  We present the simplest model to which one can apply the supersymmetric
Hubbard operators recently introduced. For the atomic model, $H=-E_d X_{00}$,
where $X_{00} = |0><0|$ 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 = 2
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.
