Interacting electrons and quenched randomness: From the mesoscopic Kondo problem to the depeleted Kondo lattice

The physics of strong electron-electron interactions in quantum many body physics is an exciting subject. An understanding of these effects is crucial for explaining the unusual behavior of many materials. These materials are interesting both from a fundamental view point and for application to future technologies. One of the simplest many-body phenomena is the interaction of a local "impurity" spin with a sea of electrons, the so-called "Kondo Impurity Problem," discovered about half a century ago. In the last two decades there has been a huge revival in the study of "Kondo" type phenomena. Experiments on two types of quantum materials have been responsible for this revival. On the one hand, fabrication of nano-meter scale electronic devices that act as "Kondo" impurities have provided flexible and tunable realizations of quantum impurities. On the other hand, the discovery of "Kondo" lattice materials (or "Heavy-Fermion" metals) have provided a window onto the physics of metals with a dense concentration of "Kondo" impurities. This thesis is concerned with physical problems relevant to both these types of materials.; Part I of this thesis is concerned with manifestations of the interplay of quantum coherence and the Kondo effect in nano-structures. The effect of this interesting competition can be observed in a variety of nano-devices. In some set-ups the Kondo temperature itself becomes a mesoscopic quantity (Chapter 3), exhibiting huge sample to sample fluctuations, while in other set-ups the definition of the Kondo temperature itself becomes the question of interest (Chapter 4). We have also developed a new method to simulate general quantum impurity models using quantum Monte-Carlo methods (Chapter 5) and applied it to study the universal Kondo physics that takes place when the screening bath is placed in a finite-size box. Finally, we have proposed a new experimental set-up in which the finite-size spectrum of many interesting quantum impurity problems, including the Kondo problem, can be observed. By general arguments we have predicted the form of the finite-size spectrum for many quantum impurity problems that can be realized in current experiments (Chapter 6).; In Part II of this thesis we examine the connection between the Fermi-liquid ground state of the Kondo Impurity Problem and the Fermi-liquid ground state of the Kondo Lattice: it has been possible very recently to fabricate metals (e.g. CexLa 1-xCoIn5) in which the concentration of "Kondo" spins is varied in a controlled way; the positions of the "Kondo" spins however are random. This results in an interesting interplay of quenched randomness and strong electron-electron interactions. We find that while the Fermi-liquid state is well defined in the two limiting concentrations of "Kondo" spins, at intermediate values there are strongly inhomogeneous phases with incomplete screening of the "Kondo" spins, likely to be characterized by non-Fermi-liquid exponents.