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

The transport properties of disordered solids have been the subject of much work since at least the 1950s, but with a new burst of activity during the 1980s which has survived up to the present day. There have been numerous reviews of a more or less specialized nature. The present review aims to fill the niche for a non-specialized review of this very active area of research. The basic concepts behind the theory are introduced with more detailed sections covering experimental results, one-dimensional localization, scaling theory, weak localization, magnetic field effects and fluctuations.

https://iopscience.iop.org/article/10.1088/0034-4885/56/12/001/pdf

Localization: theory and experiment

We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

The theory and properties of randomly disordered crystals and related physical systems

Powdered samples of the type Ce1−xRExO2−y, where RE=La, Pr, Nd, Eu, Gd, and Tb, are synthesized over the range 0≤x≤0.5 starting from nitrate solutions of the rare earths. X‐ray diffraction and Raman scattering are used to analyze the samples. These compounds, at least in the low doping regime and for strictly trivalent dopants, form solid solutions that maintain the fluorite structure of CeO2 with a change in lattice constant that is approximately proportional to the dopant ionic radius. The single allowed Raman mode, which occurs at 465 cm−1 in pure CeO2, is observed to shift to lower frequency with increasing doping level for all the rare earths. However, after correcting for the Gruneisen shift from the lattice expansion, the frequency shift is actually positive for all the strictly trivalent ions. In addition, the Raman line broadens and becomes asymmetric with a low frequency tail, and a new broad feature appears in the spectrum at ∼570 cm−1. These changes in the Raman spectrum are attributed to O va...

Raman and x‐ray studies of Ce1−xRExO2−y, where RE=La, Pr, Nd, Eu, Gd, and Tb

We develop a theory for dielectric response of water‐saturated rocks based on a realistic model of the pore space. The absence of a percolation threshold manifest in Archie’s law, porecasts, electron‐micrographs, and general theories of formation of detrital sedimentary rocks indicates that the pore spaces within such rocks remain interconnected to very low values of the porosity ϕ. In the simplest geometric model for which the conducting paths remain interconnected, each grain is envisioned to be coated with water. The dielectric constant of the assembly of water‐coated grains is obtained by a self‐consistent effective medium theory. In the dc limit, this gives Maxwell’s relation for conductivity σ of the rock σ=2σwϕ/(3-ϕ), where σw is the conductivity of water. In order to include the local environmental effects around a grain, a self‐similar model is generated by envisioning that each rock grain itself is coated with a skin made of other coated spheres; the coating at each level consists of other coate...

A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads

A formal study is made of the normal modes in a disordered alloy having substitutional defects, with explicit treatment of isolated defects and pairs of defects. Particular attention is paid to "multiple occupancy" of a site by more than one defect, which requires kinematic corrections. In the process, two new and quite general approaches to the disordered-alloy problem are presented. The first, a $t$-matrix approach, is shown to be equivalent to a second self-consistent diagrammatic approach. The second method is used to calculate formally the corrected and fully self-consistent scattering by pairs of defects, and is shown to be an extension of the self-consistent calculations by Soven and Taylor. It is found that the proper way to perform a self-consistent calculation is not to insert the full propagator into the self-energy parts, but to insert the full propagator minus the contribution containing scattering sites already explicitly in the parts.

Pair Effects and Self-Consistent Corrections in Disordered Alloys

The connection between Taylor's recent self-consistent calculations of phonon frequency spectra in disordered alloys and those of Davies and Langer in the usual self-consistent-field approximation is established. It is found that Taylor's results are equivalent to summing the same diagrams summed by Davies and Langer, but with proper compensation against multiple occupancy of sites by defects. The recently developed technique of Elliot, Aiyer, and Leath, which self-consistently adjusts the diagram rules to obtain proper compensation, is used. This method is quite general and clearly can be used for other self-consistent calculations is disordered alloys.

P. L. Leath

Papers

The theory and properties of randomly disordered crystals and related physical systems

Pair Effects and Self-Consistent Corrections in Disordered Alloys

Self-Consistent-Field Approximations in Disordered Alloys