Field Theory Approach to Quantum Impurity Problems ∗

Quantum impurity problems occur in various areas of physics. Some examples are: an impurity spin in a metal (the Kondo problem), the two-impurity Kondo problem, a defect or impurity spin in a spin chain, tunnelling through a barrier in a quantum wire proton-monopole scattering (the Callan-Rubakov effect). Some common features of these problems are: 1. gapless “bulk” excitations (ie. far from the impurity or in the absence of an impurity the excitations are gapless), 2. essentially harmonic behavior away from the impurity (Fermi or Luttinger liquids), 3. the impurity is localized and may carry quantum-mechanical degrees of freedom (eg. the two spin states of the impurity spin), 4.perturbation theory in the impurity-bulk coupling may be infrared divergent, 5. the problems are either initially defined in one dimension or can be mapped into one dimension (ie. some sort of partial wave expansion can be made and only a finite number of partial waves are important). The first and last points imply that the systems are equivalent to (1 + 1)-dimensional conformal field theories with the impurity or defect at the origin of the one-dimensional position space. In the imaginary time path-integral formulation, we have a boundary or defect line at (τ, 0). See Figure 1. John Cardy recently developed [1] a boundary conformal field theory to treat two-dimensional classical critical systems with boundaries. We have extended his technique to deal with the (1 + 1) dimensional quantum case, allowing for the possibility of a dynamical defect [2-14]. In the next section we review a simple and venerable example to provide some motivation: the local Fermi liquid theory of the Kondo effect, developed by Nozières [15], following earlier ideas of Wilson and Anderson. In Section 3 we discuss our general approach to quantum impurity problems. In Section 4 we present some of our results on