Simulation of strongly quantum-degenerate uniform electron gas using the pseudo-fermion method

Abstract

For strongly quantum-degenerate systems at finite temperatures, the fermion sign problem remains the major obstacle to first-principles simulations. In this work, we apply the recently proposed pseudo-fermion method - designed to overcome the sign problem - to strongly quantum-degenerate uniform electron gases. We find that the pseudo-fermion method can efficiently and highly accurately infer the energy of the uniform electron gas while being free from the fermion sign problem. For example, in the strongly quantum-degenerate regime where RPIMC fails (33 spin-polarized electrons at the density parameter $r_s = 0.5$), the relative deviation between the pseudo-fermion method and the exact CPIMC result is only 0.6%. In particular, the pseudo-fermion method bridges the gap where neither CPIMC nor RPIMC can accurately simulate the regime $1 \le r_s \le 2$ at the reduced temperature $θ= 0.0625$. This work demonstrates that the pseudo-fermion method opens a new pathway for studying strongly quantum-degenerate systems in a sign-problem-free manner.

Figures (7)

• Figure 1: For $\beta=1,a=10^{-10}$: (a) the sign factor $X(\beta,a,\lambda)$; (b) blue crosses: $E_{pf}(\lambda)$, red line: $E_{pf}(\lambda=0)$, black circles: $E_f(\lambda)$; (c) blue crosses: $\tilde{E}_f$, black circles: $E_f$; (d) relative deviation between $\tilde{E}_f$ and $E_f$.
• Figure 2: $X(\beta,\lambda,M)$ for $N=4$, $r_s=0.5$ and $\theta=0.0625$. Black crosses with error bars correspond to $\lambda=1$, while the red crosses correspond to $\lambda=0$. The inset displays the ratio $X(\beta,\lambda=1,M)/X(\beta,\lambda=0,M)$ for $M$ up to 8.
• Figure 3: For the case of $N=4$, $r_s=0.5$, and $\theta=0.0625$, the red and black crosses with error bars represent the simulation results of the average energy per pseudo-fermion at different $M$ for $\lambda=0$ and $\lambda=1$, respectively.
• Figure 4: For the case of $N=4$, $r_s=0.5$, and $\theta=0.0625$: (a) The black crosses with error bars represent $\delta E_{pf}(\lambda=1,M)/N$ at different $M$ values. The black curve shows the fit using the function $f(M)$. (b) The black crosses with error bars represent the inferred fermion energy at different $M$ values, and the black curve is the corresponding fit using $f(M)$. The red line indicates the exact result from CPIMC CPIMC-4, and the two blue dashed lines correspond to a $0.5\%$ upward and downward shift of the red line.
• Figure 5: For the case of $r_s=0.5$ and $\theta=0.0625$, (a) shows the energy per fermion for different particle numbers as obtained by the pseudo-fermion method (blue crosses with error bars) and CPIMC CPIMC-4 (red dot). For the case of $r_s=1.0$ and $\theta=0.0625$, (b) shows the energy per fermion for different particle numbers as obtained by the pseudo-fermion method (blue crosses with error bars), CPIMC CPIMCCPIMC-4 (red dots), and RPIMC Brown (yellow dot with error bar).