Two-Electron Correlations in the Metallic Electron Gas

Abstract

We present high-precision ab initio calculations of the four-point vertex function for the three-dimensional uniform electron gas using variational diagrammatic Monte Carlo. From these results, we extract Landau parameters that demonstrate a density-driven crossover from underscreening to overscreening. Guided by our numerical data, we propose a charge-based Kukkonen--Overhauser effective interaction within the local-density approximation, supplemented by a small s-wave correction (sKO$^+$), which accurately captures the electron--electron scattering amplitude. Using our numerically determined scattering amplitude, together with the sKO$^+$ ansatz, we compute the electron-electron contribution to the thermal resistivity, demonstrating excellent agreement with experimental measurements in simple metals.

Figures (4)

• Figure 1: The electron-electron contribution to the thermal resistivity $W_{ee}$ as a function of density parameter $r_s$ in the metallic regime derived through our numerical results (blue circles) through the variational diagrammatic Monte Carlo (VDMC) method and some theoretical models as the random phase approximation (RPA), s-p approximation transportcoeff1969MacDonald_1980, the original Kukkonen-Overhauser interaction PhysRevB.20.550 (KO$^\pm$) and the one with charge channel only (KO$^+$). For comparision, we plot the experimental data (black circle) for some simple metals Cook1972Cook1973Cook1979Uher2004. The solid red line presents our proposed sKO$^+$ model, the KO$^+$ model with s-wave corrections, which shows good agreement with the VDMC results and the experimental measurements. The inset shows the results of the electron-electron scattering rate.
• Figure 2: Numerical results of the symmetric (left panel) and antisymmetric (right panel) Landau parameters versus $r_s$, which confirms the rapid convergence of the Legendre expansion. The linear decrease of $F_0^s$ with $r_s$, which approaches $-1$ at $r_s^c \approx 5.7$, reveals the crossover of the Coulomb screening effect from underscreening to overscreening Takada_rsc3d. For comparison, the orange dash-dot line shows the $F_0$ calculated from the RPA vertex. The violet dash-dot line presents the dimensionless static exchange-correlation (XC) kernel $f_{\rm XC}(q=0)N_F$ for the uniform electron gas derived from the PW92 energy functional PW92, which demonstrates the approximate equality of the XC kernel and the zeroth Landau parameter. The corresponding numerical data are listed in Table S1 of the Supplemental Material.
• Figure 3: Angle-resolved Landau quasiparticle interaction for $r_s =1,5$ with parallel and antiparallel spin. The dotted-dashed line represents the bare exchange Coulomb interaction. The larger deviation from the bare interaction for $r_s=5$ indicates the Coulomb screening increases as the electron density decreases.
• Figure 4: The dimensionless two-electron scattering amplitude $A^{\sigma\sigma'}(\theta, \phi)$ for parallel ($\uparrow\uparrow$, top row) and anti-parallel ($\uparrow\downarrow$, bottom row) spins at $r_s = 1$ and $r_s=5$ at $T/T_F = 0.025$. Panels (a-d) show the forward scattering channel ($\phi=0$). Panels (e-f) show the Cooper channel ($\theta=\pi$). Our VDMC results (symbols) are compared against serval theoretical models. We also plot the sKO$^+$ conjectured through analyzing the deviation of our VDMC results and the KO$^+$ interaction. which shows good agreement with the numerical results. For $r_s = 5$, numerical results at $T/T_F = 0.05$ are also ploted. The divergence of the scattering amplitude at the two effectively low temperature in the Cooper channel is consistent with the expected logarithmic divergence of the four-point vertex function as $T\to 0$PhysRevB.106.L220502.