Many-Body Effects and Further Theory

Abstract

For Jellium, due to momentum conservation, we re-label the diagram 11.10 d) involving the bubble π₀ as in Figure 12.1 and (11.73) becomes

$$ - i\pi _0 (q,\alpha ) = - 2\smallint \frac{{d^3 m}} {{(2\pi )^3 }}\smallint \frac{{d\beta }} {{2\pi }}ig_0 (m,\beta )ig_0 (m + q,\beta + \alpha ) $$

(1)

and (11.76) becomes, restoring ℏ,

$$ \pi _0 (q,\alpha ) = 2\smallint \frac{{d^3 m}} {{(2\pi )^3 }}\left[ {\frac{{f_m (1 - f_{m + q} )}} {{\hbar \alpha + \varepsilon _m - \varepsilon _{m + q} + i\eta }} - \frac{{f_{m + q} (1 - f_m )}} {{\hbar \alpha + \varepsilon_{m} - \varepsilon_{m + q} + i\eta }}} \right] $$

((12.2))

with

$$ \begin{gathered} \operatorname{Re} \pi _0 (q,\alpha ) = 2\sum {_m } \frac{{f_m - f_{m + q} }} {{\hbar \alpha + \varepsilon _m - \varepsilon _{m + q} }}, \hfill \\ \operatorname{Im} \pi _0 (q,\alpha ) = - 2\pi \sum {_m } f_m (1 - f_{m + q} )\delta (\hbar \alpha + \varepsilon _m - \varepsilon _{m + q} ). \hfill \\ \end{gathered} $$

((12.3))

So, the self-energy (11.86) becomes

$$ - i\sum (\omega ) = \smallint \frac{{d^3 q}} {{(2\pi )^3 }}\smallint \frac{{d\alpha }} {{2\pi }}ig^0 (k - q,\omega - \alpha )( - iV_q )^2 ( - i)\pi _0 (q,\alpha ) $$

((12.4))

and since $V_{q}=\frac{4\pi e^{2}}{q^{2}}$ , this diverges at small q. The electron gains self-energy by exciting the medium and then re-adsorbing the excitations, but the process runs out of control for the long-wavelength ones.What is going wrong at long distances? It is the Coulomb interaction V, which is causing the divergence by its long range, but should actually be replaced by a shorter ranged screened interaction W.