  We consider non-equilibrium phenomena in a very simple model that displays a
zero-temperature first-order phase transition. The quantum Ising model with a
four-spin exchange is adopted as a general representative of first-order
quantum phase transitions that belong to the Ising universality class, such as
for instance the order-disorder ferroelectric transitions, and possibly
first-order T = 0 Mott transitions. In particular, we address quantum quenches
in the exactly solvable limit of infinite connectivity and show that, within
the coexistence region around the transition, the system can remain trapped in
a metastable phase, as long as it is spatially homogeneous so that nucleation
can be ignored. Motivated by the physics of nucleation, we then study in the
same model static but inhomogeneous phenomena that take place at surfaces and
interfaces. The first order nature implies that both phases remain locally
stable across the transition, and with that the possibility of a metastable
wetting layer showing up at the surface of the stable phase, even at T=0. We
use mean-field theory plus quantum fluctuations in the harmonic approximation
to study quantum surface wetting.
