Conformal Field Theory Approach to Quantum Impurity Problems

Abstract

There are several problems in condensed matter (and particle) physics in which a local degree of freedom interacts with a gapless (critical) continuum. Some of these are intrinsically 1 dimensional (1D): quantum wires, quantum Hall effect edge states, quantum spin chains. Others can be reduced to 1D because the bulk excitations are harmonic and can be expanded in spherical (or other) harmonics giving a radial (1D) problem: the Kondo problem (1,2,3, ... impurities), X-ray edge singularities, the monopole-baryon system (Callan—Rubakov effect). These problems typically exhibit infrared singularities causing the breakdown of perturbation theory in the impurity-bulk interactions. A powerful method to study the long distance, long time behaviour is the renormalization group (RG) together with boundary conformal field theory (BCFT), which I developed with various collaborators, especially Andreas Ludwig [1,2]. Other useful methods are the Bethe ansatz [3,4], exact integrability [5], the study of special models which are fully harmonic (called the Toulouse limit in the Kondo problem) [6] and numerical techniques [7]. In these lectures I will discuss the Kondo problem (including its multi-channel generalizations). This is the model for which BCFT has been most useful so far. New applications of these techniques may arise as experiments on nanostructures progress.