Classical and quantum theories of proton disorder in hexagonal water ice

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.