Supersymmetric SP(N) spin representation for heavy fermion systems

In heavy fermion materials, the character of the spin correlations change radically between the antiferromagnetic and fermi liquid region of the phase diagram. In the latter, the spin behaves as a bosonic object, condensing into magnetic order, which we traditionally describe using a Holstein-Primakoff or Schwinger bosons, whereas in the the fermi liquid phase, the spin binds to the conduction electrons to form a composite heavy fermion, usually described by Abriskosov fermions. Past work [1] developed a supersymmetric representation of SU(N) spin operators. Here we analyze the supersymmetry of SP(N) symplectic spin operators, which provides us with the capability of studying antiferromagnetic and superconducting order. As a warm up problem, we show how this formalism can be applied to a two site Kondo model, coupled via a Heisenberg coupling. [1] P. Coleman, C. Pepin and A. Tsvelik, Phys. Rev. B 62, 3852 (2000).