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