Discussion about the two choice of regularizers

Notes/Notes.tex

@@ -248,6 +248,7 @@ In addition, the one-body hamiltonian has no first order contribution so
 \end{align}
 After integration, using the initial condition $E_0^{(2)}(0)=0$, we obtain
 \begin{equation}
+  \label{eq:SRG_MP2}
   E_0^{(2)}(s) = \frac{1}{4} \sum_{i j} \sum_{a b} \frac{\aeri{ij}{ab}^2}{\Delta_{ab}^{ij}}\left(1-e^{-2s (\Delta_{ab}^{ij})^2}\right)
 \end{equation}
 
@@ -655,7 +656,7 @@ The constant is determined as
 \end{align}
 Which finally gives
 \begin{align}
-  F_{pq}^{(2)}(s) &= -\sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v)}{(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2 + (\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
+  F_{pq}^{(2)}(s) &= \sum_{r,v} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2 (\epsilon_r^{(0)} \pm \Omega_v)}{(\epsilon_p^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2 + (\epsilon_q^{(0)} - \epsilon_r^{(0)} \pm \Omega_v)^2} W^{(0)}_{p,(r,v)} \notag \\
                   &\times W^{(0),\dagger}_{(r,v),q}\left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s}   e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right) \notag
 \end{align}
 
@@ -663,9 +664,9 @@ Which finally gives
 
   The expression for the GF(2) case is
 \begin{align}
-  F_{pq}^{(2)}(s) &= - \frac{1}{2} \sum_{ija} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{ij}^a}{(\epsilon_p^{(0)}  - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)}  - \Delta_{ij}^a)^2} \aeri{pa}{ij}\notag \\
+  F_{pq}^{(2)}(s) &= \frac{1}{2} \sum_{ija} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{ij}^a}{(\epsilon_p^{(0)}  - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)}  - \Delta_{ij}^a)^2} \aeri{pa}{ij}\notag \\
                   &\times\aeri{qa}{ij}  \left(1 - e^{-(\epsilon_p - \Delta_{ij}^a)^2s}   e^{-(\epsilon_q - \Delta_{ij}^a)^2s}\right) \notag \\
-  & - \frac{1}{2} \sum_{iab} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{i}^{ab}}{(\epsilon_p^{(0)}  - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)}  - \Delta_{i}^{ab})^2} \aeri{pi}{ab}\notag \\
+  & + \frac{1}{2} \sum_{iab} \frac{\epsilon_{p}^{(0)} + \epsilon_{q}^{(0)} - 2\Delta_{i}^{ab}}{(\epsilon_p^{(0)}  - \Delta_{ij}^a)^2 + (\epsilon_q^{(0)}  - \Delta_{i}^{ab})^2} \aeri{pi}{ab}\notag \\
                   &\times\aeri{qi}{ab}  \left(1 - e^{-(\epsilon_p - \Delta_{i}^{ab})^2s}   e^{-(\epsilon_q - \Delta_{i}^{ab})^2s}\right) \notag
 \end{align}
 
@@ -808,8 +809,40 @@ One of the con of the static approximation is that we loose information about th
 However, SRG allows us to stop at a finite value $s$ corresponding to a renormalized coupling but the coupling is still present.
 Therefore the satellites can still be observed due to the non-linearity induced by the renormalized coupling.
 
+%=================================================================%
+\section{Discussion on the choice of regularizers}
+\label{sec:regularizers}
+%=================================================================%
 
-% =================================================================%
+Before discussing the regularizers in MBPT, we start by analyzing behavior of the second order SRG correction to the energy derived in Eq.~(\ref{eq:SRG_MP2}). The equation is written here again for readability and also we used $s = 1/\Lambda^2$ where $\Lambda$ is the characteristic cutoff energy.
+\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 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.
+\begin{align}
+  \label{eq:GW_selfenergy_regularized}
+  (\Sigma_c^{\GW}(\omega,\Lambda))_{pq} &= \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \Omega_{i,v}^{\dRPA} - \ii \eta}\left(1-e^{-2\left( \frac{\omega - \Omega_{i,v}^{\dRPA} - \ii \eta}{\Lambda} \right)^2}\right) \notag \\
+  &+ \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \Omega_{a,v}^{\dRPA} + \ii \eta}\left(1-e^{-2\left( \frac{\omega - \Omega_{a,v}^{\dRPA} + \ii \eta}{\Lambda} \right)^2}\right) \notag
+\end{align}
+Similarly to the SRG-MP2 case, here we see that for small $\Lambda$ only the divergent denominators are removed.
+These denominators are precisely the one responsible of the discontinuities which explain the good results of regularized GW to remove discontinuities seen in Enzo's paper.
+Enzo also showed that for larger $\Lambda$ the quasiparticle energies are still smooth yet less accurate.
+This is because in addition to the divergent denominators we are removing more and more terms from the self-energy which leads to worse energies.
+
+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{Towards second quantized effective Hamiltonians for MBPT?}
 \label{sec:second_quant_mbpt}
 %=================================================================%