Momentum distribution of the uniform electron gas at finite temperature: Effects of spin polarization

We carry out extensive direct path integral Monte Carlo (PIMC) simulations of the uniform electron gas (UEG) at finite temperature for different values of the spin-polarization $\xi$. This allows us to unambiguously quantify the impact of spin effects on the momentum distribution function $n\left(\mathbf{k}\right)$ and related properties. We find that interesting physical effects like the interaction-induced increase in the occupation of the zero-momentum state $n\left(\mathbf{0}\right)$ substantially depend on $\xi$. Our results further advance the current understanding of the UEG as a fundamental model system, and are of practical relevance for the description of transport properties of warm dense matter in an external magnetic field. All PIMC results are freely available online and can be used as a benchmark for the development of methods and applications.

Figure 1

Schematic illustration of the off-diagonal configuration space $Z_{\mathbf{r},{\mathbf{r}}^{\prime };\sigma }$ including a single open trajectory with different start and end points $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$. Adapted from Ref. [88] with the permission of the authors.

Figure 2

Polarization dependence of the spin-resolved density parameter $r_s^{\sigma }$ (top panel) and reduced temperature ${\theta }^{\sigma }$ (bottom panel) for $r_s=2$ and $\theta =2$ (using the Fermi energy of the unpolarized system as a reference).

Figure 3

Momentum distribution of the spin-polarized ($\xi =1$) electron gas for ${\theta }^{\uparrow }=1$. The dashed black curve shows results for the ideal Fermi gas [Eq. (5)], and the data points correspond to our PIMC data for different $r_s$. The left and right panels shows $n\left(\mathbf{k}\right)$ on a linear and logarithmic scale, respectively.

Figure 4

Top: $r_s$ dependence of $n\left(0\right)$ for the spin-polarized UEG with ${\theta }^{\uparrow }=1$. Bottom: $r_s$ dependence of the exchange–correlation part to the kinetic energy (per particle) $K_{x}c$ [Eq. (16)], with the red circles being our PIMC results and the dotted green line having been computed from Eq. (17) using the parametrization by Groth et al. [68].

Figure 5

Momentum distribution function of the spin-polarized ($\xi =1$) UEG for $N=33$ electrons. Top row: $r_s=4$ and ${\theta }^{\xi =1}=1$; center row: $r_s=40$ and ${\theta }^{\xi =1}=1$; bottom row: $r_s=40$ and ${\theta }^{\xi =1}=0.5$. Left column: $n\left(\mathbf{k}\right)$, with the red circles, green crosses, and yellow crosses corresponding to our direct PIMC results, restricted PIMC results from Ref. [86], and modified RPIMC results (see text), respectively. Right column: Relative difference between $n\left(\mathbf{k}\right)$ and $n_0\left(\mathbf{k}\right)$ in percent, cf. Eq. (18).

Figure 6

Top: System-size dependence of the momentum distribution of the UEG at $r_s=10, \theta =2$, and $\xi =2/3$. The left and right columns correspond to the spin-up and -down electrons, respectively. Bottom: Relative deviation between PIMC data for $n^{\sigma }\left(\mathbf{k}\right)$ and the ideal Fermi distribution $n_0^{\sigma }\left(\mathbf{k}\right)$ in percent, cf. Eq. (18). Note that we use the Fermi wave number of the unpolarized system $k_F^{\xi =0}$ as a reference.

Figure 7

Off-diagonal density matrix $n\left(s\right)$ [cf. Eq. (4)] of the UEG at $r_s=10, \theta =2$, and $\xi =2/3$. The left and right panels show results for spin-up and spin-down electrons, and the red circles, green crosses, and yellow stars have been obtained for $N=18, N=36$, and $N=60$, respectively.

Figure 8

Top: System-size dependence of the momentum distribution of the UEG at $r_s=2, \theta =2$, and $\xi =2/3$. The left and right columns correspond to the spin-up and -down electrons, respectively. Bottom: Relative deviation between PIMC data for $n^{\sigma }\left(\mathbf{k}\right)$ and the ideal Fermi distribution $n_0^{\sigma }\left(\mathbf{k}\right)$ in percent, cf. Eq. (18). Note that we use the Fermi wave number of the unpolarized system $k_F^{\xi =0}$ as a reference.

