SRG GW and GF2 ready to be implemented (modulo some typos...)

Notes/Notes.tex

@@ -551,11 +551,11 @@ In the following, upper case indices correspond to the 2h1p and 2p1h sectors whi
 \end{align}
 The differential equation for the coupling blocks can be solved in the GF(2) case because in this case $\bC{}{}$ is diagonal (see Appendix~\ref{sec:diagC}).
 However, in the general case this matrix differential equation is not trivial to solve.
-In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_p$.
+In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we only need to solve the above equations for $\bF{}{} = \epsilon_r$.
 \begin{align}
   \label{eq:matrixdiffeq}
-  \dv{\bV{}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \\
+  \dv{\bV{r}{(1)}}{s} &= \bV{}{(1)} (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \\
-  \dv{\bV{}{(1),\dagger}}{s} &=  (2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger} 
+  \dv{\bV{r}{(1),\dagger}}{s} &=  (2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2) \bV{}{(1),\dagger} 
 \end{align}
 
 \begin{itemize}
@@ -577,7 +577,7 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
   \end{equation}
   
 \item \textbf{GW}
-  In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
+  In order to solve the matrix differential equation Eq.~(\ref{eq:matrixdiffeq}), we need to diagonalize $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ which is a polynomial in $\bC{}{(0)}$ so the polynomial admits the same eigenvectors $\bC{}{(0)}$.
   The elements of the matrix $\bC{}{(0)}$ in the GW case are given in Eq.~(\ref{eq:GW_unfolded}) and can be written equivalently as
    \begin{align}
     \label{eq:GW_unfolded}
@@ -594,23 +594,23 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
    \begin{align}
      \Omega_{(i,v)} &= \epsilon_i - \Omega_v & \Omega_{(a,v)} &= \epsilon_a + \Omega_v 
    \end{align}
-   Therefore the eigenvalues of $2 \epsilon_p \bC{}{(0)} - \epsilon_p^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_p \Omega_{(q,v)} - \epsilon_p^2 - \Omega_{q,v}^2 = -(\epsilon_p - \Omega_{(q,v)})^2$.
+   Therefore the eigenvalues of $2 \epsilon_r \bC{}{(0)} - \epsilon_r^2\mathbb{1} - (\bC{}{(0)})^2$ are $2 \epsilon_r \Omega_{(q,v)} - \epsilon_r^2 - \Omega_{q,v}^2 = -(\epsilon_r - \Omega_{(q,v)})^2$.
    And finally the analytical expressions for the GW coupling blocks at first order are
    \begin{align}
-     \bV{}{\hhp,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s})\bU^{-1}\bV{}{\hhp,(1)}(0) \\
+     \bV{r}{\hhp,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s})\bU^{-1}\bV{r}{\hhp,(1),\dagger}(0) \\
-     \bV{}{\pph,(1)}(s) &= \bU \text{diag}(e^{-(\epsilon_p - \Omega_{(a,v)})^2s})\bU^{-1}\bV{}{\pph,(1)}(0)
+     \bV{r}{\pph,(1),\dagger}(s) &= \bU \text{diag}(e^{-(\epsilon_r - \Omega_{(a,v)})^2s})\bU^{-1}\bV{r}{\pph,(1),\dagger}(0)
    \end{align}
 
  Therefore the downfolded SRG self-energy is
  \begin{align}
-   &\bSig(\omega)^{\hhp} = \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) \\
+   &\bSig(\omega)^{\hhp}_r = \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) \\
-   &\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_p - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
+   &\text{diag}(\frac{1}{\omega - \epsilon_i + \Omega^{(v)}}) \text{diag}(e^{-(\epsilon_r - \Omega_{(i,v)})^2s}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag \\
-   &= \bV{}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_p - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{}{\hhp,(1)}(0))^{\mathsf{T}} \notag
+   &= \bV{r}{\hhp,(1)}(0) \bU^{\hhp} \text{diag}(\frac{e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}}{\omega - \epsilon_i + \Omega^{(v)}}) (\bU^{\hhp})^{-1} (\bV{r}{\hhp,(1)}(0))^{\mathsf{T}} \notag
  \end{align}
- Renaming the index $p$ as $r$ and using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
+Using a similar matrix product as in Appendix~\ref{sec:downfolding}, we finally obtain
  \begin{align}
    \label{eq:SRGGW_selfenergy}
-   \Sigma(\omega)_{pq} &= \sum_{(i,v)} \frac{M_{ip}^{(v)}M_{iq}^{(v)}}{\omega - \epsilon_i + \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(i,v)})^2s} \notag \\
+   (\Sigma(\omega)_r^{\GW})_{pq} &= \sum_{(i,v)} \frac{M_{ip}^{(v)}M_{iq}^{(v)}}{\omega - \epsilon_i + \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(i,v)})^2s} \notag \\
    &+ \sum_{(a,v)} \frac{M_{ap}^{(v)}M_{aq}^{(v)}}{\omega - \epsilon_a - \Omega^{(v)}}e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}
  \end{align}
  where the $\pph$ part has been obtained by an analog derivation.
@@ -621,7 +621,7 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
    A compeltely analog derivation gives
    \begin{align}
   \label{eq:SRGGF2_selfenergy}
-  \Sigma_{pq}^{GF(2)}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
+  (\Sigma_r^{\GF2})_{pq}(\omega) &= \frac{1}{2} \sum_{ija} \frac{\aeri{pa}{ij}\aeri{qa}{ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} e^{-2(\epsilon_r - \Delta_{ij}^a)^2s}\notag \\
   &+ \frac{1}{2} \sum_{iab} \frac{\aeri{pi}{ab}\aeri{qi}{ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} e^{-2(\epsilon_r - \Delta_{i}^{ab})^2s} \notag  
 \end{align}
 
