  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.
