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 discusses instabilities of the Fermi-liquid state of conduction electrons in metals with particular emphasis on magnetic quantum critical points. Both the existing theoretical concepts and experimental data on selected materials are presented; with the aim of assessing the validity of presently available theory. After briefly recalling the fundamentals of Fermi-liquid theory, the local Fermi-liquid state in quantum impurity models and their lattice versions is described. Next, the scaling concepts applicable to quantum phase transitions are presented. The Hertz-Millis-Moriya theory of quantum phase transitions is described in detail. The breakdown of the latter is analyzed in several examples. In the final part experimental data on heavy-fermion materials and transition-metal alloys are reviewed and confronted with existing theory.

http://export.arxiv.org/pdf/cond-mat/0606317

Fermi-liquid instabilities at magnetic quantum phase transitions

http://myweb.rz.uni-augsburg.de/~eckern/WEH-287/ABSTRACTS/ruitenbeek.pdf

Quantum properties of atomic-sized conductors

In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/NRG/BULLA=numerical_renormalization_group_method_for_quantum_impurity_systems.pdf

Numerical renormalization group method for quantum impurity systems

We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a Luttinger liquid implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.

https://iopscience.iop.org/article/10.1088/0034-4885/58/9/002/pdf

One-dimensional Fermi liquids

Starting from the most general form of the Anderson hamiltonian, the behaviour of magnetic impurities in real metals is considered, taking into account the orbital structure of the local impurity electrons, crystal field and spin orbit splittings. The analysis is carried out in an atomic limit, in which the impurity has a well defined integer valency (a Schrieffer Wolff transformation is then valid). The main steps of a scaling procedure are described in detail. As the temperature goes down, the excited states of the ground state configuration decouple one after the other. The hierarchy of these decouplings, and their interplay with Kondo singularities are analyzed. When a Fermi liquid picture applies as T → 0, the number of independent parameters may be reduced considerably using symmetry and universality arguments which bypass the numerical description of the crossover region. That first part sets a language in which to describe specific problems. We apply that language to the case where the atomic ground state is an orbital singlet. In the absence of anisotropies, the only parameters are the impurity spin S and the number of orbital channels n. We show that an anomalous fixed point occurs at finite coupling when n > 2 S. That fixed point is unstable with respect to anisotropies. The scaling trajectories are discussed for a cubic crystal field for several choices of valencies. The universality of the low temperature behaviour is clarified. A similar analysis is carried out when the atomic ground state only has one electron (or hole). The influence of crystal field and spin orbit interactions is analyzed — and their relevance to the Kondo crossover and to universality is ascertained.

/pdf/kondo-effect-in-real-metals-dfd7qkweqy.pdf

Kondo effect in real metals

Methods introduced by Keldysh to treat non-equilibrium problems are applied to strong, localized time-dependent perturbations in metals. After having introduced the formalism, we treat simple examples which are linked to the dynamics of atoms near surfaces : ionization probabilities of atoms leaving a metallic surface and friction coefficients of atoms near a metallic surface.

/pdf/localized-time-dependent-perturbations-in-metals-formalism-4fk6yu70uo.pdf

Localized time-dependent perturbations in metals : formalism and simple examples

On etudie la structure electronique des impuretes de transition dans les alliages desordonnes a faible concentration. On en deduit une classification des proprietes magnetiques de ces alliages. Un modele d'interaction magnetique entre impuretes est propose, pour rendre compte des proprietes antiferromagnetiques de certains alliages (Cu Mn).

https://hal.archives-ouvertes.fr/jpa-00236009/document

Propriétés magnétiques des alliages dilués. Interactions magnétiques et antiferromagnétisme dans les alliages du type métal noble-métal de transition

Les fortes variations d'intensite de la resonance magnetique nucleaire dans les alliages dilues sont dues a la variation de densite electronique Δρ creee a grande distance par les impuretes dissoutes. Aux noyaux distants de r d'une impurete, Δρ a la forme : Cte × cos ([FORMULE]) et donne lieu a un gradient de champ ayant la meme variation avec la distance et une constante dependant fortement du caractere s, p ou d des fonctions d'onde au niveau de Fermi du metal pur. Les differents cas (alliages d'Al, Cu) etudies experimentalement sont discutes sur la base de cette theorie. Des conclusions analogues ont ete obtenues simultanement par Kohn et Vosko [18].

/pdf/effets-quadrupolaires-dans-la-resonance-magnetique-nucleaire-4q087lzhuu.pdf

Effets quadrupolaires dans la résonance magnétique nucléaire des alliages dilués

The change and broadening of the Knight shift of silver produced by various solute element. are explained by the fluctuations in electronic density produced at long range by each solute atom. The change in Knight shift observed for Cd, In, Sn and Sb solutes are in satisfactory agreement with those computed using a free electrons approximation for the matrix. From the sign and the values of the changes observed for Pd and Pt solutes, it in concluded that most of the screening arises there from an emptying of their d shells especially for Pt. Finally the change and broadening of the Knight shift of the solute element. are explained in the same way.

Annie Blandin

Papers

Kondo effect in real metals

Localized time-dependent perturbations in metals : formalism and simple examples

Propriétés magnétiques des alliages dilués. Interactions magnétiques et antiferromagnétisme dans les alliages du type métal noble-métal de transition

Effets quadrupolaires dans la résonance magnétique nucléaire des alliages dilués

On the knight shift of alloys