  We discuss the problem of proton motion in Hydrogen bond materials with
special focus on ice. We show that phenomenological models proposed in the past
for the study of ice can be recast in terms of microscopic models in close
relationship to the ones used to study the physics of Mott-Hubbard insulators.
We discuss the physics of the paramagnetic phase of ice at 1/4 filling (neutral
ice) and its mapping to a transverse field Ising model and also to a gauge
theory in two and three dimensions. We show that H3O+ and HO- ions can be
either in a confined or deconfined phase. We obtain the phase diagram of the
problem as a function of temperature T and proton hopping energy t and find
that there are two phases: an ordered insulating phase which results from an
order-by-disorder mechanism induced by quantum fluctuations, and a disordered
incoherent metallic phase (or plasma). We also discuss the problem of
decoherence in the proton motion introduced by the lattice vibrations (phonons)
and its effect on the phase diagram. Finally, we suggest that the transition
from ice-Ih to ice-XI observed experimentally in doped ice is the
confining-deconfining transition of our phase diagram.
