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\underline{Exercise N4}

SU($N)$ Anderson impurity model in the limit of infinite Coulomb 
interaction ($ U = \infty$) at the impurity site. 





Consider the Hamiltonian:


\[
\begin{array}{l}
 H = H_{b} + H_{f} + H_{I} \\
 H_{b} = {\sum\limits_{kn} {}} \varepsilon _{k} c_{kn}^{\dag}  c_{kn} \\
 f_{n}^{\dag}  \equiv X_{n0} \\
 H_{f} = \varepsilon _{f} {\sum\limits_{m} {}} f_{m}^{\dag}  f_{m} =
\varepsilon _{f} {\sum\limits_{m} {}} X_{mm} \\
 H_{I} = {\sum\limits_{m} {}} V_{k} c_{km}^{\dag}  X_{0m} + V_{k}^{\ast}
X_{m0}c_{km}^{} \\
 \\
 \end{array}
\]



Compute the first three moments of $\langle \langle f_{} (\omega )f_{}^{\dag
} (\omega )\rangle \rangle $

Namely


\[
\begin{array}{l}
 \langle {\left\{ {X_{n0} X_{0n}}  \right\}}\rangle \\
 \langle {\left\{ {X_{n0} [H,X_{0n} ]} \right\}}\rangle \\
 \langle {\left\{ {[H,X_{n0} ][H,X_{0n} ]} \right\}}\rangle \\
 \end{array}
\]



What is their physical meaning?



%================

\underline{Exercise N5}

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


\begin{equation}
\label{eq1}
\sigma (\omega ) = -{\frac{{1}}{{V}}}{\frac{{1}}{{\omega}} }Im\Pi _{jj}^{R}
(\omega ),
\end{equation}



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


\[
\begin{array}{l}
 \Pi _{jj}^{R} (\omega ) = {\left\langle {{\left\langle {j,j} \right\rangle
}} \right\rangle} (\omega ) \\
 j = {\sum\limits_{i = 1}^{N} {}} {\frac{{p_{i}}} {{m}}}e \\
 \end{array}
\]



Prove the $f$-sum rule:
\[
\int_{- \infty}^{\infty} \frac{d \omega}{\pi} \sigma(\omega) = \frac{n e^2}
{m},
\]
where $n = N/V$.


\underline {Hints:}

Express (\ref{eq1}) 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.





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