range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We also present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

No abstract

We present an exact mapping of the Hubbard model in infinite dimensions onto a single-impurity Anderson (or WollA model supplemented by a self-consistency condition. This provides a mean-field picture of strongly correlated systems, which becomes exact as d~. We point out a special integrable case of the mean-field equations, and study the general case using a perturbative renormalization group around the atomic limit. Three distinct Fermi-liquid regimes arise, corresponding to the Kondo, mixed-valence, and empty-orbitals regimes of the single-impurity problem. The Kondo resonance and the satellite peaks of the single-impurity model correspond to the quasiparticle and Hubbard-bands features of the Hubbard model, respectively.Despite intensive theoretical work, the physics of strongly correlated fermions still contains numerous unsolved problems, even in its simplest formulation such as the single-band Hubbard model. In particular, there is no widely accepted mean-field theory which becomes exact in some limit. In a number of statistical-mechanics problems (including spin glasses, fully frustrated models, and lattice gauge theories) the mean-field theory obtained by taking the limit of large space dimensionality provided great insights. I n a pioneering paper, ' Metzner and Vollhardt pointed out that this limit is also of great interest for quantum many-body models, which simplify remarkably while retaining the main features, making their physics nontrivial.In this paper, we construct a meanfield picture of the Hubbard model, which becomes exact as d ee and unravels a connection with the singleirnpurity Anderson model. This analogy elucidates many features of the Hubbard model in infinite dimensions. A different mean-field approach has been recently proposed by Van Dongen and Vollhardt for the Falicov-Kimball model.As usual in statistical mechanics, the limit d~must be taken while scaling the parameters in a definite way, in order to avoid the situation that a single term in the Harniltonian dominates all others. For the Hubbard model on a d-dimensional hypercubic lattice with nearest-neighbor hopping t;, the on-site U need not be scaled, while t;j must be scaled as I/Wd in order to keep both the kinetic and potential energy per site finite. Specifically, we shall choose H= P (C; Cj +H.c.)+U+n;ln;1. 2jg tij)e , i With this scaling, the model remains an itinerant system with correlations. The free (V=0) density of states (DOS) acquires a Gaussian form' in the d~limit: D(s) = I/Jxe '. It displays band tails extending from -to +, but in many instances, the effective bandwidth is given by the variance of the density of states. Metzner and Vollhardt's observation stimulated a number of subsequent works, concentrating mainly on the study of variational wave functions and weak-coupling expansions, which simplify enormously in d =. Furthermore, in a remarkable piece of work, Brandt and Mielsch (BM) obtained the exact solution of the infinite-dimensional Falicov-Kimball model, a simplified Hubbard model in which only on...

Strong electronic correlations are often associated with the proximity of a Mott insulating state. In recent years however, it has become increasingly clear that the Hund's rule coupling (intra-atomic exchange) is responsible for strong correlations in multi-orbital metallic materials which are not close to a Mott insulator. Hund's coupling has two effects: it influences the energetics of the Mott gap and strongly suppresses the coherence scale for the formation of a Fermi-liquid. A global picture has emerged recently, which emphasizes the importance of the average occupancy of the shell as a control parameter. The most dramatic effects occur away from half-filling or single occupancy. The theoretical understanding and physical properties of these 'Hund's metals' are reviewed, together with the relevance of this concept to transition-metal oxides of the 3d, and especially 4d series (such as ruthenates), as well as to the iron-based superconductors (iron pnictides and chalcogenides).

16 pages, 5 figuresWe propose a systematic procedure for constructing effective models of strongly correlated materials. The parameters, in particular the on-site screened Coulomb interaction U, are calculated from first principles, using the GW approximation. We derive an expression for the frequency-dependent U and show that its high frequency part has significant influence on the spectral functions. We propose a scheme for taking into account the energy dependence of U, so that a model with an energy-independent local interaction can still be used for low-energy properties

We compute the thermodynamic properties of the Sachdev-Ye-Kitaev (SYK) models of fermions with a conserved fermion number, Q. We extend a previously proposed Schwarzian effective action to include a phase field, and this describes the low temperature energy and Q fluctuations. We obtain higherdimensional generalizations of the SYK models which display disordered metallic states without quasiparticle excitations, and we deduce their thermoelectric transport coefficients. We also examine the corresponding properties of Einstein-Maxwell-axion theories on black brane geometries which interpolate from either AdS 4 or AdS 5 to an AdS 2 × R 2 or AdS 2 × R 3 near-horizon geometry. These provide holographic descriptions of non-quasiparticle metallic states without momentum conservation. We find a precise match between low temperature transport and thermodynamics of the SYK and holographic models. In both models the Seebeck transport coefficient is exactly equal to the Q-derivative of the entropy. For the SYK models, quantum chaos, as characterized by the butterfly velocity and the Lyapunov rate, universally determines the thermal diffusivity, but not the charge diffusivity.

A theory of the metal-insulator transition in vanadium dioxide from the high- temperature rutile to the low- temperature monoclinic phase is proposed on the basis of cluster dynamical mean-field theory, in conjunction with the density functional scheme. The interplay of strong electronic Coulomb interactions and structural distortions, in particular, the dimerization of vanadium atoms in the low-temperature phase, plays a crucial role. We find that VO2 is not a conventional Mott insulator, but that the formation of dynamical V-V singlet pairs due to strong Coulomb correlations is necessary to trigger the opening of a Peierls gap.

Using t2g Wannier-functions, a low-energy Hamiltonian is derived for orthorhombic 3d 1 transitionmetal oxides. Electronic correlations are treated with a new implementation of dynamical mean-field theory for non-cubic systems. Good agreement with photoemission data is obtained. The interplay of correlation effects and cation covalency (GdFeO3-type distortions) is found to suppress orbital fluctuations in LaTiO3, and even more in YTiO3, and to favor the transition to the insulating state.