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

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Two soluble models of an antiferromagnetic chain

The generalization of the Local Density Approximation (LDA) method for the systems with strong Coulomb correlations is presented which gives a correct description of the Mott insulators. The LDA+U method is based on the model hamiltonian approach and allows to take into account the non-sphericity of the Coulomb and exchange interactions. parameters. Orbital-dependent LDA+U potential gives correct orbital polarization and corresponding Jahn-Teller distortion. To calculate the spectra of the strongly correlated systems the impurity Anderson model should be solved with a many-electron trial wave function. All parameters of the many-electron hamiltonian are taken from LDA+U calculations. The method was applied to NiO and has shown good agreement with experimental photoemission spectra and with the oxygen Kα X-ray emission spectrum.

https://iopscience.iop.org/article/10.1088/0953-8984/9/4/002/pdf

First-principles calculations of the electronic structure and spectra of strongly correlated systems: the LDA+ U method

In 1955, a phenomenological theory of ferromagnetism was well established and had been corroborated by a considerable amount of experimental data. However, there were problems in the phenomenological theory of the dynamics of the magnetization field. The Landau-Lifshitz equation for damping of the motion of the magnetization field could not account for the large noneddy-current damping in thin Permalloy sheets. The problem undertaken herein is a reformulation of the theory in a way that is more consistent with the theory of damping in other physical systems in order to be able to take large damping into account.

A phenomenological theory of damping in ferromagnetic materials

It has been found recently that in ferromagnetic resonance experiments performed in inhomogeneous rf exciting fields at a fixed frequency, absorption of power takes place at a number of distinct magnetic fields. This is ascribed to the existence of long-wavelength modes of oscillation of the ferromagnetic sample. The mode spectrum of spheroids is examined for the case, which may often hold in practice, where exchange and electromagnetic propagation can be ignored simultaneously.

Magnetostatic Modes in Ferromagnetic Resonance

The anomalous behavior of the temperature dependence of the susceptibility and specific heat of the intermetallic compound FeSi is reexamined from both an experimental and a theoretical point of view. It is shown that a consistent set of experimental data is obtained below a temperature of about 700\ifmmode^\circ\else\textdegree\fi{}K. An interpretation of the thermodynamic properties is given, based on a model that requires the existence of correlated magnetic excited states only a few hundred degrees above the nonmagnetic ground state. Other, more conventional models are also explored, and the reasons for their rejection are discussed.

Paramagnetic Excited State of FeSi

Orbach's integral equation which leads to the value of the ground-state energy of an anisotropic antiferromagnetic linear chain of spins, $S=\frac{1}{2}$, has been solved. The result is then expanded in powers of the anisotropy parameter. In this form it corresponds to the result of a perturbation calculation, the transverse part of the Hamiltonian being the perturbation. The rapid convergence of the energy series even for the isotropic case and the adequate convergence of that for the short range order, suggests that the result given by perturbation theory for the sublattice magnetization may also be satisfactory.

Antiferromagnetic Linear Chain

A general formulation is presented for the calculation of static field distributions caused by randomly placed impurities in a lattice. Specific evaluations are carried out for the case of combined Ruderman-Kittel-Kasuya-Yosida (RKKY) and dipolar interactions between magnetic impurities and host nuclei. In this case the line shape is shown to approach a Lorentzian in the limit of great dilution, in agreement with earlier theories for the dipolar case. Analytic expressions for the half-widths are given. The Lorentzian shape is shown to be a consequence of the ${R}^{\ensuremath{-}3}$ range dependence (i.e., to hold only in this case) and to hold also for an arbitrary distribution of impurity-moment values. These results are corroborated by explicit machine computations for the concentration range $0.01\ensuremath{\le}c\ensuremath{\le}1.0$ at.%, in which no approximations of a serious nature are introduced. The linewidth law for combined RKKY and dipolar coupling is derived and verified. This technique is applied to the problem of line narrowing due to resistivity damping of the RKKY oscillations, obtaining a damping length greater by a factor of \ensuremath{\sim}2 than that originally deduced by Heeger, Klein, and Tu. The self-damping of RKKY oscillations by the Mn impurities in $\mathrm{Cu}\mathrm{Mn}$ is estimated to be significant for $c\ensuremath{\gtrsim}1$ at.%. Further applications are discussed.

L. R. Walker

Papers

Magnetostatic Modes in Ferromagnetic Resonance

Paramagnetic Excited State of FeSi

Antiferromagnetic Linear Chain

Nuclear-resonance line shapes due to magnetic impurities in metals

Ferromagnetic resonance line width in insulating materials