@@ -643,6 +643,57 @@ In practice, we often resort to the \GOWO~or \evGW~schemes which implies that we
 The two first equations can be solved by simple integrations.
 The two last equations admit the same solutions as the first order coupling blocks differential equations with different initial conditions.
 
+We focus on $\bF{}{(2)}$ because it is the only second order block contributing to the second order quasiparticle equation.
+We are still in the case where $\bF{}{(2)} = \epsilon_r^{(2)}$ 
+
+\begin{align}
+  \dv{\epsilon_r^{(2)}}{s} &= \epsilon_r^{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\epsilon_r^{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger}  \\
+                             &= 2\bV{}{(1)} \left(\epsilon_r^{(0)} \mathbb{1} -  \bC{}{(0)}   \right) \bV{}{(1),\dagger} \notag \\
+                             &=  2\bV{}{\hhp,(1)} \left(\epsilon_r^{(0)} \mathbb{1} -  \bC{}{\hhp,(0)}   \right) \bV{}{\hhp,(1),\dagger} \notag \\
+                           &+ 2\bV{}{\pph,(1)} \left(\epsilon_r^{(0)} \mathbb{1} -  \bC{}{\pph,(0)}   \right) \bV{}{\pph,(1),\dagger} \notag
+\end{align}
+We focus on the $\hhp$ part
+\begin{align}
+  &= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1} \left(\epsilon_r^{(0)} \mathbb{1} -  \bC{}{\hhp,(0)}   \right) \bU \notag \\
+  &\times e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
+  &= 2\bV{}{\hhp,(1)}(0) \bU e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} (\epsilon_r^{(0)} \mathbb{1} - \boldsymbol{\Omega}) e^{-(\epsilon_r^{(0)} - \boldsymbol{\Omega})^2s} \bU^{-1}\bV{}{\hhp,(1),\dagger}(0) \notag \\
+\end{align}
+\begin{itemize}
+\item \textbf{GW}
+  In the GW case this evaluates to
+  \begin{align}
+    &= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag
+  \end{align}
+  The other part is analog and we obtain
+  \begin{align}
+  \dv{\epsilon_r^{(2)}}{s} &= 2 \sum_{(i,v)} M_{ir}^{(v)}M_{ir}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(i,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
+  &+ 2 \sum_{(a,v)} M_{ar}^{(v)}M_{ar}^{(v)}\left(\epsilon_r^{(0)} - \Omega_{(a,v)} \right)e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} \notag 
+\end{align}
+which can be integrated as
+\begin{align}
+  \epsilon_r^{(2)}(s) &= - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(i,v)})^2s} \notag \\
+                      &- \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} e^{-2(\epsilon_r^{(0)} - \Omega_{(a,v)})^2s} + \text{Cte} \notag
+\end{align}
+The constant is determined as
+\begin{equation}
+  \epsilon_r^{(2)}(0) = 0 = - \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(i,v)}} - \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r^{(0)} - \Omega_{(a,v)}} + \text{Cte} \notag
+\end{equation}
+Which finally gives
+\begin{align}
+  \epsilon_r^{(2)}(s) &=  \sum _{(i,v)} \frac{M_{ir}^{(v)}M_{ir}^{(v)}}{\epsilon_r - \Omega_{(i,v)}} \left(1 - e^{-2(\epsilon_r - \Omega_{(i,v)})^2s}\right) \notag \\
+                      &+ \sum _{(a,v)} \frac{M_{ar}^{(v)}M_{ar}^{(v)}}{\epsilon_r - \Omega_{(a,v)}} \left(1 -  e^{-2(\epsilon_r - \Omega_{(a,v)})^2s}\right)  \notag
+\end{align}
+
+\item \textbf{GF(2)}
+
+  The expression for the GF(2) case is
+  \begin{align}
+  \epsilon_r^{(2)}(s) &=  \sum _{ija} \frac{\aeri{ra}{ij}^2}{\epsilon_r ^{(0)}- \Delta_{ij}^a} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{ij}^a)^2s}\right) \notag \\
+                      &+ \sum _{iab} \frac{\aeri{ri}{ab}^2}{\epsilon_r^{(0)} - \Delta_{i}^{ab}} \left(1 -  e^{-2(\epsilon_r^{(0)} - \Delta_{i}^{ab})^2s}\right)  \notag
+\end{align}
+  
+\end{itemize}
+
 % =================================================================%
 \section{Towards second quantized effective Hamiltonians for MBPT?}
 \label{sec:second_quant_mbpt}