\chapter{Spectral Representation of Green's functions}
\label{app:spectral}
In this appendix we will derive the spectral representation of the
multiorbital Green's function. Our starting point is the definition of
the Green's function for complex time, $\tau$:
\begin{eqnarray}
  \label{eq:GreensFunctionDefinition}
  G_{pq}(\tau) &=& -\langle T_{\tau} c_p(\tau)c_q^+(0)\rangle 
  \nonumber \\
  &\overset{\tau>0}{=}& {-1 \over Z}
  \trace\left\{e^{\beta H}e^{\tau H}c_p e^{-\tau H}c_q^+\right\}
  \nonumber \\
  &=& {-1 \over Z}\sum_{n}
  \langle n|e^{\beta H}e^{\tau H}c_p e^{-\tau H}c_q^+|n\rangle
  \nonumber \\
  &=& {-1 \over Z}\sum_{nm}e^{(\tau-\beta)E_{n}-\tau E_{m}}
  \langle n|c_p|m\rangle\langle m|c_q^+|n\rangle
\end{eqnarray}
This expression is now Fourier transformed to Matsubara frequency axis
and noting that $i\omega$ is Fermionic we get:
\begin{eqnarray}
  G_{pq}(i\omega) &=& \int_{0}^{\beta}G_{pq}(\tau)e^{i\omega \tau}
  \nonumber \\
  &=& {1 \over Z}\sum_{nm}
  \langle n|c_p|m\rangle\langle m|c_q^+|n\rangle
  {e^{-\beta E_n}+e^{-\beta E_m} \over i\omega-E_m+E_n}.
\end{eqnarray}
The retarded and the advanced Green's functions are now obtained by
replacing the Matsubara frequency, $i\omega$ by $\omega+i0$ and
$\omega-i0$ respectively.  Using the fact that we can write:
\begin{equation}
  {1 \over \omega\pm i0+\omega_0} = {\cal P}{1 \over \omega+\omega_0}
  \mp i\pi\delta(\omega+\omega_0)
\end{equation} 
we get,
\begin{eqnarray}
  G_{pq}^{R}(\omega) &=& {1 \over Z}\sum_{nm}
  \langle n|c_p|m\rangle\langle m|c_q^+|n\rangle
  \left(e^{-\beta E_n}+e^{-\beta E_m}\right)
  \nonumber\\
  &&\times
  \left({\cal P}{1 \over \omega-E_m+E_n}
    - i\pi\delta(\omega-E_m+E_n)\right)
\end{eqnarray}
and similarly,
\begin{eqnarray}
  G_{pq}^{A}(\omega) &=& {1 \over Z}\sum_{nm}
  \langle n|c_p|m\rangle\langle m|c_q^+|n\rangle
  \left(e^{-\beta E_n}+e^{-\beta E_m}\right)
  \nonumber\\
  &&\times
  \left({\cal P}{1 \over \omega-E_m+E_n}
    + i\pi\delta(\omega-E_m+E_n)\right).
\end{eqnarray}
Let us now look at the complex conjugate of the matrix elements of the
retarded Green's function.
\begin{eqnarray}
  (G_{pq}^{R}(\omega))^* &=& {1 \over Z}\sum_{nm}
  \langle m|c_p^+|n\rangle\langle n|c_q|m\rangle
  \left(e^{-\beta E_n}+e^{-\beta E_m}\right)
  \nonumber\\
  &&\times
  \left({\cal P}{1 \over \omega-E_m+E_n}
    + i\pi\delta(\omega-E_m+E_n)\right)
  \nonumber\\
  &=& G_{qp}^{A}(\omega).
\end{eqnarray}
Thus we see that the advanced Green's function is the Hermitian
conjugate of the retarded one.  From the representation above we see
that the Green's function can be written as
\begin{equation}
  G_{pq}(z) = \int{\rho_{pq}(\epsilon) \over z-\epsilon}
\end{equation}
where the so called spectral density matrix, $\rho_{pq}$ is given by:
\begin{equation}
  \rho_{pq}(\epsilon) = {1 \over Z}\sum_{nm}
  \langle n|c_p|m\rangle\langle m|c_q^+|n\rangle
  \left(e^{-\beta E_n}+e^{-\beta E_m}\right)
  \delta(\epsilon-E_m+E_n).
