done with parquet notes for now
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@ -47,7 +47,12 @@
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\newcommand{\si}{\sigma}
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\newcommand{\la}{\lambda}
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\newcommand{\ii}{\mathrm{i}}
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\newcommand{\up}{\uparrow}
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\newcommand{\dw}{\downarrow}
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\newcommand{\upup}{\up\up}
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\newcommand{\updw}{\up\dw}
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\newcommand{\dwup}{\dw\up}
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\newcommand{\dwdw}{\dw\downarrow}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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@ -145,8 +150,8 @@ with
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\end{gather}
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One must be very careful not to double-count diagrams in the pp sector.
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The complete ($\Gamma$ and $\Gamma_P$) and irreducible vertices ($\Gamma^\pp$, $\Gamma^\ph$, and $\Bar{\Gamma}^\ph$) are usually spin adapted.
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For example, $\Gamma^\ph$ is written as a sum of its density ($\Gamma^\ph_\text{d}$) and magnetic ($\Gamma^\ph_\text{m}$) components, while $\Gamma^\pp$ is decomposed in its singlet ($\Gamma^\ph_\text{s}$) and triplet ($\Gamma^\ph_\text{t}$) components.
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The complete ($\Gamma$ and $\Gamma_P$) and irreducible vertices ($\Gamma^\pp$, $\Gamma^\ph$, and $\Bar{\Gamma}^\ph$) are usually spin adapted (see below).
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For example, $\Gamma^\ph$ is written as a sum of its density ($\Gamma^\ph_\text{d}$) and magnetic ($\Gamma^\ph_\text{m}$) components, while $\Gamma^\pp$ is decomposed in its singlet ($\Gamma^\pp_\text{s}$) and triplet ($\Gamma^\pp_\text{t}$) components.
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Similar expressions can be found for $\Bar{\Gamma}^\ph$ (crossing relation), and the approximated forms of $\Gamma$ and $\Gamma_P$ (but usually violate crossing conditions).
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@ -230,7 +235,7 @@ Now, we focus on the self-energy which can be written as a non-scattering and a
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\begin{equation}
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\Sigma(11') = \Sigma_1(11') + \Sigma_2(11')
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\end{equation}
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where, after various manipulations, on gets
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where, after various manipulations, one gets
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\begin{equation}
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\begin{split}
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\Sigma_2(11')
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@ -254,9 +259,157 @@ where, after various manipulations, on gets
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Thanks to the spin adaptation, we have the following picture: the electrons interacting through the exchange of four varieties of composite ``bosons'' made of electrons and holes which themselves strongly interact through the exchange of other bosons: density, magnetic, singlet-pair, and triplet-pair fluctuations.
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Thanks to the crossing symmetry of the complete vertices, the parquet equations automatically build in the nonlinear coupling between dressed electron and boson excitations necessary for full consistency.
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We now explicitly treat the spin degrees of freedom starting from the ph vertex functions that we decompose in density and magnetic components.
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We hence remove the spin variable from the composite variable $1 \equiv (\br_1,t_1)$.
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Defining the density and magnetic operators
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\begin{gather}
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d(12) = \frac{1}{\sqrt{2}} \qty[ \cre{1\up}\ani{2\up} + \cre{1\dw}\ani{2\dw} ]
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\\
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m(12) = \frac{1}{\sqrt{2}} \qty[ \cre{1\up}\ani{2\up} - \cre{1\dw}\ani{2\dw} ]
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\end{gather}
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we have
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\begin{gather}
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\Gamma_\text{m}^{+1}(12;34) = \Gamma_{\updw,\updw}(12;34)
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\\
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\Gamma_\text{m}^{-1}(12;34) = \Gamma_{\dwup,\dwup}(12;34)
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\end{gather}
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and
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\begin{gather}
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\Gamma_\text{d}(12;34) = \Gamma_{\upup,\upup}(12;34) + \Gamma_{\upup,\dwdw}(12;34)
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\\
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\Gamma_\text{m}^{0}(12;34) = \Gamma_{\upup,\upup}(12;34) - \Gamma_{\upup,\dwdw}(12;34)
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\end{gather}
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Hence, $\Gamma_\text{m} \equiv \Gamma_\text{m}^{0} = \Gamma_\text{m}^{\pm1}$.
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Now, let us decompose the pp vertex functions into their singlet and triplet components in very much the same way.
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Again, the 3 triplet components
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\begin{gather}
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\Gamma_\text{t}^{+1}(12;34) = \Gamma_{\upup,\upup}(12;34)
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\\
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\Gamma_\text{t}^{-1}(12;34) = \Gamma_{\dwdw,\dwdw}(12;34)
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\\
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\Gamma_\text{t}^{0}(12;34) = \Gamma_{\updw,\updw}(12;34) - \Gamma_{\updw,\dwup}(12;34)
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\end{gather}
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are equal and abbreviated as $\Gamma_\text{t}$ while the singlet component reads
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\begin{equation}
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\Gamma_\text{s}^{0}(12;34) = \Gamma_{\updw,\updw}(12;34) + \Gamma_{\updw,\dwup}(12;34)
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\end{equation}
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We recover the fact that the singlet component is symmetric under the exchange of space-time labels, while the triplets are antisymmetric, i.e.
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\begin{gather}
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\Gamma_\text{s}(12;34) = + \Gamma_\text{s}(12;43)
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\\
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\Gamma_\text{t}(12;34) = - \Gamma_\text{t}(12;43)
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\end{gather}
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We are now ready to write down the parquet equations in a spin-adapted basis.
