# Quantum Monte Carlo Simulation of Hydrogen at High Pressure

Quantum Mechanics is the fundamental theory on which our present understanding of the physical world is based. During the last century the “Founding Fathers” have developed the theory and established its fundamental character, and now we are facing the problem of using the laws to predict the behavior of systems of many particles in a wide range of physical conditions, from the ultra-cold states of matter to the extreme conditions of temperature and pressure found in the interiors of planets. This is a phenomenal challenge that can only be attacked by numerical methods.

The current description for extended systems of nuclei and electrons is based on a self-consistent mean field theory which is successful for many applications but has some well known limitations. In the specific case of hydrogen at the interesting physical conditions, related to planetary science, to high pressure experiments and to energy applications, this theory is not accurate enough and new fundamental developments are required.

The main goals of a project under the leadership of Carlo Pierleoni of the University of L’Aquila, Italy, were: 1) to elucidate important aspects of the hydrogen phase diagram related to the pressure-induced molecular dissociation and metallization. In particular a) the occurrence of two coexisting liquid phases (one molecular and insulating, the other monatomic and metallic) and the related reentrant melting line of the molecular solid; b) the possible existence of a liquid ground state of metallic hydrogen, a new state of matter. 2) to improve the treatment of electronic correlation by developing algorithms based on Quantum Monte Carlo (QMC) methods. QMC is much more accurate than the mean-field theory but requires at least an order of magnitude more computer resources. However it is highly parallelizable and scales extremely well on many thousands of core, therefore is ideal for Tier-0 machines like Hermit.

The main results achieved are: 1) a new prediction of the liquid-liquid transition line with Coupled Electron-Ion Monte Carlo (a QMC-based method) including a proper treatment of nuclear quantum effects. These results are to date the most accurate predictions of this phase transition. Our results will be a reference for future experiments and also a benchmark for better approximations within the mean-field framework. 2) Benchmark studies of the mean-field method with several approximation which has been the subject of three publications. 3) Evidence for a stable crystalline low temperature phase of metallic hydrogen which rules out the ground state liquid.

The study will be very useful for the large community of scientists that uses the approximate theory to better understand its limitations. More in general the new QMC algorithms will constitute the method to overcome the mean-field treatment of electronic correlations in the future.

The project was made possible through the Partnership for Advanced Computing in Europe, PRACE, which allocated 24 million core hours of computing time on GCS supercomputer Hermit of HLRS Stuttgart to the project.

The team was international with a nearly equal number of participants from Italy and from the USA, and one additional member from France. The same team also benefitted from other computer time allocations within the INCITE-USA program.

Source: © University of L’Aquila, Italy

*Figure 1. Phase diagram of hydrogen at high pressure. The four experimentally observed phases of the molecular crystal are reported at low temperature (I,II,III,IV). The reentrant nature of the molecular melting line is shown with experimental data below 200GPa and self-consistent theory predictions (black line and yellow downward triangles and yellow dashed line). The liquid-liquid phase transition predictions from several theories are reported by almost vertical lines. Full squares are from Coupled Electron-Ion Monte Carlo (CEIMC), our QMC based method, both with classical protons (black squares) and with quantum protons (red squares). Black upward triangles are predictions from the self-consistent theory with the standard approximation for the exchange-correlation functional (GGA-PBE) while downward triangles are predictions with an improved approximation (vdW-DF2). Note how wide is the scatter of self-consistent theory predictions with different approximations. Also the melting line of the metallic monatomic crystal (Cs-IV) at higher pressure (P>400GPa), as reported independently by two groups, is shown. This prediction rules out the existence of a stable liquid state at low temperature.*

Carlo Pierleoni

Dipartimento di Scienze Fisiche e Chimiche

Universita' dell'Aquila

Via Vetoio, 10 I-67100 L'AQUILA/Italy

e-mail: carlo.pierleoni@aquila.infn.it

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