We study the magnetic anisotropy effects on the localized sub-gap excitations induced by quantum impurities coupled to a superconducting host. We establish the ground-state phase diagrams for single-channel and twochannel high-spin Kondo impurities; they unveil surprising complexity that results from the (multi-stage) Kondo screening in competition with the superconducting correlations and the magnetic anisotropy splitting of the spin multiplets. We discuss the possibility of detecting the Zeeman splitting of the sub-gap states, which would provide an interesting spectroscopic tool for studying the magnetism on the single-atom level. We also study the problem of two impurities coupled by the Heisenberg exchange interaction, and we follow the evolution of the sub-gap states for both antiferromagnetic and ferromagnetic coupling. For sufficiently strong antiferromagnetic coupling, the impurities bind into a singlet state that is non-magnetic, thus the sub-gap states move to the edge of the gap and can no longer be discerned. For ferromagnetic coupling, some excited states remain present inside the gap.

The explanation of heavy-fermion superconductivity is a long-standing challenge to theory. It is commonly thought to be connected to nonlocal fluctuations of either spin or charge degrees of freedom and therefore of unconventional type. Here we present results for the Kondo-lattice model, a paradigmatic model to describe heavy-fermion compounds, obtained from dynamical mean-field theory which captures local correlation effects only. Unexpectedly, we find robust s-wave superconductivity in the heavy-fermion state. We argue that this novel type of pairing is tightly connected to the formation of heavy quasiparticle bands and the presence of strong local spin fluctuations.

We study the zero-temperature properties of the Kondo lattice model within the dynamical mean-field theory. As an impurity solver we use the numerical renormalization group. We present results for the paramagnetic case showing the anticipated heavy-fermion physics, including direct evidence for the appearance of a large Fermi surface for antiferromagnetic exchange interaction. Allowing for the formation of a Néel state, we observe at finite doping an antiferromagnetic metal below a critical exchange interaction, which shows a crossover from a local moment antiferromagnet with a small Fermi surface for weak exchange coupling to a heavy-fermion antiferromagnet with a large Fermi surface for increasing exchange. Including lattice degrees of freedom via an additional Holstein term we observe a significant suppression of the Kondo effect, leading to a strongly reduced low-energy scale. For too large electron-phonon coupling we find a complete collapse of the heavy Fermi liquid and the formation of polarons.

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We study the Kondo lattice model with additional attractive interaction between the conduction electrons within the dynamical mean-field theory using the numerical renormalization group to solve the effective quantum impurity problem. In addition to normalstate and magnetic phases we also allow for the occurrence of a superconducting phase. In the normal phase we observe a very sensitive dependence of the low-energy scale on the conductionelectron interaction. We discuss the dependence of the superconducting transition on the interplay between attractive interaction and Kondo exchange.