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

Abstract

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-Hückel 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.

Figures (5)

• Figure 1: (a) Interpenetrating diamond structure. (b) The hydrogen configurations restricted to a single diamond lattice in the ice-VII phase. The white spheres correspond to hydrogen atoms and the red spheres correspond to oxygen atoms. In this phase, there is no long-range order. (c) An example of the hydrogen configuration in the ice-VIII phase. In the configuration shown, the system possesses a macroscopic polarization along the $z$-direction and the other diamond lattice has an opposite polarization. (d) The ice-X phase. The hydrogen atoms occupy the symmetric midpoints between oxygen atoms.
• Figure 2: (a) An illustration of the relationship between the diamond lattice and the pyrochlore lattice. (b) The spin representation of the hydrogen configuration in the ice-VII phase. Each tetrahedron satisfies the "two-in, two-out" configuration but may possess a different local magnetic moment. (c) The ice-VIII phase. Each tetrahedron satisfies the ice rules and shares an identical macroscopic magnetic moment aligned along the $z$-direction. (d) The ice-X phase. The hydrogen atoms are localized at the symmetric midpoints of the oxygen-oxygen bonds, which corresponds to the $S^z=0$ state on all pyrochlore sites.
• Figure 3: Monte Carlo results for the transformation between ice-VII and ice-X ($J'=0$). (a) The $\Delta$ dependence of the $S^z=0$ occupation $m_\mathrm{X}$ at $T=0.3$ and $L=16$. (b) The system size dependence of the specific heat $c$ and (c) the susceptibility $\chi_\mathrm{X}$ at $T=0.3$. The peak heights saturate as $L$ increases, indicating a continuous crossover rather than a phase transition. (d) The peak positions of the second-order derivatives of the free energy for $L=4$. Blue points show peaks of $c$, while orange points represent peaks of $\chi_\mathrm{X}$. The green lines indicate the phase transition boundaries in the monopole-free limit Pandey2025.
• Figure 4: Monte Carlo results for $J' = 0.02$ and $T = 0.2$. System size dependence of (a) the squared ferromagnetic order parameter $m_{\mathrm{VIII}}$ (circles) and the $S^z = 0$ occupation $m_{\mathrm{X}}$ (squares), and (b) the specific heat $c$. The insets show zoomed-in views near the phase transition point at $\Delta \simeq 0.86$.
• Figure 5: Phase diagram for $J'=0.02$ and $L=4$. The color maps represent (a) the ferromagnetic order parameter $m_{\mathrm{VIII}}$ and (b) the $S^z=0$ occupation $m_\mathrm{X}$. White circles denote peak positions of the specific heat $c$, and the red star represents the first-order transition point at $T=0$ arising from a level crossing. At $T=0$, the level crossing point is shifted to $\Delta_c = J + 2J' = 1.04$ due to the energy gain from the next-nearest-neighbor interaction. The red line, drawn as a guide to the eyes, represents the expected first-order phase transition boundary separating the ice-VIII phase from the other phases.