  The Hartree-Fock approximation to the many-fermion problem can break exact
symmetries, and in some cases by changing a parameter in the interaction one
can drive the Hartree-Fock minimum from a symmetry-breaking state to a
symmetry-conserving state (also referred to as a ``phase transition'' in the
literature). The order of the transition is important when one applies the
random phase approximation (RPA) to the of the Hartree-Fock wavefunction: if
first order, RPA is stable through the transition, but if second-order, then
the RPA amplitudes become large and lead to unphysical results. The latter is
known as ``collapse'' of the RPA. While the difference between first- and
second-order transitions in the RPA was first pointed out by Thouless, we
present for the first time non-trivial examples of both first- and second-order
transitions in a uniform model, the interacting shell-model, where we can
compare to exact numerical results.
