Review of techniques in the large-$N$ expansion for dilute magnetic alloys

Magnetic ions embedded in a nonmagnetic host exhibit a number of unusual low-temperature properties known collectively as the Kondo effect. Conventional small-parameter expansions for the Kondo effect are plagued by infrared divergences, and the theory of dilute magnetic alloys has for many years centered on nonperturbative techniques. This paper reviews a novel perturbative approach to the magnetic alloy problem based on expansion in $\frac{1}{N}$, with $N$ the ionic angular momentum degeneracy. This approach is analogous to perturbative expansions in statistical mechanics and field theory based on an integer-valued parameter (such as the number of spin components or the number of colors). The large-$N$ expansion reproduces the essential features of the Kondo effect at $O\left(1\right)\mathrm{}$ and appears to yield convergent (or asymptotic) expressions for ground-state properties. In contrast with previous nonperturbative approaches, the expansion provides information on dynamic, as well as static, properties. The evidence for convergence of the expansion is reviewed, and large-$N$ calculations are compared with exact results for static properties. A number of independent, but essentially equivalent, approaches to the large-$N$ expansion have been developed during the last five years. These techniques are reviewed pedagogically, and their relative strengths and weaknesses emphasized. A guide to notation in the recent literature is provided.