diff --git a/Notes/parquet.tex b/Notes/parquet.tex index 1c62b9a..f29aade 100644 --- a/Notes/parquet.tex +++ b/Notes/parquet.tex @@ -36,6 +36,7 @@ \newcommand{\br}{\boldsymbol{r}} \newcommand{\bx}{\boldsymbol{x}} \newcommand{\bI}{\boldsymbol{1}} +\newcommand{\cK}{\mathcal{K}} % orbital energies \newcommand{\eps}{\epsilon} @@ -116,8 +117,8 @@ with $\Gamma(12;34) = - \Gamma_P(14;32)$ and $\Gamma_P(12;34) = - \Gamma_P(21;34 Both the crossing and antisymmetry properties stem from the Pauli exclusion principle. -The full vertex function $F$ contains the fully irreducible terms produced by the input $\Lambda$ (which is fully irreducible) and the reducible part from the repeated scattering terms (iterations). -At the two-body level, there are three different channels for the reducible vertices: one pp channel $\Gamma^\ph$, and two ph (horizontal and vertical) channels, $\Bar{\Gamma}^\ph$ and $\Gamma^\ph$. +The full vertex function $F = \Gamma + \Gamma_P$ contains the fully irreducible terms produced by the input $\Lambda$ (which is fully irreducible) and the reducible part from the repeated scattering terms (iterations). +At the two-body level, there are three different channels for the reducible vertices: one pp channel $\Gamma^\pp$, and two ph (horizontal and vertical) channels, $\Bar{\Gamma}^\ph$ and $\Gamma^\ph$. For the pp case, reducible diagrams are defined as the ones that can be separated into two pieces by cutting two fermion lines. ph irreducibility is tricker as two channels exist. @@ -375,6 +376,7 @@ Finally, the crossing relations become \\ \Gamma_\text{m}(12;34) = - \frac{1}{2} \Gamma_\text{s}(14;32) - \frac{1}{2} \Gamma_\text{t}(14;32) \end{gather} + %%%%%%%%%%%%%%%%%%%%%%%%% \section{Fluctuation-exchange approximation} %%%%%%%%%%%%%%%%%%%%%%%%% @@ -411,6 +413,111 @@ The first flavour includes the single-fluctuation-exchange diagrams while the se When the vertex functions are spin-diagonalized omitting the Aslamazov-Larkin contributions, we recover the parquet equations where one sets $\Lambda^\irr = \Gamma^\ph = \Gamma^\pp = v$. It has been shown that the Aslamazov-Larkin diagrams deterioetas the accuray of the two-body vertices obtained via FLEX. +%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Self-consistent $GW\Gamma$} +%%%%%%%%%%%%%%%%%%%%%%%%% +The infamous Hedin equations are +\begin{align} + \Gamma(12;3) & = \delta(12) \delta(13) + \fdv{\Sigma(12)}{G(45)} G(46) G(75) \Gamma(67;3) + \\ + P(12) & = - \ii G(23) G(42) \Gamma(34;1) + \\ + W(12) & = v(12) + v(13) P(34) W(42) + \\ + \Sigma(12) & = \ii W(13) G(14) \Gamma(42;3) + \\ + G(12) & = G_0(12) + G_0(13) \Sigma(34) G(42) +\end{align} + +Hedin's equations tell us that the self-energy is +\begin{equation} + \Sigma(12) = \ii G(13) W(14) \Gamma(432) +\end{equation} +with $1 \equiv (\br_1,t_1)$, where the vertex function at the $n$th iteration is given by +\begin{equation} +\label{eq:Gamma} + \Gamma^{(n+1)}(123) + = \delta(12) \delta(13) + + \cK^{(n)}(1245) G(46) G(75) \Gamma(673) +\end{equation} +where we have introduced the interaction kernel +\begin{equation} + \cK^{(n)}(1234) = \fdv{\Sigma^{(n)}(12)}{G(34)} +\end{equation} +The self-consistency introduces additional diagrams at each iteration and corresponds to a resummation of the many-body interaction. +A closure must be imposed via truncation at finite order or restriction on the diagrammatic topology. +Of course, the case where $\cK = 0$ corresponds to the well-known $GW$ case with $\Gamma(123) = \delta(12) \delta(13)$ and +\begin{equation} + \Sigma^{(0)}(12) = \ii G(12) W(12) +\end{equation} + +Because $\Sigma = \ii GW\Gamma$, we have +\begin{equation} +\begin{split} + \cK + & = \ii \fdv{(GW\Gamma)}{G} + \\ + & = \underbrace{\ii \fdv{G}{G}W\Gamma}_{\cK_G} + + \underbrace{\ii G\fdv{W}{G}\Gamma}_{\cK_W} + + \underbrace{\ii GW\fdv{\Gamma}{G}}_{\cK_\Gamma} +\end{split} +\end{equation} + +At lowest order in $\Gamma$, i.e. $\Gamma = 1$, we have +\begin{equation} + \cK^{(0)}_G(1234) = \ii W(12) \delta(13) \delta(24) +\end{equation} +By substituting this expression into Eq.~\eqref{eq:Gamma}, we get +\begin{equation} + \Gamma_G^{(1)}(123) = \ii G(13) W(12) G(23) +\end{equation} +yielding the following self-energy +\begin{equation} + \Sigma^{(1)}(12) + = \ii G(13) W(14) \qty[ \delta(32) \delta(42) + + \ii G(34) W(32) G(42) ] +\end{equation} +This term brings a SOSEX-like correction, i.e., exchange to the $GW$ equations. + +Now let us check the term $\cK_W$. +Here again, we can stay at the lowest order in $W$, i.e., +\begin{equation} + W(12) = v(12) - \ii v(13) G(34) G(43) W(42) +\end{equation} +which gives +\begin{multline} +\label{eq:Gamma_W} + \Gamma_W^{(1)}(123) + = v(14) G(12) W(25) \Big[ G(45) G(34) G(53) + \\ + + G(54) G(35) G(43) \Big] +\end{multline} +This term brings the direct part of the pp and eh $T$-matrix terms. + +The last term, $\cK_\Gamma$, is more tricky, as it requires a non-trivial vertex as an input. +\begin{equation} +\begin{split} + \fdv{\Gamma(123)}{G(45)} + = & \fdv{\qty[\cK(1267) G(63) G(37)]}{G(45)} + \\ + = & \fdv{\qty[\cK(1267)]}{G(45)} G(63) G(37) + \\ + & + \cK(1247) G(37) \delta(35) + \\ + & + \cK(1265) G(63) \delta(34) +\end{split} +\end{equation} + +By setting $\cK = \cK_G = \ii W$ and $\fdv*{\cK}{G} = 0$, we get +\begin{multline} + \Gamma^{(2)}_\Gamma(123) = - W(14)G(15)W(25) \Big[ G(53)G(34)G(42) + \\ + + G(54)G(43)G(32) \Big] +\end{multline} +which is analogous to Eq.~\eqref{eq:Gamma_W} upon exchange. +hence, this term brings the exchange part of the pp and eh $T$-matrix terms. +Topological novel diagrams are exclusively introduced by this term. + %%%%%%%%%%%%%%%%%%%%%%%% \acknowledgements{ This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}