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-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 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.

Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions

An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

The superconducting transition temperature is calculated as a function of the electron-phonon and electron-electron coupling constants within the framework of the strong-coupling theory. Using this theoretical result, we find empirical values of the coupling constants and the "band-structure" density of states for a number of metals and alloys. It is noted that the electron-phonon coupling constant depends primarily on the phonon frequencies rather than on the electronic properties of the metal. Finally, using these results, one can predict a maximum superconducting transition temperature.

Transition temperature of strong-coupled superconductors.

The Renormalization group and the epsilon expansion

Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Statistical physics of social dynamics

Quantum Monte Carlo simulations of a quarter filled 1-D Hubbard model show that the onsite Coulomb interaction suppresses 2p F charge density fluctuations. Simulations of the 2 and 3-D Hubbard model indicate that the p-wave pairing response is suppressed by U. These results suggest we take another look at our theoretical understanding of these many-body systems.

Two Questions Raised by Simulations of the Hubbard Model

The anomalous dimension index η assumes its critical value for small momenta and vanishes for large momenta. We provide, in the form of a quadrature, a momentum dependent η(k) which interpolates between the above limits. High- and low-momentum expansions for the spectral function are obtained, leading to a Pade approximant from which η(k) is determined, thereby giving a crossover critical correlation function.

Crossover critical correlation function

In recent years there has been increasing interest in the nonequi-librium properties of superconducting systems. In a broad sense, of course, all transport properties involve nonequilbirium (or metastable) states, and hence all of applied superconductivity depends upon these phenomena. However, we know from the fluctuation dissipation relation that linear transport coefficients can be determined by an analysis of equilibrium fluctuations. Thus, for some time efforts focused upon the nearly equilibrium situation and the problems of equilibrium fluctuations. Current interest in nonequilibrium superconductivity is associated with superconductors which are strongLy driven so that the quasipartides, phonons, and the pair field may be significantly different from their equilibrium values. It is this type of system, the “hot superconductor”, which will be discussed in this paper.

“Hot Superconductors”: The Physics and Applications of Nonequilibrium Superconductivity

We summarize recent density matrix renormalization group studies of stripes and pairing in the t-J model. These results imply that the ground state of this model is striped, with weak pairing. However, with the introduction of a next nearest neighbor hopping t’, the stripes are destabilized, and a strongly paired state can arise.

Dmrg Studies of Stripes and Pairing in the t-J Model

We present a variational treatment of the ground state of the 2-leg t-J ladder, which combines the dimer and the hard-core boson models into one effective model. This model allows us to study the local structure of the hole pairs as a function of doping. A second order recursion relation is used to generate the variational wave function, which substantially simplifies the computations. We obtain good agreement with numerical density matrix renormalization group results for the ground state energy in the strong coupling regime. We find that the local structure of the pairs depends upon whether the ladder is slightly or strongly dopped.

Douglas J. Scalapino

Papers

Two Questions Raised by Simulations of the Hubbard Model

Crossover critical correlation function

“Hot Superconductors”: The Physics and Applications of Nonequilibrium Superconductivity

Dmrg Studies of Stripes and Pairing in the t-J Model

The Dimer-Hole-RVB State of the 2-Leg t-J Ladder: A Recurrent Variational Ansatz