trying some stuff...

Notes/Notes.tex

@@ -528,7 +528,7 @@ Recalling that $\bHod{0} = \bO$ and $\bHd{1} = \bO$, we derive
 
 \begin{align}
 \dv{\bF{}{(2)}}{s}  &= \bF{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bF{}{(0)} - 2 \bV{}{(1)}\bC{}{(0)}\bV{}{(1),\dagger} \\
-  \dv{\bC{}{(2)}}{s}  &= \bC{}{(0)}\bV{}{(1)}\bV{}{(1),\dagger} + \bV{}{(1)}\bV{}{(1),\dagger}\bC{}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \\
+  \dv{\bC{}{(2)}}{s}  &= \bC{}{(0)}\bV{}{(1),\dagger}\bV{}{(1)} + \bV{}{(1),\dagger}\bV{}{(1),}\bC{}{(0)} - 2 \bV{}{(1)}\bF{}{(0)}\bV{}{(1),\dagger} \\
   \dv{\bV{}{(2)}}{s} &= 2 \bF{}{(0)}\bV{}{(2)}\bC{}{(0)} - (\bF{}{(0)})^2\bV{}{(2)} - \bV{}{(2)}(\bC{}{(0)})^2 \\
   \dv{\bV{}{(2),\dagger}}{s} &= 2 \bC{}{(0)}\bV{}{(2),\dagger}\bF{}{(0)} - \bV{}{(2),\dagger}(\bF{}{(0)})^2 - (\bC{}{(0)})^2\bV{}{(2),\dagger} 
 \end{align}
@@ -818,7 +818,7 @@ Before discussing the regularizers in MBPT, we start by analyzing behavior of th
 \begin{equation}
   E_0^{(2)}(s= 1/\Lambda^2) = \frac{1}{4} \sum_{i j} \sum_{a b} \frac{\aeri{ij}{ab}^2}{\Delta_{ab}^{ij}}\left(1-e^{-2\left(\frac{\Delta_{ab}^{ij}}{\Lambda}\right)^2}\right)
 \end{equation}
-For $s=0$ the SRG-MP2 energy is equal to the MP2 one while  for $s \to \infty$ the SRG-MP2 energy goes to zero.
+For $\Lambda \to 0$ the SRG-MP2 energy is equal to the MP2 one while  for $\Lambda  \to \infty$ the SRG-MP2 energy goes to zero.
 For a finite value of $s$, hence a finite energy cutoff $\Lambda$, then the term of the sum with $\Delta_{ab}^{ij} < \Lambda$ are almost zero. Therefore a small cutoff removes only the divergent $1/\Delta_{ab}^{ij}$.
 
 A similar analysis can be done about the regularized correlation self-energy introduced by Monino and Loos. Here we discuss only the GW self-energy but without loss of generality.
@@ -834,13 +834,52 @@ This is because in addition to the divergent denominators we are removing more a
 
 Finally, we discuss the renormalized correlation self-energy introduced in this work.
 \begin{align}
-  \label{eq:GW_selfenergy_renormalized}
      (\Sigma_c^{\GW}(\omega,\Lambda))_{pq} &= \sum_{iv} \frac{W_{pi,v}^{\GW}W_{qi,v}^{\GW}}{\omega -\Omega_{i,v}^{\dRPA} - \ii \eta}e^{-\left( \frac{\epsilon_p - \Omega_{i,v}^{\dRPA} }{\Lambda} \right)^2}   e^{-\left( \frac{\epsilon_p - \Omega_{i,v}^{\dRPA} }{\Lambda} \right)^2}    \notag \\
      &+ \sum_{av} \frac{W_{pa,v}^{\GW} W_{qa,v}^{\GW}}{\omega - \Omega_{a,v}^{\dRPA} + \ii \eta} e^{-\left( \frac{\epsilon_p - \Omega_{a,v}^{\dRPA}}{\Lambda} \right)^2}e^{-\left( \frac{\epsilon_q - \Omega_{a,v}^{\dRPA} }{\Lambda} \right)^2}  \notag
  \end{align}
  In this case the situation is reversed, \ie the divergent denominators will be the last removed when $\Lambda$ is increased.
  Therefore the renormalized self-energy seems not to be the good strategy to remove discontinuities.
  However, it defines SRG-PT2 approximations to the quasiparticle energies which have the same pros as the SRG-MP2 discussed above.
+
+%=================================================================%
+\section{An alternative partitioning designed for discontinuities}
+\label{sec:discontinuities}
+%=================================================================%
+
+As we have seen before the SRG scheme studied so far has been designed to renormalize the quasiparticle in the ``right way'', \ie by handling correctly the divergent denominators in the static energy expression.
+However, doing so we do note handle correctly the renormalization of the self-energy.
+The aim of this section is to find a partitioning that do it the other way around with respect to the previous one.
+The idea to obtain this is to start from the full Hamiltonian and use a perturber that remove the coupling, this gives
+\begin{equation}
+  \bH(0) =
+  \begin{pmatrix}
+    \bF{}{} & \bV{}{}\\
+    \bV{}{\dagger} & \bC{}{} 
+  \end{pmatrix}
+  + \lambda
+  \begin{pmatrix}
+    \bO & -\bV{}{} \\
+    -\bV{}{\dagger} & \bO
+  \end{pmatrix}
+   = \bH_{\text{d}}(0) + (1-\lambda)\bH_{\text{od}}(0)
+ \end{equation}
+ We define $\lambda' = 1 - \lambda$, hence we can expand it like this
+ \begin{align}
+   \bH(s) & = \bH'^{(0)}(s) + \lambda' \bH'^{(1)}(s) + \lambda'^2 \bH'^{(2)}(s) + \cdots
+   \\
+   \bF{}{}(s) &= \bF{}{'(0)}(s) +\lambda' \bF{}{'(1)}(s) + \lambda'^2 \bF{}{'(2)}(s) + \cdots
+   \\
+   \bC{}{}(s) & = \bC{}{'(0)}(s) + \lambda' \bC{}{'(1)}(s) + \lambda'^2 \bC{}{'(2)}(s) + \cdots
+   \\
+   \bV{}{}(s) & = \bV{}{'(0)}(s) + \lambda' \bV{}{'(1)}(s) + \lambda'^2 \bV{}{'(2)}(s) + \cdots
+ \end{align}
+ We can use the expansion in terms of $\lambda$ and transform them to $\lambda^'$ and then identify with the expressions above, for example for $\bF{}{}$
+ \begin{align}
+   \bF{}{}(s) & = \bF{}{(0)}(s) + (1 - \lambda') \bF{}{(1)}(s) + (1 - \lambda')^2 \bF{}{(2)}(s) + \cdots \\
+              &= \qty( \bF{}{(0)}(s)  +  \bF{}{(1)}(s)  +  \bF{}{(2)}(s) + \cdots) \notag  \\
+              &+ \lambda'\qty(-\bF{}{(1)}(s) - 2 \bF{}{(2)}(s) + \cdots) \notag \\
+              &+ \lambda'^2 \qty(\bF{}{(2)}(s) + \cdots) \notag 
+ \end{align}
  
 %=================================================================%
 \section{Towards second quantized effective Hamiltonians for MBPT?}