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

The structures, the phase diagrams, and the appearance of a neutron resonance signaling an unconventional superconducting state provide phenomenological evidence relating the cuprates, the Fe-pnictides/chalcogenides as well as some heavy fermion and actinide materials Single- and multi-band Hubbard models have been found to describe a number of the observed properties of these materials so that it is reasonable to examine the origin of the pairing interaction in these models In this review, based on the experimental phenomenology and studies of the pairing interaction for Hubbard-like models, it is proposed that spin-fluctuation mediated pairing is the common thread linking a broad class of superconducting materials

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A common thread: The pairing interaction for unconventional superconductors

We present a formalism for carrying out Monte Carlo calculations of field theories with both boson and fermion degrees of freedom. The basic approach is to integrate out the fermion degrees of freedom and obtain an effective action for the boson fields to which standard Monte Carlo techniques can be applied. We study the structure of the effective action for a wide class of theories. We develop a procedure for making rapid calculations of the variation in the effective action due to local changes in the boson fields, which is essential for practical numerical calculations.

Monte Carlo calculations of coupled boson-fermion systems. I

The case for dx2 − y2 pairing in the cuprate superconductors

Low-energy quasiparticle scattering, recombination, and branch-mixing lifetimes and phonon pair-breaking and scattering lifetimes are calculated for superconductors. The quasiparticle calculations relate these lifetimes to the low-frequency behavior of ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})$. Results are obtained using the low-frequency approximate form ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})=b{\ensuremath{\Omega}}^{2}$, with $b$ determined from electron tunneling measurements. For the strong-coupling superconductors Pb and Hg, the full tunneling form for ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})F(\ensuremath{\Omega})$ is used. The phonon lifetimes are shown to depend on ${\ensuremath{\alpha}}^{2}(\ensuremath{\Omega})$. Results are compared with experiment.

Quasiparticle and phonon lifetimes in superconductors

We discuss the problems that arise in the numerical simulation of many-electron systems when the measure of the functional integrals is not positive definite. We present theoretical arguments and numerical data which indicate that the expectation value of the sign of the measure decreases exponentially as the inverse temperature \ensuremath{\beta} increases, unless the measure is forced to be positive by an explicit symmetry. We therefore conclude that a recent proposal for dealing with the sign problem due to Sorella et al. Leads to an uncontrolled approximation. In the cases we have studied it is a good approximation for the ground-state energy, and we present a method for calculating the correction needed to make it exact. However, for some physical quantities, such as the d-wave pair field susceptibility, the neglect of signs can yield misleading results.

Douglas J. Scalapino

Papers

A common thread: The pairing interaction for unconventional superconductors

Monte Carlo calculations of coupled boson-fermion systems. I

The case for dx2 − y2 pairing in the cuprate superconductors

Quasiparticle and phonon lifetimes in superconductors

Sign problem in the numerical simulation of many-electron systems