Approximate mapping of the two-impurity symmetric Anderson model in the local-moment regime to a classical problem

Using a path-integral formulation for the partition function the two-impurity symmetric Anderson model in the local-moment regime is mapped onto a problem of a classical four-component "Coulomb" gas interacting with a logarithmic potential. This can be viewed alternately as a problem of a general four-state spin system with inverse-square interaction in one dimension. The scaling equations for this model are derived and their implications for the properties of the Anderson model are pointed out. This mapping, although approximate, provides an explicit and physically interesting realization of a class of general $n$-state inverse-square spin systems in one dimension.