Coulomb-hole and screened exchange in the electron self-energy at finite temperature

We present a finite temperature generalization of the COHSEX approximation to the quasiparticle electron self-energy, with explicit comparisons of the static and dynamic screened exchange (SEX) and Coulomb-hole (COH) contributions for the homogeneous electron gas. Consistent with the zero-temperature behavior, we show that the static SEX approximation agrees well with the fully dynamic SEX at all temperatures and densities. In contrast, dynamic corrections to the COH approximation are always significant, especially for high energy unoccupied states. Notably the COHSEX quasiparticle self-energy is complex valued at finite temperature even at the Fermi level.

Figure 1

Results for the SEX (top) and COH (bottom) contributions to the static (dashed) and full dynamic (solid) quasiparticle self-energy vs $k/k_F$ at various temperatures $T/T_F$ for the homogeneous electron gas for $r_s=2.0$: (from blue to red) the curves denote $T/T_F=0.01$, 0.5, 1.0, and 2.0. Note that the COH contribution is always significantly larger in magnitude than the SEX and persists to higher $k/k_F$, while the SEX becomes for unoccupied states above $k_F$.

Figure 2

The real part of static COHSEX approximation (dashed) vs fully dynamic self-energy (solid) for various densities: (from blue to orange) $r_s=1.0$, 2.0, 3.0, 4.0 at $T/T_F=0.01$ (top) and 1.0 (bottom). Note that the high temperature behavior is considerably smoother than near $T=0$ and the change in behavior beyond about $k=k_p$ reflecting the onset of plasmon excitations. For reference the LDA-$v_{\text{xc}}\left(T\right)$ [5, 17] for each $r_s$ are added as circles at $k_F$.

Figure 3

Fully dynamic COH (solid) and SEX (dashed) contributions to the imaginary part of the quasiparticle correction to the DFT energy ${\varepsilon }_k$ vs $k/k_F$ for $r_s=4.0$, for various temperatures: (from blue to red) $T/T_F=0.01$, 0.5, 1.0, and 2.0. Note that the dominant contribution comes from the COH terms, while the dynamic contributions to the SEX from particle-hole excitations are significant only near $k_F$.

Figure 4

The electron energy-loss function $L(q,\omega )=|\mathrm{Im}{\epsilon }^{-1}(q,\omega )|$ for $r_s=2.0$ at various temperatures: (from blue to red) $T/T_F=0.01$, 0.5, 1.0, and 2.0 for momentum $q/k_F=0.5$ (top) and 2.0 (bottom).

Figure 5

Real (solid) and imaginary (dashed) parts of the renormalization constant $Z_k$ vs $k/k_F$ for $r_s=4.0$ at (blue to red) $T/T_F=0.01$, 0.5, 1.0, and 2.0.

Figure 6

Dynamic correction to COHSEX self-energy $\mathrm{\Sigma }(k,E_k)-{\mathrm{\Sigma }}_k^{\text{COHSEX}}$ at the Fermi level $k_F$ as a function of $r_s$: from blue to red, the curves denote $T/T_F=0.01$, 0.5, 1.0, and 2.0.