Quantum replica approach to the underscreened Kondo model

We extend the Schwinger boson large N treatment of the underscreened Kondo model in a way that correctly captures the finite elastic phase shift in the singular Fermi liquid. The new feature of the approach is the introduction of a flavor quantum number with K possible values associated with the Schwinger boson representation. The large N limit is taken maintaining the ratio k=K/N fixed. This approach differs from previous approaches in that we do not explicitly enforce a constraint on the spin representation of the Schwinger bosons. Instead, the energetics of the Kondo model cause the bosonic degrees of freedom to “self-assemble” into a ground-state in which the spins of K bosons and N-K conduction electrons are antisymmetrically arranged into a Kondo singlet. With this device the large N limit can be taken in such a way that a fraction K/N of the Abrikosov-Suhl resonance is immersed inside the Fermi sea. We show how this method can be used to model the full energy dependence of the singular Abrikosov-Suhl resonance in the under-screened Kondo model and the field-dependent magnetization.