Continuous crossover between high-pressure ice phases VII and X driven by monopole screening: a model study

The proton-disordered molecular phase of water ice (ice-VII) and its ultrahigh-pressure non-molecular phase (ice-X) share identical macroscopic crystal symmetry (space group $Pn\bar{3}m$). This raises a fundamental thermodynamic question: are they distinct phases separated by a singularity, or are they adiabatically connected via a continuous crossover? To resolve this paradox, we investigate the finite-temperature phase diagram of high-pressure ices VII and X, as well as VIII, the proton-ordered phase that emerges at lower temperatures, using an effective classical spin-$1$ Blume-Capel model on the pyrochlore lattice. Through Monte Carlo simulations, we demonstrate that within this model, the transformation between the states corresponding to ice-VII and ice-X lacks a thermodynamic singularity, as characterized by non-divergent and non-coinciding peaks in the specific heat and susceptibility associated with the $S^z=0$ occupation. We attribute this continuous crossover behavior to the topological fragility of the hydrogen-bond network: the thermal proliferation of point-like monopole excitations (violations of the ice rules) induces Debye-H\"{u}ckel screening of the emergent gauge field, destroying the topological Coulomb phase at any finite temperature. In contrast, the destruction of the proton-ordered ice-VIII phase involves spontaneous symmetry breaking and remains a first-order phase transition. Our findings provide a microscopic rationale that reconciles the macroscopic crystallographic symmetries of dense ice with its underlying topological properties.