Quantum motion shapes the melting line of hydrogen at extreme pressure

When we think of a solid, we usually imagine arrays of atoms frozen in a perfect lattice. Quantum mechanics allows for a fundamentally different kind of matter. The uncertainty principle requires that atoms have significant quantum jitteriness, known as zero-point motion (ZPM). The collective effect of ZPM on a macroscopic object can fundamentally change its properties: it is this quantum jitteriness that prevents helium from ever freezing at atmospheric pressure—even at absolute zero temperature. In fact, it requires around 25 atmospheres of pressure to get Helium to freeze at 0 K, indicating the extent of the excess kinetic energy because of the ZPM in helium atoms. Such substances in which the ZPM becomes key to the macroscopic behavior are called quantum solids or liquids, with helium and hydrogen serving as the quintessential examples of such systems. In our paper, we took some of the first steps towards quantitatively understanding the macroscopic behavior of these quantum states of matter by predicting the melting line of molecular hydrogen using and not using the ZPM of the protons.

In most materials, the ZPM of the atoms is negligible compared to thermal fluctuations. This means we can get away with not accounting for quantum effects at normal temperatures and pressures. This is why the engineers constructing your HVAC or safeguarding water lines during winter can work accurately with classical physics. Extreme pressure, however, forces the atoms closer together and brings the quantum mechanics to the fore. Our research showed that for molecular hydrogen, at pressures larger than 150 GPa, the ZPM begins to have large effects on the macroscopic thermodynamics—lowering the melting temperature by as much as 100 K. To put this into perspective, if you were to carry the weight of the entire US population on your body, then you would experience about 150 GPa of pressure.

To establish our results, we studied the melting line—the boundary between a solid and a liquid as a function of pressure and temperature. First, we determined a purely classical melting line for molecular hydrogen that ignored the ZPM of the protons. We conducted a series of first-principle simulations using a two-phase configuration (see the first image below) at various temperatures and pressures. In this approach, we constructed a simulation cell of solid hydrogen in one half of the cell and liquid hydrogen in the other half. We then identified the equilibrium points where one phase outcompetes the other, thus obtaining upper and lower bounds of our melting line at various pressure-temperature points. This two-phase setup was necessary to overcome any surface energy barriers, as it provides an interface over which freezing or melting can occur. Additionally, we calculated the latent heat of melting by running single-phase simulations of homogeneous liquid and solid hydrogen separately at the same temperatures within the upper and lower bounds we previously identified. The latent heat is directly related to the slope of the melting line through the Clausius-Clapeyron equation. Thus, we obtained the melting line by fitting a smooth curve constrained within the upper and lower bounds identified by the two-phase simulations and having the slopes calculated by the single-phase simulations.

Once we constructed the melting line using classical protons, we added the associated free energy of the ZPM of the protons to smoothly deform our classical melting line to approach the true quantum-hydrogen melting line. We ran multiple single-phase simulations with increasing amounts of ZPM and computed how the classical melting line (the solid line in the figure below) shifts as we turn up the ZPM (to obtain the dotted line). Then, we re-computed the latent heat, this time including the ZPM to obtain the correct slopes. With the shift in melting line and the correct slopes in hand, we fit an accurate melting curve.

Our research confirmed prior results that molecular hydrogen remains a stable solid even at temperatures above 1600 K. Accessing this temperature range at the enormous pressures required is beyond the reach of current experiments. Our simulations currently provide the only window into a detailed understanding of these intrinsically quantum systems.