Variational solution of the $T$-matrix integral equation

We present a variational solution of the $T$-matrix integral equation within a local approximation. This solution provides a simple form for the $T$ matrix similar to Hubbard models but with the local interaction depending on momentum and frequency. By examining the ladder diagrams for irreducible polarizability, a connection between this interaction and the local-field factor is established. Based on the obtained solution, a form for the $T$-matrix contribution to the electron self-energy in addition to the $\mathit{GW}$ term is proposed. In the case of the electron-hole multiple scattering, this form allows one to avoid double counting.

Figure 1

Feynman diagrams for $T_{\sigma {\sigma }^{\prime }}^{e\text{−}e}$ (a) and $T_{\sigma {\sigma }^{\prime }}^{e\text{−}h}$ (b) in coordinate space. The $T$ matrix is shown by the shaded square. The wiggly lines signify the dynamically screened Coulomb interaction $W$. The solid lines with arrows represent the Green function $G$.

Figure 2

Feynman diagrams for ${\Gamma }_{\sigma {\sigma }^{\prime }}^{e\text{−}e}$ (a) and ${\Gamma }_{\sigma {\sigma }^{\prime }}^{e\text{−}h}$ (b) in momentum space.

Figure 3

A diagrammatic representation of the trial solution ${\tilde{\Gamma }}_{\sigma {\sigma }^{\prime }}^{\alpha }\left(Q\right)$, Eq. (16), shown for the $e\text{−}e$ (up-directed arrow on the right-hand part of the bubble $K_{\sigma {\sigma }^{\prime }}^{\alpha }$) and $e\text{−}h$ (down-directed arrow) cases.

Figure 4

Feynman diagrams for the irreducible polarizability $P$ in coordinate space. The RPA bubble (on the left) and the ladder diagrams (on the right) expressed in terms of the $T$ matrix (shaded square) are represented here.

Figure 5

Feynman diagrams for the direct (a) and exchange (b) terms of the electron self-energy.

Figure 6

Four redundant diagrams originated from the $T^{e\text{−}e}$ matrix (left column) and from the $T^{e\text{−}h}$ matrix (right column).

Figure 7

The imaginary part of the electron self-energy $\mathrm{Im}\Sigma [\mid \mathbf{q}\mid ,ϵ\left(\mathbf{q}\right)]$ of the electron gas as a function of momentum $\mid \mathbf{q}\mid$ at $r_s=2.07$ (aluminium) and $r_s=4.86$ (potassium). $\mathrm{Im}{\Sigma }^{\mathit{GW}}$ and $\mathrm{Im}{\Sigma }^{T^{e\text{−}h}}$ are shown by dashed and dotted lines, respectively. Solid line represents the sum of these terms. Insets: the electron lifetime $\tau$ for the corresponding $r_s$ values, versus the excitation energy $E-E_F$. Dashed (solid) line shows $\tau$ obtained from $\mathrm{Im}{\Sigma }^{\mathit{GW}}\left(\mathrm{Im}[{\Sigma }^{\mathit{GW}}+{\Sigma }^{T^{e\text{−}h}}]\right)$. $ϵ\left(\mathbf{q}\right)$ is the free electron energy and $q_F\left(E_F\right)$ is the Fermi wave vector (energy).