Exercise N5

The a.c. conductivity of a system of interacting electron is given by

$$\sigma (\omega ) = -{\frac{{1}}{{V}}}{\frac{{1}}{{\omega}} }Im\Pi _{jj}^{R} (\omega ),$$ (1)

where $\Pi _{jj}^{R} (\omega )$ is the retarded current-current correlation function

$$\begin{array}{l} \Pi _{jj}^{R} (\omega ) = {\left\langle {{... ...limits_{i = 1}^{N} {}} {\frac{{p_{i}}} {{m}}}e \\ \end{array}$$

Prove the $f$-sum rule:

$$\int_{- \infty}^{\infty} \frac{d \omega}{\pi} \sigma(\omega) = \frac{n e^2} {m},$$

where $n = N/V$.

Hints:

Express (1) in terms of the spectral function of the current $\rho _{jj}^{} (\omega )$which is odd.

Relate ${\frac{{\rho _{jj}^{} (\omega )}}{{\omega}} }$ to $\rho _{Pj}^{} (\omega )$, where $P$ is operator such that ${\frac{{\partial P}}{{\partial t}}} = j$. Apply the sum rule for $\int {d\omega} \rho _{Pj}^{} (\omega )$ that we learned in class.