Conclusions

In this work, we have applied path integral Monte Carlo simulations (PIMC) to hot, dense hydrogen and deuterium. Our goal was to determine the equation of state from first principles and to elucidate the high temperature phase diagram. The main focus was on explaining the structural changes that lead to a molecular, an atomic and a plasma regime at low density and exhibit a metallic regime at high density. We determine the equation of state in density and temperature range corresponding to $1 \leq r_s \leq 14$ and $5000 \leq T \leq 10^6 \,\rm K$. The wide range of possible applications include studies of the brown dwarfs and Jovian plants. Furthermore, the data can be used in chemical models in order to fit free parameters, which then allows one to obtain related properties with an improved accuracy.

We developed a variational density matrix (VDM) related to the variational methods at zero temperature. In our approach to dense hydrogen, we derived a VDM that includes interactions and describes bound states, ionization, and dissociation processes. In this approach, we used a single determinant with Gaussian orbitals. Possible extensions include improved single particle orbitals, treating the exchange terms with a higher accuracy and the consideration of correlation effects.

The VDM was developed with the motivation to replace the free particle nodes, used in the PIMC simulations so far, with a more realistic density matrix that already includes the principle physical effects in dense hydrogen. We carefully analyzed what effect the improved nodal surfaces have on the thermodynamics properties derived from PIMC simulations. The most significant changes were found in the regime of the abrupt transition to a metallic state observed using PIMC simulations with free particle nodes. Using VDM nodes, we found no evidence for a phase transition in the parameter range under consideration. It remains to be determined if the improvements in the nodes eliminated the plasma phase transition altogether or if it was shifted to temperatures below 5000 K.

An important part of our research was the comparison with laser shock wave experiments. We performed a detailed analysis in the relevant density range of the various approximations entering into PIMC simulations including different time steps and system sizes as well as the type of nodal surface being employed. None of them had a significant effect on the comparison, which showed that PIMC simulation increased compressibility of $4.3 \pm 0.1$ on the shock Hugoniot and cannot reproduce the experimental findings of values of about $6 \pm 1$. Further theoretical and experimental work will be needed to resolve this discrepancy.

Furthermore, we extended the restricted path integral method to the sampling with open path in order to calculate off-diagonal elements of the fermionic many-body density matrix. As a first application to the electron gas, we could show how the momentum distribution of interacting system changes from Maxwell-Boltzmann type at high temperature to a Fermi-Dirac distribution at low temperature. In the path integral formalism, this process is governed by an increasing number of permuting paths, which contribute with different signs to the averages. In the future work, we want to advance the method to lower temperatures, study its scaling behavior and compare with zero temperature calculations.

We also used the off-diagonal sampling technique to determine the natural orbitals of hydrogen at low density. The remaining challenge is how this method can be applied to a many-body system at high density while keeping the computational demand at a reasonable level.