Approximation for discrete Fourier transform and application in study of three-dimensional interacting electron gas

The discrete Fourier transform is approximated by summing over part of the terms with corresponding weights. The approximation reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency with several orders without losing accuracy. As an example, we apply the algorithm to study the three-dimensional interacting electron gas under the renormalized-ring-diagram approximation where the Green’s function needs to be self-consistently solved. We present the results for the chemical potential, compressibility, free energy, entropy, and specific heat of the system. The ground-state energy obtained by the present calculation is compared with the existing results of Monte Carlo simulation and random-phase approximation.

Figure 1

(Color online) $\Pi \left(x\right)$ as a function of $x$ at parameters $p=5$ and $p={10}^5$. Solid lines represent the exact function. Circles are the numerical results obtained using the present summation rule.

Figure 2

(Color online) $g\left(\tau \right)$ as a function of $\tau$ (normalized by $\beta =1/T$) at parameters $T/\xi =0.01$, 0.1, and 0.3. Solid lines represent the exact function. Symbols are the numerical results obtained using the present summation rule.

Figure 3

(Color online) Diagrammatic expressions for “free energy” functional $\Phi$. The line represents the Green’s function and the wavy line is the Coulomb interaction.

Figure 4

(Color online) Function $rG(r,\tau )$ at $T/E_F=0.05$ and $r_s=5$.

Figure 5

(Color online) Real part self-energy ${\Sigma }_r(k,i{\omega }_n)$ at $T/E_F=0.05$ and $r_s=5$.

Figure 6

(Color online) Imaginary part self-energy ${\Sigma }_r(k,i{\omega }_n)$ at $T/E_F=0.05$ and $r_s=5$.

Figure 7

(Color online) Chemical potential $\mu$ as a function of temperature $T$ for parameters, from top, $r_s=1$, 2, 3, 4, 5, 6, 8, 10, 15, 20, 25, and 30. The inset is a zoom in $\mu -T$ picture at $r_s=5$.

Figure 8

(Color online) Inverse compressibility ${\kappa }^{-1}$ as a function of $r_s$ at $T/E_F=0.01$. The inset shows $\mu$ as a function of $r_s$ at $T/E_F=0.01$.

Figure 9

(Color online) Energy $\Delta \epsilon$ and free energy $\Delta f$ per particle as functions of $T$ at $r_s=1$, 2, 5, and 10.

Figure 10

(Color online) Ground-state correlation energy ${\epsilon }_c$ (in atomic unit) per particle as functions of $r_s$. The present calculation (circles) is compared with RPA (squares) and MC (solid circles).

Figure 11

(Color online) Ground-state energy $\epsilon$ (in atomic unit) per particle as functions of $r_s$ for two-dimensional electron gas. The present calculation (circles) is compared with RPA (squares) and MC (solid circles). The diamonds are the RRDA results of the previous numerical calculation.

Figure 12

(Color online) Entropy $s$ per particle as functions of $T$ at $r_s=1$, 5, 10, 20, and 30.

Figure 13

(Color online) Specific heat $C$ normalized by $C^F$ of free electrons as functions of $r_s$ at $T/(e^2/a)$ $=$ 0.01, 0.02, 0.05, and 0.1. The dashed line at small $r_s$ is the Gell-Mann high-density expansion [32].