Electronic and Nuclear Quantum Effects on the Ice XI/Ice Ih Phase Transition

We study the isotope effect on the temperature of the proton order/disorder phase transition between ice XI and ice Ih, using the quasiharmonic approximation combined with \textit{ab initio} density functional theory calculations. We show that this method is accurate enough to obtain a phase transition temperature difference between light ice (H$_2$O) and heavy ice (D$_2$O) of 6 K as compared to the experimental value of 4 K. More importantly, we are able to explain the origin of the isotope effect on the much debated large temperature difference observed in the phase transition. The source of the difference is directly linked to the physics behind the anomalous isotope effect on the volume of hexagonal ice that was recently explained in [Phys. Rev. Lett. 108, 193003 (2012)]. These results indicate that the same physics might be behind the isotope effects in transition temperatures between other ice phases.