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

Monte Carlo results for the frequency dependent conductivity [sigma][sub 1]([omega]), the angular resolved photoemission spectral weight [ital A]([ital p],[omega]), and the electron momentum distribution [l angle][ital n][sub [ital p]][r angle] are calculated for a half-filled Hubbard model with the on-site Coulomb interaction [ital U] equal to the bandwidth 8[ital t] We find that even for [ital U]=8[ital t], a spin-density-wave approximation provides a sensible description of this data and hence a useful picture of the electronic degrees of freedom in the insulating state

Electronic properties of the insulating half-filled Hubbard model.

For the traditional low-${T}_{c}$ superconductors, the superconducting condensation energy is proportional to the change in energy of the ionic lattice between the normal and superconducting state, providing a clear link between pairing and the electron-ion interaction. Here, for the $t\ensuremath{-}J$ model, we discuss an analogous relationship between the superconducting condensation energy and the change in the exchange energy between the normal and superconducting states. We point out the possibility of measuring this using neutron scattering and note that such a measurement, while certainly difficult, could provide important evidence for an exchange interaction-based pairing mechanism.

Superconducting condensation energy and an antiferromagnetic exchange-based pairing mechanism

We present density-matrix renormalization group results for the ground state properties of two-leg Hubbard ladders. The half-filled Hubbard ladder is an insulating spin-gapped system, exhibiting a crossover from a spin liquid to a band insulator as a function of the interchain hopping matrix element. When the system is doped, there is a parameter range in which the spin gap remains. In this phase, the doped holes from singlet pairs and the pair field and the “4kF” density correlations associated with pair-density fluctuations decay as power laws, while the “2kF” charge density wave correlations decay exponentially. We discuss the behavior of the exponents of the pairing and density correlations within this spin-gapped phase. Additional one-band Luttinger liquid phases which occur in the large interband hopping regime are also discussed.

The ground state of the two-leg Hubbard ladder a density-matrix renormalization group study

A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a two-dimensional Hubbard model with a near neighbor hopping $t$ and an onsite Coulomb interaction $U$. The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as $(\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{x}--\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{k}_{y})$ over most of the Brillouin zone and its frequency dependence is determined by the $S=1$ particle-hole continuum which for large $U$ varies as several times $J$. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by ${(2J)}^{\ensuremath{-}1}$. The strength of the pairing interaction measured by the size of the $d$-wave eigenvalue peaks for $U$ of order of the bandwidth $8t$. It is found to increase as the system is underdoped.

Pairing interaction in the two-dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation

Bardeen and Schrieffer discuss details of the interactions of electrons and phonons in a given metal sought in deviations of observed properties from the original BCS predictions. This chapter analyses these deviations and their relationship to the electron-phonon interaction in metals. Theoretical results for a number of the thermodynamic and the dynamic properties of the strong-coupling superconductors are shown in good agreement with the observed values. This agreement gives support to the underlying theory and allows the deviations from the original BCS predictions used in obtaining information on the electron-phonon interaction. Since the determination of the physical properties of the strong-coupling superconductors and the relation of the electron-phonon interaction to these properties is based upon the self-energy equations, it is essential to develop some physical insight into their structure. Although electron tunneling experiments provide the most direct probe of the internal dynamics of the strong-coupling superconductors, there are a number of other measurements which yield information on the electron-phonon interaction.

Douglas J. Scalapino

Papers

Electronic properties of the insulating half-filled Hubbard model.

Superconducting condensation energy and an antiferromagnetic exchange-based pairing mechanism

The ground state of the two-leg Hubbard ladder a density-matrix renormalization group study

Pairing interaction in the two-dimensional Hubbard model studied with a dynamic cluster quantum Monte Carlo approximation

The Electron-Phonon Interaction and Strong-Coupling Superconductors