A. Zawadowski

The ordinary single-channel Kondo model consists of one or more spin-½ local moments interacting antiferromagnetically with conduction electrons in a metal. This model has provided a paradigm for understanding many phenomena of strongly correlated electronic materials, ranging from the formation of heavyfermion Fermi liquids to the mapping of a one-band model in the cuprate superconductors. The simplest extension of this ordinary Kondo model in metals which yields exotic non-Fermi-liquid physics is the multichannel Kondo impurity model in which the conduction electrons are given an extra quantum label known as the channel or flavour index. In the overcompensated regime of this model, nonFermi-liquid physics is possible, in contrast with the single-channel model. We overview here the multichannel Kondo impurity model candidates most extensively studied for explaining real materials, specifically the two-level system Kondo model relevant for metallic glasses, nanoscale devices and some doped semiconductors, and the quadrupolar and magnetic two-channel Kondo models developed for rare-earth and actinide ions with crystal-field splittings in metals. We provide an extensive justification for the derivation of the theoretical models, noting that, whenever the local impurity degree of freedom is non-magnetic, a two-channel Kondo model must follow by virtue of the magnetic spin degeneracy of the conduction electrons. We carefully delineate all energy and symmetry restrictions on the applicability of these models. We describe the various methods used to study these models along with their results and limitations (multiplicative renormalization group, numerical renormalization group, non-crossing approximation, conformal field theory and Abelian bosonization), all of which provide differing and useful views of the physics. We pay particular attention to the role that scale invariance plays in all these theoretical approaches. We point out in each case how various perturbing fields (magnetic, crystalline electric, electric field gradients and uniaxial stress) may destabilize the non-Fermi-liquid fixed point. We then provide an extensive discussion of the experimental evidence for the relevance of the two-level system Kondo model to metallic glasses and nanoscale devices, and of the quadrupolar and magnetic two-channel models to a number of heavyfermion-based alloys and compounds. We close with a discussion of the extension of the single-impurity models which comprise the main focus of this review to other systems (Coulomb blockade), multiple impurities and lattice models. In the latter case, we provide an overview of the relevance of the two-channel Kondo lattice model to non-Fermi-liquid behaviour and exotic superconductivity in heavy-fermion compounds and to the theoretical possibility of odd-frequency superconductivity, which is realized (for the first time) in the limit of infinite spatial dimensions for this model.

A simple model for the depinning and dynamics of a sliding charge-density wave is presented. The model consists of viscously damped motion in a periodic potential and accounts for the main features of the nonlinear conductivity, noise, and ac-dc coupling in the charge-density-wave state of Nb${\mathrm{Se}}_{3}$ at low temperature. Comparison with experimental data suggests that the depinning process is associated with the displacement of the charge-density wave by approximately one period $\ensuremath{\lambda}$.

The properties of 3d transition metal impurities in simple metal hosts are summarized. There is a short discussion of the basic microscopic models, then the Anderson model is treated in some detail. The next section considers the s-d model and the Kondo effect. The main experimental features are reviewed and finally the question of dilute alloys is considered.

The conduction electron density of states nearby single magnetic impurities, as measured recently by scanning tunneling microscopy (STM), is calculated, taking into account tunneling into conduction electron states only. The Kondo effect induces a narrow Fano resonance in the conduction electron density of states. The line shape varies with the distance between STM tip and impurity, in qualitative agreement with experiments, but is very sensitive to details of the band structure. For a Co impurity the experimentally observed width and shift of the Kondo resonance are in accordance with those obtained from a combination of band structure and strongly correlated calculations.

The general form of the interaction between tunneling two-level systems (TLS) and conduction electrons is discussed for metallic glasses. The particular form of the Hamiltonian is given in the case where only a single atom tunnels between two positions. There are two couplings corresponding to the two basic scattering processes: In the first one, the tunneling atom does not change position; the second process is the conduction-electron-assisted tunneling process. The two coupling parameters are estimated. The difference in the angular dependence of these couplings on the directions of the incoming and of the outgoing electrons is responsible for the appearance of logarithmic corrections in the scattering amplitude. Scaling equations are derived for the couplings in terms of changing the bandwidth cutoff. It is shown that the scaling equations lead to especially strong coupling in two conduction-electron scattering channels which are linear combinations of the $s$-, $p$-, and $d$- like spherical wave functions. The Hamiltonian scales to a spin $S=\frac{1}{2}$ antiferromagnetic Kondo Hamiltonian, which indicates the formation of a "bound state," where the motions of the tunneling atom and of the conduction-electron screening cloud around the TLS are strongly correlated; thus the Friedel oscillations follow the tunneling atom. The crossover temperature, below which the correlation becomes especially strong, is determined in the leading logarithmic approximation.