  It has been known since the pioneering work of Bernal, Fowler and Pauling
that common, hexagonal (Ih) water ice is the archetype of a frustrated material
: a proton-bonded network in which protons satisfy strong local constraints -
the "ice rules" - but do not order. While this proton disorder is well
established, there is now a growing body of evidence that quantum effects may
also have a role to play in the physics of ice at low temperatures. In this
Article we use a combination of numerical and analytic techniques to explore
the nature of proton correlations in both classical and quantum models of ice
Ih. In the case of classical ice Ih, we find that the ice rules have two,
distinct, consequences for scattering experiments - singular "pinch points",
reflecting a zero-divergence condition on the uniform polarization of the
crystal, and broad, asymmetric features, coming from its staggered
polarisation. In the case of the quantum model, we find that the collective
quantum tunnelling of groups of protons can convert states obeying the ice
rules into a quantum liquid, whose excitations are birefringent, emergent
photons. We make explicit predictions for scattering experiments on both
classical and quantum ice Ih, and show how the quantum theory can explain the
"wings" of incoherent inelastic scattering observed in recent neutron
scattering experiments [Bove et al., Phys. Rev. Lett. 103, 165901 (2009)].
These results raise the intriguing possibility that the protons in ice Ih could
form a quantum liquid at low temperatures, in which protons are not merely
disordered, but continually fluctuate between different configurations obeying
the ice rules.
