  The stationary points of the potential energy function V are studied for the
\phi^4 model on a two-dimensional square lattice with nearest-neighbor
interactions. On the basis of analytical and numerical results, we explore the
relation of stationary points to the occurrence of thermodynamic phase
transitions. We find that the phase transition potential energy of the \phi^4
model does in general not coincide with the potential energy of any of the
stationary points of V. This disproves earlier, allegedly rigorous, claims in
the literature on necessary conditions for the existence of phase transitions.
Moreover, we find evidence that the indices of stationary points scale
extensively with the system size, and therefore the index density can be used
to characterize features of the energy landscape in the infinite-system limit.
We conclude that the finite-system stationary points provide one possible
mechanism of how a phase transition can arise, but not the only one.