\end{equation}
and by looking at the expressions for the real and the advanced Green's 
function we see that the spectral density can be written as:
\begin{equation}
  \rho_{pq}(\epsilon) = {-1 \over 2\pi i}
  \left[G_{pq}^{R}(\epsilon)-G_{pq}^{A}(\epsilon)\right]
  = {-1 \over 2\pi i}
  \left[G_{pq}^{R}(\epsilon)-\left(G^{R}\right)^{+}_{pq}(\epsilon)\right].
\end{equation}
We should note here that in contrast to the single orbital spectral
densities that we are used to from regular many body theory, the
spectral density matrix is not necessarily real and positive, but as
we can see it is Hermitian and its diagonal values are all real and
positive.

Using the spectral representation it is now easy to compute the complex
time Green's function:
\begin{equation}
  G_{pq}(\tau) = -\int_{-\infty}^{\infty}d\epsilon\rho_{pq}(\epsilon)
  e^{-\epsilon\tau}\left(
    \theta(\tau)f(-\epsilon)-\theta(-\tau)f(\epsilon)
  \right).
\end{equation}  

Finally let us see how we can relate the complex time derivative of the
Green's function to expectation values like $\langle
(\partial_{\tau}c_p)(\tau)c_q^+(0)\rangle$.  To do this we note first that:
\begin{alignat}{1}
  &{1 \over Z}
  \trace\left\{e^{\beta H}(\partial_{\tau}A)(\tau)B(0)\right\}
  = {1 \over Z}
  \trace\left\{e^{\beta H}e^{\tau H}[H,A]e^{-\tau H}B(0)\right\}
  \nonumber\\
  = &{-1 \over Z}
  \trace\left\{e^{\beta H}e^{\tau H}Ae^{-\tau H}[H,B]\right\}
  = {-1 \over Z}
  \trace\left\{e^{\beta H}A(\tau)(\partial_{\tau}B)(0)\right\}.
\end{alignat}
Now since there is no time dependence in the $B$ operator we see that:
\begin{alignat}{1}
  &\frac{\partial^{n}}{\partial \tau^{n}}\left({1 \over Z}
  \trace\left\{e^{\beta H}A(\tau)B(0)\right\}\right)
  = {1 \over Z}
  \trace\left\{e^{\beta H}(\partial^{n}_{\tau}A)(\tau)B(0)\right\}
  \nonumber\\
  = &{(-1)^{q} \over Z}
  \trace\left\{e^{\beta H}(\partial^{(n-q)}_{\tau}A)(\tau)
    (\partial^{q}_{\tau}B)(0)\right\}
  = {(-1)^{n} \over Z}
  \trace\left\{e^{\beta H}A(\tau)
    (\partial^{n}_{\tau}B)(0)\right\}.
\end{alignat}
We see now that for $\tau\in\,]0,\beta[$ we have
\begin{alignat}{1}
  \left\langle
    (\partial^{m}_{\tau}c_{p})(\tau)
    (\partial^{n}_{\tau}c^{+}_{q})
  \right\rangle   
  & = (-1)^{n+1}\frac{\partial^{(m+n)}}{\partial \tau^{(m+n)}}G_{pq}(\tau)
  \nonumber\\
  \left\langle
    (\partial^{m}_{\tau}c^{+}_{q})(\tau)
    (\partial^{n}_{\tau}c_{p})
  \right\rangle   
  & = (-1)^{n}\frac{\partial^{(m+n)}}{\partial \tau^{(m+n)}}G_{pq}(-\tau)
\end{alignat}
Strictly speaking these equations do not hold at $\tau=0$ where the Green's
function has a discontinuity and the time derivative would therefore be a
delta-function at zero.  The expectation values however do not have any such 
pathologies.  As we know the value of the Green's function at zero is
arbitrary and depends on definition and is usually defined as the limit as one
approaches zero from either the positive or the negative direction.  The same
should of course hold for its derivative, i.e.~the derivative is computed in
an open interval that does not include zero and the value at zero depends on 
definition, and has not delta-function.  If we understand the derivatives in
that sense the equations above hold for all $\tau$.








