Supersymmetric spin operators

We develop a supersymmetric representation of spin operators which unifies the Schwinger and Abrikosov representations of SU(N) spin operators, allowing a second-quantized treatment of representations of the SU(N) group with both symmetric and antisymmetric character. By applying this to the SU(N) Kondo model, we show that it is possible to develop a controlled treatment of both Magnetism and the Kondo effect within a single large N expansion. 78.20.Ls, 47.25.Gz, 76.50+b, 72.15.Gd Typeset using REVTEX 1 I. MOTIVATION FOR A NEW SPIN REPRESENTATION Recent experiments on quantum phase transitions in heavy fermion materials have led to a debate about how magnetism condenses out of the metallic state at absolute zero. Certain heavy fermion materials can can be tuned between the magnetic and the paramagnetic state through the use of pressure or chemical doping. The quantum critical point which separates these two phases is of great current interest, in part because materials in its vicinity may become fundamentally new kinds of metal5–7. Heavy fermion materials contain a dense lattice of magnetic moments; conventional wisdom assumes that the spins of the local moments are magnetically screened and of no importance to the magnetic quantum critical point8–10. Recent neutron data contradict this viewpoint, by showing that the spin correlations at the quantum critical point are critical in time, but local on an atomic scale4,11–13, suggesting that unscreened local moments emerge from the metallic state at the quantum critical point. If it is indeed true that the magnetic quantum critical points involve local moment physics, then a new theoretical approach is required. Traditionally, heavy fermion physics is modeled using a Kondo lattice Hamiltonian, describing the interaction between a bath of conduction electrons and an array of local moments. One of the well-developed theoretical methods for approaching this model is the large N expansion15–19, where the idea is to use a generalization of the quantum mechanical spin operators, in which the underlying spin rotation group is generalized from SU(2) to SU(N). The utility of this method derives from the fact that in the limit N → ∞, it provides an essentially exact, analytic treatment of the Kondo lattice problem. Unfortunately the way this procedure is carried out at present, magnetic interactions are suppressed as a 1/N correction, beyond the horizon for a controlled computation. In this paper, we show how we can overcome this shortcoming by the use of a supersymmetric spin representation for local moments. The theoretical description of interacting local moments poses a fundamental problem: the Pauli Spin operator S does not satisfy a Wick’s decomposition theorem, which pre-empts