Low-density phase diagram of the three-dimensional electron gas

Variational and diffusion quantum Monte Carlo methods are employed to investigate the zero-temperature phase diagram of the three-dimensional homogeneous electron gas at very low density. Fermi fluid and body-centered cubic Wigner crystal ground-state energies are determined using Slater-Jastrow-backflow and Slater-Jastrow many-body wave functions at different densities and spin polarizations in finite simulation cells. Finite-size errors are removed using twist-averaged boundary conditions and extrapolation of the energy per particle to the thermodynamic limit of infinite system size. Unlike previous studies, our results show that the electron gas undergoes a first-order quantum phase transition directly from a paramagnetic fluid to a body-centered cubic crystal at density parameter $r_{\text{s}}=86.6\left(7\right)$, with no region of stability for an itinerant ferromagnetic fluid.

Figure 1

DMC energy against logarithm of Gaussian exponent $C$ for ferromagnetic and antiferromagnetic bcc Wigner crystals (a) at $r_{\text{s}}=100$ and (b) at $r_{\text{s}}=125$, using SJ trial wave functions. The energy has been extrapolated to zero time step and infinite system size in each case. The solid curves show quadratic fits to the energies of ferromagnetic and antiferromagnetic crystals as functions of $ln\left(C\right)$, while the dashed curve shows a quartic fit. The horizontal line in (a) shows the twist-averaged paramagnetic Fermi fluid energy (with the dotted lines indicating error bars), while the vertical line shows the Gaussian exponent given by Eq. (7).

Figure 2

TA SJB-DMC energy against inverse of system size for a paramagnetic Fermi fluid at $r_{\text{s}}=100$. Also shown are the effects of applying the first and second terms of Eq. (10), and the effects of fitting various FS scaling laws to the resulting data.

Figure 3

SJ-DMC energy against inverse of system size for a ferromagnetic bcc Wigner crystal at $r_{\text{s}}=100$. Also shown are the effects of applying the first term of Eq. (10) and of subtracting an estimate of the center-of-mass kinetic energy, together with fits of $E\left(N\right)=E\left(\infty \right)+b{N}^{-1}$ to the resulting energy data.

Figure 4

TA Fermi fluid SJ-DMC energies as functions of system size $N$ at different spin polarizations $\zeta$. The dotted line represents the extrapolation to infinite system size. These SJ-DMC energies have not been extrapolated to zero time step.

Figure 5

TA SJB-DMC energies against system size $N$ for different spin polarizations $\zeta$. The dotted lines represent extrapolations to infinite system size. The DMC energies were not extrapolated to zero time step.

Figure 6

SJ-DMC energy against the inverse of system size $N$ for the crystal phase. The dotted line shows the extrapolation to infinite system size. $\zeta =0$ and 1 are the spin polarizations of the system (antiferromagnetic and ferromagnetic crystals, respectively). The DMC energies have been extrapolated to zero time step.

Figure 7

Extrapolated estimates of the electronic charge densities of antiferromagnetic (AF) and ferromagnetic (F) bcc Wigner crystals plotted along a straight line in the [100] direction. Results are shown for two density parameters in the vicinity of the crystallization density. The charge densities have been obtained by extrapolated estimation (twice the DMC charge density minus the VMC charge density), which largely removes errors that are linear in the error in the trial wave function. Furthermore, the charge densities have been extrapolated to infinite system size, assuming the FS error goes as $N^{-1}$. The shaded regions indicate one standard error about the mean.

Figure 8

TA SJB-DMC energy against time step $\tau$ for the paramagnetic ($\zeta =0$) and ferromagnetic ($\zeta =1$) fluid phases at different system sizes $N$.

Figure 9

TA SJB-DMC energies against system size $N$ for different spin polarizations $\zeta$. The DMC energies are extrapolated to zero time step. The dotted line represents the extrapolation to infinite system size.

Figure 10

SJ-DMC energy against time step for bcc Wigner crystals at (a) $r_{\text{s}}=80$ and (b) 100. Results are shown for both antiferromagnetic and ferromagnetic crystals. The system size is $N=64$ electrons. The target population is varied in inverse proportion to the time step.

Figure 11

(Left) SJ-DMC spin polarization energy of the 3D-HEG multiplied by $r_{\text{s}}^{3/2}$ against density parameter $r_{\text{s}}$. (Right) SJ-DMC spin polarization energy of the 3D-HEG multiplied by ${r_{\text{s}}}^{3/2}$ against spin polarization $\zeta$ at various densities.

Figure 12

Correlation energies obtained in SJ-DMC calculations for the fluid phase (symbols), together with the fit to Eq. (1) of the main text (lines). Error bars on the QMC data are shown, but are smaller than the symbols. The DMC energies were not extrapolated to zero time step.

Figure 13

(Left) SJB-DMC relative energies of fluid phases of the 3D-HEG are plotted against $r_{\text{s}}$ for different spin polarizations $\zeta$ in the thermodynamic limit of infinite system size. (Right) SJB spin polarization energy of the 3D-HEG multiplied by ${r_{\text{s}}}^{3/2}$ at various densities. The DMC energies were not extrapolated to zero time step.

Figure 14

Correlation energies obtained in SJB-DMC calculations for the fluid phase (symbols), together with the fit of Eq. (14) (lines). Error bars on the QMC data are shown, but are smaller than the size of the symbols.

Figure 15

Energies per particle of the Fermi fluid and Wigner crystal phases at low density. The Madelung energy of the bcc lattice has been subtracted off and the resulting energies have been rescaled by $r_{\text{s}}^{3/2}$ to highlight the differences between phases.

Figure 16

Twist-averaged SJB-VMC and SJB-DMC energies per electron $E$ against spin polarization $\zeta$ at $r_{\text{s}}=50$ for $N=54$ electrons. Our results were obtained using a face-centered cubic simulation cell. The results reported in Ref. [11] [“PRE (2002)”] may have been obtained in a simple cubic simulation cell, although the difference between the ground-state energies in face-centered and simple cubic cells is small, as shown in Table 11.

Figure 17

Energy per electron as a function of spin polarization $\zeta$ at $r_{\text{s}}=50$. The energies are obtained using extrapolation from TA energies at $N=54$ and $N=108$ to infinite system size.