diff --git a/Notes/Notes.tex b/Notes/Notes.tex index 38680c6..3aa862e 100644 --- a/Notes/Notes.tex +++ b/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} %=================================================================%