Phase Coherence Phenomena in Disordered Superconductors

Mechanisms of quantum phase coherence heavily influence spectral and transport properties of weakly disordered normal conductors. Such effects are manifest in weak and strong localization effects, and characteristic fluctuation phenomena. Over the past thirty years, theoretical progress in elucidating the mechanisms of quantum phase coherence in weakly disordered structures has been substantial: By now a consistent theory of weakly interacting disordered structures has been developed (For a review, see e.g., Refs. [1–3]). At the same time, considerable experimental effort has been directed towards the exploration of the influence of phase coherence effects on the quasi-particle properties of disordered superconductors. Again, attempts to develop a consistent theory have enjoyed great success. By now a reliable theory of the weakly interacting superconducting system has been formulated. Yet, a complete description of the phenomenology of the disordered superconductor in the presence of strong interaction effects has yet to be established. The continuing developments and refinements of experimental techniques continue to present fresh challenges to theoretical investigations. On this background, the aim of these lecture notes is to selectively review the recent development of a quasi-classical field theoretic framework to describe phase coherence phenomena in disordered superconductors. This approach, which is motivated by the parallel formulation of the theory of the normal disordered system, presents average properties of the superconductor as a quantum field theory with an action of non-linear σ-model type. The limited scope of these lectures does not permit an extensive review the many applications of this technique. Instead, to illustrate the impact of quantum phase coherence phenomena on the quasi-particle properties of the disordered superconducting system, and the practical application of the field theoretic scheme, the final part of these notes will be devoted to a study of the magnetic impurity system.