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

N. E. Bickers

N. E. Bickers

  • Institute for Theoretical Physics, Santa Barbara, California 93106

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Rev. Mod. Phys. 59, 845 – Published 1 October, 1987

DOI: https://doi.org/10.1103/RevModPhys.59.845

Abstract

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 1N, 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(1) 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.

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Rev. Mod. Phys. 59, 845– Published 1 October, 1987

    © 1987 American Physical Society

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