Figure 9

Momentum distribution function $n^{\sigma }\left(\mathbf{k}\right)$ for ${\theta }_{\xi =0}=2$. Top left: $r_s=0.5$; top right: $r_s=2$; bottom left: $r_s=4$; bottom right: $r_s=10$. The colors distinguish different values of the spin-polarization $\xi$, and the dashed black curves show the corresponding results for the ideal Fermi gas $n_0^{\sigma }\left(\mathbf{k}\right)$. Finally, the crosses and diamonds correspond to spin-up and spin-down electrons, respectively. Note that we use the Fermi wave number of the unpolarized system $k_F^{\xi =0}$ as a reference.

Figure 10

Total momentum distribution function $n\left(\mathbf{k}\right)$ at ${\theta }_{\xi =0}=2$. Left: $r_s=0.5$; right: $r_s=10$. The colors distinguish different values of the spin-polarization $\xi$, and the corresponding dotted curves show results for the ideal Fermi gas $n_0^{\sigma }\left(\mathbf{k}\right)$. Note that we use the Fermi wave number of the unpolarized system $k_F^{\xi =0}$ as a reference.

Figure 11

Top: Spin-resolved momentum distribution function $n^{\sigma }\left(\mathbf{k}\right)$ for $r_s=2$. Left: ${\theta }_{\xi =0}=1$; right: ${\theta }_{\xi =0}=4$. The colors distinguish different values of the spin polarization $\xi$ and the dashed black curves show the corresponding results for the ideal Fermi gas $n_0^{\sigma }\left(\mathbf{k}\right)$. Finally, the crosses and diamonds correspond to spin-up and spin-down electrons, respectively. Bottom: Corresponding total momentum distributions $n\left(\mathbf{k}\right)=n^{\uparrow }\left(\mathbf{k}\right)+n^{\downarrow }\left(\mathbf{k}\right)$. Note that we use the Fermi wave number of the unpolarized system $k_F^{\xi =0}$ as a reference.

Figure 12

Density dependence of the occupation at zero momentum $n\left(0\right)$ of the UEG at ${\theta }_{\xi =0}=2$. Top panel: Direct PIMC data for the total momentum distribution $n\left(0\right)=n^{\uparrow }\left(0\right)+n^{\downarrow }\left(0\right)$ for $\xi =0$ (red circles), $\xi =1/3$ (black stars), and $\xi =1$ (green crosses). The horizontal lines depict the corresponding ideal values $n_0\left(0\right)$. Bottom panel: Direct PIMC results for $\xi =1/3$, black stars: $n\left(\mathbf{k}\right)$; red circles: $n^{\uparrow }\left(\mathbf{k}\right)$; green crosses: $n^{\downarrow }\left(\mathbf{k}\right)$.

Figure 13

Density dependence of the exchange–correlation part of the kinetic energy $K_{x}c$ at ${\theta }_{\xi =0}=2$ for different values of the spin polarization $\xi$ evaluated from the parametrization of $f_{x}c$ by Groth et al. [68] via Eq. (17).

Figure 14

Polarization dependence of the occupation at zero momentum for $r_s=2$ (top), $r_s=10$ (center), and $r_s=50$ (bottom) at ${\theta }_{\xi =0}=2$. The red circles, green crosses, and black stars correspond to the spin-up component $n^{\uparrow }\left(0\right)$, spin-down component $n^{\downarrow }\left(0\right)$, and the total distribution function $n\left(0\right)$, respectively. The corresponding lines show the respective ideal Fermi distribution function.

Figure 15

Polarization dependence of the exchange–correlation contribution to the kinetic energy $K_{x}c$ at ${\theta }_{\xi =0}=2$ evaluated from the parametrization of $f_{x}c$ by Groth et al. [68] via Eq. (17). The different lines correspond to various values of the density parameter $r_s$.