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The ordinary single channel Kondo model consists of one or more spin 1/2 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 heavy fermion fermi liquids to the mapping to 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 flavor index. In the overcompensated regime of this model non-Fermi liquid physics is possible, in contrast to 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 energetic and symmetry restrictions on the applicability of these models. We describe the various 1 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 of these theoretical approaches. We point out in each case how various perturbing fields (magnetic, crystalline electric, electric field gradients, 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, nanoscale devices, and of the quadrupolar/magnetic two-channel models to a number of heavy fermion 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 sytems (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 behavior 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.

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