We present an explicit solution of the problem of two coupled spin-1/2 impurities, interacting with a band of conduction electrons. We obtain an exact effective bosonized Hamiltonian, which is then treated by two different methods (low-energy theory and mean-field approach). Scale invariance is explicitly shown at the quantum critical point. The staggered susceptibility behaves like ln(${\mathit{T}}_{\mathit{K}}$/T) at low T, whereas the magnetic susceptibility and 〈${\mathbf{S}}_1$⋅${\mathbf{S}}_2$〉 are well behaved at the transition. The divergence of C(T)/T when approaching the transition point is also studied. The non-Fermi-liquid (actually marginal-Fermi-liquid) critical point is shown to arise because of the existence of anomalous correlations, which lead to degeneracies between bosonic and fermionic states of the system. The methods developed in this paper are of interest for studying more physically relevant models, for instance, for high-${\mathit{T}}_{\mathit{c}}$ cuprates.

• Received 28 June 1993

DOI:https://doi.org/10.1103/PhysRevB.48.13833