\begin{displaymath}H_{0} = \sum\limits_{k\sigma} \varepsilon_{k} c^{\dagger}_{k \sigma} c_{k \sigma}\end{displaymath}


\begin{displaymath}N = \sum\limits_{k\sigma} c_{k \sigma}^{\dagger} c_{k \sigma}\end{displaymath}


\begin{displaymath}\mid \psi_{0} \rangle = \prod\limits_{k < k _{F}}
c_{k \uparrow}^{\dagger} c_{k \downarrow}^{\dagger} \mid 0 \rangle\end{displaymath}


\begin{displaymath}\rho_{q} \equiv \sum\limits_{k \sigma} \ \ c_{k+q \sigma}^{\dagger} c_{k \sigma}\end{displaymath}


\begin{displaymath}{\Delta^\dagger}{q} \equiv \sum\limits_{k} c_{- k+q \uparrow}^{\dagger} c_{k \downarrow}^{\dagger}\end{displaymath}

a) zero temperature.
Calculate the Fourier transform of $G_{\rho}(t,q)$, $G_{p}(t,q)$ in time.


\begin{displaymath}G_{\rho}(t,q) = \langle T(\rho_{q}(t) \rho_{-q}(0)) \rangle\end{displaymath}


\begin{displaymath}G_{p}(t,q) = \langle T(\Delta_{q}(t){\Delta^\dagger} _{q}(0))\rangle\end{displaymath}

write an expression it will involve a k - integration consider the case of $d$ dimensions can you evaluate some of the integrals in d=1,2, or 3?

Problem 2 Repeat the first problem at finite temperature Finite temperature - if $\tau$ is the imaginary time it is convenient to use Fourier series

\begin{displaymath}G_{\rho}(i \nu_{n},g)= \int_{o}^{\beta} e^{i \nu_{n}\tau} G_{\rho}(\tau,q)d \tau\end{displaymath}


\begin{displaymath}G_{p}(i \nu_{n},q) = \int_{o}^{\beta} e^{i \nu_{n}} \tau G_{p}(\tau,q) d \tau.\end{displaymath}

Evaluate them in 3 and 2 dimensions.

Now $G_{\rho}$ and $G_{p}$ are defined by


\begin{displaymath}G_{\rho}(\tau,q)=
{1\over Z} tr e^{-\beta(H_{o} -\mu N)}\rho_{q}({\tau})\rho_{-q}(o) \end{displaymath}


\begin{displaymath}G_{\rho}(\tau,q)= {1\over Z} tr e^{-\beta(H_{o} -\mu N)}\Delta_{q}({\tau})
{\Delta^{\dagger}}{q}(o) \end{displaymath}

Problem 3: Evaluate the following Matsubara sums

\begin{displaymath}T\sum\limits_{n}\frac{e^{i\omega_n 0^+}}{i\omega_n -\varepsil...
...\limits_{n}\frac{e^{i\omega_n 0^-}}{i\omega_n -\varepsilon_k} .\end{displaymath}

Interpret the result why do the two sums differ