Harmonic phase in polar liquids and spin ice

Many liquid or liquid-like states remain stable down to temperatures well below the interaction energy scale, where mean-field theory predicts an ordering transition. In magnetism, correlated states such as spin ice and the spin liquid have been described as Coulomb phases, governed by an emergent gauge principle. In the physical chemistry of polar liquids, systems that evade mean field order have, in contrast, been described by Onsager's theory of the reaction field. Here we observe that in the low-temperature limit, Onsager's theory may be cast as a prototypical theory of the Coulomb phase. However at finite temperature, it describes a distinct geometrical state, characterised by harmonic functions. This state, labelled here the 'harmonic phase', is shown to occur experimentally in spin ice, a dipolar lattice system. It is suggested to be relevant to more general dipolar liquids.