Comparison of Variational and Free Particle Nodes

In this section, we are going to compare thermodynamic properties derived from PIMC simulations of deuterium with free particle nodes with those using VDM nodes. The effect of the nodal surfaces is largest in the region of a high degree of electronic degeneracy, found at high density and low temperature. It should be noted for low densities e.g. $r_s \geq 3$, electrons become bound and localized before a significant degree of degeneracy is reached. This can be seen in the phase diagram in Fig.  and in the histogram of the permutation probabilities in Fig. . For this reason, we focus in the discussion on the two densities corresponding to $r_s=2$ and $1.86$ where the electronic degeneracy becomes very important for $T \leq 31\,250\,\rm {K}$.

Figure: Difference in the internal energy per atom from PIMC simulations with VDM and free particle nodes vs. temperature using $N_P=32$, $\tau^{-1}=2 \cdot 10^6 \rm{K}$, $n_{\rm A}=1$, and $n_{\rm E}=2$.

In Fig. , a comparison of the internal energy from simulations with FP and VDM nodes is shown. Simulations with VDM nodes lead to lower internal energies than those with FP nodes. The differences become smaller with increasing temperature since the both density matrices are exact in this limit. Since the free energy $F$ is the integral of the internal energy over temperature, one can conclude that VDM nodes yield to a smaller $F$ and hence, are the more appropriate nodal surface.

Figure: Histogram of the permutation probability from PIMC simulations with VDM nodes as in Fig.  for free particles nodes.

In the following, we will discuss the revised phase diagram shown in Fig. . First of all, the nodal surfaces are not important for low densities e.g. $r_s \stackrel{\scriptstyle>}{\scriptscriptstyle\sim}\:3$. Therefore, we can copy the low density PIMC results using free particle nodes that determine the location of the molecular and atomic regime. Furthermore, we compare the histograms of the permutation probability shown in Fig.  with those from free particle nodes in Fig. . Generally, one finds that the permutation probability is reduced for VDM nodes and that the area of metallic regime in the temperature-density plane has shrunk considerably. This can be understood as follows. FP nodes are generated from maximally delocalized orbitals. Therefore, they could favor delocalized states and allow more permutations. VDM nodes include bound states and lead to more localized orbitals, which consequently reduces the fraction of permuting electrons. A comparison of the distribution of the cycle length can be found in Fig. . However, it should be noted that there is the possibility that the VDM leads to too localized orbitals in the limit of high density, as indicated by the too small width of the Gaussians orbitals shown in Fig. 3.11.

Figure: The proton-proton pair correlation functions from PIMC simulations with FP nodes (solid lines) and VDM nodes (dashed lines) at $r_s=1.86$ are shown in the left column for different temperatures, which are indicates on the right. The right graphs show the corresponding electron permutation cycle distributions ($\Diamond$ for FP nodes and $\circ$ for VDM nodes). The VDM nodes show a gradual increase in the number of molecules and a lower permutation probability while FP nodes show a more abrupt change in the molecular fraction and a higher number of permutations, which suggests more delocalized electrons.

In the next step, we calculated the pressure as a function of temperature for $r_s=1.86$ using different time steps as in Fig. . Within the statistical error bars, results from PIMC simulations with VDM nodes do not show a region where the pressure drops down with increasing T, $\left.\frac{dP}{dT}\right\vert _{V^{\!\!\!\!\!\!\!\;\,^{^\diamond}}}<0$. This shows that the results in regime near the PPT are strongly affected by the type of nodes being used. This also raises some concern on how reliable the PIMC simulations are in this regime. However, one can proceed as is done in QMC at $T=0$, where one takes those set of nodes which leads to the lowest internal energy. At finite temperature, we used the free energy argument above and concluded that the VDM nodes are more reliable. They do not seem to predict a first order PPT in the particular region of density and temperature shown in Fig. . However, there is the possibility that the PPT has been shifted to temperatures below $5000\,$K and that the metallic phase exhibits again a low temperature boundary, which is characterized by a PPT similar to the metallic regime predicted using free particle nodes as shown in Fig. . On the other hand, one can say that the PPT might have disappeared because we employed VDM nodes that includes a reasonable description of bound states as well as free particle states and therefore, we got rid of the imbalance that the free particle favored the metallic regime.

The PPT predicted by free particle nodes also predicted a change in the number of molecules. In Fig. , the proton-proton pair correlation functions as well as the permutation cycle distributions are shown for series of simulations for different temperatures at $r_s=1.86$. The two functions were chosen in order to characterize the effect of the two different nodal surfaces. At high temperatures such as $T=125\,000\,\rm {K}$, simulations with FP and VDM nodes give identical results because the nodes are equivalent in this limit. At this temperature, hydrogen is composed of strongly interacting gas of free protons and electrons with a moderate degree of electronic degeneracy ( $T_F=168\,090\,\rm {K}$). With decreasing temperature, the degeneracy increases, which can be inferred from cycle distributions. At $31\,250\,\rm {K}$, small deviations between FP and VDM nodes emerge. At half the temperature, the differences have increased further, which consequently lead to a different state with different proton-proton pair correlations. Simulations with VDM nodes predict a significant molecular fraction and a smaller degeneracy, while FP results show no peak in the proton-proton pair correlation function in combination with a larger degeneracy. In the FP case, most of the electrons are delocalized and occupy states similar to a free Fermi gas. VDM nodes predict that the electrons are in more localized states, which leads to the molecular binding.

Figure: Comparison of the cycle length distribution (probability of an electron being involved in a permutation cycle of the length i) in a PIMC simulation of hydrogen with $32$ protons and $16$ electrons of each spin state at $T=10\,000\,\rm{K}$ and $r_s=1.86$ ( $T_F=168,090\,\rm{K}$).

At $7812\,\rm {K}$ one finds big differences in the proton-proton pair correlation functions since FP nodes predict a metallic state and VDM nodes a more molecular structure. At $5000\,\rm {K}$ both are rather similar because according the FP nodes, one has crossed the phase boundary, the pressure has risen and molecules are formed. VDM nodes, on the other hand, predict a smooth transition.

From Fig. , it can be deduced that the discussed differences in the number of permutations are indeed a consequence of the type of nodes rather than a result of using a too large time step. Going from $\tau^{-1}=2 \cdot 10^6 \rm {K}$ to $8 \cdot 10^6 \rm {K}$ reduces the number of permutations slightly because the nodal surfaces are enforced more accurately, which prevents some permutations from occurring. However, the differences to FP results are significantly larger.