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It has been known since the pioneering work of Bernal, Fowler and Paulingthat 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 wellestablished, there is now a growing body of evidence that quantum effects mayalso have a role to play in the physics of ice at low temperatures. In thisArticle we use a combination of numerical and analytic techniques to explorethe nature of proton correlations in both classical and quantum models of iceIh. 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 thecrystal, and broad, asymmetric features, coming from its staggeredpolarisation. In the case of the quantum model, we find that the collectivequantum tunnelling of groups of protons can convert states obeying the icerules into a quantum liquid, whose excitations are birefringent, emergentphotons. We make explicit predictions for scattering experiments on bothclassical and quantum ice Ih, and show how the quantum theory can explain the"wings" of incoherent inelastic scattering observed in recent neutronscattering experiments [Bove et al., Phys. Rev. Lett. 103, 165901 (2009)].These results raise the intriguing possibility that the protons in ice Ih couldform a quantum liquid at low temperatures, in which protons are not merelydisordered, but continually fluctuate between different configurations obeyingthe ice rules.

Classical and quantum theories of proton disorder in hexagonal water ice