We present the simplest model to which one can apply the supersymmetric Hubbard operators recently introduced P. Coleman, C. P{é}pin, J. Hopkinson, Phys. Rev. B62, 3852 (2000). For the atomic model, H = -E_d X₀₀, where X₀₀ = |0\rangle \langle 0| is a Hubbard operator and E_d is the energy of the localized spin level, we show how one can develop exact solutions for the entropy and heat capacity as a function of temperature. With this gold standard we are able to develop a controlled approximation scheme to field theoretically treat the SUSY approximation at the level of mean field + Gaussian corrections and test its accuracy against the widely used slave boson and slave fermion approximations. We find that in addition to slave boson and slave fermion solutions, a new class of solutions exists in the physical case Q=1, N=24 which can be properly treated by neither previously existing approach. The phase diagram generated by the mean field saddle-point bears a superficial resemblance to the V-shaped phase diagram common to systems close to a quantum critical point and may provide a natural starting point for investigations of strongly correlated models capturing this physics.

PACS numbers: 71.27.+a, 71.10.Hf, 75.20.Hr, 75.40.--s