  Many body models undergoing a quantum phase transition to a broken-symmetry
phase that survives up to a critical temperature must possess, in the ordered
phase, symmetric as well as non-symmetric eigenstates. We predict, and
explicitly show in the fully-connected Ising model in a transverse field, that
these two classes of eigenstates do not overlap in energy, and therefore that
an energy edge exists separating low-energy symmetry-breaking eigenstates from
high-energy symmetry-invariant ones. This energy is actually responsible, as we
show, for the dynamical phase transition displayed by this model under a sudden
large increase of the transverse field. A second situation we consider is the
opposite, where the symmetry-breaking eigenstates are those in the high-energy
sector of the spectrum, whereas the low-energy eigenstates are symmetric. In
that case too a special energy must exist marking the boundary and leading to
unexpected out-of-equilibrium dynamical behavior. An example is the fermionic
repulsive Hubbard model Hamiltonian H. Exploiting the trivial fact that the
high energy spectrum of H is also the low energy one of -H, we conclude that
the high energy eigenstates of the Hubbard model are superfluid. Simulating in
a time-dependent Gutzwiller approximation the time evolution of a high energy
BCS-like trial wave function, we show that a small superconducting order
parameter will actually grow in spite of the repulsive nature of interaction.