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Defining
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\begin{gather}
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\Phi_\text{d/m}(12;34) = \qty[ \Gamma_\text{d/m}^\ph (\bI - G^\ph \Gamma_\text{d/m}^\ph)^{-1} G^\ph \Gamma_\text{d/m}^\ph ](12;34)
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\\
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\Psi_\text{s/t}(12;34) = \qty[ \Gamma_\text{s/t}^\pp (\bI - G^\pp \Gamma_\text{s/t}^\pp)^{-1} G^\pp \Gamma_\text{s/t}^\pp ](12;34)
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\end{gather}
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we get
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\begin{align}
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\begin{split}
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\Gamma_\text{d}^\ph(12;34)
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= \Lambda_\text{d}^\irr(12;34)
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& - \frac{1}{2} \Phi_\text{d}(42;31)
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- \frac{3}{2} \Phi_\text{m}(42;31)
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\\
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& + \frac{1}{2} \Psi_\text{s}(41;32)
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+ \frac{3}{2} \Psi_\text{t}(41;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{m}^\ph(12;34)
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= \Lambda_\text{m}^\irr(12;34)
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& - \frac{1}{2} \Phi_\text{d}(42;31)
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+ \frac{1}{2} \Phi_\text{m}(42;31)
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\\
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& - \frac{1}{2} \Psi_\text{s}(41;32)
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+ \frac{1}{2} \Psi_\text{t}(41;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{s}^\pp(12;34)
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= \Lambda_\text{s}^\irr(12;34)
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& + \frac{1}{2} \Phi_\text{d}(24;31)
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- \frac{3}{2} \Phi_\text{m}(24;31)
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\\
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& + \frac{1}{2} \Psi_\text{s}(14;32)
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- \frac{3}{2} \Psi_\text{t}(14;32)
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\end{split}
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\\
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\begin{split}
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\Gamma_\text{t}^\pp(12;34)
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= \Lambda_\text{t}^\irr(12;34)
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& + \frac{1}{2} \Phi_\text{d}(24;31)
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+ \frac{1}{2} \Phi_\text{m}(24;31)
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\\
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& - \frac{1}{2} \Psi_\text{s}(14;32)
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- \frac{1}{2} \Psi_\text{t}(14;32)
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\end{split}
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\end{align}
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Check out this nice symmetry!
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Now that we have the parquet equations written in a convenient spin-adapted form, the last step is to write down the self-energy via the Dyson-Schwinger equation.
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Doing so displays the four varieties of composite bosons mentioned previously:
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\begin{equation}
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\label{eq:Sig2_sa}
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\begin{split}
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\Sigma_2(11')
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= \frac{1}{2} \Bigg\{
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& - \frac{1}{2} G(76) \qty[
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\frac{1}{2} \Lambda_\text{d}^\irr G^\ph v_\text{d} + \frac{3}{2} \Lambda_\text{m}^\irr G^\ph v_\text{m} ](17;1'6)
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\\
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& + G(67) \qty[ \frac{1}{2} \Lambda_\text{s}^\irr G^\pp v_\text{s} + \frac{3}{2} \Lambda_\text{t}^\irr G^\pp v_\text{t} ](17;1'6)
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\Bigg\}
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\\
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& - G(76) \qty[ \frac{1}{2} \Phi_\text{d} G^\ph v_\text{d} + \frac{3}{2} \Phi_\text{m} G^\ph v_\text{m} ](17;1'6)
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\\
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& + G(67) \qty[ \frac{1}{2} \Psi_\text{s} G^\pp v_\text{s} + \frac{3}{2} \Psi_\text{t} G^\pp v_\text{t} ](17;1'6)
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\end{split}
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\end{equation}
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Finally, the crossing relations become
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\begin{gather}
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\Gamma_\text{m}(12;34) = - \frac{1}{2} \Gamma_\text{d}(42;31) + \frac{1}{2} \Gamma_\text{m}(42;31)
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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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The FLEX approximation is a variant of the Baym-Kadanoff approximation and then focuses on the self-energy itself.
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This lack of two-body self-consistency limits quantitative accuracy of FLEX compared to parquet.
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In FLEX, one sets $\Lambda^\irr = \Gamma^\ph = \Gamma^\pp = v$.
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Equation \eqref{eq:Sig2_sa} then becomes
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\begin{equation}
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\label{eq:Sig2_sa}
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\begin{split}
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\Sigma_2(11')
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= \frac{1}{2} \Bigg\{
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& - \frac{1}{2} G(76) \qty[
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\frac{1}{2} v_\text{d} G^\ph v_\text{d} + \frac{3}{2} v_\text{m} G^\ph v_\text{m} ](17;1'6)
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\\
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& + G(67) \qty[ \frac{1}{2} v_\text{s} G^\pp v_\text{s} + \frac{3}{2} v_\text{t} G^\pp v_\text{t} ](17;1'6)
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\Bigg\}
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\\
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& - G(76) \Big[
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\frac{1}{2} v_\text{d} (\bI - G^\ph v_\text{d})^{-1} (G^\ph v_\text{d})^2
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\\
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& + \frac{3}{2} v_\text{m} (\bI - G^\ph v_\text{m})^{-1} (G^\ph v_\text{m})^2
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\Big](17;1'6)
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\\
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& + G(67) \Big[
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\frac{1}{2} v_\text{s} (\bI - G^\pp)^{-1} (G^\pp v_\text{s})^2
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\\
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& + \frac{3}{2} v_\text{t} (\bI - G^\pp)^{-1} (G^\pp v_\text{t})^2
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\Big](17;1'6)
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\end{split}
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\end{equation}
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Because FLEX is a parquet with non-self-consistent two-body vertices, their expressions are a bit different and two flavours exist.
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The first flavour includes the single-fluctuation-exchange diagrams while the second flavor includes also the so-called Aslamazov-Larkin diagrams.
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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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\acknowledgements{
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